# WittmannBlack - Mathematical actions

- Mathematical actions as procedural resources:

An example from the separation of variables - EMBODIED MATHEMATICAL IMAGINATION AND COGNITION (EMIC) WORKING
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- Setting and participants
- Methods
- Selected conversations
- Emergent context: a factoring discussion
- The significance of indexical language for collaborative learning
- Conclusion
- References
- A Linguistic and Narrative View of Word Problems in Mathematics Education
- MathExamples
- Contexts, Goals, Beliefs, Mathematics*
- and Learning
- THE CRUCIAL ROLE OF NARRATIVE THOUGHT

PHYSICAL REVIEW SPECIAL TOPICS—PHYSICS EDUCATION RESEARCH 11, 020114 (2015)

**Mathematical actions as procedural resources:**

An example from the separation of variables

An example from the separation of variables

Michael C. Wittmann

Department of Physics and Astronomy, Maine Center for Research in STEM Education,

University of Maine, Orono, Maine 04469, USA

Katrina E. Black

Department of Physics, Michigan Technical University, Houghton, Michigan 49931, USA

(Received 29 September 2014; published 23 September 2015)

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Students learning to separate variables in order to solve a differential equation have multiple ways of correctly doing so. The procedures involved in separation include division or multiplication after properly grouping terms in an equation, moving terms (again, at times grouped) from one location on the page to another, or simply carrying out separation as a single act without showing any steps. We describe student use of these procedures in terms of Hammer’s resources, showing that each of the previously listed procedures is its own “piece” of a larger problem solving activity. Our data come from group examinations of students separating variables while solving an air resistance problem in an intermediate mechanics class. Through detailed analysis of four groups of students, we motivate that the mathematical procedures are resources and show the issues that students must resolve in order to successfully separate variables. We use this analysis to suggest ways in which new resources (such as separation) come to be.

DOI: 10.1103/PhysRevSTPER.11.020114

**I. INTRODUCTION**

As part of a project to describe mathematical problem solving in physics from a resources perspective [1–6], we have looked at students’ understanding of variables [7], understandingofintegrationconstants[8],theuseofmultiple methods for finding unknown integration constants [9,10], and students’ choice of epistemic games [11] when carrying outintegration.Whilecarryingoutthesestudies,weregularly observed students struggling with algebra in some contexts and not in others. There is an extensive literature on the teaching and learning of algebra (as summarized in Refs. [12,13]), but little work has been done in physics. One area in which students struggle with algebraic manipulation is the separation of variables, used, for example, to isolate terms in a first order differential equation so that one can integrate the two sides of an equality independently. We have found that students who are otherwise strong in their algebraic skills (as observed in classroom activities or on examinations) have difficulties when separating variables. In this paper, we use the discussion of students resolving their difficulties to extend our understanding of student reasoning in a physics class.

We model reasoning in terms of a theoretical framework focused explicitly on variability of students’ reasoning in

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. |

PACS numbers: 01.40.Ha

problem solving: Hammer’s resources [1]. This framework has been valuable in understanding conceptual [3–6] and epistemological [14–16] reasoning as well as the transfer of knowledge [2]. We extend the framework to include mathematical actions, which we call “procedural resources” [9]. We show that mathematical actions meet more than one of the definitions of resources (two of which are discussed below). This enriches our understanding of the use of mathematics in physics, suggesting both instructional and analytical tools to expand our knowledge of learning in physics.

To make our case, we look at just one step in a long problem in which students needed to describe the motion of a thrown object. As part of the problem, students are required to separate variables. The task is a relatively simple one, as long as one understands that the derivative (in this case, dv=dt) can be treated as a fraction. We do not observe students having problems with that aspect of the problem. Instead, they struggle with how to group terms appropriately in order to create an equation with variables isolated on each side of the equation. (Other aspects of our work investigate what happens after this point, as students begin to integrate [9–11].)

1554-9178=15=11(2)=020114(13) Published by the American Physical Society |

We begin this paper with a discussion of the resources framework, including a brief discussion of local coherence and compilation of resources. This is followed by a description of our research setting and analysis methods. Next, we present data from four groups discussing the separation of variables in an air resistance problem. Finally, we discuss our results and their meaning within the resources framework and their implications for instruction and curriculum design. Our central point is that the mathematical actions discussed in this paper can be thought of as procedural resources.

**II. MODELING STUDENTS****’ ****IDEAS USING THE RESOURCES FRAMEWORK**

The reasoning resources framework proposed by Hammer [1] and extended by Hammer, Redish, and others [2–6,17,18] was introduced as a kind of “knowledge in pieces” approach to thinking [18–24] with a specific emphasis on problem solving in physics. We think of knowledge pieces as schemalike units of thinking which are activated in a given setting to reason about a given topic. Most of the “pieces” discussed in the physics education research literature (see, for example, Refs. [5,21,23]) describe these as basic, “primitive” ideas which are developed early in life and used in later life to make sense of complicated new situations.

Several kinds of small-scale conceptual knowledge have been described in the literature, including “agents” [18], “facets of knowledge” [20], “phenomenological primitives” (p-prims) [21], “mathematical forms” [22,25], and “intuitive rules” [24]. A discussion of the differences between these different descriptions of knowledge is beyond the scope of this paper. In our work, we have used Hammer’s “resources” [1]. Hammer describes them as elements of a problem solving activity that can be of any cognitive scale. We make an analogy to printing a document from a computer. One would need a printer, drivers, and a program from which to print the document. Each of these—the program, the driver, or the printer—is a resource when attempting to print the desired document. They are of different natures (software and hardware) and scales (a driver is a small piece of code in the operating system, specific to the printer; a program is a larger piece of code independent of the operating system, where many programs can print to the printer). They are each necessary in their own way, and solving the problem of printing requires that all be used in the correct order. Finally, they are all useful, though they might be useless until used in conjunction with other resources.

Resources, like most schemalike theories, are helpful in organizing our description of knowledge into “chunks” of information, consistent with what we know about working memory. Working memory is fast but limited [26]; it can hold only a few items at a time, and those only for a few seconds (see, for example, Ref. [27]). However, the “items” in working memory may encode considerable structure. Resources, as described above, are one way of describing the at times highly encoded structure of chunks of knowledge. Even very basic knowledge pieces might have considerable substructure. A knowledge piece like dying away, a phenomenological primitive (p-prim) described by DiSessa [21], requires an object, a property of the object, a time-dependent change in the property, and that the change be a gradual lessening. This is clearly not a primitive idea when treated analytically, but dying away is used readily and easily to describe the sound of a rung bell and the motion of an object given a shove across a floor. In this paper, we discuss student reasoning in terms of the procedural resources of problem solving that eventually come to be an easily usable resource. We provide examples of the resources in our data and discuss creation of new resources after.

Before we are able to do so, we must first determine whether the mathematical actions we observe students engaged in can be thought of as procedural resources. We look to the descriptions of resources given in the literature. Sayre and Wittmann [28] describe resources as “individual reusable thoughts” that can exist in two states (activated in a situation or not) and are connected to other thoughts, while having some level of internal structure that is perhaps apparent to the user, but possibly not. We show below that each of these criteria is met by our procedural resources. We also consider Scherr’s analysis of the pieces model [17], in which knowledge pieces are mutually independent, context dependent, and indeterminate. We show that these criteria are also met in our analysis of procedural resources.

The resources framework allows for the idea that resources can develop at any time in one’s learning career and can be applied usefully in many different settings. Ideas about resource creation include compilation [29] of a locally coherent set of resources [2] and reification of processes into mental objects [30–32]. Our work adds to the discussion of the resources framework by providing examples of students who have not yet compiled the procedural resource of separating, suggesting that there are multiple ways of doing so.

In the discussion, we revisit these points and discuss the advantages of using a resources framework to explain student mathematical problem solving in physics.

**III. RESEARCH DESIGN**

The work described in this paper was part of a larger project to study student use of mathematics in advanced physics courses at a large state university [7–11]. Part of our larger project was to understand student use of mathematics in an advanced physics class and develop learning materials to help these students in their learning. Thus, our attention was focused on student use of mathematics, their difficulties and strengths, and the ways in which these were manifested in group learning activities.

In this section, we describe the classroom setting, including a description of the student population studied. We describe the problem we studied and its possible solutions. We describe how the sample of students studied was chosen, and end with a discussion of constraints on our data collection and its effects on our analysis.

- Classroom setting

The study took place in a sophomore-level mechanics course in which students are required to have a course in differential equations as a corequisite. Class size was between 12 and 20 students. The framework of the mechanics course was primarily Newtonian, with only brief discussions of Lagrangian mechanics and the variational principle. The topic of first order linear differential equations typically came in the first few weeks of the course.

Students in the course used the Intermediate Mechanics Tutorials [33–35] in about a third of their classes. In these activities, students work to combine their conceptual and mathematical knowledge of mechanics using guided questions. These small group learning activities are based on the Tutorials in Introductory Physics [36] as well as the Activity-Based Tutorials [37,38].

- Student population

Students taking the Intermediate Mechanics course exhibited a large range of experience and ability in mathematics and physics. Most were physics or engineering physics majors or minors. Students had typically completed introductory physics courses in mechanics, electromagnetism, and optics. Some had a course in modern physics, and many were taking a course in relativity in parallel. Although Intermediate Mechanics was a sophomore level course, the students in this study were fairly equally divided among sophomores and juniors, with some seniors included, leading to a disparity in mathematics backgrounds—about half had already taken a course in differential equations and half took it concurrently. Notably, there was no relationship between a student’s class standing (sophomore, junior, or senior) and their mathematics class; some sophomores had already taken differential equations, and some seniors were taking it concurrently. A total of over 50 students were observed as part of the research described in this paper. About 20 are represented in the groups described in this paper, though not all spoke and are therefore not all named in the episodes described below.

In early parts of the Intermediate Mechanics course, students solved first-order, separable differential equations (e.g., mdv=dt ¼ −bv). For many students, this was an introduction to the method of separation of variables, since their corequisite differential equations course had not yet introduced the topic. However, mastery of the algebra required to separate variables was expected of all students, given the mathematics pre- and corequisites for the course. Separation of variables was simply an extension of algebra that included the treating a differential term (dv=dt) as a fraction.

- The problem and its solutions

As part of our regular classroom instruction, we assigned students several problems on a group quiz given after

*F *= *ma *= *mg bv*^{2}

*dv*

*m *= *mg bv*^{2 }*dt*

*Transformcoordinates*

*dv*

*mv *= *mg bv*^{2 }*dx*

FIG. 1. Whensolvinga typical air resistance problem,all student groups sought to separate variables in the last equation shown.

several weeks of instruction on air resistance problems and first order linear differential equations. We asked students to start from Newton’s second law to solve the problem of a large object (such as a beach ball) being thrown vertically downward (see the first line of Fig. 1) and to find the velocity of the ball as a function of position. We told them to treat down as positive and that the object was beginning at greater than terminal velocity. They were given no other information, and had to know, for example, that an object of this size would best be described with v^{2 }drag forces acting on it. (In one episode below, students did not make this assumption, but this does not affect our data for the purposes of this paper.) All of the observed students successfully moved from the first line to the last line of equations shown in Fig. 1.

The focus of this paper is on what they did at that point. All the groups recognized that the task was to separate variables. To indicate this, we start our transcripts with a student statement to that effect, when possible.

To correctly separate the variables, one might group the terms on the left, (mg−bv^{2}), and then divide both sides of the equation by the grouped term and multiply both sides of the equation by dx. This would move all the terms of x to the right side of the equation and all the terms of v to the other. Or, one might take the metaphorical language of the previous sentence (“move all the terms”) and enact it through gestures (as discussed in Ref. [39]). That, or simply write down the equation with separated variables and be done with it.

In the data discussed below, we show that students use several procedural resources. First, there is grouping, in which terms are combined into a single new term. Second, there are the mathematical procedures of subtraction, multiplication, and division. Finally, there is the possibility of moving, which may also require grouping and may involve other actions like pointing and dragging. These gestural elements of moving are not the point of this paper, and are discussed elsewhere [39] in more detail.

We note that all groups eventually accomplished the same task successfully. But, not all immediately solved the problem correctly. In the process of discussing possible solutions, students evaluated the usefulness of the different resources, as will be shown below.

**Choice of student sample**

To observe student reasoning during the group quiz, we video recorded all students as they worked in small groups of 3 or 4. The students were videotaped with permission, in part to allow the instructor to review their work after the quiz was completed (and assign partial credit where needed) and in part for the purposes of research on how students discussed mathematics in a testing environment. Cameras were placed on tripods at a distance that would allow all the students in the group to be filmed. Because of a lack of cameras, only one camera per table was used.

We observed three kinds of student interactions in our groups. Some groups did not talk while separating variables, so we have no information on what they were doing or their reasoning as they did it. Other groups talked, but did not discuss their reasoning; we give one example, below. The groups we analyzed were those which described their reasoning when isolating the variables in a simple equation.

Video data were collected over the course of three years of instruction. In year 1, we recorded all five groups as they worked on this problem. Of these, two talked about separation of variables in detail. In year 2, one of the groups provided an excellent example of procedures we had seen in classroom settings but had not yet seen in that detail in the group quiz. In year 3, a further group stood out as providing a unique way of discussing these procedures. We analyzed these four groups as an illustration of the kinds of procedures that students might use when solving this kind of problem. Other groups, not analyzed in this paper, showed similar reasoning or talked less than these particular groups.

**Constraints on our data selection and analysis**

Small class sizes allowed us to record video data of every group, but restricted us to a small number of groups each year. We gathered data from multiple years in order to have sufficient data to examine patterns, but we do not claim to have observed all possible problem-solving methods or to know what methods may be most common in a broad student population. Rather, as discussed below, our observations begin to build a list of procedural resources that at least some number of students brought to these classes; further research would be required in order to uncover other resources or to study their prevalence among different populations.

Because we gathered data from multiple years of instruction, we increased the variability in students’ backgrounds, the specifics of instruction, and other variables. For example, during the time of this study, one author (M. C. W.) taught the course twice and the other (K. E. B.) taught the course once. By combining these data into a single study, we have chosen to leave out course-specific variables in our discussion of students’ mathematics use, and instead focus on very general behaviors shared by all the studied groups.

In addition, collecting data during an authentic group activity, as opposed to a group interview, meant that not all groups verbalized their actions as they worked through the targeted task. Some looked off each other’s paper without talking at any point, others worked independently and arrived at the same point without discussion, and others discussed only the answer they arrived at, not how they arrived at it. A consequence is that we do not know what all the students in the course did, only those that spoke about their reasoning. The motivations for speaking (such as a strong student needing to explain to a weaker one what is happening at a particular point) might affect the data since such discourse might be more like the explanation of a solution and less the invention of a solution.

Third, the lack of cameras (we had only 5 available to us) meant that we only had one camera to use per group. These cameras were far enough away from the table to include all group members, but were typically too far to allow us to read what students wrote on their papers. Also, they used a screen resolution that did not allow for zooming in more closely while still being able to read what students wrote. This affected our data from year 1; we describe our best interpretations of student work in the episodes below. In years 2 and 3, we asked students to use a whiteboard and markers as they discussed, and we were able to see more of their writing during their discussions.

**F. Analysis of data**

Our first data were selected by one author (K. E. B.) while reviewing the year 1 results from all groups. She noticed that students’ actions varied from group to group while solving this particular part of the problem. After finding three of the groups to be of interest (while two did not talk), she brought the video of these groups to our 15-person research group meeting. At that point, she presented her preliminary interpretation of the data and a detailed discussion followed. Her first interpretation was not this final version but instead focused on the use of mathematical terminology, did not focus on the gestures students used, and was not yet connected to a resources interpretation. Further interpretations were suggested and discussed, and the analysis was refined as it was put into the context of the larger study. Throughout, we used interaction analysis as described by Jordan and Henderson [40] and a modified version of grounded theory [41,42] in which repeated discussion allowed for the emergence of categories and descriptions of the data that were agreed upon by all participants.

In our choice of data, we came to rely on two different types of observation. We began by focusing on student discourse, attending to students’ word choice and their inflection. However, student use of some phrases, like “move this over there” or “this whole term,” were ambiguous until we examined the video to describe both the context of the activity and the gestures students used at those moments. For a review of gesture analysis [43–47] as it is used in the separation of variables, we refer the reader to a previous publication [39].

In this paper, both discourse and gesture analysis were used to provide evidence for the claim that students’ mathematical actions can be thought of as procedural resources.

**IV. RESOURCE USE WHEN SEPARATING VARIABLES**

We observed different ways in which students in a group activity combined the procedural resources of grouping, moving, multiplication, and division, and some debate about subtracting, as well. In this section, we present several episodes (transcripts and description of events) to illustrate variations in student actions; in the following discussion, we analyze the consequences of variations in student activities and thinking.

We suggest that for a typical physicist, separating variables in this problem might be a necessary but trivial step in working through the problem. This appeared to be the case for one of the stronger students in the class, alias Phil. While solving the problem, Phil did not verbalize any explanation of his steps; he simply arrived at an answer, as follows:

- Phil: OK, now we want to separate variables. We have mv dv over mg minus bv squared equals [makes noises while writing] uhh, dx.

(We number lines of transcript sequentially in this paper to help in the discussion of group interactions.)

Though we cannot see what is on the page, we hear him speak the correct answer, arrived at with no explicit algebraic steps indicated. We believe that students eventually arrive at this level of expertise, using the procedure as a single resource, much like addition and multiplication become basic resources (e.g., primitive [21]) to a student. We return to this point in the discussion.

**A. Sarah divides and multiplies to separate**

For three students, Sarah, Paul, and Tim, separating variables was not trivial. These students worked individually on the problem while occasionally comparing their solutions. Sarah first silently completed the separation of variables on her own paper and then helped her classmates Paul and Tim to catch up. In this episode, we attend primarily to Sarah as she interacts first with Phil and later with Tim. We italicize words in the transcript which we refer to later in the discussion or analyze in terms of resources.

- Sarah: Alright, so where are you guys at?

- Paul: I’m still trying to separate it.

- Sarah: OK, um, one of the easiest ways is dividing the entire thing by this side, and then multiplying both sides by dx.

- Paul: Yeah, yeah.

- long pause—more than a minute

- Sarah: Ok, alright. What are you doing right now?

- Tim: Alright, so we’re at this point, right? Weve got mv dv dx is equal to mg minus bv squared.

- Sarah: Mm-hm.

- Tim: Alright. So what you’re doing is just bringing this dx over?

- Sarah: You bring the dx over and then divide both sides by this entire, um, expression. So it becomes, um, mv dv over mg minus bv squared.

We analyze line 4 first. Sarah responded to Paul’s line 3 statement about separating variables in the equation by naming two procedures—dividing the whole equation (“the entire thing”’) by one side of the equation (“this side”) to get all the velocity terms on one side and then multiplying by dx to get the position terms on the other side of the equation. When she said “the entire thing”, she underlined both the left and right sides of the equation in a single gesture, and when she said “this side”, she circled the mg−bv^{2 }on the right side of the equation. We interpret this to mean that she was carrying out algebraic steps (dividing) on both sides of the equals sign, being careful to define by grouping what she was dividing with. For Sarah responding to Paul, separation happened by grouping one side of the equation, followed by dividing and then multiplying.

Later in the episode, Sarah helped Tim with his work as Paul observed their work silently (Paul had not “separated it” after his comment in line 5). In line 8, Tim stated the equation that all seemed to be working with. In line 10, he suggested something different from Sarah, namely, “bringing this dx over” to the other side of the equation. In line 11, Sarah affirmed that this idea of bringing over was equivalent to the multiplication she described to Paul earlier (or at least consistent with her act of multiplying), and then repeated her statement about division. As before, Sarah’s ambiguous words about “this entire… expression” were accompanied by a vigorous circling gesture (during the word “entire”) which we believe represents the procedure of grouping the several terms she was referring to (Tim’s “mg minus bv squared” and not “the entire thing” that she described when talking to Paul). She concluded line 11 with a statement of the separated equation, mvdv=ðmg−bv^{2}Þ ¼ dx.

In both cases, the grouping procedure preceded the division procedure; mathematically speaking, it was a necessary step before division could be used to successfully separate variables. But, in both cases, the verbal and gestural evidence of grouping came after the use of “division” in speech.

As with Phil (line 1), these students wished to separate the terms in the equation, but they described many more procedures to do so. We see Sarah grouping and dividing and then multiplying to arrive at a separated equation. We

FIG. 2. Sarah’s procedure of separation contains multiple procedures. These include formal mathematical operations and explicit activity to group terms in the equation appropriately. She also indicates an equivalence between a description of moving and the mathematics of multiplication.

also see Tim moving terms by “bringing … over” a term from one side to the other. (We discuss other examples of moving in more detail below.)

Through her interactions with her group, Sarah shows that one way of separating is to think in terms of mulitplying and dividing while being careful about grouping terms that are to be part of the division. We represent the collection of Sarah’s resources in a graph where the process of separation (which for Phil was a single action) in this example is made up of multiple procedures (see Fig. 2). In the graph, we indicate that grouping precedes dividing by using an arrow that the one leads to the other. We also indicate that Tim’s moving by “bringing… over” is equivalent to Sarah’s multiplying by using a double line like an equals sign.

**B. Students move to separate**

In a separate episode, two students, Simon, who was in control of the whiteboard marker, and Dan, who dominated the conversation, discussed separation of variables quite differently from Phil or Sarah. (Two other students in the group did not speak during this episode.) We have described the use of gesture in this episode elsewhere [39]. Here, we discuss the procedural resources used by the students. As before, we mark words we will return to by italicizing them. We use [square brackets] to describe gestures essential to understanding the text.

- Simon: So then we’re gonna shuffle things around

- Dan: Yes. dx over m 14 Simon: d what? dx over 15 Dan: dx over m 16 Simon: You mean… d… d?

- Dan: I just did this whole thing now, this exact same problem.

- Simon: Are you moving dx over there?

- Dan: Yeah, you move dx over there, and them [his fingers bracket the mg−c
_{2}v^{2 }on the white board] over there and this [m] over there

- Simon: Let’s do it like this first.

- Dan: It doesn’t matter…

- Simon: m v dv equals m g minus c two v-squared dx 23 Dan: Yeah, if you want to write down that step. 24 Simon: And then we get the v on the other side

- Dan: We move this [fingers bracket mg−c
_{2}v^{2}] over there and that [points at dx] over here.

- Simon: Move the one with v over there?

- Dan: Yeah. Move this whole term [fingers bracket again] over there.

- Simon: mv over mg minus c two v squared equals dx. Even as Simon very formally wrote out all the steps required to separate variables, Dan simply “move[d]” terms around on the white board. But, Dan and Simon showed a different set of procedures than Sarah did in talking to Paul and Tim. They moved terms, in response to Simon’s desire to shuffle things around, but never spoke of dividing or multiplying.

There were several moments of ambiguity which suggest that Simon was unsure of what to do until Dan clarified. For example, in line 26, Simon asked which terms to move, and asked if he should only move the one term with v. Dan responded by making clear he meant the whole term, and Simon then accurately wrote down “mg minus c two v squared” (line 28). The gestures used by Dan at this point were consistent with other gestures he made in line 19: he bracketed the desired terms with fingers on either side. This bracketing gesture helped make sense of “them” (line 19) and “this whole term over there” (line 27) and stood in contrast to him simply pointing at “this” (line 19) when referring to the m in the equation.

Like Sarah, Dan was using gestures to indicate the grouping of terms.

Where Sarah had grouped, divided, and showed an equivalence between multiplication and moving (Fig. 2), Dan was focusing only on grouping and moving without namingmathematicaloperations.WhereTimhadonlymade an equivalence between multiplication and moving, Dan used moving to carry out the procedures of both multiplication (the dx in line 19) and division (the mg−c_{2}v^{2 }and m later in line 19).

As before, we make a graph of the resources used by one student in the group. We describe Dan in Fig. 3, because he does most of the work here. In the resource graph, we see that separating only consists of grouping which precedes moving, like with Sarah.

**C. Subtraction isn****’****t a good choice**

Our third episode of students separating variables contains a debate about which procedures are to be used. The students know that they must get all the terms of one sort to one side of the equation and all the others to the other side, but they are not sure how to do so. The students arrive at the correct answer, but their work is interesting to us because

FIG. 3. Dan’s resource of separation contains the procedures of grouping and moving.

they explicitly evaluate the use of an incorrect procedure and decide not to use it.

- Jared: Now we can separate, right?

- Keith: Yes.

- Jared: Do, we, do we want to pull the whole entire term over, or—like divide by an entire– 32 Keith: I think we want to divide– 33 Brian: What term are we gonna have on one side? Just the v?

- Jared: I think we want to pull this entire term over, divide by it.

- Brian: Yeah.

- Jared: Cause if, cause if– 37 Ann: Well, when–

- Jared: Because if you subtract you’re gonna have a zero, so there’s no way to separate the variables after that.

- Jared: (after a silence) So it’s going to be m over mg minus bv squared dv dx equals one. Which is why we wanted to, because we wanted to have the one on the other side instead of a zero. So m dv over mg minus bv squared equals dx. Right?

Like Sarah and Dan, Jared used gestures to group terms and talked about moving terms, though he was more like Sarah in that he also talked about division, which Dan had not talked about.

In line 31, Jared introduced the idea of division, a step that was supported by Keith in line 32. Jared was clear that he was talking about “the whole entire term” (line 31), indicating the “mg minus bv squared” (from line 39) by drawing exaggerated parentheses in the air above his sheet of paper. This stands in contrast to Brian’s question, in line 33, about which terms to divide by, the “term [with] just the v?” Brian’s question was akin to Simon’s question to Dan in lines 26 and 27. Jared responded by repeating his desire (like Sarah and Dan in their interactions) to use “this entire term”, (line 34).

We note that Jared, in line 34, drew an equivalency between “pull[ing] this entire term over” and “divid[ing] by it.” For Jared in this situation, moving was like division, but not like multiplication, as it was with Dan.

Later, Jared contrasted the procedure of pulling or division with the procedure of subtraction (line 38), which would lead to a zero and an inability to continue separating variables. We note that subtraction leading to a zero requires everything on that side of the equation to

FIG. 4. Jared’s procedure of separation included grouping and division and an explicit statement that subtraction was an inappropriate procedure.

be subtracted, not just the v term, telling us that Jared was working with the grouped set of terms on the right side of the equation. Jared elaborated on his thinking about subtraction in line 39, talking about needing a “one on the other side instead of a zero.” Having a 1 allowed for the implicit procedure (multiplication or moving is not specified) of the dx moving from the denominator on the left to the numerator on the right. A zero would not have allowed that (e.g., multiplying anything by zero leads to zero, so subtraction must be the wrong step to take).

We represent Jared’s use of separation in the graph shown in Fig. 4. As before, grouping is needed before one can divide, which in this case is equivalent to moving by “pulling over” from one side of the equation to the other. In contrast to Sarah, though, Jared suggests grouping before he mentions dividing, in line 31. Subtraction is considered, but it would lead to a “zero” which we describe as a canceling out because Jared describes it as ending the possibility of further work (in line 38). The idea is not pursued further by Jared or the others. Finally, though the dx term ended up on the other side of the equation from where it began, the group does not discuss multiplication or moving for it to have done so. As a result, multiplication is not part of the graph.

**D. ****Subtraction ****and ****division ****without ****grouping**

In the final episode of this paper, students used the same resources as have been named before, but some considered subtraction and division without also considering grouping. In this example, students are working with a slightly different equation, mdv=dt ¼ mg−cv, valid for v-dependent drag forces. Because of the length of the full discussion of separation of variables, we highlight only a one minute-long segment of the discussion before the students resolved their problems.

In our episode, Charlie was scribe, setting up the situation, while Bill and Derrick were the ones debating what to do. As the episode started, Charlie had written an incomplete equation on a whiteboard, but his head blocked the written work from our camera.

- Charlie: That’s why I tried to get the cv over there

- Bill: It could be one over, it could be one… one over, cv and one over… no…

- Bill: (a short moment later) Um, leave cv, leave it right where it is, move this [pointing to mg] over, take this [pointing to the dt in dv=dt] and put it over there and that would be one over cv.

- Derrick: Could you divide it by this whole side [brackets mg−cv with his fingers] and multiply it by dt? 44 Bill: Just subtract the mg

- Derrick: Really? [His fingers bracket mg-cv again]

- Charlie: Wait, I’m trying to do, like, two different things at once, but let me just erase, let me do it, let me do it!

Charlie, who started talking about the cv alone, recognized by the end that the two ideas (Bill’s and Derrick’s) needed to be resolved; we do not quote from the lengthy discussion that followed but note that they did, eventually, follow Derrick’s suggestion.

We note similarities in this interaction and the previous ones. As a reminder: Phil used separation without explanation, Sarah grouped terms she manipulated mathematically while assisting Paul and Tim, Simon and Dan grouped and moved terms without using any mathematical language, and Jared considered subtraction while grouping, moving, and dividing to separate variables. Bill used moving to describe both multiplication (line 42) and subtraction (as we draw the parallel between line 42 and 44). Like Brian (line 33) and Simon (lines 24 and 26), Charlie and Bill were not grouping terms and instead were working with either the cv or the mg alone. In contrast, in lines 43 and 45, Derrick suggested dividing by “this whole side” with gestures that clearly indicated that he meant both terms, like Sarah (lines 4 and 11), Dan (lines 19, 25, and 27), and Jared (lines 31 and 34). Like Sarah (lines 4 and 11), Derrick grouped with gestures, indicated the grouping by talking about “this whole side”, and multiplied, rather than moved, to have the differential term from the denominator on one side be in the numerator on the other side of the equation.

From this example, we observe the distinct procedures of grouping, subtraction, division, and multiplication. We observe Bill struggling to divide by cv (lines 41 and 42, leading to the one over cv) or subtract by mg (line 44, and possibly also 42) while not grouping. Also, we observe Derrick grouping and dividing and multiplying, without using the idea of subtraction.

We summarize Bill and Derrick’s very different approaches in two graphs in Fig. 5. In Bill’s graph, we indicate the implied equivalence between moving and multiplication (of dt) as well as the more explicit equivalence of moving and subtracting.

FIG. 5. Bill’s two uses of moving without grouping are in contrast to Derrick’s use of multiplication and division after grouping.

V. DISCUSSION

In these four episodes, as well as the single statement by Phil in line 1, we observe a variety of mathematical actions including subtraction, multiplication, and division, as well as moving and grouping. All have been used with the purpose of separation of variables, a necessary step in the particular problem that students were solving.

In this section, we discuss why these different mathematical actions can be thought of as procedural resources, highlight the importance of the grouping resource, and discuss the issue of expertise in separation. Finally, we discuss implications for teaching.

A. Theoretical issues around procedural resources

Earlier, we provided two different sets of criteria that help define resources. We find that the mathematical actions described in this paper are consistent with these criteria.

**1. Mathematical actions as procedural resources**

Consistent with Sayre and Wittmann [28], we find ample evidence that the mathematical actions are “individual” and “reusable” parts of problem solving. They were not only used by multiple students, but by individual students at multiple times during a single episode. We find support for the idea of activation, namely, that the ideas were active in a given setting. We have ample evidence from our classes to know that these students could all group, divide, and so on. We find that some of these ideas were activated and some were not, in this setting. Furthermore, we find that the actions were connected to other actions, such as when grouping was named as a necessary element before one could divide. Finally, we observe that there was some level of internal structure to a given action. Where Phil may have had a single action of separation, the other students indicated that several resources were needed; we can think of separation as being made up of these resources, and return to this point below. In a second example, the resource of moving seemed to have elements of pointing and dragging associated with it. By Sayre and Wittmann’s criteria, then, these mathematical actions are resources.

Consistent with Scherr’s analysis of the knowledge-inpieces model [17], we also find ample evidence that these mathematical actions were mutually independent, contextdependent, and indeterminate. They were mutually independent as can be seen from the ways that the students used the individual actions in a variety of orders and invoked them separately from each other. Each group did something slightly different with these few mathematical actions. They were context-dependent in that some actions were useful in this setting, but others (for example, subtraction) were not going to lead to a solution of the problem. Finally, there was nothing inherently incorrect about these mathematical actions. Subtraction is subtraction, regardless of context. The idea itself was not correct or incorrect, only its use.

By naming mathematical actions “procedural resources,” we expand the discussion of the use of mathematics in physics and allow some of the machinery associated with the resources framework to be associated with this area of study. We can talk about transfer, for example, while connecting to previous analysis of these issues [2]. We can analyze activation and how it relates to epistemological concerns [14–16]. In sum, identifying mathematical moves as procedural resources allows us to carry out analysis based on previous work on resources.

**2. Modeling resources as parts of resources**

One particular way we can make use of the resources framework is to help make sense of the scale and grain size of these procedural resources and how resources come together to form new resources. Here, we address Hammer et al. as they explain, “locally coherent sets of resources may, over time, become established as resources in their own right” [2], seeking to bring detail to their statement.

The data suggest that the students’ locally coherent set of resources were different. For Phil, there was a single resource (as there probably was for Sarah and Dan, as well, were they not working in a group setting). Sarah and Derrick used gestures to group and mathematical procedures to reorganize their equation. Dan used only gestures to both group and move. Jared used gestures and mathematics, but took the additional step of explicitly excluding the procedure of subtraction from the act of separation (at least in this context, because in others it might be perfectly reasonable). It is possible that Phil’s single action of separation is simply the reification [31,32] of locally coherent sets of resources.

The variety of local coherences suggests that there is no single path toward learning separation of variables, and that many different paths can lead to separation as a resource in its own right. We do not have the data to determine how the use of these resources develops in time. Would Jared eventually act like Sarah, who might later act like Dan, until they are all acting like Phil? Our data do not provide an answer. Further work would be needed to determine the details of locally coherent resources becoming resources of their own.

Our work suggests that coherence may be determined by one’s group interactions, since Sarah and Dan both seemed to engage in discussion only because of their group members. Further study would be required in this area of resource creation.

Finally, our work suggests that resource compilation, reification, or creation is a process that requires further study. We cannot expect experts to carefully carry out every step of every problem; at some point, facility with new ideas is needed. At the same time, we expect experts to be able to “unpack” resources. We suggest that Sarah and Dan are engaged in such an unpacking, as they interact with their groups.

B. The importance of particular resources: Grouping

Grouping played an essential role in all four episodes Students in each episode (Tim, Simon, Brian, Bill, Charlie) questioned which terms one must move or divide by. The task of moving all of one variable to one side of the equation would suggest that one only move the terms that include that variable. Moving both the mg and the bv^{2 }together is not the obvious choice, unless one thinks ahead in the problem. The particular reason that grouping is needed is because the typical next step (of multiplying by the dx) would lead to mixed terms if one only subtracted or divided by bv^{2}.

Some students talked about dividing or moving before they defined what they were dividing by or moving. Sarah (lines 4 and 11) and Derrick (line 43) talked about dividing and only then about grouping. Jared (line 31 and 24) wanted to “pull the whole term over,” so the pull preceded the definition of what he was pulling. Maybe these students knew what they were moving or pulling or dividing by before they spoke. Our data cannot give us further information.

A different solution lies in a different order of operations. Rather than moving the mg−bv^{2 }term first, Dan created a mathematical solution that created a grouped term to manipulate next. Dan (line 19) moved the dx first, which basically required grouping as it came to multiply ðmg−bv^{2}Þ, creating parentheses which indicated the grouping that happened (as Simon wrote in line 22). The difference in order made the problem of identifying grouped terms easier.

We note that grouping typically happened with a gesture. Dan and Derrick used their thumb and a finger to bracket mg−bv^{2}, indicating what they were about to move or divide by. Sarah circled the two terms. Jared indicated parentheses around the terms, much like Simon wrote. For all these students, gesture were an important act in grouping. From our data, we cannot tell if this was because of the role of gesture in explanations in a group environment or if it is related to something more fundamental about the role of gesture in our thinking.

C. Resources and expertise

We have already mentioned the issue of unpacking an idea as a type of expertise and suggested that Sarah and Dan showed this level of expertise. In this section, we discuss further issues that relate to expertise.

Given that all the groups successfully separated variables, we note large differences in their ease of carrying out the procedure. Phil flew through the task. Dan was only slowed down by Simon’s writing. Sarah could have moved on easily, were it not for helping Paul and Tim. Jared’s group struggled. Derrick’s group was mired in the longest discussion of all these groups, mostly because they were not grouping terms correctly.

Our discussion of these observations is tied to the number of ideas they had to keep in mind, and the limited ability of working memory to consider too many ideas at once. Derrick’s group was caught by not having included a necessary resource. Jared’s group had many questions and procedures to consider: what is the consequence of subtraction? a canceling that is not desired; is pulling over the same as division? and what do we divide by? To think of all those things and consider them carefully would slow any group down. Sarah, in contrast, had a clear task to convey and did so twice. Phil just solved the problem and moved on, treating the act of separation as a single procedure.

We expect of our students that they learn to solve problems quickly, including carrying out certain mathematical steps automatically. This might be a kind of quick pattern matching, or it might be the reification of locally coherent sets of resources into a single resource. If we think of resources as chunks of working memory, as suggested above, we suggest that a single chunk of separation can be worked with easily (as Phil does), but that the use of additional procedural resources requires additional work and slows students down (as Dan and Sarah seem to be slowed by their groups or Jared by his many considerations). Of course, part of the slowness comes from having to name and elaborate what is meant by the procedures.

A different kind of expertise can be found in Jared, in a way that is not observed in the other students. Jared considered more than one method for solving the problem and was able to account for why this method was incorrect. That kind of thinking required that he be working with the sub-pieces of separation rather than the resource itself. This second definition of expertise is about the ability to analyze situations in detail and understand not just what works but also what doesn’t.

The resources analysis provided in this paper allows for multiple forms of expertise to be discussed within the theoretical framework.

D. Implications for teaching

In this paper, we have looked closely at students solving one part of a many part problem. The differences in student actions suggest some of the difficulty of teaching students about a topic like separation of variables. Given the importance of the topic in intermediate and advanced physics (including when solving partial differential equations), and given the role of algebraic manipulation in all of physics, the insights gathered from this paper have implications for our teaching.

First, applying a resources perspective to mathematical actions expands the implications of pedagogical suggestions made in the literature [23,48]. On a theoretical level, we have not engaged with the implication that mathematical actions are thoughts, an issue which would require a longer exploration of work in the learning sciences (see, for example, the work of Nemirovsky and collaborators [49–52]). Taking this perspective would provide us with a foundation for modeling group interactions and group reasoning, a useful tool when our data come from authentic classroom group activities, like in this paper. On a more concrete level, we should expect to find variety in students’ mathematical work much like we find it in their conceptual reasoning, and we should also be providing them with problems to solve in which a greater variety of mathematical reasoning is possible.

Second, students have a multitude of ways of correctly solving this problem. Taken more generally, physics problems typically have multiple solutions. As an instructor, it is important to recognize all these solution pathways and to recognize their value and their shortcomings. It may be that one is more effective than the other, such as Dan effectively multiplying by dx before dividing by mg−bv^{2}. Knowing what to suggest to a student at a given moment requires an understanding of the variety of student thinking as has been provided in this paper.

Third, our students had the most difficulty with grouping terms. Where other procedural resources in this paper had formal mathematical names (subtract, divide, etc.), grouping has associated mathematical representations (using parentheses correctly) but does not have as formal a name. Simply being aware that students struggle with this idea might be of help to an instructor, or a more direct intervention might be developed.

VI. CONCLUSION

The mathematical procedure of separation of variables is common to upper division physics and is a common tool used when solvingfirst order differential equations.Students begin to learn the procedure in their post-introductory courses. In our study, the first course in which students used the idea in physics was Intermediate Mechanics.

In this paper, we have argued that there are multiple procedural resources that can be brought into problem solving, and that these resources are used in different combinations by different groups of students. We consider those resources which are procedural, rather than conceptual, because we can more easily observe the use of a specific procedure than we could infer the concepts a student might have when carrying it out. These procedural resources fulfill the criteria set forth by Sayre and Wittmann [28] and Scherr [17].

We have discussed the need for creating new resources (or compiling or reifying existing networks of resources) as a way of making problem solving more efficient, an expertlike behavior. We have suggested that the episodes we present are pathways to the creation of this new resource, separation, and have shown one example (Phil, in line 1) of what the use of this resource might look like. We have also shown what it looks like when students are not using a single resource to separate variables, including when they are evaluating which resources to use and choosing not to use some. In the process, we have discussed how multiple forms of expertise can be described within a resources framework.

Extensions of this work might be found in several areas. We have suggested two. First, if we consider mathematical actions as a kind of thought, we raise questions about the nature of knowledge in a social and material context, a topic we have discussed previously [39,53,54]. Second, an awareness of the variability in students’ mathematical thinking is an essential element of advanced physics instruction. In particular, focusing on the utility of a resource, rather than its correctness, allows for a refinement of our model of teaching [48]. Speer and colleagues have investigated teacher understanding of the ideas that students bring to the college-level classroom for example [55,56]. For instance, a student solving a similar problem on an examination (as described in Ref. [57]) used the variational principle to solve this problem—a perfectly valid and correct approach, though one rarely used by physics faculty, in our experience. Other extensions might be to investigate the use of procedural resources over time and in new contexts, to study the interactions of procedural and epistemological resources, or to define how procedural resources are parts of larger networks of ideas, such as in coordination class theory [5,58,59].

Using the resources framework, we are able to make sense of students’ differences while also showing what they have in common. Our contribution, then, is both to the knowledge of students’ uses of mathematics in a physics classroom and to the development of a model of knowledge that plays a central role in our research community.

ACKNOWLEDGMENTS

A previous version of this analysis was published as Ref. [9]. In returning to that analysis, we have refined several points. Our thanks to Brandon Bucy and Kate Hayes for assistance in videorecording and Lauren BarthCohen, Dan Capps, and Laura Millay for their help in preparing this manuscript. This work was supported in part by the National Science Foundation under Grants No. DUE-0441426, No. DUE-0442388, and No. REC0633951, and No. MSP-0962805.

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**EMBODIED MATHEMATICAL IMAGINATION AND COGNITION (EMIC) WORKING **

GROUP | ||

Mitchell J. Nathan | Erin R. Ottmar | Dor Abrahamson |

Univ. of Wisconsin, Madison | Worcester Polytechnic Inst. | Univ. of California, Berkeley |

MNathan@wisc.edu
| EROttmar@wpi.edu | dor@berkeley.edu |

Caroline Williams-Pierce | Candace Walkington | Ricardo Nemirovsky |

Univ. at Albany, SUNY | Southern Methodist Univ. | San Diego State Univ. |

CWilliamsPierce@albany.edu | CWalkington@smu.edu | Nemirovsky@mail.sdsu.edu |

*Embodied cognition is growing in theoretical importance and as a driving set of design principles for curriculum activities and technology innovations for mathematics education. The central aim of the EMIC (Embodied Mathematical Imagination and Cognition) Working Group is to attract engaged and inspired colleagues into a growing community of discourse around theoretical, technological, and methodological developments for advancing the study of embodied cognition for mathematics education. A thriving, informed, and interconnected community of scholars organized around embodied mathematical cognition will broaden the range of activities, practices, and emerging technologies that count as mathematical. EMIC builds upon our 2015 working group, and investigations in formal and informal education and workplace settings to bolster and refine the theoretical underpinnings of an embodied view of mathematical thinking and teaching, while reaching educational practitioners at all levels of administration and across the lifespan. *

Keywords: Classroom Discourse, Cognition, Informal Education, Learning Theory

**Motivations for This Working Group **

Recent empirical, theoretical and methodological developments in embodied cognition and gesture studies provide a solid and generative foundation for the establishment of **an Embodied Mathematical Imagination and Cognition (EMIC) Working Group **for PME-NA. The central aim of EMIC is to attract engaged and inspired colleagues into a growing community of discourse around theoretical, technological, and methodological developments for advancing the study of embodied cognition for mathematics education, including, but not limited to, studies of mathematical reasoning, instruction, the design and use of technological innovations, learning in and outside of formal educational settings, and across the lifespan.

The interplay of multiple perspectives and intellectual trajectories is vital for the study of embodied mathematical cognition to flourish. Partial confluences and differences have to be maintained throughout the conversations; this is because instead of being oriented towards a single and unified theory of mathematical cognition, EMIC strives to establish a philosophical/educational “salon” in which entrenched dualisms, such as mind/body, language/materiality, or signifier/signified are subject to an ongoing and stirring criticism. A thriving, informed, and interconnected community of scholars organized around embodied mathematical cognition will broaden the range of activities and emerging technologies that count as mathematical, and envision alternative forms of engagement with mathematical ideas and practices (e.g., De Freitas & Sinclair, 2014). This broadening is particularly important at a time when schools and communities in North America face persistent achievement gaps between groups of students from many ethnic backgrounds, geographic regions, and socioeconomic circumstances (Ladson-Billings, 1995; Moses & Cobb, 2001; Rosebery, Warren, Ballenger & Ogonowski, 2005). There also is a need to articulate evidence-based findings and principles of embodied cognition to the research and development communities that are looking to generate and disseminate innovative programs for promoting mathematics learning through movement (e.g., Petrick Smith, King, & Hoyte, 2014). Generating, evaluating, and curating empirically validated and reliable methods for promoting mathematical development and effective instruction through embodied activities that are engaging and curricularly relevant is an urgent societal goal.

**The EMIC Working Group: A Brief History **

The first meeting of the EMIC working group took place in East Lansing, MI during PME-NA 2015. It has a somewhat longer origin, however, growing out of several earlier collaborative efforts to review the existing literature, document embodied behaviors, and design theoretically motivated interventions. One early event was the organization of the 2007 AERA symposium, “Mathematics Learning and Embodied Cognition” (Nemirovsky, 2007). This and other gatherings led to a funded NSF “catalyst” grant to explore a Science of Learning Center, which was to involve scholars from multiple institutions and countries. Though unfunded, those SLC efforts shaped a subsequent 6-year NSF-REESE grant, “Tangibility for the Teaching, Learning, and Communicating of Mathematics,” starting in 2008. Interest from the International PME community in this topic grew, and led to special issues of *Educational Studies in Mathematics* (2009), *The Journal of the Learning Sciences *(2012), and an NCTM 2013 research pre-session keynote panel,“Embodied cognition: What it means to know and do mathematics,” along with a series of academic presentations, book chapters, and journal articles, as well as several masters’ theses and doctoral dissertations. By now, several research programs have formed to investigate the embodied nature of mathematics (e.g., Abrahamson 2014; Alibali & Nathan, 2012; Arzarello et al., 2009; De Freitas & Sinclair, 2014; Edwards, Ferrara, & Moore-Russo, 2014; Lakoff & Núñez, 2000; Radford 2009), demonstrating a “critical mass” of projects, findings, senior and junior investigators, and conceptual frameworks to support an on-going community of likeminded scholars within the mathematics education research community.

It was within this historical context that approximately 22 members of PME-NA 2015 came together for three 90-min sessions of semi-structured activities. On **Day 1,** the organizers engaged attendees in some of the body-based math activities used in their research on proportional reasoning and geometry. We discussed how embodied theories are advancing our understanding of mathematical thinking, and how these ideas are shaping a new class of educational interventions. During **Day 2**, we used hands-on activities to expand our own understanding of topology. We then built on the emerging rapport among the group to hold a facilitated discussion of the potential intellectual benefits of forming a self-sustaining Working Group on embodied cognition, along with the necessary infrastructure it would need to maintain. Several concrete proposals led to the list of Future Steps on **Day 3**. However, before we tackled those matters, participants began the session doing math games and activities in small groups, including Spirograph, Set, Rush Hour, Tangrams, and Mastermind. We reflected on how some games and activities draw people into rich mathematical thinking and actions, and how we naturally engage in math through these activities. Day 3 culminated in an organized list of Future Steps, with some working group members assigned to specific tasks.

Since our first meeting at PME-NA 2015 **our accomplishments** include:

1.! Creating a contact list with names and emails of attendees (*n* = 22) and other interested scholars who could not attend PME-NA 2015 (*n* = 25);

2.! Developing a group website using the Google Sites platform to support ongoing interactions throughout the year

3.! Joint submission of an NSF DRK-12 by members who first met during the 2015 EMIC sessions

4.! Some senior members joining a junior member’s NSF ITEST grant proposal

5.! Submitting a proposal for the continuation of the EMIC WG to PME-NA 2016

6.! Examining the potential for an NSF Research Coordination Network (RCN)

**Focal Issues in the Psychology of Mathematics Education **

Emerging, yet still influential, views of thinking and learning as embodied experiences have grown from several major intellectual developments in philosophy, psychology, anthropology, education, and the learning sciences that frame human communication as multi-modal interaction, and human thinking as multi-modal simulation of sensory-motor activity (Clark, 2008; Hostetter & Alibali, 2008; Lave, 1988; Nathan, 2014; Varela et al., 1992; Wilson, 2002). These views acknowledge the centrality of both unconscious and conscious motor and perceptual processes for influencing conscious awareness, and of embodied experience as following/producing pathways through social and cultural space. As Stevens (2012, p. 346) argues in his introduction to the *JLS* special issue on embodiment of mathematical reasoning, it will be hard to consign the body to the sidelines of mathematical cognition ever again if our goal is to make sense of how people make sense and take action with mathematical ideas, tools, and forms.

Four major ideas exemplify the plurality of ways that embodied cognition perspectives are relevant for the study of mathematical understanding: (1) Grounding of abstraction in perceptuomotor activity as one alternative to representing concepts as purely amodal, abstract, arbitrary, and self-referential symbol systems. This conception shifts the locus of “thinking” from a central processor to a distributed web of perceptuo-motor activity situated within a physical and social setting. (2) Cognition is for action. This tenet proposes that things, including mathematical symbols and representations, are understood by the actions and practices we can perform with them, and by mentally simulating and imagining the actions and practices that underlie or constitute them. (3) Mathematics learning is always affective: there are no purely procedural or “neutral” forms of reasoning detached from the circulation of bodily-based feelings and interpretations surrounding our encounters with them. (4) Mathematical ideas are conveyed using rich, multimodal forms of communication, including gestures and tangible objects in the world.

Alongside these theoretical developments have been technical advances in multi-modal and spatial analysis, which allow scholars to collect new sources of evidence and subject them to powerful analytic procedures, from which they may propose new theories of embodied mathematical cognition and learning. Just as the “linguistic turn” in the social sciences was largely made possible by the innovation that enabled scholars to collect audio recordings of human speech and conversation *in situ*, growth of interest in multi-modal aspects of communication have been enabled by high quality video recording of human activity (e.g., Alibali et al., 2014; Levine & Scollon, 2004), motion capture technology (Hall, Ma, & Nemirovsky, 2015; Sinclair, 2014), and developments in brain imaging (e.g., Barsalou, 2008; Gallese & Lakoff, 2005).

**Plan for Active Engagement of Participants **

Our formula from PME-NA 2015 proved to be effective: By inviting participants into math activities at the beginning of each session, we were rapidly drawn into those very aspects of mathematics that we find most rewarding. Facilitated discussions (and we now have many effective members who can trade off in this role!) then help us all to “pull back” to the theoretical and methodological issues that are central to advancing math education research. Within this structure of beginning with mathematical activities and facilitated discussions, on **Day 1 **we plan to introduce our new website, demonstrate the online resources for building sustained community, and revisit and further develop the items listed in our Future Steps, including assigning roles to EMIC members. On

**Day 2**, we will discuss concrete goals and products. One example is the NSF Research Coordination

Network (RCN), as a potential compliment to the PME-NA Working Group. The RCN is not intended to promote any one particular research program, but rather to build the networked community of international scholars from which many fruitful lines of inquiry can emerge.

Commensurate with the aims of the RCN, we will explore ways to share information and ideas, coordinate ongoing or planned research activities, foster synthesis and new collaborations, develop community standards, and in other ways advance science and education through communication and sharing of ideas.

This sharing and coordination will continue into **Day 3**. One proposed activity is to perform a live concept mapping activity that is displayed for all participants to explore the range of EMIC topics and identify common conceptual structure. Harkening back to the four major ideas that we developed earlier, sample seed topics for organizing this activity will be explored, such as:

1.! Grounding Abstractions

a.! Conceptual blending (Tunner & Fauconnier, 1995) & metaphor (Lakoff & Núñez,

2000)

b.! Perceptuo-motor grounding of abstractions (Barsalou, 2008; Glenberg, 1997)

c.! Progressive formalization (Nathan, 2012; Romberg, 2001) & concreteness fading

(Fyfe, McNeil, Son, & Goldstone, 2014)

d.! Use of manipulatives (Martin & Schwartz, 2005)

2.! Cognition is for Action: Designing interactive learning environments for EMIC

a.! Development of spatial reasoning (Liu, Uttal, Marulis, & Newcombe, 2008)

b.! Math cognition through action (Abrahamson, 2014; Nathan et al., 2014)

c.! Perceptual boundedness (Bieda & Nathan, 2009)

d.! Perceptuomotor integration (Nemirovsky, Kelton, & Rhodehamel, 2013)

e.! Attentional anchors and the emergence of mathematical objects (Abrahamson &

Sánchez–García, in press; Abrahamson, Shayan, Bakker, & Van der Schaaf, in press)

f.! Mathematical imagination (Nemirovsky, Kelton, & Rhodehamel, 2012)

g.! Students’ integer arithmetic learning depends on their actions (Nurnberger-Haag, 2015).

3.! Affective Mathematics

a.! Modal engagements (Hall & Nemirovsky, 2012; Nathan et al., 2013)

b.! Sensuous cognition (Radford, 2009)

4.! Gesture and Multimodality

a.! Gesture & multimodal instruction (Alibali & Nathan 2012; Cook et al., 2008;

Edwards, 2009)

b.! Bodily activity of professional mathematicians (Nemirovsky & Smith, 2013)

c.! Simulation of sensory-motor activity (Hostetter & Alibali, 2008; Nemirovsky & Ferrara, 2009)

Finally, we will introduce the EMIC website (see Figure 1) and invite members to join, and to encourage their interested colleagues to email Caro at cwilliamspierce@albany.edu for access. On this website, we have a list of members with their emails and bios, information about our PME-NA presence, and short personal introduction videos. We’ve also created a space for members to share information about their research activities – particularly for videos of the complex gesture and action-based interactions that are difficult to express in text format. In addition, we have a common publications repository to share files or links (including to ResearchGate or Academia.edu publication profiles, so members don’t have to upload their files in multiple places). At our 2015 working group, some junior members expressed particular interest in this literature support for their pending theses, while more senior members were eager to share and organize the emerging body of work on embodied math education. We’ve also linked the Google Sites platform directly to a Google Group, so members can participate in online forums (or the linked listserv), and discuss cutting edge topics, share in-progress working papers for review, or advertise for conferences, special issues, or other EMIC-relevant opportunities.

**Figure 1. **The EMIC website landing page serves as one of the ways EMIC members can share information about themselves and their work, support a common paper repository, post relevant announcements, and coordinate emerging collaborations.

**Follow-up Activities **

Even prior to our first anniversary, we have already seen a great deal of progress. This is perhaps best exemplified by coming together of the EMIC website and this proposal submission, which draws across multiple institutions. We envision an emergent process for the specific follow-up activities based on participant input and our multi-day discussions. At a minimum, we will continue to develop a list of interested participants and grant them all access to our common discussion forum and literature compilation. Those that are interested in the NSF RCN plan will work to form the international set of collaborations and articulate the intellectual topics that will knit the network together. One additional set of activities we hope to explore is to introduce educational practitioners at all levels of administration and across the lifespan to the power and utility of the EMIC

perspective. We thus will strive to explore ways to reach farther outside of our young group to continually make our work relevant, while also seeking to bolster and refine the theoretical underpinnings of an embodied view of mathematical thinking and teaching.

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** **

Educ Stud Math (2013) 82:169–181

DOI 10.1007/s10649-012-9428-4

Algebraic manipulation as motion within a landscape

Michael C. Wittmann & Virginia J. Flood &

Katrina E. Black

Published online: 14 September 2012

# Springer Science+Business Media B.V. 2012

Abstract We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of embodied cognition and use conceptual metaphors such as the path-source-goal schema and the idea of fictive motion. We find that students solving the problem correctly and efficiently do not use overt mathematical language like multiplication or division. Instead, their gestures and ambiguous speech of moving are the only algebra used at that moment.

Keywords Gesture analysis . Algebra . Embodied cognition . Physics . Differential equations . Separation of variables

1 Introduction

In this paper, we show that students treat an equation as if consisting of physically grabable objects that can be moved around on the page. This movement is not in place of or representative of the formal algebra that uses overt mathematical terminology; it is the algebra, with the end result that the variables are arranged as desired. Our context is second-year university students solving first order linear differential equations in a physics course. They often use the method of separation of variables to arrive at two integrals that can be integrated in order to find the solution to the problem. How they go about separating

M. C. Wittmann : K. E. Black

Department of Physics and Astronomy, University of Maine, Orono, ME, USA e-mail: mwittmann@maine.edu

M. C. Wittmann : V. J. Flood

Center for Research in STEM Education, University of Maine, Orono, ME, USA

the variables—namely, by moving them from location to location on a whiteboard—is the topic of this paper.

A formal approach to simplifying algebraic equations includes subtracting something from both sides of the equation, dividing both sides by a quantity, and so on, until one has rewritten the equation in a form that suits one’s purposes. Kieran (1992, 2007) has summarized the research on how students simplify equations. These activities include the formal steps of transposing (i.e., changing signs when changing sides of the equality) and carrying out the same operation on both sides of an equation.

It is also possible to think more informally about moving terms in an equation from place to place until they are arranged differently from before. Particularly when talking of separating variables in a first order linear differential equation, people might speak of gathering terms with like variables on each side of the equation. This would be consistent with separating objects in the real world, sorting them into and placing them at different locations. Though words like separating, gathering, moving, and arranging are meant metaphorically, we show in this paper that they are consistent with how two students practice mathematics. Our results extend the discussion of gesture, conceptual metaphors, and embodied cognition in mathematics education.

Our study takes place as part of a larger attempt to understand the use of mathematics in university physics courses. Students in a physics course should have a facile understanding of algebraic manipulation and, at the appropriate level, calculus and differential equations. Students’ understanding of differential equations has been explored by many researchers (Camacho-Machín, Perdomo-Díaz, & Santos-Trigo, 2012; Rasmussen, 2001; Rowland & Jovanoski, 2004; Soon, Lioe, & Mcinnes, 2011; Stephan & Rasmussen, 2002) but the emphasis in this work has not been on the algebraic manipulation required to separate variables when finding solutions for separable first order differential equations. Instead, the focus has been on common student difficulties, classroom practices, interpretation of the individual mathematical terms, and the nature of solutions which are not constants but themselves functions.

Here, we seek to understand the ways that physics students accomplish the algebraic manipulation needed to actually separate variables. We analyze one episode of students’ successful separation of variables. Our discussion extends the work summarized by Kieran (1992, 2007) by adding detailed microanalytic ethnographic observations and interpretations of the methods by which students simplify and manipulate algebraic equations.

2 Movement in an algebraic space

To establish the central claim of our paper, that students do algebra by rearranging objects on the page using gestures, we first summarize previous work on conceptual metaphor, including the theories of image schemas and fictive motion. We then review ties between gesture and talking and thinking about space, and describe the spatial treatment of mathematical equations.

In Where Mathematics Comes From, Lakoff and Núñez (2000) propose that metaphor is “the basic means by which abstract thought is made possible” (p. 39). We adopt their perspective of embodied cognition: that all human concepts, including mathematical concepts, are based in the perceptual motor system experiences we have while interacting with the world around us. Image schemas and specific conceptual metaphors are components of the metaphorical thinking that allow us to consider complex mathematics. In this paper, we begin with the metaphor of mathematical entities as physical objects and show how gestures indicate the movement of these objects as the algebraic manipulation of equations.

Mathematical entities can be spoken of as if they were physical objects. Successful mathematicians sometimes discuss abstract mathematical ideas as if they were material bodies (Sfard, 1991, 1994). Sfard writes, “Seeing a mathematical entity as an object means being capable of referring to is as if it was a real thing, a static structure existing somewhere in space and time.” (1994, p. 4) We take particular notice of the idea of a “real thing… existing somewhere in space and time.” For the purposes of this paper, we think of mathematical objects as taking up space, having a location, and being able to move—in other words, that they are treated like physical objects. We provide evidence for this interpretation below.

The motion of mathematical objects can be described by the source-path-goal image schema (Lakoff & Núñez, 2000), which includes a trajector, its source location, a trajectory, and a goal destination. One of the ways source-path-goal schemas are manifested is in the fictive motion metaphor. Len Talmy (1996) first proposed fictive motion metaphors to describe linguistic expressions in which everyday static scenes are described with terms that denote motion through a landscape. For example, when saying, “the road runs along the coast,” the road is treated as if it is moving, though it is not. The road doesn’t “run” in any literal sense of the word, but it remains close to the coast along its path. Similarly, saying, “the trail goes through the desert,” one imagines traveling on a trail, but the trail itself is of course static in the desert. (Both examples are from Matlock, 2004). In each case, the dynamic nature of these scenes is imaginary and cannot be interpreted literally. Núñez (2004) explains fictive motion as “a fundamental embodied cognitive mechanism through which we unconsciously (and effortlessly) conceptualize static entities in dynamic terms.” Examples of fictive motion metaphors are prevalent in mathematics. The statements “sin 1/x oscillates more and more as x approaches zero,” or “g never goes beyond 1,” involve motions which are not actually described by the mathematics (Núñez, 2004).

Past work has highlighted the presence of fictive motion words in student explanations of strategies for transforming and solving algebra equations. Staats and Batteen (2009) observed that students use verbs related to motion as resources for expressing mathematical ideas (p. 67). In their discussion of indexical language in algebra transformations, they suggested that student statements like, “you slide that 12 over to the other side” reflect a target and a path (Staats & Batteen, 2010). Staats and Batteen’s analysis focused on elements of speech, while our work explores the rich interplay of students’ speech, bodily motion, and interaction with the environment, with a focus on gestural versions of sliding, sometimes when the accompanying speech is ambiguous.

There is a strong connection between gesturing and verbal communication about spatiomotoric representations (Emmorey & Casey, 2001). Results by Lavergne and Kimura (1987) suggest that gestures are more common when subjects talk about spatial information (walking around a campus), as compared to neutral topics (names and occupations of family members) or verbal topics (recounting of a recent telephone conversation). Feyereisen and Havard (1999) found that representational gestures occurred most frequently when subjects recalled and explained motor tasks and second most frequently during explanations of visual (spatial) information. In another study, participants asked to think aloud in mechanical reasoning problems gestured while giving explanations in 90 % of the problems (Hegarty, Mayer, Kriz, & Keehner, 2005). Rauscher, Krauss and Chen (1996) used narratives about Wile E. Coyote cartoons to show that during “spatial content phrases,” gesturing was nearly five times more frequent than it was for nonspatial phrases (ca. 0.5 vs. 0.1 gestures per word). In sum, gestures commonly arise during verbal communication about spatial information, and occur more frequently when people discuss spatial relationships.

Gestures even arise spontaneously as people solve spatial visualization problems when they are not engaged in verbal communication. Students solving gear rotation problems (read to them aloud) silently stared at their moving hands before answering questions about the direction of gear rotation (Schwartz & Black, 1996). Another study that asked students to compare complex shapes in different orientations (Chu & Kita, 2011) found that students produced silent or “co-thought” gestures more often for difficult rotations than easier ones (240° vs. 90°). Based on these results, we believe that gesturing during spatial problem solving is a natural and common occurrence.

Experiencing and interpreting inscriptions like algebra equations may involve a kind of spatial reasoning. Kirshner and Awtry (2004) presented evidence that as students initially learn algebra, they are immediately receptive to the visual structure (the typical spacing and positioning) of algebraic rules, independently of the declarative content of those rules. For example, students learn to rely on the closer spacing of two multiplied terms (the 4x in 4x+2) as a signifier of precedence over the further spaced additive term. Similarly, even subtle space changes in the positioning of arithmetic sentences have been found to affect the order of how people choose to perform operations (Landy & Goldstone, 2007). Erlwanger (1973) stated that Benny’s famously discussed erroneous math rules appeared to rely mostly on spatial elements, the “awareness of patterns printed on a page,” rather than on mistaken operations (cited in Kirshner & Awtry, 2004).

Radford and Puig (2007) proposed that simplifying equations requires diagrammatic thinking based in spatio-sensual reasoning:

In perceiving the signs as parts of a diagram or “a skeleton-like sketch” stressing “relations between its parts” (Stjernfelt, 2000, p. 363), a definite shift of attention occurs: attention moves from the verbal meaning of signs to the shapes of the expressions that they constitute. This shift leads one to see the equation as an iconic, spatial object. (p. 156)

The authors provide justification for this interpretation in the analysis of a classroom episode where a student treats an equation as a situated spatial object. In our analysis in this paper, we extend the idea to look at individual elements of the equation as physical and movable objects.

The kinds of gestures people use to describe solutions to spatial problems depend on the physical affordances of the environment in which they work. Cook and Tanenhaus (2009) found that gestures correspond with specific types of actions that were used for spatial problem solving. When participants recounted how they solved the Tower of Hanoi problem, their gestures were different depending on whether they had solved it with a physical puzzle or a computer simulation. In the computer simulation, it was possible to click and drag, while in real life, the pieces had to be lifted up over the pegs. The trajectories of gestures were more arced for people who had solved the puzzle with the traditional pieces. If terms in algebraic equations are treated as physical objects moving along a path, student gesturing during their manipulation would be expected and may serve as evidence of the spatial nature and treatment of algebraic equations.

While the path and the motion of mathematical objects is metaphorical, enacting this event by hand is a tangible experience. Understanding the cognitive process of how an algebraic manipulation is accomplished necessitates paying careful attention to how the body interacts with the environment. Nemirovsky (2003) explains,

The actions one engages in mathematical work, such as writing down an equation, are as perceptuo-motor acts as the ones of kicking a ball or eating a sandwich; elements of, say, an equation-writing act and other perceptuo-motor activities relevant to the context at hand are not merely accompanying the thought, but are the thought itself as well as the experience of what the thought is about. (p. 109)

Hutchins (Hutchins, 1995; Hutchins & Palen, 1997) argues that the perceptual-motoric experience of gesture within the context of an inscription can be described as a form of thinking and understanding distributed over an interaction with the external space of the environment.

In the analysis to come, we approach speech and gesture in a manner closely aligned with views proposed by Radford (2009), Nemirovsky (2003, Nemirovsky & Ferrara, 2009) and Hutchins (Hutchins & Palen, 1997). We reject the idea that gestures are indicators or reflections of internal mental representations. They are, instead, actual components of thinking, inseparable from the environment in which they occur and the body that carries them out. Specifically, we advocate for Radford’s framework of “a sensuous conception of thinking—one in which gestures and bodily actions are not the ephemeral symptoms announcing the imminent arrival of abstract thinking, but genuine constituents of it” (p. 123). Making sense of students’ algebraic manipulations and what happens in the complex moments of a mathematical operation enacted by hand is incomplete without considering the action itself as part of cognition.

To review, we have developed a framework that suggests how source-path-goal image schemas, the idea of fictive motion, and the interpretation of elements of equations as spatial objects explain how gestural manipulations of mathematical terms constitute elements of mathematical thought. Gestures should be common when thinking about spatial information. Particular gestures, sometimes accompanied by speech indicating motion, reveal the trajectories that students move mathematical objects along. In sum, we propose that students rearranging algebraic equations to separate variables treat mathematical terms as physical objects that move within a landscape of the surface on which the equation is written.

3 The mathematical problem

The episode described below requires that students take Eq. 1 and manipulate it to arrive at Eq. 2. The details of the origin of this equation are not important for the analysis that follows, though it is useful to point out that this is an air resistance problem in a physics class, and one point of the exercise is a separation of variables such that all terms involving v and dv are on one side of the equation, while all terms involving dx end up on the other side of the equation. The quantities m and c_{2 }are constants (mass and the air damping constant). Once separated, each side of the equation can be integrated independently, allowing for a function of v to describe the velocity of the object as a function of vertical position, x. (The names of the variables in Eqs. 1 and 2 were determined by the students described below; we might have used y instead of x and b instead of c_{2}).

dv mv ¼ mg c | ð^{1}Þ |

dv mv ¼ dx mg c | ð^{2}Þ |

The data come from videotaped group quizzes that took place in a sophomore-level physics course taught by the first author. Students were familiar with group work on guided inquiry worksheets (Ambrose, 2004; Wittmann & Ambrose, 2007) but had not been in a group testing environment before. Students worked in small groups of 3 or 4 students to solve the given problem. They had arrived at Eq. 1 as part of their own analysis, and recognized that they needed to get to Eq. 2.

Students in the class are typically physics majors, though some mathematics, engineering, education, or computer science students also take the course. Typically half the class is coenrolled in a course on differential equations, while the other half has taken the course already. By the time of the group quiz, students co-enrolled in the differential equations course had already studied first order differential equations and the solution method of separation of variables. Because this is relatively advanced university-level mathematics, we expect a high degree of fluency with manipulation of algebraic statements; we expected students to easily derive Eq. 2 from Eq. 1. Instead, many groups struggled.

Group quizzes create a high-stakes environment in which students typically discuss their thinking more openly than in ungraded group learning activities in the classroom (Black & Wittmann, 2007, 2009). Unlike group interactions involving homework help or in-class group learning activities that are part of instruction, the quizzes are to be graded. Thus, students might be motivated to do their very best work and to argue coherently when they do not agree with the methods of their classmates. At the same time, we might expect that some students do not speak up because the pressure to get the right answer creates a group dynamic in which the best students speak. Furthermore, we expect to observe students in conflict over whether to write down steps of a derivation so that they get partial credit compared to quickly arriving at the correct solution so that they can get on to the next problem. Over a several-year period, many groups were videotaped. We discuss in detail the results of one group as they solve for Eq. 2. This group was chosen because of the richness of their interaction, but many other groups were observed to engage in the same discussion, using similar gestures.

4 Gestures and spatial arrangement

To help the reader with the detailed analysis that follows, we provide a brief description of the common gestures used in solving the problem and of the spatial arrangement in which the students are doing their work.

Two gestures, grabbing and sliding, are shown in Fig. 1. Grabbing or bracketing (Fig. 1a) is used to indicate a group of terms (such as mg − c_{2}v^{2 }in Eq. 1). The gesture consists of using the thumb and a finger to “pinch” around two grouped terms. Often, the thumb and the middle finger are used, and the pointing finger is free for the next move, namely sliding the bracketed equation to a new location on a page or a white board (Fig. 1b). Individual terms, such as dx or m, can be pointed to and slid, as well. Sometimes, when the location where the term ends up is far away, there is less of a slide to a given location and more of a fling in the direction, ending with pointing to its location on the board.

Two additional pieces of evidence set a slide apart from simply two consecutive deictic gestures of pointing first to one location and then the other. First, and most importantly, students talk about “moving from here to there.” They are describing a trajectory as they make the gesture. In addition, the trajectory itself indicates a specific path being taken. A slide is often carried out so that the finger does not get in the way of other notated elements on the board at which the pointing is taking place. So, sliding a bracketed term from one side

Fig. 1 Gestures during algebraic manipulation. a Grabbing or bracketing, (b) sliding or flinging

of the equation to the other might include a curving trajectory so as to avoid the equal sign, as shown in Fig. 1b.

We schematically represent the space in which students work in Fig. 2. We give locations to different regions in this space, A–G. This convention helps us describe the equation, though we are careful to note that students do not refer to their work in this way. This is a descriptive representation for our own analysis.

In this form, Eq. 1 has mv in location A, dv in location B, dx in location C, and mg − c_{2}v^{2 }in locations D and E. (We ignore the role of the minus sign between regions D and E because the minus sign plays no role in student gestures in this problem, but mg and − c_{2}v^{2 }are treated separately at times, so we wish to show them in different regions on the page.) In Eq. 2, the contents of C have been switched with the contents of D and E. As will be shown below, students suggest other possible ways of writing the equation, creating the need for locations F and G in our diagram.

5 Mathematical terms as physical objects that move along a path

In this section, we give a detailed description of two students discussing how to separate variables (e.g., rewrite Eq. 1 to the form of Eq. 2). In the next section, we engage in a broader discussion of the results.

We present the dialogue of two men, Dan and Simon. The two other group members are silent during this interaction. (As noted above, some group members may not have spoken because the graded group quiz suggested that the strongest group members do the most work, and Dan and Simon were excellent students.) In the transcript, we highlight both the dialogue (sometimes accompanied by descriptions of gestures) as well as the mathematical equation students are looking at or referring to. We then analyze the discourse, the gestures,

Fig. 2 The spatial areas in which point and sliding occur

and, where necessary, the mathematical equation. Turns at talk are numbered for reference in later analysis. Turn 16 is shown in Fig. 3.

- Simon: So then we’re gonna shuffle things around —(Simon points to locations C and E on the whiteboard.)
- Dan: Yes. dx over m3. Simon: d-what? dx over-
| Students begin with: mv dx G empty in our spatial representation. |

4. Dan: dx over m | Points at dx in location C |

- Simon: You mean…d… d?
- Dan: I just did this whole thing now-
| Points at C |

7. Simon: Or are you moving 8. Dan:-this exact same problem | Moves hand from C to E |

9. Simon: Are you moving dx over there? | Points to C and slides to F |

10. Dan: Yeah, you move dx over there (points to C and slides to E), and m over there (points to A and slides to G) and this (brackets mg − c_{2}v^{2 }in D and E) over there (slides to C). | vdv dx Dan wants 2 2 ¼ occupied but E and F are empty. |

- Simon: Let’s do it like this first,
- Dan: It doesn’t matter…
| Simon begins erasing a part of the board previously written on |

- Simon: m v dv is mg minus c-two v-squared dx
- Dan: Yeah, if you want to write down that step.
| Simon writesparentheses as shown. In the spatial diagram,mvdv ¼ ðmg c regions A, B, D, E, and F are occupied, and C and G are now empty |

15. Simon: And then we get the v on the other side | Simon’s hand moves back and forth between E and C at this point (with c_{2}v^{2 }in region E) |

- Dan: We move this (brackets D and E) over there(slides to C) and that (points at A) over here (slides to G).
- Simon: Move the one with v (points at E) over there? (slides to C)
| Dan moves mg − c_{2}v^{2 }to under mvdv and m to under dx. |

18. Dan: Yeah… Move thisss (brackets D and E) whole term—over there (points to Cº) | Dan moves mg − c_{2}v^{2 }to under mvdv |

19. Simon: mdv over mg minus c-two v-squared equals dx. | Simon writes mvdv 2 ¼dx mgc_{2}v |

The episode began with Simon using the language of moving. In turn 1, there is talk of shuffling, while pointing at different locations. In turns 7 and 9, he asked Dan about moving. In turn 1, there was much pointing, and in turns 7 and 9, he repeatedly pointed from region C to E. (Notably, distinguishing between regions D and E is difficult at this point, and we do not lend much support to hypotheses dependent on differences between the two locations).

Dan responded in turn 10 with a complicated mixture of speech and gesture. The mathematical terms dx and m are named, while mg − c_{2}v^{2 }was referred to as “this.” Each

Fig. 3 Turn 16. Dan is bracketing a term while saying “We move this” (Fig. 1a) and just before saying “over there” (Fig. 1b). Simon sits at the near left with marker in hand

term was moved, first by being pointed to, then slid. With the terms dx and m, the connection between what was pointed at and what was named was clear. With the expression mg − c_{2}v^{2}, fingers bracketed the expression before it was slid to a new location. Each movement brought the term to what would be its correct location—because Dan did not have the marker, he could not write down what he wished to have at this point. But, from his gestures, the desired equation was provided. The entire turn takes about 2 s.

After some back and forth (in turns 12 and 14, Dan sounded exasperated that Simon didn’t write the full solution immediately), a new equation was written on the whiteboard. Simon again used a language of motion, saying “we get the v on the other side.” In other observations, we have heard students talk about “bringing” something from one side of the equation to the other. In both these cases, getting and bringing, students use a spatial metaphor, as if objects are going from one place to another. The objects are the mathematical terms.

In turn 16, Dan made another bid for his solution. Because the dx was in the location he desired (on the right side of the equation), he only had two gestures to make. They were nearly identical to the previous gestures. He bracketed mg − c_{2}v^{2 }and called it “this” before moving it “over there.” He pointed to m and called it “that” before moving it “over here.” His use of “here” and “there” is consistent with how we spatially describe our surroundings, namely that “here” is closer to him than “there.” We also note that he did not refer to any mathematical terms by name. They were only “this” and “that.”

Simon made a similar gesture to Dan’s as he asked about only the first operation, moving to the location under the mdv on the left side of the equation. But, he did not bracket—he only pointed to the c_{2}v^{2}. He again used the verb “move,” but his description of what to move was unclear, because he referred to “the one with v.” This was ambiguous, because he might have meant the whole term, (mg − c_{2}v^{2}), which was now written in parenthesis on the page, or he might have meant the c_{2}v^{2 }by itself. Because he only pointed, and did not bracket both terms, we believe he meant only the c_{2}v^{2}.

Dan seemed to respond to this ambiguity in his response in turn 18. He referred to “this whole term” while bracketing (mg − c_{2}v^{2}) and spoke with an emphasis on the word “this,” as well as a lengthening of the letter s, as in “thissss.” In addition, he called it the “whole term” rather than just “this,” as he had done in turns 10 and 16. His language and the bracketing make clear what he was referring to. He then slid his finger to the location under mdv as he said “over there.” At this point, Simon wrote out the equation, the m never moved, and they went on to integrate both sides of the equation.

6 Discussion

In the discussion that follows, we suggest that Simon and Dan’s process of separating variables in algebraic equations treats mathematical terms as physical objects, and that these objects move in a landscape of the surface the equation is written on. Several examples from the episode support our assertion.

Consider when students gesture as they say, “move this over there and that over here.” The gesture tells us what moves (bracketing), how it moves (the path traced), and where it moves to (the final position of the hand). The speech indicates the term’s spatial relationship to the speaker: how near to or far the locations are. We see this combination of speech and gesture as evidence of the spatial, diagrammatic treatment of an algebraic equation where the inscription exists in a physically defined space. This landscape has defined regions and physical structure that requires particular types of motion to navigate through, as evident below.

We observe that students slide terms on the page so that the path of the moving term does not intersect other stationary terms. As Dan moves (mg − c_{2}v^{2}), he avoids the equal sign, suggesting that what he is moving is being treated as a physical object and that it exists as a physical object even at intermediate positions as it travels along its path. Consistent with Cook and Tanenhaus (2009), we see that students’ sliding gestures are often arced, further indicating to us that they are treating the mathematical terms like physical objects in a landscape. The students make sure to avoid other terms that might be in the way, just as Tower of Hanoi problem solvers’ gestures reflected the constraints of physical pegs after working with the physical situation (and not the computer simulation).

It is important to note what Simon and Dan are not doing as they solve this mathematical task. Though Simon tried to write down several steps in the equation, neither he nor Dan ever used overt mathematical terminology in this episode. Dan and Simon regularly use indexical language (this, that, here, there) (Staats & Batteen, 2010) to describe terms in the equation and their locations on the page. Notably, they never refer to formal mathematical operations such as dividing or subtracting as they transform the equation. Only when reading an equation (during an intermediate step between Eq. 1 and Eq. 2) did one of them say “minus,” and only while reading the equation itself. The “minus” was not treated as an operation. Still, Simon and Dan achieve their goal of separating variables in a way that allowed them to integrate both sides of the equation. The resulting, transformed equation is syntactically correct.

For Simon and Dan, the gesture, the action, is the actual operation of separating terms. The mathematical operations are manifested in the paths of the objects that exist in the motion of the gestures. In the formal mathematics, of course, there is no path. The motion performed by the students can be described as metaphorical or fictive in this regard—there are no objects and no motion, but the mathematical terms end up at another location on the board.

If one considers the conceptual metaphor of the mathematical terms as physical objects (Lakoff & Núñez, 2000; Sfard, 1994) existing in defined space, their motion seems reasonable. Objects move along defined paths, after all. We think of this path as being part of a source-path-goal schema (Lakoff & Núñez, 2000). Our results indicate that this schema can apply to the symbolic written mathematical terms in an equation.

By adopting Radford’s sensuous conception of thinking (Radford & Puig, 2007) to appreciate the ways that Dan and Simon perform mathematics, we find support for Nemirovsky’s (Nemirovsky, 2003; Nemirovsky & Ferrara, 2009) proposal that perceptuo-motor activities engaged in during mathematics are more than just accompaniments to formal mathematical thoughts. To fully understand Dan and Simon’s process of transformation, it is necessary to attend to the rich interplay of metaphor and bodily motion present in their practice. Our detailed description of events demonstrates the ways in which local actions of gesture in an environment are constitutive elements of doing mathematics.

7 Conclusions

In sum, we find evidence that students work with mathematical terms in an equation as if they are manipulating physical objects along a path through a landscape on a surface. Where Staats and Batteen (2010) described how transforming and solving algebra equations can involve a sense of motion for students as indicated by their speech, we have shown evidence that fictive motion can be indicated by gestures, as well. Additionally, we have illustrated how fictive motion can relate to mathematical terms, adding to the interpretations of graphs or of functions that has been previously described (Núñez, 2004). Extending the work of Staats and Batteen as well as Núñez requires that we add physicality to the concept of a mathematical object (Sfard, 1994).

Our analysis contributes to an understanding of algebraic simplification of equations (Kieran, 1992, 2007) by showing how two students rearrange and transform an equation without talking about formal mathematical operations. Kieran (1992, 2007) summarizes the formal and informal ways that students simplify equations, including the idea of transposing terms (where a term changes sign as it changes sides of an equation) or the more general action of performing the same operation on both sides of an equation. We have added to this discussion by describing a unique student method of algebraic simplification: moving terms through gesture to achieve the mathematical step of division. In conclusion, we argue that this manipulation of algebraic terms by hand constitutes mathematical thought.

We believe future work addressing the roles of the body in algebraic practices should investigate the mechanisms that allow for gestures to lead to the correct answer. Dan and, for the most part, Simon manipulate the mathematical terms such that they end up where they should. Why are gestures to incorrect spatial regions not made? In other work (Black, 2010), we have suggested that conceptual blending (Edwards, 2009; Fauconnier & Turner, 2002; Núñez, 2005, 2010; Turner & Fauconnier, 1995) can describe how information from the mathematical formalism influences the gestural activity.

We are also curious to know when and how students develop the idea of using the motion of mathematical terms as physical objects as a way of simplifying algebraic equations. We find, informally, that physics and mathematics professors working problems or deriving equations at the board talk and gesture as if physically moving terms. Discussions with colleagues have highlighted how readily they think of this movement as the mathematics. If experts in physics and mathematics readily use the skills described in this paper, and the literature on students’ simplification of algebraic equations does not discuss it (Kieran, 1992, 2007), then we offer two questions for further exploration: First, when does this skill develop, and in what ways? This developmental question might be answered by close observation of students’ problem solving at different school levels. Second, how does the skill develop in different populations of practitioners, such as physicists, engineers, and mathematicians? We could imagine studying experts in different settings and observing how they move mathematical terms when working at the board during a class, for example.

We believe that further microanalysis of people’s successful mathematical activities can help us understand these and other issues. Our multimodal approach, analyzing gesture, speech, and the details of what is written down, provides a rich, detailed description of the mathematical practice of algebraic manipulation. We find that students accurately separate variables in an equation by gesturing as if the terms of the equation are physical objects that can be moved to the desired locations.

Acknowledgements The work described in this paper was supported in part by National Science Foundation grants DUE-9455561, DUE-0442388, DRL-0633951, and DUE-0941191. The authors thank Rachel E. Scherr, Eric Brewe, and Evan Chase for their valuable input and feedback during preparation of this manuscript. We further thank three anonymous reviewers for their very careful reading and helpful suggestions. Data were gathered by Katrina E. Black at the University of Maine and form part of her Ph.D. dissertation (Black, 2010).

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FLM Publishing Association

Poetic Lines in Mathematics Discourse: A Method from Linguistic Anthropology Author(s): Susan K. Staats

Source: For the Learning of Mathematics, Vol. 28, No. 2 (Jul., 2008), pp. 26-32

Published by: FLM Publishing Association

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**The poetics of argumentation: the relevance of conversational repetition for two theories of emergent mathematical reasoning**

# The poetics of argumentation: the relevance of conversational repetition for two theories of emergent mathematical reasoning

Susan Staats

College of Education and Human Development, University of Minnesota, Minneapolis, MN, USA

ABSTRACT ARTICLE HISTORY

Poetic structures emerge in spoken language when speakers repeat Received 15 June 2016 grammatical phrases that were spoken before. They create the Accepted 6 March 2017

potential to amend or comment on previous speech, and to

KEYWORDS convey meaning through the structure of discourse. This paper

Mathematical discourse;

considers the ways in which poetic structure analysis contributes argumentation; poetic to two perspectives on emergent mathematical reasoning: structures

Toulmin’s model of argumentation and Martin, Towers, & Pirie’s theory of collaborative coactions in multi-speaker discourse. Poetic structures appear in varied argument types and at varied educational levels. They appear to facilitate speakers’ expression of warrants, backings, qualifications, and coactions.

Introduction

A common characteristic of conversation is that people repeat each other. Repeating the phrases of a prior speaker is documented widely. Repetition occurs in conversation and in single speakers’ discourse (Tannen, 1989), in folktales and storytelling (Hymes, 1981), in classroom talk (Staats, 2008; Wortham, 2006) and in teachers’ identity narratives (Oslund, 2012). In these anthropologically grounded examples, repetition is usually called a poetic structure, following Roman Jakobson’s treatment of oral poetry (1960). Aligning one’s phrases with others’ serves functions for both speakers and listeners. Repetition can validate another speaker’s contribution, it can confirm participation, and it can make interpretation and production of new sentences faster and easier (Tannen, 1989). A key aspect of repetition in conversation is that it is emergent—it arises in conversation without explicit agreement or awareness of speakers. Such a widespread, multifunctional behaviour that is fundamental to the exchange of ideas is likely to be implicated in collaborative learning.

In current mathematics education research, two broad research areas address the emergent quality of collaborative classroom reasoning. First, Toulmin’s theory of practical argumentation has inspired models of collective, classroom construction of arguments and proofs (Knipping, 2008; Krummheuer, 1995, 2007; Toulmin, 1958). Second, the theory of improvisational coaction highlights the distributed nature of mathematical discovery through conversation (Martin & Towers, 2009; Martin, Towers, & Pirie, 2006).

© 2017 British Society for Research into Learning Mathematics

The goal of this paper is to demonstrate that central discursive moves in argumentation, such as warranting, justifying, qualifying, and the collaborative adopting and amending of conjectures, are often conveyed through the structure of speech, through poetic structures, as much as through the lexical dimension of speech.

The paper opens with examples of poetic structures, recommendations for representing repetitions in transcriptions, and commentary on the treatment of poetic structures in several academic traditions. Following this, I review two theories of emergent mathematical reasoning, Toulmin’s model of argumentation, and perspectives on the development of collective mathematical understanding advanced by Martin, Towers and Pirie. I consider the contribution of poetic structures to argumentation in postgraduate students’ mathematical monologues in Inglis, Mejia-Ramos, and Simpson (2007). I also identify poetic structures in a highly collaborative mathematical conversation drawn from Martin and Towers’ paper on the co-construction of mathematical insight (2009), and indicate the ways in which coactions can be expressed through poetic structures.

Identifying and representing poetic structures

Poetic structures can involve either syntactic or lexical elements, or both. In this paper, I focus on repetition in which some grammatical structure is repeated, because this usually involves a statement that is long enough to express a conjecture or a method. In the following excerpt, for example, a first-year university student explained her solution method for −4w + 4 = 3 − 2w to her classmates (Staats & Batteen, 2010, p. 50). The class had arrived at this equation while solving an equation composed of rational expressions. Shaniqua narrated the final steps as the instructor wrote her equations on the board.

Selection 1

**1 Shaniqua: So we gotta try to get all the like terms on one side and 2 all the other terms on the other, 3 so we can plus 2w on both sides which gives you**

- Then you gotta minus 4 and then you get

- and then you got,

**6 so you can divide −2 which gives you w is equal to,**

- I don’t know if it’s negative ½ or positive ½.

- Mindy: Positive ½.

- Shaniqua: So that’s our answer.

Selection 1 illustrates sentences that have grammatical repetition but not full lexical repetition. Shaniqua’s explanation relies on four sentences that are similar to each other: the sentence over lines 1 and 2, at line 3, at line 4 and over lines 6 and 7. Each of these four sentences repeats a general grammatical structure:

Pronoun/modal auxiliary verb/main verb/secondary verb/equation as direct object

(we; you)/(gotta; can)/(try; plus; minus; divide)/(get; gives)/(current equation)

The pronouns alternate between we and you. The modal auxiliary verbs are gotta and can, and the main verbs are try, plus, minus and divide. Modal auxiliary verbs qualify their main verbs with a manner of knowing or level of certainty (Palmer, 1979). A secondary verb get or give introduces the equations that result from algebraic manipulation. In addition to this broad repetition, there is a secondary level of organisation, in which each of the four combinatorial subject-verb arrangements of we, you, gotta, and can also occurs.

Though it’s impossible to know why Shaniqua spoke in this highly patterned way, it certainly lends a sense of cohesion to her solution statement. The scholarship of David Pimm and Tim Rowland allows us to consider more deeply the pragmatic force of this patterning. The pronoun we can represent coercive or authoritative qualities of a discourse community (Pimm, 1987) and the pronoun you can represent a sense of generalisation (Rowland, 2000). Rowland suggests that modal verbs are important in mathematical discourse because they represent a “propositional attitude”, positioning the speaker “towards the factual content of what s/he says” (2000, p. 64). Shaniqua’s modalisation based on gotta and can shifts between a sense of obligation and of option or agency. It is through her robust poetic structure that Shaniqua most clearly shows her confidence in algebraic procedure, but the repetition also allows her to offer quiet colourings of the work of algebra, a world in which one has agency to choose a problem-solving pathway even while participating within an academic community.

It is well-known among discourse researchers that no form of transcription is objective (Herbel-Eisenmann, 2014; Ochs, 1979). Every layout represents the researchers’ theoretical choices, with some features of the spoken sound highlighted and others omitted. A poetic structure layout must present the repetitions that the researcher considers important, but it must also respond to the degree of repetition and the width of the printed page. In choosing a layout for selection 1, I tried to arrange the basic syntactic unit of pronoun / verb phrases / equation as direct object onto one line with subject/verb phrases in one column and the equations in another column. I identified the secondary level of repetition in the verbs and pronouns with several styles of underlining. Underlining highlights a stretch of discourse that the researcher thinks coheres together and that is repeated, even if there may be some variable words inside of it. The verb phrases with gotta … get received single underlining and the verb phrases with can … gives received double underlining. Indentation helps to place grammatically similar phrases into columns, and in cases like selections 2 and 3, to indicate a stanza of lines that shows very strong repetition.

Poetic structures in social, cognitive and linguistic perspectives

The study of grammatical repetition has a fairly complex academic history. In a wellknown study from sociolinguistics, Dutch shopkeepers demonstrated conversational repetition (Levelt & Kelter, 1982, p. 89, original translation). When asked, “What time does your shop close?”, shopkeepers tended to answer, “five o’clock”. The question “At what time does your shop close?” prompted the answer “at five o’clock”, maintaining the prepositional phrase across question and answer. Tannen reports similar conversational turns, for example, when a woman teases her friend about sharing a soft drink called Tab (1989, p. 57, original underlining and emphasis, layout slightly modified).

Marge: Do you want to split a Tab?

Kate: Do you want to split MY Tab? (laughter)

In Marge and Kate’s exchange, repetition influences the emergent quality of meaning in conversational interaction. When Marge uses the indefinite article in a Tab, she presupposes that the soft drink is a novel object in the conversation. This allows Marge to claim a position of generosity by offering to share the drink. However, a new meaning emerges when Kate repeats her entire sentence with a single shift to the possessive pronoun in my Tab. With a single word, Kate suggests that the drink, in fact, had a prior presence in their shared world, and that Kate is the legitimate authority over its dispensation. Neither sentence alone captures the full meaning of the exchange. The discursive structure of the repetition is the smallest unit that allows an adequate analysis of the scenario.

Sentence-level repetitions inspired intensive experimental work in psycholinguistics as a means to understand the mental representation of language (Bock, 1986; Branigan & Pickering, 2004). Experiments typically measure the tendency to repeat a grammatical construction after being exposed to or “primed” with one of two possible options. For example, the double object prime sentence: The police handed the judge the cake tends to create increased preference to say The pilot sold the teacher the book rather than the direct object alternative: The pilot sold the book to the teacher. This experimental work suggests that at least some forms of grammatical repetition are explained by cognitive processes; other cases of repetition, like the soft drink example, may be explained additionally or alternatively by social functions of communication.

A second academic tradition in the study of repetition was greatly influenced by the work of linguist Jakobson (1960). For Jakobson, whenever an addresser conveys information to an addressee, several communicative functions are active. The referential function specifies relevant elements of the context, the phatic function involves cues about when the speaker holds and relinquishes a turn at speech, and the emotive function represents the sentiments of the speaker. Jakobson describes the poetic function of communication in rather mathematical terms: “The poetic function projects the principle of equivalence from the axis of selection into the axis of combination” (Jakobson, 1960, p. 358, original italics). The axis of combination refers to the ordering of words over time, in a spoken comment or in a sentence, so that there are particular positions to be filled. The principle of equivalence refers to the act of selecting from among words that are similar in some way—in terms of meaning, grammatical role, rhythm, potential for alliteration, and so on, and inserting them from this axis of selection into an appropriate location in the axis of combination. In Tannen’s soft drink example above (1989, p. 57), when Marge asks, Do you want to split a Tab?, she establishes an axis of combination. Kate uses the principle of equivalence to notice that a Tab is similar to my Tab, which she projects into her poetic play on Marge’s words. For Jakobson, the key role of the poetic function is to draw attention to the form of a message; this self-reference is the central feature of spoken and written language that makes it poetic. The poetic structures that are described below involve similar small shifts in one of the components of a grammatically ordered comment about mathematics.

Two theories of emergent mathematical reasoning

**Toulmin’s model of argumentation**

Toulmin’s model of argumentation offered a theory of reasoning that sought to be widely relevant. This model intends to explain informal ways in which a social group comes to accept a conclusion, without using formal logic, through six components: datum, warrant, backing, modal qualifier, rebuttal and conclusion (based on 1958, p. 104). The core of an argument is the connection between the foundational information, the “datum” of a question, to a conclusion. Toulmin’s model became an influential alternative to formal logic through its attention to several qualities of reasoning—warrants, backings and qualifications—that facilitate this connection and make it convincing to others. In the following commentary, we will consider the role of poetic structures in presenting warrants, backings and qualifications (Figure 1).

A warrant is a statement that connects a datum to a conclusion, which validates reaching the conclusion given that one accepts the datum. Warrants are often implicit and unstated, and they often hold force only in restricted conditions, for example, subject to qualification. In contrast, the backing is a “field dependent” statement that refers most directly to a warrant, and that asserts the validity of using the warrant in a particular social, professional or disciplinary context. Krummheuer, for example, considered finger counting as a backing for a child’s warrant on counting by grouping (Krummheuer, 1995, 244). In an adult domain, enumeration would be handled differently, for example, with survey data serving as a backing for warrants. Distinguishing backings and warrants can be difficult, but it is an important consideration for mathematics educators because a student’s argument may rely on mathematical methods that would not necessarily be recognized as persuasive or intelligible in other contexts.

Two components of Toulmin’s model that are sometimes neglected in analysis of informal argumentation are modal qualifiers and rebuttals (Inglis et al., 2007). Linguistically, modal qualifiers express a speaker’s epistemic orientation to an assertion, a level of confidence in an assertion. This can be accomplished through several linguistic means, including adverbs like possibly or probably, and modal auxiliary verbs like should, can, or must (Palmer, 1979). Inglis et al. (2007) argue that modal qualification is an important sign of mathematical sensitivity, particularly while students are in the process of developing a proof or an opinion about conjectures.

In Toulmin’s original model, if an argument involves a modal qualifier, then this component may be linked to a rebuttal, which indicates conditions under which a warrant is not a valid means of reaching a conclusion to an argument. In Toulmin’s model, a rebuttal is not a statement of opposition to an argument. Instead, it supports an emerging argument and its movement from datum to conclusion. A rebuttal recognises that there are some situations in which the movement from datum to conclusion is not justified, but if these conditions are not present, the conclusion can indeed be reached. It will be

Figure 1. Toulmin’s diagram of an argument. The components of the argument are D, the datum; C, the conclusion; W, the warrant; B, the backing; Q, the modal qualifier; and R, the rebuttal.

useful later to distinguish between arguments in which conflicting conjectures are in play, and arguments in which rebuttals are used to reach a conclusion.

Several researchers have applied Toulmin’s model to arguments that arise within mathematics classrooms in naturalistic ways, as a type of ethnography (Knipping, 2008; Krummheuer, 1995, 2007). Krummheuer (1995) considers classroom argumentation as a collaborative, social construction that takes place through speakers’ adjustment of their positions. This paper highlights cases when the adjustments of argumentation are conducted through the shared phrasing of conversational repetition.

Researchers have identified several difficulties in applying Toulmin’s model of argumentation to mathematical conversation. Toulmin may not have intended the model to make sense of emergent arguments, but rather, only complete ones (Simpson, 2015). Alternative typologies of warrants have been proposed; the blurred distinction between warrant and backing and the potential for different readers to create varied argumentation diagrams have been exposed (Nardi, Biza, & Zachariades, 2012; Simpson, 2015). The current discussion will be most useful to readers who accept close analysis of emergent mathematical arguments.

**Collective construction of arguments**

Another prominent approach to understanding collaborative argumentation is the body of work emerging from the research group of Martin, Towers and Pirie (Martin & Towers, 2009, 2015; Martin et al., 2006; Towers & Martin, 2014). Although several of their insights resonate with poetic structure analysis, the central connection is through their theory of improvisational coaction (Martin & Towers, 2009; Martin et al., 2006). In this view, collective understanding is constantly renegotiated. Coaction occurs when students build upon other’s previous mathematical work, so that a collective understanding—correct or not—can be shared, then dismantled and then re-collected by the group. Improvisational coaction is

a process through which mathematical ideas and actions, initially stemming from an individual learner, become taken up, built upon, developed, reworked and elaborated by others, and thus emerge as shared understandings for and across the group (Martin & Towers, 2009, p. 4).

This research group draws upon scholarship in musical and verbal improvisation to develop a framework for describing mathematical improvisation in collaborative conversations (Martin & Towers, 2009; Martin et al., 2006). Their perspective on improvisation captures the sense of unpredictability, emergence, and reference to prior expression that is also fundamental to poetic structure analysis. They find that four features of improvisation are relevant to mathematical collaboration. First, significant insights are spread across the comments of multiple speakers, and may never be articulated fully by an individual. Another characteristic of improvisational coaction is “collectively building on the better idea” (Martin & Towers, 2009, p. 15). If the group recognises a new idea as potentially useful, they may adopt it into the ongoing investigation. This decision may be implicit rather than explicit, a third feature of improvisational coaction termed “listening to the group mind” (Martin & Towers, 2009, p. 15). The collective nature of improvisational coactions also involves “an interweaving of partial fragments of images” (Martin & Towers, 2009, p. 14). Individuals may contribute parts of an idea, but the collective discussion of the group weaves them together into a more coherent commentary on the mathematical task.

While all four of these dimensions of improvisational coaction may be present in particular poetic structures, the characteristic of combining image fragments is always relevant. Grammatical repetition always involves incorporation of a prior comment into a new one. In multi-speaker conversations, the other dimensions of improvisation are likely to be in play as well.

Like Krummheuer and Knipping’s uses of Toulmin, the improvisational coaction model derives from a model of individual thinking. The theory of coaction extends the Pirie–Kieren analysis of dynamic, non-linear growth of mathematical understanding (Pirie & Kieren, 1994). Martin and Towers also acknowledge the significance of distributed intelligence perspectives on learning (Cobb, 1998; Cobb & Yackel, 1996), but they wish to emphasise the dynamic nature of mathematical conversations, and their focus is on smaller time segments of conversations. Martin and Towers make central use of Pirie and Kieren’s concept of image making, image having, and property noticing, three of the early stages in the growth of mathematical understanding, and recast them as collective mathematical activities (Martin & Towers, 2015; Towers & Martin, 2014). In collective image making, several students contribute to concrete activities to develop initial concepts about a mathematical object. In collective image having, students use an image to investigate a problem more deeply without returning to their initial concept building activities. In collective property noticing, multiple students collaborate to express a more general attribute of the mathematical object.

Importantly, Martin and Towers note that collectively developed images and properties are not just visual or pictorial ones, but may be “any ideas the learner may have about the topic, any mental representations” (2009, p. 2). In selection 1, for example, Shaniqua’s poetic structure expresses the sense that every algebraic operation in solving this linear equation is similar to every other operation—she is in an image having stage for solving linear functions. This paper will assert that a phrase that is repeated and modified by a second speaker is a non-visual image that plays an important role in making and having mathematical ideas.

Perspective on selection and interpretation of discourse samples

Most of the samples discussed below are extensions of analyses published by other researchers. The most important selection criterion was to identify discourse samples with relatively little teacher or researcher presence, to avoid an explicit relationship of guidance and authority that might influence discursive repetition. In extending the analysis of argumentation to include the relevance of poetic structures, I tried to acknowledge and as much as possible adopt the position of the original researchers. I took this as an ethical requirement of re-analysing others’ data—I assumed that their analysis and interpretive intuition is stronger than those of researchers who were not involved in the primary research.

Applying Toulmin’s model to mathematical discourse can be hindered when a student has not spoken a particular component of the argument. Qualifiers, backings, even warrants may have been left unspoken. I prefer to instantiate components of Toulmin’s model with particular spoken phrases. Some components of Toulmin’s model may simply not be present or constructable. I prefer this partial application of Toulmin to an approach that relies on researcher-developed statements that allow the model be fully realised.

Poetic structures in mathematical monologues: warrants, backings and qualifications

Several of our first examples are drawn from a study on qualification in mathematical argumentation (Inglis et al., 2007). The authors of this study recorded postgraduate students as they evaluated number theory conjectures to be true or false, surrounding the definitions: given an integer n, if the sum of divisors is 2n, the integer is said to be perfect; if the sum of divisors is greater than 2n, the integer is abundant; if the sum of divisors is less than 2n, the integer is deficient.

Selection 2

Chris responded to the conjecture: If p_{1 }and p_{2 }are primes, then p_{1}p_{2 }is not abundant (Inglis et al., 2007, p. 7). Just before line 1 in the transcript rendering below, Chris decided that this conjecture is “probably” true, and he checked it for pairs 2 and 3; and 5 and 97. The authors develop a complete Toulmin diagram that includes all six components, but that highlights the student’s use of a modal qualifier, probably. The authors include the following phase of Chris’ statement as the warrant of the argument (Inglis et al., 2007, p. 8, layout modified, line numbers added). Here, the poetic structure occurs across lines 2, 3 and 4: I know (feel) / this statement / is (should be) true / for (large, small, middle) p_{1}, p_{2}.

1 In other words,

**2 I know that this statement is true for large p**_{1}**, p**_{2}**; 3 I know it’s true for small p**_{1}**, p**_{2}**;**

- so I feel therefore that it should be true for p
_{1}, p_{2 }in the middle.

- Umm, but I might have to do some work to show that.

Inglis et al. (2007) consider this as an inductive warrant type or argument because the student has performed calculations for small values and for large values, and the student uses this to persuade (lines 2–4). Warrants and backings are the most important part of an argument, because if they are unspoken, there is little else that can count as a justification of the argument. It is notable, then, that the most insightful part of the student’s commentary—the warrant—was conveyed through a poetic structure. Here, the repeated sentence structure expresses the sense that the conjecture holds for a breadth of cases, and legitimises the plan to prove the conjecture. In this way, the warranting character of lines 2 through 4 is not merely conveyed by the content of the isolable words, but by the poetic structure itself.

In addition to expressing qualification, poetic structures seem to be very well-suited for expressing the warrants for inductive arguments, focusing on specific cases and alluding to a broader conclusion. Importantly, though, repetition does not mean that the argument is correct or complete. Analysis of poetic structures in mathematical discourse should be taken as a means of highlighting the authentic nature of student reasoning and insight, which may represent various degrees of completeness and correctness.

Chris did not use the poetic structure rigidly – there are small shifts, for example, from is to should be, and from the adjectives large and small to the prepositional phrase in the middle, and the shift from knowing to feeling. The shifts have some importance in tracking the quality of Chris’ argument—the shift from knowing to feeling may help him position his evaluation of the conjecture as tentative. The shift from large/small to in the middle could be explained in several ways, for example, avoiding a grammatically awkward alternative middle p_{1}, p_{2}. Another possibility is that ending the sentence with in the middle focuses attention on a range of numbers that still carries some uncertainty for the student. As important as modal qualifiers like “probably” are, students signal their level of certainty extensively through larger units of discourse. Each of these poetic shifts work together to convey a qualification.

Selection 3

In the next selection, another postgraduate mathematics student responded to a number theory conjecture: If n and m are abundant, then n + m is abundant (Inglis et al., 2007, p. 12, layout modified, line numbers added). The student’s initial, and correct, argument is that the conjecture is false. As the final comments in this selection suggest, he decided to try to prove the conjecture, but eventually identified counterexamples. In this selection, there are several distinct types of repetition, and so I use indentation and several types of underlining to draw attention to them.

- I think, going on instinct, it’s probably false …
- […] saying whether something is abundant

**3 is to do with its divisors,**

4 so it’s to do with things that divide it, 5 it’s to do with multiples.

- And then,

- when you add two numbers together, it doesn’t necessarily mean

- that any properties of the divisors stay the same. I mean, like, I don’t know,

- when you add 3 and 5, 3 and 5 have certain divisors, but 8 has completely

- different divisors. Umm, but you never know.

- So, but abundant is a very sort of wide statement, so, I mean, intuitively you’d expect to 12 apply to roughly half of all numbers, so maybe it’s not so absurd to think they would, err, 13 that would hold. So I’ll try.

Inglis et al. (2007) identify this argument as one centred on a structural-intuitive warrant, characterised by “using observations about, or experiments with, some kind of mental structure” (p. 12). In their analysis, the structural intuitive warrant is that the student asserts that divisors of a sum are not related to divisors of integers, a statement which occurs in lines 7–8. Divisors have certain properties—a mental structure—and the experiment with 3, 5, and 8 calls the conjecture into question. This interplay between mental representation and experimentation makes this a structural-intuitive argument. A different graduate student made a similar comment with very concise poetic phrasing: “because the factors of n + m don’t really have anything to do with the factors of n or m” (Inglis et al., 2007, p. 13, underlining added).

There are a couple of plausible interpretations of the role poetic structures play in this structural-intuitive argument. The strongest repetition is in the stanza of lines 3–5, starting with is to do with its divisors. This stanza is preparatory thinking to the warrant in lines 7–8. The warrant itself is repeated as a poetic structure in lines 9–10 as the student experiments with numerical examples. The poetic lines thus explore the nature of the mathematical structure—divisors of sums and products, so that they become the experimental or observational component of the structural-intuitive argument.

In the repetition of lines 7–8 and 9–10: when you add (two numbers) … I don’t know, the experimentation or exemplification is tied to the hedging quality of I don’t know. As in selection 2, poetic structures contribute to qualification of the argument. But in lines 11–13, the student shifts out of the exemplification strategy, and in this transition, he uses fewer poetic structures. One way to interpret this shift away from repetition is that the student has decided to reverse the direction of his initial argument—that the conjecture is false—and so the existing string of poetic structures is less useful for the future proof attempt. The repeating themes of multiples, divisors and examples did not uncover structures or relationships that could lead to a proof of the conjecture’s truth, and they assist in warranting the conjecture’s falseness. Continued repetition does not support the new direction of reasoning.

So far, we’ve seen that poetic structures lend themselves to expressing patterns, trials, and examples in both inductive and structural-intuitive arguments. In the following selection, a postgraduate mathematics student responded to the conjecture that if n is perfect, then kn is abundant for any integer k (Inglis et al., 2007, p. 15, layout modified, line numbers added). The conjecture is true for integers larger than 1.

Selection 4

**1 OK, so if n is perfect, then kn is abundant, for any k … […] 2 Yeah so if n is perfect, and 3 I take any p**_{i }**which divides this n,**

- then afterwards the sum of these p
_{i}s is 2n.

- This is the definition.

**6 Yeah ok, so actually 7 we take kn, 8 then obviously 9 all kp**_{i }**divide kn, 10 actually,**

11 we sum these and we get 2kn.

In this portion of the commentary, the student was close to achieving a proof for the conjecture for integers n larger than 1. He only needed to consider further that 1 is also a divisor of kn, so that the sum of divisors of kn is at least 2kn + 1. Inglis et al. (2007, p. 16) consider this as a deductive warrant argument type, because “the conclusion follows necessarily from the data”. The deductive character of this argument is conveyed primarily through two poetic structure transformations.

Line 1 summarises the argument with an If … then structure; the phrase if n is perfect is the datum and then kn is abundant is the conclusion. This structure was repeated in lines 2 to 3 to accommodate stating the definition of a perfect integer, using the expanded, transformed structure If … take … divide … then … sum … is. This new form of the poetic structure was repeated again in lines 6 to 11 with small modifications, as take … divide … sum … get. This new repetition mostly conserves the verbal structure of the previous one, and in so doing, conserves relationships among the mathematical concepts. Maintaining the verbs (mostly) allows the noun positions to shift as the coefficient k is verbally distributed: p_{i }becomes kp_{i}, n becomes kn and 2n shifts to 2kn. This repetition culminates in a final poetic structure that provides the outline of a proof. As in previous selections, the support for the argument, warrants or backings according to one’s interpretive preference, emerge through conversational repetition.

As in selection 2, the poetic structure is prominent, but is not expressed rigidly. In line 7, we take kn instead of taking all kp_{i}, which would have more perfectly replicated line 3. From line 3 to line 7, p_{i }and kn occupy parallel discursive positions but they seem to represent different levels of analysis mathematically. How can we interpret this fluidity of topic at a mathematically important moment? Generally, we cannot expect natural discourse to roll out in a consistently mechanical manner. Still, the most important mathematical relationships seem to be the divide … sum relationships, that is, to establish a parallel comparison between the p_{i}s that divide n and sum to 2n, and to establish a parallel comparison with kp_{i}s that divide kn and sum to 2kn. The introductory we take kn (line 7) initiates the topic that the conjecture must address, and then preserves the poetic structure as a way to apply the definition of abundant, decomposing the new integer kn into divisors kp_{i}.

Overall, then, the student started with the conjecture, then repeated this if/then structure in order to introduce a definition, and then repeated it again to outline a proof. By transforming the datum into a definition and then into a proto-proof, the speaker uses poetic structures to preserve relationships among mathematical ideas. Maintaining a close verbal structure across the conjecture, definition and warranting commentary may be a means of reducing doubt. Discourse structure, beyond the mere words, facilitated the student’s first version of a deductive argument.

Poetic structures in collaborative discussions: coactions and competing arguments

In Martin and Towers’ analysis of collaborative coactions, future primary school teachers use a geoboard to classify triangles as scalene, isosceles and equilateral (2009, p. 7, layout modified). Their discussion compares the side lengths of a right triangle. On the geoboard, the hypotenuse consists of two line segments from a vertex to an interior vertex to another vertex, just like each leg of the triangle (see Figure 2, based on Martin & Towers, 2009, p. 7). Throughout much of the conversation, Mary and Shauna believe that the triangle

Figure 2. Geoboard representation of a triangle.

might be equilateral, because each side is “two pegs away.” Hilary believes that the hypotenuse has a different length compared to the legs.

Selection 5

- Mary: But, like that distance should be the same as that [pointing to the horizontal and vertical sides] if they’re, wouldn’t these dots all be equal distance?

- Hilary: Yeah, but distance [pause] yeah that’s what I’m thinking, this should be the same as this [horizontal and vertical sides].

- Mary: So, this distance [hypotenuse], this one would be then, yeah right, don’t you think?

- Hilary: Yeah, but they’re not … see look.

- Shauna: [Laughs]

- Mary: This one right here. No, this one down here, where this is, “cos this is one peg

away and this is one peg away. [Referring to the distance from vertices to centre

peg on the hypotenuse.]

- Hilary: But that’s not the same. 24. Mary: Oh, but …

25. Hilary: Okay, but if you have a square. See this is a square […] so if you have a square. This is two and this is two and this is two and this is two [indicating distances between horizontal and vertical pairs of pins]. But that is not two [indicating the hypotenuse of the triangle] and that’s what the triangle is. […]

- Shauna: This one looks equal. […]

- Shauna: Because it’s two and two and two. [Indicating the length of each side of the triangle in terms of the number of dots, rather than actual units of length.]

As the conversation continues, Hilary recalls the Pythagorean Theorem (line 67), and with this information, the group is able to resolve the question at line 85, concluding that An equilateral can’t have a right angle in it (p. 12). In the following discussion, we will consider the association between Martin and Towers’ coactions and conversational repetition. We will also consider the role repetitions could play if one analyses the conversation using Toulmin’s model.

**Coactions through poetic structures in the geoboard conversation**

Current work on collective coactions does not identify the types of evidence that indicate a coaction occurs, but rather, accepts that readers will notice the referential connections between students’ spoken ideas. Attention to conversational poetic structures can make coaction analysis more concrete. In lines 17 to 25 of this collaborative discussion, there is substantial repetition—both speaker-internal repetition and repetition across speakers —especially between Mary and Hilary. In Mary’s first comment, she establishes a phrase that distance should be the same as that. This phrase sets into motion a series of repetitions that allow the students to express their fundamental disagreements about their mathematical images. The repetitions rely on small changes such as that to this (lines 17 and 18); should be to would be (lines 18 and 19) to this is one peg away (line 22) and this is two (line 25). Though Hilary seems persuasive at line 25, Shauna later suggests that the triangle is really equilateral, by saying This one looks equal … Because it’s two and two and two. (lines 27–28). Emphasising repetition and deemphasising speaker, we could portray the conversation as:

that distance should be the same as that

but the distance […] this should be the same as this

So, this distance […] this one would be then ‘cos this is one peg away and this is one peg away […]

This is two and this is two and this is two and this is two […] But that is not two

This one looks equal …

Because it’s two and two and two

This rendering of the conversation highlights the transformation of that should be into this is into it’s. The repetitions suggest that the students are intently trying to bring their mathematical models of the geoboard triangle into alignment with each other. As the repeated phrases become shorter, the students seem to agree on the features that they need to look at in order to resolve the question, but not on how to visualise or interpret these features. The emergent, collective argument, focusing on several proposed images of distance, has a prominent poetic structure.

Just as repetition does not signal correctness, nor does it signal agreement. Poetic structure coactions in selection 4 demonstrate that shared phrasing does not necessarily mean that the mathematical image is shared fully. Instead, students have agreed upon a verbal sign, and they have agreed to work towards a shared interpretation of it. Repetition is a signal that coaction in emerging, collective mathematical thinking, is happening.

**Competing arguments through poetic structures in the geoboard conversation**

We can also analyse the collective nature of the students’ mathematical reasoning in terms of Toulmin’s model. Despite Hilary’s suggestion in line 25 that the hypotenuse and the sides have different lengths, an argument emerges that the triangle is equilateral. We can identify components of this incorrect argument with the datum, warrant/backing and conclusion at lines 17–18; lines 22 and 28; and line 27 respectively.

Mary expressed the datum for this argument, that the horizontal and vertical sides of the figure have equal length, in line 17: That distance should be the same as that, and Hilary repeated it in line 18, this should be the same as this. Hilary and Mary used similar phrases to express a shared datum. From this datum, Mary and Shauna argue for their false conclusion, which Shauna states directly at line 27, saying This one looks equal. Hilary argues that the side lengths are not all equal, which she states in line 20, … but they’re not. Hilary’s comment is not a rebuttal in the sense that Toulmin uses the term, because a rebuttal supports the current argument. Instead, Hilary’s comment at line 20 is better understood as a statement of a conclusion for an alternative argument. The components of Toulmin’s model for these two arguments, the Mary/Shauna argument and the competing Hilary argument, are shown in Table 1.

Just as in earlier selections, poetic structures play a prominent role in the emerging supportive statements for the arguments—the warrants or backings. In this multi-speaker

Table 1. Toulmin model for two competing arguments.

Datum | Support: warrant or backing | Conclusion |

Mary and Hilary: The horizontal and vertical sides of the figure have equal length. “That distance should be the same as that” | Mary and Shauna: Distances that are “one peg away” are of equal length; or distances that are two pegs away, “This is two”, are of equal length. “This is one peg away and this is one peg away” “Because it’s two and two and two.” | Shauna: The triangle is equilateral. “This one looks equal.” |

Hilary: Distances that are “two pegs away” could be different. “This is two and this is two and this is two and this is two … But that is not two” | Hilary: The hypotenuse has a different length compared to the legs. “Yeah, but they’re not … see look” |

conversation, though, poetic structures link the competing warrants/backings of the competing arguments. All speakers use poetic structures to establish agreement on what the conflict is, and to create contrasting support for the two arguments.

In line 25, for example, Hilary supports her conclusion (the lengths of the legs and the hypotenuse are different) through a poetic structure transformation of Mary’s line 22. Mary’s comment at 22 of … this is one peg away and this is one peg away warrants her conclusion that side lengths are all the same, and at line 25, the sentence becomes Hilary’s … This is two and this is two and this is two and this is two … But that is not two. Hilary transforms Mary’s warrant in order to warrant (or back) a different argument. At line 28, Shauna transforms Hilary’s line 25 with … Because it’s two and two and two, using Hilary’s words to reassert her commitment to her own argument. Each of the lines 22, 25 and 28 are attempts to support the speakers’ arguments, and the later ones are clearly poetic structure transformations of the earliest one. The earliest of these supportive statements, Mary’s line 22, is a poetic transformation of lines 17 and 18 about side length distances being the same, which was the shared datum for both arguments.

I’ve left the interpretation open as to whether these supportive statements are warrants or backings, to acknowledge the likelihood that different readers would interpret them differently. My own preference is to consider Mary and Shauna’s poetic structures as warrants, and Hilary’s as a backing. When Hilary’s comments … This is two and this is two and this is two and this is two … , she seems to use a field dependent, mathematical approach of introducing side lengths and a figure that are not present in the original question; this seems to be a mathematical problem-solving strategy. By identifying all side lengths as … this is two … and contrasting this with the hypotenuse length … But that is not two, Hilary seems to build what Toulmin called an analytical argument, in which the backing directly includes the conclusion as a special case (Toulmin, 1958). Overall, though, the supports for the arguments, whether one categorises them as warrants or backings, are highly facilitated by poetic structures.

In this conversation, poetic structures convey almost all of the mathematical thinking. The speakers use closely related poetic structures to agree upon the datum, to pose competing arguments, and to offer conflicting support—warrants or backings—for the arguments. Just as in selections 2–4, repetition is deeply engaged in establishing the warrants, backings, qualifications and more generally, the support for moving from datum to conclusion.

Conversational repetition is important for describing emerging, collective arguments because it allows speakers to use comparatively few words to signal topics on which they agree or disagree. Nearly all of the coactions in this conversation are literally co-phrases.

Conclusion

Students engaged in emergent mathematical reasoning can convey central components of their arguments, warrants, backings and coactions, through poetic structures. All of the warrant types described by Inglis et al. (2007)—structural-intuitive, inductive and deductive warrants—co-occurred with prominent poetic structures. During a multi-speaker mathematical task, poetic structures were the primary discursive means of presenting coactions (Martin & Towers, 2009). Speakers at many educational levels—from primary to postgraduate—appear to use poetic structures to express their mathematical thinking across all argument types (Staats, 2008; Inglis et al., 2007).

In part, poetic structures facilitate argumentation because they allow speakers to investigate the interplay of similarities and differences. A poetic structure retains a previous part of an image while another part shifts. In selections 1–4, identifying the similarity among things was important to the speakers. In selection 1, Shaniqua’s highly repetitive cadence expressed her sense of image having (Pirie & Kieren, 1994). In selections 2–4, students were engaged in image making, but used poetic structures to express a particular level of similarity and difference. In selection 3, for example, the poetic structure: [It] is to do with its divisors … things that divide it … with multiples allows the speaker to consider alternative concepts that represent nearly the same thing. In selection 2, the poetic structure stem of I know that this statement is true for … allows the speaker to suggest that seemingly different numbers in fact may behave in the same way, in the environment of this conjecture. In selection 4, the initial statement in the poetic structure was a definition. The student expanded the units of the definition slightly to outline the beginnings of an algebraically oriented proof. His uncertainty about the conjecture’s truth may have compelled him to use a poetic structure to stay close to the relationships within the definition. In selection 5, though, Hilary first established similarity This is two and this is two and this is two and this is two … so that she could then forcefully assert the contrast But that is not two. This ability to express gradations of similarity and difference allows poetic structures to explore the kinds of precision that mathematical thinking requires.

A broader accounting for the association of emergent argumentation and grammatical repetition may derive from Toulmin’s distinction between warranting and field dependent backings, ambiguous though this may be. Warrants are less field dependent than backings, so that warrants are components of reasoning that may be widely accepted across disciplines or social settings. As a tool of ordinary conversation for creating a sense of legitimation, coherence or distinction, poetic structures can function as an informal warrant. Poetic structures place concepts into relationships of similarity and difference, and through repetition, assert the impression that these relationships are inevitable.

Quotes of students’ mathematical speech are commonplace in research literature, and in most cases, interpretation of speech is conducted through word-level decoding. Rarely are any discourse features recorded other than the words. And certainly, vast research insights have been garnered through this attention to words and not discursive form. Still, mathematics is a discipline committed to the study of abstract form. Mathematics education is relatively committed to the idea that mathematical meaning can be expressed through multiple representations—not just the “declarative sentence” of an equation, but through graphs, diagrams, and for some, proofs-without-words, in which the meaning is expressed through a non-verbal image. Mathematics education researchers can consider poetic structure analysis as a new form of representation for understanding how mathematical discourse amplifies, qualifies, and advances the word-level message.

Disclosure statement

No potential conflict of interest was reported by the author.

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# Linguistic indexicality in algebra discussions

Susan Staats^{a}^{,}∗, Chris Batteen^{b}

a r t i c l e i n f o Keywords: Mathematical communication Discourse Indexical language Social learning | a b s t r a c t In discussion-oriented classrooms, students create mathematical ideas through conversations that reflect growing collective knowledge. Linguistic forms known as indexicals assist in the analysis of this collective, negotiated understanding. Indexical words and phrases create meaning through reference to the physical, verbal and ideational context. While some indexicals such as pronouns and demonstratives (e.g. this, that) are fairly wellknown in mathematics education research, other structures play significant roles in math discussions as well. We describe students’ use of entailing and presupposing indexicality, verbs of motion, and poetic structures to express and negotiate mathematical ideas and classroom norms including pedagogical responsibility, conjecturing, evaluating and expressing reified mathematical knowledge. The multiple forms and functions of indexical language help describe the dynamic and emergent nature of mathematical classroom discussions. Because interactive learning depends on linguistically established connections among ideas, indexical language may prove to be a communicative resource that makes collaborative mathematical learning possible. © 2010 Elsevier Inc. All rights reserved. |

^{a }Department of Postsecondary Teaching and Learning, 178 Pillsbury Drive SE, University of Minnesota, Minneapolis, MN 55455, USA ^{b }Institute of Linguistics, English as Second Language, and Slavic Language and Literatures, University of Minnesota, Minneapolis, MN 55455, USA

**Introduction**

In contemporary mathematics classrooms, students verbalize, negotiate and evaluate mathematical ideas. Students speak to each other for a variety of purposes including monitoring the relationship of one’s actions to those of others, reacting to the assumptions of others, and taking claims into account (Voigt, 1998, pp. 201–202). Relating to the ideas of others is a fundamental activity in mathematical collaboration. Given the prominence of discussion in reform mathematics classes, it is surprising that we know relatively little about the linguistic mechanisms that make this type of learning possible. How can mathematics students learn from each other by talking when they have only partial knowledge of the subject? Many modes of communication assist students as they construct mathematical ideas together, but speaking is regarded by many teachers and researchers as an indispensible dimension of classroom learning. To understand how collaborative learning is possible, therefore, we must identify linguistic resources that allow students to share understandings, negotiate disagreements or to focus each others’ attention on a single issue.

In order to be understood in this densely relational setting, students who offer mathematical opinions must coordinate their ideas with the ongoing action by relating it to some aspect of their learning: previous statements from students or the teacher; graphs and equations on the chalkboard; the textbook; personal written work; or knowledge that has not yet entered the conversation. These learning situations are dynamic, constructed from moment to moment by the language choices of participants.

∗ Corresponding author. Tel.: +1 612 625 7820; fax: +1 612 625 0709.

E-mail addresses: staats@umn.edu (S. Staats), batteenc@umn.edu (C. Batteen).

0732-3123/$ – see front matter © 2010 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmathb.2010.01.002

Linguistic anthropology positions these verbal expressions as sociocultural activities that express and establish interpersonal relationships. Verbal expression must be interpreted within a setting that includes physical characteristics, social beliefs and actions, and knowledge that may be shared differentially across participants. This situational complexity and the varied scholarly traditions that require nuanced analysis of speech defy unified definitions of the context of speech (Goodwin & Duranti, 1992, p. 2). Still, a descriptive perspective that underlies anthropological treatments of context is that

(t)he notion of context...involves a fundamental juxtaposition of two entities: (1) a focal event; and (2) a field of action within which that event is embedded (Goodwin & Duranti, 1992, p. 3).

Contexts of speaking, therefore, involve a figure and ground relationship between a verbal expression and a potentially complex situation of use. Neither the figure nor the ground in this “juxtaposition” can be subsumed within the other. This view of context recognizes that participants negotiate their positions relative to each other and that they influence each others’ attention and disattention to dimensions of the situation. In this way, the focal event influences relevant aspects of context. On the other hand, verbal expression is often part of the focal event, but “talk itself both invokes context and provides context for other talk” (Goodwin & Duranti, 1992, p. 7). Prior conversation can become part of the field of action that assists interpretation and that motivates future actions.

As an example of this perspective on context, consider a mathematics classroom in which students collaborate on the algebraic task of solving a logarithmic equation. A student might negotiate his or her social position by speaking early in the discussion through mathematical assertions. Students must use effective forms of referencing in order to direct each others’ attention towards contested elements of the solution and away from elements that are not relevant to the immediate discussion. If a student refers to properties of logarithms, this comment might invoke an earlier shared context of a lecture or textbook reading. This prior discourse provides a context for the speech in the ensuing collaboration by building authority for a particular proposed solution method.

Speakers manage the relationship between their expressions and this dynamic context largely through indexical language. Indexical expressions acquire their meaning from the context of their use. They bear an “existential relation with what they refer to” (Duranti, 1997, p. 17; Peirce, 1955), that is, they convey little meaning until used in a particular context, in reference to a specific element of the context (Leezenberg, 1994). In a mathematics classroom discussion, for example, if a student comments, “I like his method,” the pronouns I and his refer to specific people in the speaking context; method refers to a series of mathematical actions. The meanings of I and his cannot be determined without knowledge of the context. If this sentence were repeated in a classroom of different students, the pronouns necessarily refer to different individuals, while the referent of method could shift or not, depending on the situation. As we outline below, certain classes of words are highly indexical, however, many levels of discourse structure—phrases, forms of address, genres—can establish relationships between speaking its context. In linguistic anthropology, all of these levels of referencing are considered indexical.

In this article, we describe mathematics students’ use of indexical language to form dynamic contexts of knowledge within their classroom. We suggest that indexical language is the central feature of language that allows students to construct mathematical ideas through collaborative discussions. Analysis of indexical language may, therefore, become a useful tool for researchers who wish to trace pathways of shared knowledge and to give evidence of the collaborative nature of the discursive construction of knowledge.

To do this, we inventory all the currently known types of indexical language that students use to express mathematical ideas and actions. Very often, these indexical linguistic forms are drawn from ordinary language and they have functions that are well understood outside of mathematics education. Within mathematics classrooms, however, students appear to put commonplace indexical forms to use as they perform significant mathematical activities. As we will see, students use indexical language forms to conjecture, to coordinate mathematical speech with written inscriptions, and to reify or objectify mathematical knowledge (Cobb, 2002; Sfard, 1991). Without careful analysis of the indexical components of their meaning, the mathematical character of ordinary indexical language forms can go unnoticed. Taken together, these types of indexical language can account for much of the convergence and redirection of attention that is characteristic of collaborative mathematical learning.

**Indexicality and context**

Indexicals are the linguistic structures that allow speakers to connect ideas to situation. Indexical words are commonly understood as words that “point to” or identify aspects of context, linking “word to world,” as Rymes phrased it (2003, p. 121). Pronouns, for example, usually change their referents entirely with a change in context, these forms of indexical language are called shifters (Jakobson, 1971). Besides personal pronouns, demonstratives (e.g. this, these, that, those) and adverbs of time (e.g. now, before, yesterday, whenever, next) are the parts of speech most commonly identified as indexicals. All words have some indexical character; their meaning shifts slightly when used in a different speech situation. But some words, like pronouns, acquire most of their meaning from context (Rymes, 2003). These types of indexical language are also termed deictics some linguists and linguistic anthropologists (Hanks, 1990; Leezenberg, 1994).

Indexicality, however, can occur through speech forms that are not thoroughly recognized in the educational literature. Some verbs of motion, for example, establish an origo, or a point of reference within the context, associated with the direction of action. The word push describes motion away from an origo, and the word pull describes motion towards an origo. Often, an origo is associated with the perspective or position taken by a co-participant in a speech event (Levinson, 1996, p. 360).

However, the types of coordinate systems established by indexical verbs of motion can be quite complex, depending on the degree to which participants share knowledge of the context of speaking (Hanks, 1990, p. 47). Talmy conceptualizes verbs of motion in terms of “fictive motion,” descriptions of movement which does not actually take place, along various types of pathways (Talmy, 1999). Elsewhere, we have argued that verbs of motion occur in students’ spontaneous classroom speech because the origo anchors attention in the representation of the problem, and the motion illustrates mathematical processes like addition or subtraction (Staats & Batteen, 2009).

While solving the equation 12−1y =4, for example, a student suggested that “you slide that 12 over to the other side” (Staats & Batteen, 2009, p. 65, emphasis added). In Talmy’s perspsective, the verb slide may represent a targeting path, that is, “fictive motion (that) establishes a path along which the Agent further intends that a particular subsequent motion will travel” (Talmy, 1999, p. 220). We have suggested elsewhere that the verb slide may replace the operation subtract in this algebraic explanation (Staats & Batteen, 2009). While replacing the verb subtract with the verb of motion slide reduces the mathematical precision of the explanation, the directionality of slide illustrates the problem-solving strategy of isolating constants on the right side of the equation and variables on the left. This manner of explanation does not demonstrate the nature of the student’s understanding of the transformation of equations. It does, however, provide clues that may help listeners anticipate future problem-solving actions and better participate in the speaker’s strategy. Strategic competence is a mathematical proficiency that is distinct from procedural ability (National Research Council, 2001). Indexical verbs of motion incorporate a sense of direction that allows students to express their problem-solving strategies.

Indexical language can rely on longer discourse units, too, and it can create referential links among spoken contexts as a means of organizing a cohesive, persuasive or ideologically coherent statement. For example, suppose a university teacher asks her students, “How would you solve this problem?” The students’ forms of evidence and argumentation would be very different if the question were asked in a mathematics class or in a political science class. The disciplinary context of the classroom influences how students interpret the question; the phrase solve this problem indexes, or points toward, different solution strategies.

Revoicing is also a longer discourse unit that is indexical. When a teacher revoices a student’s comment, the teacher’s phrase refers back, or indexes, the student’s comment, and casts it in a new light. Repetition of words and phrases with slight variations can also occur in student monologues or in conversations that do not involve the teacher. These repeated, co-referential language forms are known as poetic structures in linguistic anthropology (Jakobson, 1960; Wortham, 2003, pp. 21–22). Linguist Roman Jakobson suggested that any repetition of sounds, words, rhythms or grammatical forms calls attention to the form of a message. For Jakobson, this focus on the form of a message, rather than simply the expression of referential meaning, is at the heart of poetic expression.

The poetic function, as Jacobson defines it, “projects the principle of equivalence from the axis of selection into the axis of combination” (Jakobson, 1960, p. 358, original italics deleted). In other words, there is a class of units, perhaps alliterative sounds, perhaps a grammatical phrase, that are recognizably similar, or “equivalent.” These similar units are repeated over a longer, contiguous segment of discourse, the axis of combination. The repetition creates a sense of self-reference that makes the message distinct from other types of discourse; in this sense, Jakobson considers the message to be poetic. Wortham defines poetic structures as “patterns of indexical cues” that “limit the context relevant to interpreting a set of utterances. (2003, p. 22).

In mathematics classes, poetic structures assist students as they refine conjectures in a collaborative mathematics discussion, and they provide students with a linguistic form that can represent mathematical form (Staats, 2008). Grammatical repetitions may prove to be the most important type of poetic structure in mathematics discourse, because they allow the speaker to express precise relationships among ideas. In the following statement, a student used poetic structures to explain his understanding of a population growth model (Staats, 2008, p. 27):

- We thought
- that the birth rate
- is just going to keep on growing.
- and also
- that life expectancy 6 is going to be higher. 7 So therefore

- the growth of population

- is going to increase no matter what.

The three repeating, poetic statements (lines 1–3, lines 4–6, and lines 7–9) are represented by arranging similar phrases in indented columns, following Dell Hymes (1981). The repetition of demographic variables (birth rate and life expectancy) points the listener’s attention towards the variable that occurs in the parallel position in the next phrase (growth of population). This creates an indexical relationship among these three positions in the student’s statement; the grammatical repetition creates the sense that there is a logical relationship among these variables. In this type of example, Wortham analyses repetition as a type of indexicality. He defines poetic structures as “patterns of indexical cues” that “limit the context relevant to interpreting a set of utterances. (2003, p. 22). By repeating the verb phrase is going to in each statement, the student limits or specifies the type of relationship that exists among the variables, that population growth varies directly with birth rate and life expectancy.

According to Wortham, educational research has yet to address the relevance of indexicality fully, but still, “much of everyday speech gets its meaning indexically” (2003, p. 9). In mathematics classrooms, students frequently use the indexical forms of everyday language to negotiate classroom topics and responsibilities, to narrate methods and to organize the mathematical ideas that arise in classroom discussion.

**Indexicality in education research**

**Pronouns**

Pronouns are highly indexical because they draw much of their meaning from their context of use. In classrooms, these simple words can be powerful organizers of group membership and disciplinary expectations. Pimm (1987), for example, considers the coercive qualities of the pronoun we. In a lesson on subtraction, for example, a teacher addresses students, “What do we take from the tens column? We take a ten, don’t we?” (Pimm, 1987, p. 65). In this example, the use of the term we supports the teacher’s position of authority within a community of mathematicians, conveying the message that students should conform to standard arithmetic methods. Rowland (2000) points out that students sometimes use the pronoun you to express generalization. A student was asked, for example, to construct the number twenty in any way she could imagine. She responded, “you need to go minus one add twenty one equals twenty” (Rowland, 2000, p. 110). Rowland suggests that the choice of you instead of I in this statement depersonalizes the student’s procedure, positioning it as a general method that is accessible to anyone. In another example, a student uses the pronoun it: “times can do it, can’t it, and add, and take...no, take aways can’t do it” (Rowland, 1992, p. 46). Rowland interprets this statement as reference to commutativity, a property that the student can conceptualize but cannot name.

**Presupposing and creative indexicality**

As we have seen, the words known as shifters, such as pronouns and demonstrative adjectives, derive most of their meaning from the context of an interaction. However, the negotiation of context does not depend solely on these forms of expression. Speakers refer to contexts using more complex words and phrases as well. For example, if algebra students encounter a problem involving the sum of rational functions for the first time, and a student recommends finding a common denominator, the phrase “common denominator” references knowledge acquired in a different context, that is, adding numerical fractions. The student has pointed classmates’ attention towards shared experiences that were not previously in play in the rational function discussion. The comment indexes a different domain of knowledge and thereby widens the current context of learning by providing a new line of inquiry.

Silverstein (1993) describes a framework for analyzing this kind of indexical coordination of multiple contexts. The students’ suggestion of using common denominators in a novel situation is an example of entailing, or creative, indexicality. The idea is new for this particular interaction and it changes the shared learning context to which the class can now refer. On theotherhand,whenspeakersfocusattentiononaspectsofthecontextthatarealreadyrecognizedintheongoinginteraction, they use presupposing indexicality. This type of indexicality might occur in an algebra class if a student asks, “How did she get 3x^{2}?” The term 3x^{2 }might be written on the board, on a paper, or it might be a phrase that another student just spoke. In any case, it is already part of the group’s shared knowledge, an aspect of context that has at least provisionally been recognized as relevant. Uses of the pronoun it to express mathematical generalizations through anaphoric references (to prior ideas) and cataphoric references (to ideas yet unspoken) are examples of presupposing and entailing indexicality, respectively (Rowland, 2000, pp. 101–104). Similarly, Radford (2000, 2002) has identified presupposing positional indexical phrases such as top row, bottow row and demonstratives this and that as means for students to express conjectures about diagrammatic patterns that they can imagine but that they have not physically constructed. Tracing speakers’ use of presupposing and entailing indexicality highlights the emergent, dynamic, and negotiated nature of context.

Educational research on presupposing indexicality has focused on its role in the construction of social identities through classroom interaction. Identity work often happens through the coordination of different indexical signals. For example, both a high pitch and a mispronunciation are presupposing indexes of a child’s voice (He, 2003). Imitating a person’s speech using both of these indexicals can assign a childlike identity to the person. Similarly, over a series of classes, if a teacher characterizes a student’s comments as irrelevant, joking, or disruptive, these three representations may come to presuppose one another (Wortham, 2005). If the student presents an unexpected interpretation of classroom readings, the contribution may be cast as irrelevant and disruptive rather than as a legitimate analysis.

The use of indexicals creates an unpredictable context in which social identities are negotiated. He (2003) reports a case in which a student indexed or referred to the statements of the principal of a Chinese heritage language school. By doing so, the student implicitly referenced culturally based concepts about moral authority and was thereby able to shift the teacher’s evaluation of the student’s work. Similarly, students in a phonics-based literacy program practiced new words using sentences that indexed their own experiences and interests but that contested the teacher’s pedagogical expectations (Rymes, 2003). Linguistic indexicality provides a framework for analyzing interactions at the intersection of learning and social dynamics.

**Constructing disciplinary knowledge and discourses**

These trends in indexicality research are relevant to at least three fields of interest in contemporary mathematics education. First, studies of collective understanding in mathematics and science describe ways in which students build knowledge collaboratively by listening and reacting to each other’s ideas. This type of collaborative construction of knowledge has been termed coaction:

“(C)oacting is a process through which mathematical ideas and actions, initially stemming from an individual learner, become taken up, built on, developed, reworked, and elaborated by others, and thus emerge as shared understandings for and across the group, rather than remaining located within any one individual” (Martin, Towers & Pirie, 2006, p.

156)

This interactive creativity has been described through musical metaphors in which the teacher orchestrates (Forman & Ansell, 2002; Jurow & Creighton, 2005) and students are performers who listen to each others’ themes, repeat and refine them until a collective agreement is reached (Martin et al., 2006).

Clearly, this model of an intensely interactive classroom depends a great deal on a system of linguistic referencing. The players must know how to listen to key parts of each other’s statements. A theme emerges and becomes recognized by its resonance with or repetition of a prior motif. Refining an idea requires both reference to the idea and the introduction of new elements. From a linguistic standpoint, these actions of identifying specific elements of context, repeating phrases, and introducing new ideas are handled by various types of indexicality, including demonstratives, pronouns, poetic structures, and presupposing and entailing indexicality.

Another line of research that may benefit from attention to indexicality are studies of students’ acquisition of formal ways of speaking and writing in mathematics. As students improve their fluency using the mathematics register, they learn ways of communicating and ways of knowing that are valued by mathematicians (Moschkovich, 1996, 2002; Pimm, 1987). A similar process is known in anthropology as language socialization, “the process of becoming a culturally competent member through language use in social activities” (He, 2003, p. 94). Theories of language socialization depend on analysis of indexicality because novices use it to identify aspects of context that constitute and negotiate the social positions, feelings, and knowledge of the experts who are guiding them (He, 2003; Ochs & Shieffelin, 1984; Ochs, 1990).

Lastly, the study of students’ mathematical communication can benefit from analysis of linguistic indexicality. As the discourse samples in this paper will show, students use indexical language to express mathematical actions such as presenting and refining conjectures, generalizing, representing mathematical structures, and reifying mathematical concepts and processes into short phrases that can be manipulated during higher-demand tasks. While these are specifically mathematical discursive actions, students often use indexical forms drawn from informal, everyday language to express them in classrooms conversations. Indexical language is one of the features by which ordinary language achieves some degree of precision, particularly with regard to location, movement, and structural or logical relationships. Teachers and researchers who listen for the many varieties of indexical language in students’ mathematical expressions may better be able to appreciate students’ informal attempts to speak about mathematical ideas precisely.

Linguistic indexicality is a growing area in educational research. Relationships, beliefs and topics can change, sometimes quite rapidly, in a classroom. Students and teachers must constantly reorient themselves to shifting contexts. These contexts involve both social relationships and the disciplinary content of the class. Indexical language assists in the negotiation of social identities and in the organization of ideas and beliefs about the learning topics.

**Setting and participants**

The setting for this study was two sections of an introductory algebra class at the University of Minnesota in fall 2005 serving students who had received a low score on the university math placement exam. The students were all young adults whose ages ranged from approximately 18 to 25 years. On most days, students worked together in small, informal groups followed by full-class guided discussions. The conversations below represent students’ work in full-class discussions during the first eight weeks of a fifteen-week class.

The first author was the instructor. The class was taught primarily through constructivist, guided full-class discussions in which the instructor posed mathematical questions, students offered multiple conjectures and explanations and collectively evaluated these until reaching a consensus (Bartolini Bussi, 1998). Students solved increasingly complex problems, discussed multiple conjectures, and developed rationales for all of the major results in the class (e.g. the zero exponent, exponent laws, the use of slopes in graphing, and factoring methods). When students submitted multiple conjectures, the class offered explanations for each and used polling to eliminate answers until a consensus was reached. At times, the class stated theorems and procedures formally on the board, but only after students had articulated and justified them.

**Methods**

This study was conducted as a communicatively focused ethnographic study in which the instructor was a daily participant-observer in the classroom. As a participant-observer, the instructor must be willing to take a reflexive stance, that is, to reveal the contingency of her authority in the classroom, not always fully in control, not always fulfilling the essential components of inquiry-based teaching. Several of these moments are apparent in the selected conversations. This is a limited ethnographic investigation of a specific feature of language (Eisenhart, 1988). While ethnographers do not consider their work to be objective, most of the analyses presented here rely on structural features of discourse and are guided by substantial scholarship in linguistic anthropology. The linguistic focus reduces the potential reactive effect—students were not familiar with the concept of linguistic indexicality—and improves the validity of the study (Bryman, 2004). Readers are able to judge for themselves the clarity of the connection between data and conclusions (Sanjek, 1990). Our intention is that other researchers may be able to identify similar linguistic structures in recorded mathematics conversations and develop their own interpretations guided by our discussion of indexical referencing.

The classroom conversations presented here were recorded with an iRiver mp3 player. Nearly every class meeting for the two sections algebra was recorded. Because a single recorder was used, the selections represent full-class guided discussions rather than the students’ daily small group work. A research assistant attended class daily and positioned herself so that she could take notes on student behavior and on the mathematics on the board. The research assistant also graded homework and maintained the class grade book. She had daily administrative interactions with all the students, and they knew her quite well. It is impossible to know whether her note taking was really as unobtrusive as it seemed, but this appears to be the case because most of her interactions with students were over administrative matters: “Did I turn in homework for Section 2.3?” rather than directed towards the ethnographic process.

The research assistant recorded each problem and solution method on the blackboard and tried to identify each speaker whocontributedtothesolution.Thesameresearchassistantlistenedtotherecordingsandannotatedthemin5-minintervals for time, speaker, topic, and linguistic features such as use of pronouns and other indexical words. The data collection method allowed for triangulation for the purpose of internal reliability, based on the instructor’s participant observation, ethnographic notes written by an independent observer, and annotated audio recordings.

One of the major limitations of this method is that in an inquiry-oriented math classroom, board work and solution methods are very dynamic. There are frequent erasures and substitutions of new conjectures within a problem. In most cases, the flow of a solution method can be reconstructed from the ethnographic notes and the audio recordings. Another limitation is that it is not always possible to distinguish overlapping speech. At times a comment is attributed to one speaker when several more might have made the same suggestion simultaneously. Highly ambiguous conversations are not reported here.

Our selected conversations represent relatively ordinary moments in the classroom, but they are rich in indexical language. We selected routine classroom conversations, not necessarily the major breakthroughs as when a student expresses an algebraic generalization, because a great deal of learning takes place through small, simple exchanges: a kind of classroom vérité.

We selected conversations that show a range of uses of indexical language, particularly ones that fulfill multiple functions or that create rapid shifts in the learning context to which students and the instructor must respond. The selections are intended to show the overwhelming presence of indexical language in the making of socio-mathematical meaning. We emphasize students’ use of indexical language to negotiate issues in mathematical thinking, more than the construction of social identities, although identities are in play in several selections. In these selections, indexical language allows students to negotiate topics, negotiate responsibility, and to describe mathematical processes. It is central to the dynamic and emergent nature of learning contexts in math classrooms.

**Selected conversations**

**Negotiating responsibility with presupposing indexicality**

In discussion-based mathematics classes, students have substantial control over the conversational topics. In the first selection, a student uses presupposing indexicality to negotiate a marked shift in the group’s attention. A few minutes before this exchange took place, the students had worked in small, informal groups to decide whether or not pairs of exponential expressions produce the same answer. The instructor believed that the class had reached a consensus on the answers: “different” for the first expression and “same” for the remaining two, and she began to lead discussion on a new problem at an adjoining blackboard. Joseph raised his hand and used precise indexical language to steer the class topic back to the last exponential expression.

Interaction | Boardwork | ||

1 Joseph: | Miss Teacher. | Same or different? | |

2 Inst.: 3 Joseph: | Yes, sir. Over there, you had those the same, but they’re different. (Instructor points to the first expression). | 5^{2 }+ 3^{2 }vs. (5 + 3)^{2}- − 5)
^{2 }vs. (5 − 8)^{2}
- − 7)
^{3 }vs. (7 − 9)^{3}
| different same same |

4 Joseph: | Go down one. | ||

5 Inst.: | This one here? | ||

6 Joseph: | Go down one. | ||

7 Inst.: | This one here? | ||

8 Joseph: | You got that the same. |

**Selection 1. Negotiating topic and responsibility with presupposing indexicality**

This conversational exchange relied heavily on presupposing indexical expression. Almost every word—over there, those, this one here, go down one, same, different—referred, with little explicit description, to expressions written on the board. When Joseph said Over there, you had those the same, but they’re different, the instructor did not yet understand the nature of his confusion. Both Joseph’s statement Go down one and the instructor’s response This one here? used indexicality so that the instructor could locate the source of the disagreement. Coordinating mathematical talk with written representations is a commonplace mathematical action that can be handled through indexical language.

The instructor promoted the perspective that students were in control of mathematical topics and the assessment of mathematical truth. The instructor literally performed this ideology as Joseph orchestrated her gestures. Through indexical language, Joseph and the instructor collaborated in shifting attention from one math problem to another, so that the learning context for the entire class was altered.

The negotiation in this exchange was not simply about the mathematical topic that the class was to consider. It also concerned allocating responsibilities in the classroom for shifts in topic and for possible mathematical mistakes. Joseph may have felt self-conscious about returning the focus of his classmates to a problem after another problem was underway. While student control of topic eventually becomes a routine aspect of a discussion-based class, Joseph may not have been certain of the classroom norms at this early point in the semester. He portrayed the results of the problem as a mistake of the instructor, a perspective that he asserted through his use of the pronoun you. When Joseph said You had those the same, but they’re different...You got that the same, the pronoun you is not used in the generalized sense of one that is so common in mathematics classrooms (Rowland, 2000). You in this case refers to the instructor herself. Joseph contested the instructor’s classroom ideology, suggesting that it is the instructor’s responsibility to ensure that answers are correct.

This was a moment in which the rules and values of the classroom were negotiated alongside mathematical meanings. In this scene, then, a few brief, highly indexical sentences at once negotiated classroom attention, mathematical truth, and ideologiesofresponsibility.Thesesmallphrasescanbethecentralwaythatmathematicalknowledgeandsocialrelationships are organized in a classroom.

**Contesting a bid for entailing indexicality**

In this selection, students were beginning their study of the addition of rational functions with a discussion of how to find the common denominator of (x −2) (x +2) for the expression 3/(x −2)+4x/(x +2).

Interaction | Boardwork |

1 Inst: So first let’s analyze, figure out what the common denominator is. | Simplify |

2 Keisha: Multiply x plus two times ...x minus two, yeah, there you go. | 3/(x −2)+4x/(x +2) |

- Jody: x squared.
- Joseph: That is difficult.
- Inst: Think of that ... (pointing towards the
(x–2)). That is just one whole thing. - Joseph: How? How can you think of that like that?
5.2.1. Selection 2. Negotiating an entailing indexical statement | Common denominator: (x +2) (x −2) |

In this exchange, Joseph and the instructor negotiated the issue of algebraic grouping. Joseph interpreted (x −2) as two unlike terms that must be manipulated separately. The instructor used the term that along with a gesture spanning the parentheses, to offer a new interpretation, that the expression (x −2) is also an algebraic unit that can be multiplied. When the instructor said That is just one whole thing, the indexical term that was a presupposing reference to the factor (x −2), but the predicate that followed, just one whole thing, was an entailing interpretation of the factor that had not been expressed yet in the class. Joseph did not accept this interpretation, saying How can you think of that like that? In this sentence, the first time Joseph says that, he refers to the factor (x −2). The second that refers to the instructor’s entailing interpretation that he does not accept. He repeats the presupposing/entailing form of the instructor’s explanation, and he rejects it.

The negotiation of grouping symbols is, as in the previous selection, an example of the mathematical action of coordinating talk and written mathematical representation. It is also a negotiation regarding the process of reification of mathematical processes. As students develop mathematically, concepts and processes that once required careful consideration become objectified, and become the material themselves for symbolic manipulation (Cobb, 2002; Sfard, 1991). Joseph and the instructor struggle with this issue, using words that by themselves have very little mathematical content.

This example emphasizes the importance of analyzing mathematics conversations through the multiple lenses of indexical referencing. If we think of word meanings as denotational, dictionary definitions, without accounting for indexical control of context, we will not be able to adequately describe meaning construction in mathematics classes. In his sentence How can you think of that like that?, Joseph used a single word to refer to both a mathematical representation on the board and to the instructor’s interpretation. The multiple indexical functions of the word that were not ambiguous in this exchange. Despite the lack of mathematical vocabulary, neither speaker had trouble understanding the other.

In this conversation, indexicality allowed the speakers to juxtapose ideas and take a position on them using very few words.

**Expanding context through entailing indexicality**

Students in the next selection discussed the solution of a system of equations: x =6−5y and 2x +9y =4. Sam dictated steps of the substitution method very quickly, and several students appeared to have questions about the method. In line 1, when the instructor tried to identify the line of the board work that was confusing, Sam thought that she was asking which linear function provided the solution to the system of equations.

**Selection 3. Shifting contexts with presupposing and entailing indexicality**

Sam’s response of y = 8 is a straight line that goes through the y equals 8 area uses two layers of creative, entailing indexical meaning. Sam begins his statement by referring to y = 8. This is a presupposing indexical reference to an equation on the board. By following this with the predicate is a straight line, Sam proposed a graphical interpretation that had not been raised yet during that class day. The original presupposing reference was transformed into an entailing comment that widened the problem-solving context from algebraic to graphical interpretations of systems of equations.

A more surprising use of indexicality occurred when Sam continued his sentence, y = 8 is a straight line that goes through the y equals 8 area. At this point, y = 8 shifts its function from noun—an equation on the board—to an adjective that describes a location on a graph, perhaps the y-intercept of a horizontal line, or a region surrounding the intercept. As an adjective, y = 8 indexes or specifies a location in an imagined context that has not yet been drawn on the board. In a single sentence, Sam used two entailing references. The first introduced a new context, graphical representations, and the second located a specific point, or surrounding region, in this imagined space.

In both selections 5.2 and 5.3, a single phrase (that and y = 8) was used twice in a sentence to refer to distinctly different ideas, acting first through presupposing and then entailing indexicality. In both cases, the entailing indexical reference brought new contexts into play in the problem-solving discussion. Entailing indexical references are junctures within mathematics conversations. The concise nature of indexical expression allows these shifts to be proposed very quickly. All participants, students and instructor alike, had to decide whether to accept this new context and what position they would take on it. In these examples, a denotational perspective based on dictionary meanings would be inadequate to describe the effect on learning context and classroom focus established by the same phrase in a single sentence. Attention to the action of different types of linguistic indexicality is necessary to account for these rapid and dramatic changes in a collaborative problem-solving session.

Discursively shifting contexts can represent significant forms of mathematical thinking. First of all, Sam appeared to be thinking of multiple representations associated with the equation y =8. He referenced the written form of the equation that is on the board and then used the same phrase to introduce the graphical form of the equation. Students’ ability to shift among varied representations is a sign of mathematical growth (Brenner et al., 1997). Secondly, Sam’s use of the phrase y =8 for multiple purposes—equation, graph and perhaps, y-intercept—was an opportunity to probe the class for generalizations.

By repeating the phrase y =8 for multiple conceptual contexts, Sam may have been hinting that horizontal lines always have y-intercepts that (informally) have the same name.

**Narrating solutions with deictic verbs of motion**

In the following selection, students began for the first time to solve a system of equations using the addition method (Staats & Batteen, 2009, p. 66).

Interaction | Boardwork | |

1 Jon: | Ok, make a line and... | |

2 Inst: | Wait, first. analyze it, what’s cool about this problem? | Solve the system: |

- Jon: They’re all lined up, the x and the y values and the non-variables.

- Inst: And there’s something else...

- Jon: The ys are opposites and they cross out when you drop them through. Chea.

- Inst: Yeah, good observation.

**Selection 4. Explaining a procedure with indexical verbs of motion**

Here, Jon used several indexical verbs of motion to summarize a solution procedure: lined up, cross out and drop through. Each verb phrase used the origo, or reference point, of the verb to focus attention on a particular aspect of the problem. The phrase lined up asks the listener to notice that like terms are arranged in columns (here, the origo is more of a reference line passing through a column rather than a point). These verbs may represent what Talmy calls demonstrative paths in which a “fictively moving line functions to direct or guide someone’s attention” (1999, p. 219). As we describe elsewhere, students in this class tended to offer algebraic procedures in terms of mathematical operations and resultant lines of text which the instructor wrote on the board (Staats & Batteen, 2009). The full class then evaluated the student conjecture until they reached a consensus. We argue that cross out may refer to the act of summing the y terms to get zero below the imaginary line that Jon asked the instructor to draw, and drop through may refer to the process of addition itself. The origo of each verb anchored attention in a particular location, and the motion of the verb represented mathematical processes.

The speaker used the indexicality of verbs of motion to outline a procedural strategy for an algebra problem. Each of the verb phrases line up, drop through and cross out summarize mathematical procedures and the full statement arranges these procedures into a broader sequence of action. Summarizing procedures, assigning a name to them, and arranging them in a sequence suggests that the student considers the procedures not simply as processes but as mathematical objects themselves that can be manipulated (Sfard, 1991). The indexicality of the verbs of motion provides an informal means of expressing this reified mathematical knowledge.

**Poetic structures**

Poetic structures are a more complex type of indexicality that occurs when speakers repeat words and phrases to build unique indexical signs. While repetitious speech is often devalued in ordinary conversation, Tannen argues that it serves a range of functions including demonstrating interpersonal involvement, improving the efficiency of speech, and building shared meanings among speakers (Tannen, 1989, pp. 46–53). Revoicing is a type of repetition that occurs in classrooms between teachers and students, but repetition can occur in mathematics classrooms in more complex ways, too (Staats, 2008). A single speaker can construct an inductive or deductive argument through repetition, and a group of students can construct and refine ideas by repeating each others’ statements with small transformations.

In this selection, a student solved a linear equation using a standard procedure. Shaniqua used parallelism extensively to present a series of transformations of an equation. The first repeated unit has the form we gotta (perform an action) and we get (an equation). This sentence structure was repeated in lines 1, 2 and 4 and it is highlighted with a single underline. Shaniqua used a second repeated unit in lines 3 and 6 that has the form we/you can (perform an action) which gives you (an equation), highlighted with a double underline. This two-fold repeating format gave the speaker a coherent and understandable means of describing mathematical operations and the resulting lines of algebraic text.

In the following transcript, both underlining and indentation are used for clarity (Hymes, 1981; Tannen, 1989). The algebraic equations that the student produces are shown in bold typeface. With indentation, grammatically similar phrases canbearrangedintocolumns.Thisdrawsattentiontothepatternsofspeechthatthestudentusedasexpressedherargument.

Boardwork

(6w − 1)/(2w − 1) − 5 = (2w − 3)/(1 − 2w)

After many lines of work, students arrive at the series of equations:

w

Interaction

1 Shaniqua: So we gotta try to get all the like terms on one side and

2all the other terms on the other,

- so we can plus 2w on both sides which gives you −2w + 4 = 3.

- Then you gotta minus 4 and then you get −2w =−1

- and then you got,

- so you can divide −2 which gives you w is equal to,

- I don’t know if it’s negative 1/2 or positive 1/2.

- Mindy: Positive 1/2.

- Shaniqua: So that’s our answer.

5.5.1. Selection 5. Poetic structures in an algebraic solution.

Shaniqua’s words in this explanation were spontaneous, but they displayed a tightly organized rhetorical structure that conveyed the orderliness of her method. First, in lines 1 and 2, she articulated an algebra paradigm for equations, saying that the equation has two sides and that she will separate unlike terms on opposite sides. This statement also established a sentence format of action, result, equation that she used throughout her explanation. At a broad level of discourse organization, the action, result, equation format is a poetic structure that occurred in each sentence (1/2, 3, 4, 6/7). In another sense, there were two more specific poetic structures, the gotta/get sentence format and the can/which gives you format.So,wecouldinterpretthisaseitherfourexamplesofthesameabstractpoeticstructure,ortworepetitionseachoftwo related poetic structures. Each time Shaniqua used a poetic structure, she altered the terms slightly in order to accommodate a new mathematical action or a resulting equation. These kinds of repetition are indexical because the repeated units refer back to their previous versions and create a sense of cohesion throughout the statement. For this reason, Wortham calls poetic structures “bundles of indexicality” (2003, p. 9). While the orderliness of Shaniqua’s explanation was perhaps not deliberate, the organized pattern of speaking, drawn from habits of daily language use, may have assisted her classmates in understanding and anticipating her algebraic procedure.

At the developmental algebra level, students may still use an arithmetic, rather than algebraic, understanding of the equal sign. Students may still consider the equal sign as a means of “announcing a result” of calculations, instead of an algebraic means of representing relationships between quantities or functions (Kieran, 1981; National Research Council, 2001, p. 270). By using a repeated poetic structure to express the paradigm of an equation, Shaniqua asserted the uniformity of performing mathematical actions on equivalent expressions. The repetition of grammatical and lexical units is a means of objectifying both the equivalence relationship between algebraic expressions and processes of transforming them. The use of repeated poetic structures, like the verbs of motion in selection 4, may be considered as a discursive means of representing the reification of algebraic transformations (Sfard, 1991).

**Emergent context: a factoring discussion**

In the final selection, students negotiated a dynamic mathematical context with various types of indexicality as they factor the second-degree polynomial 5x^{2}−16x +3. They used a trial-and-error method in which they guessed components of the factors and checked with binomial multiplication. The problem was not easy for them, and the final solution used bits of ideas that were posed by several different students. Presupposing indexicality, entailing indexicality, and poetic structures were the primary means through which conjectures were posed, accepted and refined. Tracing the indexical relationships among students’ contributions highlights the striking interdependence of students’ strategies and conjectures.

Inthisfull-classdiscussion,thegroupsolvedtheproblemwithverylittleteacherinteraction,evenignoringtheinstructor’s attempt to pose a leading question in line 26. Bria presented much of the explicit mathematical analysis in the problem. In general, Bria was considering ways to account for the coefficient of −16 using calculations such as 3×5+1=16. In line 33, for example, she suggested that if the coefficient in the original problem had been +16, the problem could be solved. In lines 16 through 21, Bria and Shannon continued an earlier discussion on whether prime numbers facilitate or hinder factoring. In the previous discussion, Bria had been disappointed that a polynomial had turned out to be prime after a great deal of group discussion. From this, she concluded that prime numbers, like prime polynomials, are “not nice.” Underlying the assessment of the “niceness” of prime numbers are beliefs about strategy and efficiency of problem-solving procedures. Bria and Shannon’s debate concerned the strategic aspects of mathematical proficiency (National Research Council, 2001).

In the transcript, we underline indexical phrases that reference specific mathematical conjectures, mathematical debates or that signal a student’s position on a conjecture, as in you can or you can’t. We do not highlight every example of indexical language, instead focusing on the ones that allowed students to collaboratively create a mathematical context while solving the problem. We trace the use of presupposing and entailing indexicality and poetic structures throughout the conversation. With a few exceptions, we do not highlight pronouns because these function in ways that are already understood, for example, the generic you, or it referring to “the current conjecture.”

We note significant uses of indexical language by underlining them in the transcript and describing them briefly in the right hand column of the table. In the right column, we indicate the transcript line, the indexical phrase, and the indexicality type. The first time a conjecture or other idea is introduced in the transcript, we label it as entailing indexicality, unless the phrase had been used commonly in previous conversations. Therefore, the references to prime numbers that are nice or not nice are examples of presupposing indexicality. Similarly, statements like 5x and 1x were used in previous factoring problems. The structure itself, therefore, is an example of presupposing indexicality, but the conjectures that it presents might represent either presupposing or entailing indexicality. For the sake of brevity, we note a poetic structure the first time it appears, along with a list of lines in which it is repeated. We note the poetic structure again only when it recurs in an explanation that combines several forms of indexicality. This analysis allows us to describe the ways in which students relied on linguistic indexicality to organize their problem-solving collaboration.

**Verbs of motion in the factoring discussion**

Verbs of motion played a small role in this conversation. The verb phrase is going to/gonna occurred in lines 3 and 10. The future orientation of gonna allowed students to predict the outcomes of their conjectures as they plan their problem-solving strategies. Bria used a verb of motion in a more intentional way in line 35, when she said, You could put 3. And 3 times 1 is 3x.

This use of the pronoun you was not generic, but instead, referred to the instructor. The verb of motion put was a politely assertive way for Bria to ask the instructor to write her conjecture on the board and to introduce it as a formal conjecture for the class to consider. In this case, Bria’s use of the verb of motion was a rather direct negotiation of the written mathematical context. The indexicality of this verb form allowed Bria to present a formal conjecture using the conventions of the class.

**Entailing and presupposing indexicality in the factoring discussion**

The class used entailing and presupposing indexicality to present conjectures and to collaboratively select some for further discussion. The interplay of entailing and presupposing references determined which ideas were developed during the discussion. Tracing sequences of entailing and presupposing references allows researchers to account for how the mathematical context emerges through group collaboration.

Entailing indexicality served several functions in the conversation. Presenting a conjecture for the first time is an example of entailing indexicality, as when Shannon suggested binomial factors with constants of 4 in line 28. Another use of entailing indexicality was to offer a checking method on a conjecture in line 10. Connor used entailing indexicality in this way when he explained that you’re gonna get 5x squared (line 10). Entailing indexicality also allows a student to draw attention to an aspect of a problem that has not been discussed. Bria’s comments of Not when you have 16? (line 23), and in the middle (line 33) were references to locations or mathematical representations that were on the board and that played specific mathematical roles in the factoring problem.

As one might expect, entailing conjectures that were followed by presupposing references were significant in the group’s construction of the solution, and they were more likely to be used in presupposing references several turns later. For example, Bria’s entailing reference to 16 in line 23 inspired a quick presupposing reference from Jake, more references in lines 28, 33 and contributed to the final answer after line 44. On the other hand, Jake’s entailing reference of 10 in line 27 and Connor’s suggestion of 13 in line 34 are echoed by no one, and have no clear influence on the problem. There was little verbal response to Ben’s suggestion of 4 and negative 1 in line 30, but because the instructor wrote it on the board—a kind of a written form of revoicing—Ben’s conjecture became part of the formal mathematical context of the problem. His suggestion of negative 1 may have helped Jake form the solution in line 42, which was the next reference to negative numbers.

**Poetic structures in the factoring discussion**

Repeated poetic structures were responsible for a great deal of mathematical collaboration throughout this conversation. At several points, for example, students registered their position on conjectures with the phrases you can and you can’t. In line 3, Bria introduced the most prominent poetic structure, 5x and 1, as a paradigmatic form for offering conjectures for positions in the factored form of the quadratic expression. This poetic structure was repeated in lines 4, 6–10, 28–30, 33, 42, 44, so that it became the primary discursive form for organizing conjectures. When a student used the form A and B, it was usually understood that these elements would become the constant terms of the binomial factors and that they would be checked with binomial multiplication. Shannon asked for clarification of the mathematics implied by this format (line 8), but later she used the format herself to submit conjectures.

It appears that, at least for the students who spoke during this conversation, the poetic structure came to represent both a pair of conjectures and several mathematical checking operations. In this sense, then, the A and B discourse structure is a reification in which a conjecture is identified with a verification procedure. Attending to the use of poetic structures in this way highlights the types of mathematical thinking that students assume their classmates know. As students developed this mathematical norm in their classroom, they used the indexicality of the poetic structure to coordinate a relatively complex set of connections between talk and written inscriptions, conjectures and an evaluation method, and comparisons among differing conjectures.

Poetic structures are also important in collaborative discussions because they provide a paradigm for presenting an idea that can be shifted slightly for other purposes (Staats, 2008). At times, this modification simply involved replacing one set of conjectures with another. At other points in the conversation, however, the students made adjustments to the conjecture format that introduced slightly new forms of mathematical thinking. The A and B poetic structure supported several of these shifts in meaning. In line 10, for example, Connor shifts the conjecture 5x and 1x to 5x times 1x. Only a single linking word changes, and to times. Because the new phrase maintains the sense of commutativity or interchangeability of the original one,

Table 1

Indexicality and emergent context in a factoring discussion.

Interaction and Boardwork | Indexical Expression | ||

Factor ()() | 5x^{2}−16x +3 | ||

1. | Bria: | Can’t you just do the 3xs, (unintelligible). I don’t know if that will work. | 1. The 3xs, entailing. |

2. | Instructor: | When we multiply it, will that give us 5x squared? | 2. 5x squared, entailing. |

3. | Bria: | No...No! You have to multiply them together. So then it’s not gonna cancel, so then like, well, that’s a prime number, so can’t you use just 5x and 1? | 3. Prime, presupposing. 5x and 1, presupposing (prior discussions), entailing (conjecture of 1), poetic structure. Repeated with transformations in lines 4, 6–10, 28–30, 33, 42. |

4. | Connor: | x and 5x. | |

5. | Jake: | Yeah. | |

6. | Bria: | Yeah, 5x and 1. | |

7. | Instructor: | Nice vocabulary too, nice observation. 5x and 1x. | 9. Prime, presupposing. 1 and itself, poetic structure. |

(5x) (1x) 8. | Shannon: | Probably, how do you know it’s 5 and 1? | 10. 5x times 1x, poetic structure. |

9. | Bria: | Because it’s prime. It can only be divided by 1 and itself. | 5 x squared, presupposing. |

10. | Connor: | Well, or because if you do 5x times 1x you’re gonna get 5 x squared. | |

11. | Shannon: | Bien, bien. | |

12. | Instructor: | What’d she say? | |

13. | Connor: | Bien. Spanish: “Oh, that’s good.” | |

14. | Shannon: | “OK.” | 15. 3, entailing. |

15. | Instructor: | Oh, it’s like “fine, good.” Fine! How about 3? | 16–18. Prime, presupposing. |

16. | Bria: | 3 is a prime number, too. | |

17. | Ben: | Yeah, it’s prime. | |

18. | Instructor: | Are prime numbers nice in this problem? | 19. You can’t, poetic structure, repeated with transformations in lines 25, 28, 31, 33, 35, 39, 40. |

19. | Bria: | No! They’re not. Because you can’t divide them. | |

20. | Shannon: | No, they are nice. | 23: 16, entailing. |

21. | Bria: | No, they’re not nice. | 24: 16, presupposing. |

22. | Jake: | (groans) | 25. You can’t, poetic structure. |

23. | Bria: | Not when you have 16— | |

24. | Jake: | Yeah, how do you get 16? | 16 from 3, presupposing. |

25. | Bria: | —and you can’t get 16 from 3! | |

26. | Instructor: | Well, here’s a question. OK, definitely— | 26. Here’s a question, entailing. |

27. | Jake: | Well, maybe 10? | 27. 10, entailing. |

28. | Shannon: | You can, 4 and 4. | 28. You can, poetic structure. |

29. | Ben: | 4 and negative 1. No— | 4 and 4, poetic structure, entailing (new conjecture). |

30. | Instructor: (writing) | OK, 4 and negative 1. | |

Jake: | But then if you add like terms, then you can. | 29. 4 and negative 1, poetic structure, entailing (negative sign). | |

32. | Ben: | No, I’m sure that— | 31. Add like terms, entailing, see line 44. You can, poetic structure. |

33. | Bria: | If the 3 was in the middle, if it was 3x and plus 16, then you could get it. Because you could have that— | 33. 3, you could, Presupposing (lines 1, 25 and others). Middle, entailing. 3x and plus 16, poetic structure. |

34. | Connor: | 13 | |

35. | Bria: | You could put 3. And 3 times 1 is 3x. | |

(5x 3)(1x)=5x 37. Connor: | But you need to— | 34. 13, entailing. | |

38. Bria (doubtfully): | Oh then, I guess... | 35. 3, presupposing. Put, verb of motion. 3 times 1 is 3x, poetic structure. (from line 10). | |

39. Jake: | Yeah, I think you can do it. | ||

40. Instructor: | You think you can do it? | ||

41. Bria: | If you have 3 in the middle, but then that 3 has to be multiplied by the other number, and the other number would have to be 1 and then you don’t get 15. | 41. 3, presupposing. Middle, presupposing. Other number, entailing. 1, presupposing. 15, entailing. | |

42. Jake: | You have negative 1 and negative 3, so 5x times 1x is 5x squared and 5x times 3 is 15x, and then negative 1 times x is 1x, |

Table 1 (Continued)

Interaction and Boardwork Indexical Expression |

(5x −1) (1x −3)=5x - Instructor: Is it positive? 42. Negative 1 and negative 3.
5x times 1x is 5x squared, 5x times 3 is 15x, and negative 1 times x is 1x, poetic structures. - Jake: Negative 1x. And then positive 3 and you add
like terms. (5x −1) (1x −3)=5x - Bria: Oh my gosh, he did it.
- Jake: Naw, I just took some numbers someone else
threw out. |

we interpret this as a transformation of the original poetic structure instead of an entirely independent statement. Connor transforms the conjecture paradigm to an explanation paradigm, saying if you do 5x times 1x you’re gonna get 5 x squared. Shifting the poetic structure changed the classroom context for a moment from conjecturing to explaining. Similarly, in line 9, Bria shifts the A and B format so that she can provide the definition of prime numbers, saying that It can only be divided by 1 and itself. Students repeat and modify the comments of their classmates to modify the voice and purpose of the classroom learning context.

Bria made another shift of the A and B paradigm that proved to be a central observation in the group’s problem-solving pathway. In line 33, she modified the format A and B to become 3x and plus 16, saying if it was 3x and plus 16, then you could get it. Here, Bria was imagining a new problem, in which the middle coefficient was positive, that is, 5x^{2 }+16x −3. Instead of presenting two conjectures for the factored form of the expression, she used the A and B paradigm to focus her classmates’ attention on the relationships between the factored form and the coefficients of the polynomial. Bria used this kind of positional indexicality at other moments in the discussion, for example when she referred to the middle position in the factored form and to the other number in the factored form (lines 33 and 41). Through these conversational moves, Bria communicated a sense that the class needed to attend to the coordination of the factored and simplified forms of the problem and that the problem could be solved if they could have used a 3 in the middle to obtain a 15, probably through the calculation 3×5+1=16. By line 41, Bria doubted her original conjecture. The fact that Jake answered the problem by constructing a negative 15 suggests that he was listening carefully to Bria, relying on her ideas as he developed his own. Bria’s shift from using the A and B paradigm as a conjecture format to one that coordinated different representations of the problem may have been the central insight that allowed the class to solve the problem. This comment juxtaposed the context of the problem posed on the board with the verbal context of offering conjectures. By merging various representations—visual and verbal ones—Bria proposed a new, more complex context that helped the class solve the problem.

6.4. Summary of indexicality in the factoring discussion

When students resolve a mathematics question through discussion, multiple forms of indexical language allow the group to negotiate the elements of context that they believe are important. Indexical language is used to pose conjectures, to respond with an explanation, to register one’s position on others’ thinking, and to direct attention towards particular aspects of a problem. This dynamic and emergent classroom context is a central feature of student-centered, collaborative problem solving. Students construct and control collaborative contexts through indexical language.

A notable feature of the factoring conversation is that entailing indexicality, the act of contributing a novel insight or conjecture, was not especially prominent. Most of the indexical references are presupposing or poetic structures. As annotated, the transcript in Table 1 notes 14 instances of entailing indexicality and 34 instances of presupposing indexicality (counting repetitions or transformations of poetic structures as presupposing indexicals). Relatively little novel thinking was necessary for the class to complete the problem. In some mathematical discussions, while many students may take a turn speaking, they may not interact deeply with each others’ ideas. Tracing the patterns of indexical reference in this transcript demonstrates that this class engaged each others’ ideas fairly closely. A great deal of mathematical thinking was left unspoken; students managed to complete the problem without much direction from the teacher and without extensive commentary and explanations among themselves. Of course, we do not wish to devalue the role of rich mathematical explanations. However, at times, student discussions are extraordinarily compressed. After listening closely to each others’ dailyexplanations,studentsmaymanagetoconstructideastogetherextremelyefficiently.Tracingtheindexicalrelationships of students’ statements can establish an understanding of how the group constructs a mathematical idea.

**The significance of indexical language for collaborative learning**

When students work collaboratively, they express positions on each other’s conjectures that create a constant renegotiation of the classroom learning context. Mathematics classroom discussions are richly indexical whether students are engaged at the moment in deeply mathematical investigations of generalization or in the small, ordinary steps of collaborative problem solving. Furthermore, even in conventional moments like orienting each others’ attention to the same part of the board, significant social issues can be at stake: who is the classroom authority and whether the pedagogy is accepted by students. Indexical language can handle the negotiation of mathematical beliefs and social identities within compressed exchanges of talk.

The selections presented in this article portray students using indexical language to influence collective mathematics work through concise, sometimes imaginative, sometimes very structured references to context. At times, indexical language serves social purposes in the classroom, as in selection 1, when Joseph proposed a mathematical question that concurrently managed pedagogical responsibilities. In other selections, students use indexicality to propose conjectures and to take a position on their classmates’ conjectures, to coordinate talk with written inscriptions, to express multiple representations of an algebraic concept, and in several cases, to compress or reify mathematical processes in the form of phrases or discourse structures that can be manipulated independently of specific procedural steps. In many cases, these are informal means of speaking about mathematical ideas that were understood and accepted within the classroom community.

**Indexicality in mathematical collaboration**

Analysis of indexical language in mathematics conversations can provide evidence for collective models of mathematical knowledge including distributed intelligence (Cobb, 1998), social cognition (Powell, 2006) and coaction (Martin et al., 2006). These studies all contribute cases of collective mathematical knowledge construction at varying levels of specificity, but nonetheless, there are few “detailed accounts of collective mathematics understanding as a dynamical process and as it emerges and unfolds moment-by-moment in the classroom” (Martin et al., 2006, p. 150). The selections presented in this article are attuned to the thematic stances presented in the model of coaction such as etiquette, recognition of multiple solution pathways, and supporting collective organization (Martin et al., 2006, p. 176). Still, a close “moment-by-moment” accounting of shared knowledge construction, at least as it occurs through the channel of spoken language, will probably need to rely on analysis of the connecting, pointing functions of linguistic meaning. If we listen closely to mathematical conversations, we will find that indexicality is the language of coaction.

There are two dimensions of the relationship between indexical language and collective mathematical learning. First, analysis of indexical language helps describe the dynamic and emergent nature of mathematics classroom contexts that these theories of collective mathematical learning require. Entailing and presupposing indexicality provides descriptions of novel mathematical suggestions and of prior knowledge. Verbs of motion help students animate operations, procedures, and comparisons. Poetic structures allow students to construct spontaneous discourse formats to convey their arguments, and to create convenient phrases that others can share and manipulate to express new mathematical meanings. Second, indexical language can strengthen analysis of the mediation of individual expression and collective achievement. To merely say that people learn through interaction does not say how this learning takes place: “we need to link interactional processes back to the individual’s conceptual structure” (Waschescio, 1998, p. 225). Within language, linking processes are indexical, or to phrase it another way, indexical language is the evidence that different statements are linked—that speakers are attempting to relate ideas across different statements. Describing these connections, however, requires analysis of several indexical forms—the ones we have described and perhaps others that are yet to be identified. In the factoring discussion, Jake recognized the social nature of his own breakthrough. He acknowledged the deeply collaborative character of the discussion by observing I just took some numbers someone else threw out. Our current perspective is that no single type of indexical language is adequate to describe informal mathematical expressions and shifting contexts, but that taken together, entailing and presupposing indexicality, verbs of motion, and poetic structures can account for a great deal of interactive mathematical knowledge construction.

While this discussion has primarily aimed to provide a mode of close analysis of mathematical collaboration, it also serves to focus teachers’ attention on particular discursive moves that students may use to express mathematical ideas. Students’ use of verbs of motion or repeated poetic structures may represent objectified shorthand for mathematical ideas. Listening for these types of forms may help teachers select a moment to request deeper discussion. A teacher might ask a student, for example, “When you say, ‘Go down two squares,’ what operation are you using?” If a teacher hears several students using a repeated poetic structure, it may signal a felicitous moment to shift towards a reflective discussion that articulates its meaning (Cobb, Boufi, McClain, & Whitenack, 1997) in order to determine whether other students recognize the poetic structure as a representation of mathematical knowledge.

**Indexicality in mathematical socialization**

Although indexical forms allow speakers to use ordinary language to specify significant ideas, they also can create ambiguity that listeners find confusing. For example, if a students asks, “How did you get that?,” the listener may need several speaking turns to identify the referent of that. It is somewhat surprising, then, that a great deal of productive mathematical work can be conducted with indexical language. Learning to use indexical expressions appropriately is a hallmark of socialization for many speech communities. Formal and professional mathematical communication, particularly written mathematics, seeks to remove a great deal of indexical language (Morgan, 1998). Despite the profusion of indexical forms in classroom discourse, many with productive mathematical functions, students who advance mathematically must learn to avoid indexicals in mathematical writing.

Classroom indexicality is clearly productive for students as they learn to negotiate and communicate conceptual and procedural understanding. Students’ use of indexical forms may be a significant means by which they become socialized into the foundational actions and values of the discipline such as conjecturing, evaluating, and presenting evidence. On the other hand, professional mathematicians seek to develop and present statements that are true without regard to a particular speaking context. Our aesthetic tradition is to represent the truth of a statement by establishing a “sense of decontextualized authority” (Pimm & Sinclair, 2009, p. 23). Because links to context are largely governed by indexical language, this decontextualization requires reduction and careful control of indexicality. Removing references to the immediate context is a way (although perhaps not uniquely so) to communicate statements that are true anywhere, when stated by anyone. This insures that theorems can be removed from one situation and communicated again in another without modification of their central meaning (Staats, 2009, p. 28). Viewed in this way, mathematical socialization is complex and multi-staged because students gain proficiency in some of the core values of the discipline by using communicative tools that must be erased at higher levels within the discipline.

**Conclusion**

Indexicality is a central linguistic resource for reasoned, collaborative discourse. In any classroom that emphasizes reasoning over recitation, students will need to use indexicality to develop cohesive arguments or explore relationships. The indexical qualities of demonstrative nouns and adjectives assist speakers in focusing attention in a precise manner. Pronouns help speakers negotiate relationships, responsibilities, pedagogies, and in a strictly mathematical sense, generalization. Students may use the directional aspects of verbs of motion to express change and comparison. Because math discussions develop over time, we need to make both creative and entailing references to others’ ideas. Because math addresses abstract relationships, we must have paradigmatic, poetic structures to convey them efficiently. These issues are all handled through one type or another of indexical language. The varied types of indexical language, taken together, may prove to be fundamental linguistic resources that make collaborative learning possible.

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# A Linguistic and Narrative View of Word Problems in Mathematics Education

SUSAN GEROFSKY

Jake and Jerry went on a camping trip with their motorcycles One day Jerry left camp on his motorcycle to go to the village. Ten minutes later Jake decided to go too If Jerry was travelling 30 mph and .Jake traveled .3.5 mph, how long before Jake caught up with .Jerry?

[Johnson, 1992, 228]

A person bought oranges at the rate of .36 cents a dozen, had he received 6 more for the same money they would have cost him 6 cents a dozen less How many did he buy?

[1902 Public School Leaving Examination of the Northwest

Territories, in National Council of'7éachers of Mathematics [NCTM], 1970, 419]

mere is supposed a lawe made that (for furtheryng of tyllage) every man that doth kepe shepe, shall for every 10 shepe eare and sowe one acre of grounde, and for his allowance in sheepe pasture there is appointed for every 4 shepe one acre of pasture. Nowe is there a ryche shepemaister whyche hath 7000 akers of grounde, and would gladlye kepe as manye sheepe as he myght by that statute I demaunde howe many shepe shall he kepe?

[Robert Record, Ihe Ground of Artes, 1.552, in F'auvel &

(hay, 1987, 278]

1,000 loaves of pesu .5 are to be exchanged, a half for loaves oj pesu 10, and a halffor loaves of pesu 20 How many of each will there be?

[Rhind Mathematical Papyrus, Problem 74, in Gillings, 1972, 130]

Per bür [surface unit] I have harvested 4 gur of grain. From a second bib I have han'ested 3 gur of grain The yield of the fir;st field was 8,20 more than that of the second. The area of the nvo fields were together 30,0 How large were the fields?

[Babylonian document VAI 8.389, Berlin Museum; translation from Van der Waerden, 1954, 66]

Mathematical word problems, 01 "story-ploblems", have long been a familiar feature of school mathematics. For many students, the "transformation" of word problems into arithmetic 01 algebra causes great difficulty, and a number of recent studies have addressed the linguistic and mathematical sources of that difficulty from a psychological point of view [Burton, 1991; Harel and Hoz,1990 ; Hoz and Harel, 1990; Mangan, 1989; OrmeH,1991•, Plane, 1990; Puchalska & Semadeni, 1987]

Recent curriculum policy documents such as, for example, the Cockcroft Reput in Britain [Cockcroft, 1982, para 24.3, in Prestage & Perks, 1992] and the NCTM Standards documents in the United States [NCTM, 1989, 134 - 136] call for curricular relevance and an emphasis on the generic skills of problem solving Many mathematics educators have intelpreted these imperatives as merely a call f01 more, better, and more varied word problems. A few researchers [Borasi, 1986; Pinder, 1987; Brown & Walter, 1990] have begun to question the prevalent view that "problem-solving" means exclusively the solution of word problems, and studies in ethnomathematics [Lave, 1988; Lave, 1992; Nunes, Schliemann & Carraher, 1993] have revealed that people who are successful and efficient mathematical poblem-solvers in "real-life" (i e., life outside of school) may be unable to solve school word problems with pencil and paper, even when these wold problems appear to be similar to "real-life" problems that the person is quite capable of solving

Yet word problems are firmly entrenched as a classroom tradition, particularly in North Amelican schools They figure prominently in virtually all school mathematics textbooks and in the previous school experience of those who are now teachers and curriculum developers, and continue to be used unquestioningly by most teachers of school mathematics

In this papei, I will üy to establish a description of the mathematical word problem as a linguistic genre, particularly considering its pragmatic structure Through a description of the pragmatics and discourse features of the genre, and through comparison of the word problems genre to other spoken and literary genres, I hope to find clues to the unspoken assumptions underlying its use and nature as a medium of instruction.

**Previous linguistic studies of mathematical wor**^{r}**d problems**

There have been a great many studies dating from the midseventies onward that have looked at mathematical word problems in terms of their "readability" (that is, the linguistic factors that make them easier or harder to read and understand) 01 students' ease or difficulty in translating them from "normal language" to mathematical symbolism. (See, fol example, Burton [1991] and Nesher and Katriel [19771 ) Although such studies use linguistics to analyze word problems, they presuppose the value of word problems as they are presently used in mathematics teaching and testing, and the need for students to become more proficient at solving them

Fbr the Learning of Mathematics 16, 2 (June 1996) FL M Publishing Association, Vancouver, British Columbia, Canada |

In this paper, I want to problematize the use of word problems in mathematics education I will look at word problems as a linguistic and literary genre, and describe the features of that genre. I hope, by "making them strange", to enable mathematics educators to see word problems in a new way, to make the word problem genre a conceptual object that we will be able to circle around, look at from different perspectives, and compare usefully to other conceptual objects (for example, other litaary and cultural genres)

Contemporary linguistics has gone through many changes as a discipline since its origins at the tum of the centu1Y, and many significant changes have taken place over the past fifteen years. Useful newly-linguistic areas such as pragmatics (the study of language in use in particular contexts), discourse analysis (the study of extended stretches of spoken 01 written text in terms of utterances and their relationships), sociolinguistics (the study of language and social class and power relationships), semantics (the study of the fields of word meanings), dialectology (the study of varieties of language, their distiibution and history), stylistics (the study of literary texts in terms of theil linguistic features) and genre studies (the study of "text types" i_n terms of their linguistic and contextual féatures) have recently become available to mathematics education researchers. (For example, Roth [in press] uses conversational analysis, a form of discourse analysis, to look at the sociology of math students' learning when dealing with "real-lift" problems and wold problems, and Tim Rowland [1992] and David Pimm [1987; 1995] apply new developments in pragmatics, particularly questions of deixis, or "pointing with words", to mathematics education ) In this study, I will use methods from plagmatics, discourse analysis, and genre studies to try to shed light on the nature of word problems as applications of mathematics and as stories (Readers looking for definitions and examples of specialized terminology may wish to consult the appendix on "linguistics, discourse and genre analysis" at the end of this paper )

Word problems and their three-component structure

You receive a paycheck worth $125 50 You must give 1/5 of your earnings to both the Provincial and Federal government. How much money do you have left? (Word problem written by a grade 6 student in Vancouver, reported in Menon [1993])

Most word problems, whether from ancient 01 modern sources, and including "student-generated" word problems, follow a three-component compositional structure:

- A "set-up" component, establishing the characters and location of the putative story. (This component is often not essential to the solution of the problem itself )
- An "information" component, which gives the infOrmation needed to solve the problem (and sometimes extraneous information as a decoy for the unwary)
- A question

Variations on this sfructute occur; for example, the set-up and information components are sometimes collapsed into one sentence by the use of subordinate clauses, or the information component and the question are collapsed into a single sentence by using a subjunctive "if then" structure.

Mildred Johnson addresses this structure explicitly in her interesting little instructional book, How to solve word problems- in algebra [Johnson, 1992], Her advice to students who are having trouble with the transformation of word problems into algebra includes the following:

Look fbr a question at the end of the problem This is often a good way to find what you are solving for What you are trying to find is usually stated in the question at the end of the problem Simple problems generally have two statements One statement helps you set up the unknowns and the other gives you equation information. Translate the problem from words to symbols a piece at a time [Johnson, 1992, 1 -2]

Wickelgren [1974], in another book on mathematical problem solving, also identifies three parts to mathematical problems in general, and these fit closely with the threepart structure proposed for word problems:

All the formal problems of concern to us can be considered to be composed of three types of information: information concerning givens (given expressions), information conceming operations that transform one or more expressions into one 01 more new expressions, and information concerning goals (goal expressions). [Wickelgren, 1974, 10]

In terms of the three components typical of word moblems, the first component generally contains none of the above; the second component contains the givens and sometimes the operations (although the operations are often left unstated, since word problems are usually grouped together as a practice set for a particular set of algorithms that are currently the focus of classroom teaching. 'Ihe choice of the correct algorithm from the currently-active set is considered one of the student's main responsibilities; to give the operation explicitly in the problem would be considered cheating). Ihe thild component, the question, identifies the goals of the problem.

The three-component structure of typical word problems seems, then, to be based on the structure of arithmetic algorithms or algebraic problems, rather than on the conventions of oral or written storytelling. In the case of an algeblaic word problem, the student is required to write an

algebraic equation in terms of a set of variables which are related to one another in a fixed (or "fixable") relationship that can be stated in terms of an equality 01 inequality To "solve" the algebraic problem, the student must know which term to isolate (the goal or "the unknown") This information is given (sometimes implicitly) in components two and three of algebraic word problerns, which could be paraphrased as:

Component 2: The variables 01 quantities A, B, C, , are in the following relationship (and you must deduce from the context of your lessons and the problem itself what operations are necessary to set up the equation) .

Component 3: Solve for variable (X)

Component 1 of a typical word problem is, so far as I can see, simply an alibi, the only nod toward "story" in the story problem It sets up a situation for a group of characters, places and objects that is generally irrelevant to the wiiting and solving of the arithmetic or algeblaic problem imbedded in later components In fact, too much attention to story will distract students from the translation task at hand, leading them to consider "exüaneous" factors from the st01Y rathel than concentrating on extracting variables and operations from the more mathematically-salient components 2 and 3

I think it is important to ask why the first component is included in word problems at all, 01 why this "translation" ot "transformation" exercise should be considered important for one's mathematical studies. Many writers consider such problems to be practically usefül, at least by analogy Mildred Johnson writes:

You will find certain basic types of word problems in almost every algebra book.. You can't go out and use them in daily life, or in electronics, or in nursing. But they teach you basic procedures which you will be able to use elsewhere [Johnson, 1992, l]

On the other hand, Thomas Kubala, in Practical problems in mathematics for electricians, makes direct claims for the usefulness of the problems he presents:

The student leaming electrical theory and wiring practices will find that by using PRACTICAL PROBLEMS IN MAIHEMAIICS FOR ELECTRICIANS his [sic] understanding of various mathematics principles will be reinforced because of their use in problems frequently encountered by an electrician Any student in a program of instruction in elecüicity will benefit from the use of this related problems workbook Practicing electricians who desire to improve their math skills will also find it helpfill [Kubala, 1973, 4]

It has been documented that the nature of the stories attached to the algebraic problems is relevant to students ill terms of affect and in terms of the student's willingness to try to solve the problem at all [Sowder, 1989; Pimm, 1995] It is also interesting to note that, over the course of several years, students become enculturated in the world of school mathematics, and familiar with the conventions of the wold problem genre to the extent that they are able to reproduce it. Ramakrishnan Menon [Menon, 1993] documentS word problems in canonical f01rn written by elementary school students who were asked to formulate their own mathematical questions. Menon also cites Ellerton's [1989] large-scale study of 10,000 secondary students in Ausüalia and New Zealand who, when asked to write one difficult word problem, overwhelmingly wi0te problems similar in form to those in their textbooks Jean Lave has noted,

If you ask children to make up problems about everyday math they will not make up problems about their experienced lives, they will invent examples of the genre; they too know what a word problem is [Lave, 1992, 77]

Puchalska and Semadeni [1987] comment that, while younger elementary school children often believe, naively, that the stories in st01Y problems are relevant, more experi-

enced older children know better:

Radatz [1984] points out that, during problem solving, school beginners concentrate on stories rather than on numbers; during interviews it has been found that such children often augment the story with what follows from their own knowledge or experience Older children, however, always try to reach some solution, perhaps by a trial-and-error strategy, and they often believe that nothing is unsolvable in mathematics Children with little mathematical experience fry to analyze the story more carefully, whereas older students have a specific attitude towards mathematics: It is viewed as an activity with artificial rules and without any specific relation to out-ofLschool reality [Puchalska & Semadeni, 1987, 10]

**Word problems, intentions and speech acts: Locutionary, illocutionary, and perlocutionary force and uptake**

Questions about the perceived purpose of the textbook writer or teacher in presenting word problems, or the student in solving them, relate to notions in pragmatics called locutionary, illocutionary and per locutionary force, which are in turn part of speech act theory. I L Austin, the philosopher of language who originated speech act theory, saw the need for an analysis of language in the context of interactions, so that not only the literal meaning of an utterance but its meaning as action could be considered Austin wiites,

We may be quite clear what "Shut the door" means, but not yet at all clear on the further point as to whether as uttered at a certain time it was an order; an entreaty 01 whatnot What we need besides the old doctrine about meaning is a new doctrine about all the possible forces of utterances, towards the discovery of which our proposed list of explicit performative verbs would be a very great help [Austin, quoted in Levinson, 1983, 236]

Austin identified thlee kinds of speech acts that are simultaneously performed in an utterance:

- a locutionary act: the utterance of a sentence with determinate sense and reference
- an illocutionary act: the making of a statement, offer, promise, etc in uttering a sentence, by virtue of the conventional force associated with it (or with its explicit perfbrmative paraphrase)
- a perlocutionary act: the blinging about of effects on the audience by means of uttering the sentence, such effects being special to the circumstances of the utterance [Levinson, 1983, 236]

Levinson gives the example of the sentence:

You can't do that

which has a literal meaning (its locutionary fOrce), and which may have the illocutionary force of protesting to the person being addressed, but the perlocutionary fOrce of either checking the addressee's action, or blinging the addressee to his or her senses, 01 simply annoying the person.

Levinson differentiates between the perlocutionaay force of an utterance, which is specific to the circumstances of issuance, and the consequences of an illocutionary act, which include the understanding of the illocutionary fOrce by the addressee(s). In the case of word problems, the petlocutionary force would include questions of affect in the individual learner (the intended or unintended effect of stimulating or b0iing, encouraging or discouraging, attracting or frightening or disgusting or delighting a particular learner, for example), and Levinson quotes Austin as adrnitting that perlocutionary force is often indeterminate or indeterminable [Levinson, 1983, 237] However, the uptake, or understanding of the illocutionary force in word problems by learners, deserves further consideration

Questions about the perceived purpose of the textbook writer in presenting word problems, or the student in solving them, relate to locutionary, illocutiona1Y' and perlocutionary force (that is, the literal meaning, the performative intention, and the effect upon the audience of an utterance) Applying this analysis to word problems poses some problems in terms of their locutionaw force because of problems relating to deixis (the act of pointing with words) As discussed later (in the sections on verb tense and the Chicean maxim of quality), word problems do not generally have referents—that is to say, they do not refer to "real-life" objects, people, 01 places in any but the most arbitrary way This could conceivably place word problems in the category of fiction, but I would argue that they are so deficient in the rudiments of plot, character, diamatic tension, poetic use of language, moral or social theme, etc., as to be very pool fiction at best (The metaphor "word problems as parable" will be discussed later ) For the moment, I would prefer to view mathematical probkerns as a genre unto themselves, with indeterminate locutiona1Y force

Their illocutionary force seems to be quite directly accessible to students with sufficient enculturation in the genre; it is "Solve this!" 01 "Find X!" This command brings with it certain underlying assumptions:

that "this" is solvable, that "X" can be found,

• that the word problem itself contains all the inf0Tmation needed to do this task, that no infonnation extraneous to the problem may be sought (apan from conventional mathematical operations which likely must be supplied), that the task can be achieved using the mathematics that the student has access to, that the problem has been provided to get the student to practice an algorithm recently presented in theil math course, that there is a single correct mathematical interpretation of the problem, that there is one light answer, that the teacher can judge an answer to be correct 01 incorrect, and especially, that the problem can be reduced to mathematical form—in fact, that the problem is at heart an arithmetic or algeblaic formulation which has been "dressed up" in words, and that the student's job is to "undress" it again—to transfom the words back into the arithmetic or algebra that the writer was thinking of, then to solve the problem

Students' uptake, or understanding of the illocutionary force in word problems is, I think, quite clear, and could be

paraphrased by a learner as follows:

I am to ignore component I and any story elements of this problem, use the math we have just learned to transform components 2 and 3 into the correct arithmetic ot algebraic form, solve the problem to find the one correct answer, and then check that answer with the correct answer in the back of the book or turn it in for correction by the teacher, who knows the translation and the answer

In this light, Puchalska and Semadeni's [1987] finding that children who were experienced in school math tried somehow to solve word problems which had missing, surplus or conttadictory data is not at all smprising, I contend that these children had a well-developed schema with regal d to word problems which included many facets of the genre, including its illocutionary force, and that a command to "make sense of this story in telrns of eveyday life" or to "search for deficiencies or contradictions in this problem" wele never conceived as part of that illocutionary force (for the students, or indeed, for most teachers and textbook writers)

**The question of "truth value"**

The term "truth value" was introduced into semantics by Ftege and Strawson, and was adopted from semantics into pragmatics Frege wanted to be able to evaluate the meaning of all statements in terms of a principle of bivalence— that is, if something was not true it was false, and if not false, it must be true (This principle is familiar to anyone working with mathematical proofs, where the "law of the excluded middle" allows for the possibility of proofs by contradiction )

There are problems with hege's notion of truth value, particularly when it is applied to utterances other than the propositional statements of philosophy For example, questions, i_mperatives, and exclamations cannot be assessed for truth value. The truth value (if any) of statements in the context of fiction (storytelling, novels, plays, etc) is also problematic. Lamarque and Olsen [1994, 54] give the fOllowing statements which might occur in a work of fiction: a) John worked in the fields; b) He found it tiling; c) There was once a young man who worked in the fields; d) Working in the fields is tiling—and consider that, while statements c) and d) might be assessed for fruth value if construed outside of the context of the work of fiction (unlike a) and b)), this ignores the proper contextual construal of these statements. They give "a common alternative to the falsity thesis" in dealing with statements in fiction, which is the "no-üuth-value" view of fictive utterance, partly attributed to Frege and Strawson Three versions of this view are:

I Sentences in works of fiction are neither true nor false because their (existential) presuppositions are false; 2. Sentences in works of fiction are neither true nor false because the sentences are not asserted;

3 It is inappropriate (mistaken, etc ) to ascribe truth or fålsity to sentences in works of fiction [Lamarque & Olsen, 1994, 57]

Lamarque and Olsen resort to a description of fictive utterances as pretence, or writing "as it" something were true, and distinguish three types of pretence: pretending to be , pretending to do and pretending that . It is here that I can situate word problems; they pretend that a particular story situation exists. What is more, to paraphrase Lamarque and Olsen, readers of word problems must pretend that such a situation exists, under instruction from the writer of the word problem.. Further, students "must pretend that someone is telling them" about that situation [Lamarque & Olsen, 1994, 71, authors' emphasis] The reader's response is not in terms of truth value but mimesis, yet at the same time the story is considered disposable, interchangeable with other equivalent stories, which would certainly not be the case with a work of fiction

Linguistic and metalinguistic verb tense Levinson [1983, 73 - 78] distinguishes between linguistic tense (L-tense) and metalinguistic tense (M-tense). By Ltense he means what is usually referred to as grammatical tense in a particular language; by M-tense, he means a semantic or deictic category of tense, which indicates the temporal location of an event relative to the coding time (CT) and/or receiving time (R T) of the utterance. (L evinson points out that, in the canonical situation of utterance, RT and CT are assumed to be identical, an assumption called deictic simultaneity.) In an M-tense system, we distinguish the temporal location of events in relation to CT: past refers to events prior to CT, present to events spanning C r, future to events succeeding CT, pluperfect to events prior to past events (which are themselves prior to CT), and so on M-tenses are important in sepalating the deictic features of L-tenses from their modal and aspectual features.. For example in English, L-future tenses always contain a modal element, and in a decontextualized sentence it is difficult to Imow just what balance of "futurity" and "intentionality" is indicated by modals like will, should and may in examples like the following:

I will never go hungry again

John should speak to her tomorrow

You may have visitors on Saturday morning

Some languages like Chinese may not have morphological verb tenses markers (and so may lack L-tense in this sense), and yet, as Levinson says, "we can confidently assume that there are no languages where part of an Mtense system is not realized somewhere in time-adverbials or the like, not to mention the implicit assumptions of Mpresent if no further specification is provided " [Levinson, 1983, 78]

Looking at examples of word problems from current

British Columbia math textbooks, I found that determining

M-tense in mathematical word problems is problematic. The difficulty is strongly linked to the lack of truth value in word problems—that is, their flouting of the Gricean maxim of quality. Although several patterns of L-tense typically appear in word problems, their M-tense seems to remain consistent. For example, in many word problems the first two sentences use L.-present and the third L-future:

A truck leaves town at 10:00 a m travelling at 90 km/h A car leaves town at 11:00 am travelling at 110 krn/h in the same direction as the truck At about what time will the car pass the truck? [Alexander et al., 1989, 297]

[A truck leaves: L- present

A car leaves: L,-present

The car will pass the truck: L-future]

A second type uses L -past 01 L-present consistently in all three sentences.

A ladder is unsafe if it makes an angle of less than 15^{0 }with a wall. A 10-m ladder is leaned against a wall, with the foot of the ladder 3 m from the wall. Is it safe? [Ebos et al , 1990, 350]

[ A ladder is: L-present

If it makes: L-present

A ladder is leaned: L-present

Is it: L-present]

(This particular word problem is also interesting for its consistent use of passive, agentless sentences—there are no people in it )

A great number of anomalies can be fOund which combine L -tenses in a self-contradictory way, that is, in a way that confiadicts the usual use of L -tense in English, where the statements are assumed to have fruth-value and the event is assumed to take place in a stable deictic relationship to coding time (CT):

Each elephant at the Young Elephant I raining Centre in Pang-ha, 'Ihailand eats about 250 kg of vegetation in a day *How much would 43 elephants eat in I day? 1 week?

[Alexander et al, 1989, 35]

[Each elephant eats: L-present

*43 elephants would eat: L-future

If we accepted the truth of the first statement (and it certainly sounds convincing, since we're given the name and location of the Elephant I raining Centre) we would expect "How much do 43 elephants eat" in the second sentence ]

I think that the most sensible interpretation of the unstable tempolal, locational and personal deixis in these word problems is to interpret all of the above as M-tenseless and non-deictic (Levinson uses this analysis on such sentences as f"lwo and two is four" and "Iguanas eat ants", for example [Levinson, 1983, 77]), but having conditional or subjunctive aspect That is to say, the word problems do not actually point to a person (".Jake", ".Jerry" or "you"), place ("the Young Elephant Training Centre in Pang-ha, Thailand") or time (befOre, during or after CT) Since these ate not real places, people, or situations, there is no absolute need for logical consistency in the use of L-tense (and L -tense is often used in ways that would be considered self-contradictory in standard expository English prose) Rathel, word problems propose hypothetical situations with certain given conditions and ask for hypothetical answers.. Most word problems could be rewritten in the form: "Suppose that (some certain situation A existed ) If (conditions B, C, D, . held), then (what would be the answer to E)?"

The very inconsistency and seeming arbitrariness of Ltense choices in word problems points not only to their Mtenseless and non-deictic nature, but also to an "understanding" between writer and reader that these supposed situations do not have truth value, and that the writers' intentions and the readers' task are something other than to communicate and solve true problems. (Otherwise the meaning of these problems in terms of a Gue situation would be very difficult to decipher.) This lack of truth value can be otherwise expressed as "flouting the Gricean maxim of quality '

Flouting the Gricean maxim of quality It is my contention a feature of the word problem genre is a consistent flouting of the Glicean maxim of quality, which is to say that, as a genre, word problems have no truth value. This feature is intimately linked with, or perhaps a result of, theil deictic indeterminacy. lime deixis, as shown through metalinguistic verb tense, has no referent Personal deixis and place deixis (that is, the colrespondence between the names of persons or places and their referents) have no truth value or are irrelevant. And yet the standardized form of the genre demands that declarative statements be made about these non-existent people, places and times. Such statements may be seen to be flouting the maxim of quality ("do not say what you believe to be false")

An example from a math textbook currently in use in British Columbia:

Every year Stella rents a craft table at a local fun fair and sells the sweaters she has been making all year at home She has a deal for anyone who buys more than one sweater She reduces the price of each additional sweater by 10% of the price of the previous sweater that the person bought Elizabeth bought 5 sweaters an paid $45 93 for the fifth sweater How much did the first sweater cost? [Ebos et al , 1990, 72]

The above could be reworded as fOllows without changing its truth value (although it would be a rather odd-looking word problem, highlighting as it does one of the implicit features of the genre

Every year (but it has never happened), Stella (there is no Stella) rents a craft table at a local fun fair (which does not exist) She has a deal for anyone who buys more than one sweater (we Imow this to be fålse) She reduces the price of each additional sweater (and there are no sweaters) by 10% of the price of the previous sweater that the person bought (and there are no people, or sweaters, or prices)

The hypothetical nature of wold problems can be understood here, although it does not appear in the literal meaning of most examples of the genre. Again, Lave writes that "word problems are about aspects of only hypothetical experience and essentially never about real situations [Lave, 1992, 78] This point is brought home in the cases whele word problems seem to be referring to places, objects, or people Iqown to exist, as in the fbllowing:

A rock dropped from the top of the Leaning lower of Pisa fhlls 6 m from the base of the tower If the height of the tower is 59 m, at what angle does it lean from the vertical? [Ebos et al, 1990, 354]

The tricky part of the story problem above is the "if" (my emphasis) Celtainly the Towel of Pisa has been measured Why use the conditional form here? Is it intended to indicate that the vertical height of the tower is not stable? (This may be true—it was recently closed to visitors because increases in its "lean" had made it dangerous ) 01 is it a way of indicating that the referent for the words "the Leaning Tower of Pisa" is not the actual structure in northern Italy, but a hypothetical tower, or stick, or line segment, whose height could be set at any value (say, 59 m) and whose slope could be calculated using the given numbers and the Pythagorean Theorem? Again, the writer of the problem seems to be taking pains to say, "Here is a story, ignore this story"

**Tradition: "l did them, and my kids should do them too"**

All this leads me to a question for which I have no answer as yet, the question of the purposes of word problems as a genre They are currently used as exercises for practicing algorithms, but such practice could certainly be achieved without the use of (throwaway) stories. The claim that word problems are for practicing real-life problem solving skills is a weak one, considering that their stories are hypothetical, theil referential value is nonexistent, and unlike real-life situational problems, no extraneous information may be introduced. Nonetheless, they have a long and continuous tradition in mathematics education, and that tradition does seem to matter Pinder [1987] makes a healtfelt case from a teacher 's point of view for the non-practicality of word problems while acknowledging the strong pull of their tradition for the parent of one of her students She is discussing the following well„known word problem still current in textbooks, which dates back at least to medieval Europe and probably to Roman times:

A basin can be filled by three taps: the first fills it in sixteen hours, the second in twelve hours, and the third in eight hours How long will it take to fill it when all are going together, it at the same time the basin is being drained by a pipe which can empty it in six hours? (Problem collected by Alcuin of York (circa 790 AD) as paraphrased in F P Sylvestre's T'raité d'arithmétique, Rouen 1818 ) [Plane, 1990, 69]

A water tank has two taps, A and B Line A on the graph shows how the tank drains if only tap A is open Line B shows how the tank drains if only tap B is open

- How long does it take to drain if only tap A is open ?
- How long does it take to drain if only tap B is open?
- Use the graph to find out how long it would take to drain the tank if both taps were open [Kelly et al , 1987, 213]

Pinder writes of the father of one of her students who complained that his child wasn't being taught problems like the one above (which he had studied in school) She writes,

On reflection I realized how very stupid it was to create a problem, to be worked out by manipulating symbols, about a situation which no one in their right mind would ever create. Ihe problem was that if one filled a bath, pulled out the plug and left the taps running, how one could find how long it would take the bath to empty. My question was: what did it matter anyway? What possible use would the answer be? Could it be that one might need to know whether the bath might overflow and cause a flood? But if so, why not just turn off the taps? But perhaps they were stuck In that case surely it would be more useful for the children to learn how to turn off the water and how to locate the stopcock All in all, a pretty useless problem for children to work on; so why was that father worried that his child was not going to have to solve it? Tinder; 1987, 74 - 75]

Word problems as parables?

I eallier discounted the idea of word problems as a fictional genre, citing their paucity of plot, character, human relationship, dramatic tension and so on. But what about David Pimm's suggestion that word problems be viewed as parables? [Pimm, 1995] In approaching this metaph01, I found both supporting and contradict01Y evidence in modem literary and theological theory that dealt with parable in other contexts

A number of writers aclmowledged the non-deictic nature of parable, a feature which we have seen in the word problem genre. J H Miller writes that "all works of literature are parabolic, "thrown beside" their real meaning. They tell one st01Y but call forth something else . "Parable" is one name for this large-scale indirection characteristic of literary language, indeed of language generally I" [Miller, 1990, ix] In an essay on Parable and PerfOmative in the Gospels and in Modern L iterature, he writes,

Etymologically the word [parable] means "thrown beside", as a parabolic curve is thrown beside the imaginary line going down from the apex of the imaginary cone on the other side of whose surface the parabola üaces its graceful loop from infinity and ou! to infinity again it suggests that parable is a mode of figurative language which is the indirect indication, at a distance, of something that cannot be described directly, in literal language Secular parable is language thrown out that creates a meaning hovering there in thin air, a meaning based only on the language itself and in our confidence in it I he categories of truth and falsehood, knowledge and ignorance, do not properly apply to it [Miller; 1990, 135 - 139]

There is certainly the element of the indescribable involved in mathematical concepts, particularly those that deal with infinity, or with entities that exist perhaps only as mental images, and not in "this imperfect world" (an infinite straight line, a perfect circle, a point which has no part) Yet wouldn't the linguistic terms fol these mathematical concepts (line, circle, point) be sufficiently "parabolic" deictic terms for these referents? Nonetheless, there is some appeal in the idea of stow as a Idnd of homely way to refer to the indescribable, in the same way that religious parable speaks of spilitual matters in homely rather than theological terms Miller's reference to the irrelevance of truth-value f01 parables certainly could be seen to relate to to the word problem genre as well Still, I am somewhat dissatisfied with Miller 's fairly vague, general definitions; if the term parable can bé taken to refer to all of language and literature and practically everything, what is its meaning?

Franz Kafka, in his book Parables and Paradoxes, briefly introduces his definition of parable:

Many complain that the words of the wise are always mere parables and of no use in daily life, which is the only life we have. When the sage says: "Go over," he does not mean that we should cross to some actual place, which we could do anyhow if the labour were worth it; he means some fåbuIous yonder, something unknown to us, something too that he cannot designate more precisely, and therefore cannot help us here in the very least All these parables really set out to say merely that the incomprehensible is incomprehensible, and we Imow that already. But the cares we have to struggle with every day: that is a different matter [Kafka, 1961, 11]

Again, we can find resonance in the metaphor of word problem as parable, and Kafka's description could be lead as the complaint of those like Pinder, who value real-life problem solving over the obscure references of word problems I'm stile many perplexed mathematics students believe that "all these word problems really set out to say merely that the incomprehensible is incomprehensible, and we know that already" But word problems diffél from Kafka's palables in an important way—they stress the use of a correct method in Older to arrive at a correct answer, while Kafka's wise one cannot suggest any course of action or any expressible goal. The illocutionary force of a wold problem is an instruction to "Do this", "Solve this"; Kafka's storyteller is by no means so concrete 01 directive

Thomas Oden, wiiting about a collection of

Kierkegaard's parables, asks,

Why do we read Kierkegaard's parables, and why do they merit philosophical attention? Is it because they are like maddening puzzles daring some attempted solution? Is it because the problems they address drive to the depths of ordinary human experience? Or are they mere entertainment, revealing the comic side of human pretenses—subtle poetry, with virtually inexhaustible levels of meaning? Wherever the weight of the answer is to fall, anyone who lives with these parables for a white experiences both their power and their beauty Soon you realize that it is not you who are interpreting the parable but the parable that is interpreting you [Kierkegaard, 1978, ix]

Here I think the metaphor of word problem as parable begins to break down. Although word problems may be puzzles, and maddening ones at that, they do not dare but require solution—that is a large part of their force as speech acts. And I think it would be hard to mgue that word problems, used as they are in our schools as "disposable" exercises, could be lived with over time, and seen to have inexhaustible levels of meaning, particularly poetic meaning about the depths of human experience Nonetheless, there is the question of the durability of certain word problems, some of which appear to have been taught to scholars for centuries 01 millennia Perhaps there is something elemental 01 common to human experience in these, if we could find it, although perhaps their endurance simply speaks for the incredible conservatism of mathematiCal traditiom

Finally, in his book on the parables of Flannery O'Connor, John May quotes Dan O. Via, a modem hermeneutic scholar:

For Via the parable in the narrow sense is "a freely invented story told with a series of verbs in the past tense" (e g., the Prodigal Son, the lakents, the Unjust Steward) It is not concerned with the typical, but with "making the particular credible and probable " In the parable strictly conceived as fiction, "we have a story which is analogous to, which points to but is not identical with, a situation or world of thought outside of the story I" [May, 1976, 14]

I hus får, the analogy with word problems holds up fairly well The non-deictic use of verb tense, the concern with the credibility of particulars in a form that both does and doesn't refa to the particulars of the everyday world, all seem parallel On the other hand, May wiites,

Inasmuch as the parable in a narrow sense thrives on the

drama of human encounter as a figurative expression of the drama between God and man, it uses ordinary human language, rather than specifically theological terms, to mediate the ultimate reaches of reality to man Reflecting the historical situation of their author, the parables proclaim what it means to exist in a boundary situation, how the eschatological crisis occurs within the confines of everyday existence [May, 1976, 15]

Although word problems do reflect the historical situations of their authors, I think it would be stretching the metaphor rather far to claim that they "mediate the ultimate reaches of reality to man", that they involve eschatological crisis, 01 that they express the drama between human beings and God

Perhaps in the exercise of viewing word problems as parables, we will be able to see word in a different way that will allow us to generate new ways of using them. I suggest, too, that delineating the boundaries of the word problem genre can allow us to play with those boundaries in intelesting ways. In any case I do feel that it is important to think in new ways about the nature and purposes of word problems, about their inherent oddness and contradictions, and about our rationale for using them in school mathematics programs, rather than simply, unthinkingly visiting them upon future generations of schoolchildren

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**Appendix**

A glossary of terms used from linguistics, discourse analysis and genre analysis

Deixis: from the Greek word for "pointing" (as in "index"), refers to the process of pointing with words Deixis studies the contextual referents for demonstratives ("this", "that"), pronouns ("I", "you", "it" etc.), verb tenses, context-referential adverbs of time and place ("then", "here") and "a variety of other grammatical features tied directly to the circumstances of utterance" [Levinson, 1983, 54]

Context of utterance is here referred to in terms of pragmatic indices, co-ordinates 01 reference points FOI example, the deixis of verb tense is analyzed with reference to the time an utterance was spoken 01 wiitten ("coding time") and the time it was heard or read ("receiving time"), which may 01 may not be distinct. Demonstratives, adverbs of time and place and verb tenses are described on a continuum that ranges from closest ("maximally proximal") to furthest away ("maximally distal") from a central point of reference ("deictic centre")—for example, some dialects of American English have three adverbs referring to locations increasingly distal from a deictic centre ("here", "there" and "yonder")

Discourse analysis: This term has been adopted by a large number of disciplines, including various branches of Iinguistics, literary studies, ethnography, sociology, film studies and artificial intelligence Generally, discourse analysis refers to the structural analysis of stretches of "text" (in its broadest meaning) at a level larger than the sentence or utterance. The texts in question may range from spoken discourse in classrooms or courtrooms, to written texts like stories, novels, poems, letters or graffiti, to dialogue in film or theatre, to oral genres like storytelling, speech making, gossip, jokes and puns (see Van Dijk, [1985a]; Van Dijk [1985b]; Coulthard, [1992])

The analytic methods that fall under the general headi_ng"discourse analysis" are as heterogeneous as the texts they are used to analyze Deborah Schiffiin [1994] writes that discourse analysis is "widely recognized as one of the most vast, but also one of the least defined, areas in linguistics" [p 5], and goes on to describe six approaches currently used in discourse analysis methodology: speech act theory, pragmatics, intelactional sociolinguistics, ethnography of communication, conversational analysis and variation analysis

Genre analysis: The term "speech genre" was coined by Mikhail Bakhtin to describe "relatively stable types of utterances" [Bakhtin, 1986, 60] The notion of gente analysis has since been taken up in other areas of cultural analysis, notably film studies and literary criticism

Bakhtin stresses that genres can be analysed only by considering the whole of an utterance, including consideration of its thematic content, linguistic style (including lexical, syntactic and other grammatical features), its compositional structure, its expressiveness and its addressivity Since decontextualized words and sentences lose this quality of addressivity, a purely atomistic formal linguistic approach cannot capture the features of a genre

Gricean maxims: The study of implicature has a basis in ideas expressed by H P. Grice in a series of Harvard lectures in 1967 [Grice, 1975, 1978]. Grice looked for a set of assumptions underlying the efficient co-operative use of language The five principles he found, including a general "co-operative principle" and four "(Gricean) maxims of conversation" are listed below:

- The co-operative principle: Make your conüibution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged
- The maxim ofquality: Iry to make your contribution one that is true, specifically:

i) do not say what you believe to be false ii) do not say that for which you lack adequate evidence 3) The maxim of quantity.

- Make your contribution as infOrmative as is required for the current purposes of the exchange.
- do not make your contribution more informative than is required

- The maxim of relevance: Make your contributions relevant

- The maxim of manner: Be perspicuous, and specifically:

i) avoid obscurity, ii) avoid ambiguity, iii) be brief, and iv) be orderly

Grice's point is not that all speakers must follow these guidelines exactly, since it is obvious that no one speaks this way all the time. Rather, he says that when an utterance appears to be non-cooperative on the surface, we try to interpret it as co-operative at a deeper level. Levinson gives the following example: A: Where's Bill?

B: There's a yellow VW outside Sue's house

[Levinson, 1983, 102]

B's contribution, if taken literally, does not answer A's question, and it might seem as if B were being uncooperative and changing the topic However, if we assume that B is, at some deeper level, being cooperative and respecting the maxim of relevance, we try to make a connection between Bill's location and the location of a yellow VW, and conclude that, if Bill has a yellow VW, he may be at Sue's house.

Implicature: "provides some explicit account of how it is possible to mean more than what is actually said (ie more than what is literally expressed by the conventional sense of the linguistic expressions uttered). "[Levinson, 1983, 97] For example, Levinson gives the following example:

A: Can you tell me the time?

B: Well, the milkman has come

and paraphrases what native speakers would understand by this exchange as follows:

A: Do you have the ability to tell me the time of the present moment, as standardly indicated on a watch, and if so please do so tell me.

B: No I don't know the exact time of the present moment, but I can provide some information from which you may be able to deduce the approximate time, namely the milk-nan has come

Implicature studies the mechanisms by which speakers of a language can understand utterances' unstated relationship to context and to the speakers and listeners involved in the conversation (or to the writers and readers involved in a written exchange)

Pragmatics: In Anglo-American linguistics, pragmatics is often defined as "the study of language usage" [Levinson, 1983, 5]. This is a rather vague definition, and allows for an unintentional amount of overlap between pragmatics and other areas like sociolinguistics, psycholinguistics, etc Levinson [1983] struggles with alternative, more specific definitions, and comes up with the following possibilities:

Pragmatics is the study of those relations between language and context that are grammaticatized, or encoded in the structure of a language [p 9]

Pragmatics is the study of the relations between language and context that are basic to an account of language understanding [p 21]

Pragmatics is the study of deixis (at least in part), implicature, presupposition, speech acts, and aspects of discourse sfructure [p. 27]

Almost without exception, Western psychological theories have tended to cut experience into different and separate and often contrasting basics: thinking or feeling or acting; conation or cognition or affect; will or emotion or thought or perception. Since Old Testament times, since Plato and Aristotle, we have tended to believe that thought happens in one part of a person, feeling in another, will and action in another For Kernberg, experience comes in wholes: any encounter provides an experience which has unity and leaves an integrated memory behind, though ot course some parts of the experience may remain unconscious or preconscious or get repressed or remain unnoticed because the culture does not provide a word—a conceptlabel—for it

**Josephine Klein**

FLM Publishing Association

Contexts, Goals, Beliefs, and Learning Mathematics

Author(s): Paul Cobb

Source: For the Learning of Mathematics, Vol. 6, No. 2 (Jun., 1986), pp. 2-9

Published by: FLM Publishing Association

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The Effect of Semantic Structure on First Graders' Strategies for Solving Addition and

Subtraction Word Problems

Author(s): Erik De Corte and Lieven Verschaffel

Source: Journal for Research in Mathematics Education, Vol. 18, No. 5 (Nov., 1987), pp. 363-381

Published by: National Council of Teachers of Mathematics

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**Contexts, Goals, Beliefs, Mathematics***

PAUL COBB

*The research reported in this paper was supported by a grant from the Spencer Foundation The opinions expressed are solely those of the author and do not necessarily reflect the position of the Spencer

Foundation

The last few years have witnessed an almost exponential growth in research into students' mathematical beliefs This research emphasis has been sparked, in part, by ''hor101 st01ies^{9 }' of the apparently nonsensical things that students at all grade levels do when they attempt to solve mathematical problems.. These include examples of students with relatively sophisticated concepts who use low level, primitive methods in certain situations. Further evidence for the importance of beliefs comes from findings that demonsüate that students at higher grade levels often use superficial methods or accept "impossible" answers [ejected by children at lower grade levels.

At the primary level, for example, the relational approach to arithmetic wold problem solving typically evidenced by first graders is "replaced by the superficial analysis of verbal problems found in many older children as they attempt to decide whether to add, subtract, multiply or divide" [Carpenter, Hiebelt, & Moser, 1983, p, 55] The older children were second and third graders. Further, failure to relate tasks such as 6+7, 6+8, 6+9,

does not imply that the principle is not known, For example, some children might have refrained from using their knowledge of principles to sh01t-cut computations because they felt it was "cheating I" Indeed, a number of children seem to have interpreted looking at the used pile and using a short-cut as 'naughty" ms attitude seemed to persist despite the effbrts to countel it.. [Baroody, Ginsburg, & Waxman, 1983, pp. 167-168]

Similarly, at the high school and college level, students who are quite capable of producing formal, deductive proofS to solve geomeüy problems typically fail to generate such arguments when they are asked to solve geometry problems that involve making constructions [Schoenféld, 1985]. 'Ihese and numerous other examples indicate that students' apparently bizarre behaviors frequently cannot be accounted for solely in terms of conceptual limitations. To use Schoenfeld's phrase, there is a need to move beyond the "purely cognitive.'

The primary purpose of this paper is to advance the hypothesis that students reorganize their beliefs about mathematics to resolve problems that are primarily social rather than mathematical in origin. Thus, am suggesting that research into students' beliefs is complemented by

**and Learning**

research into the social aspect of mathematics instruction, at least at the level of classroom social interactions. I will first argue that beliefs are an essential aspect of meaning making in general and of mathematical meaning making in particular Attention will then turn to the influence of mathematics instruction as a socialization process on students' beliefs

Ihe contextuality of cognition

I will draw on work from a variety of disciplines to suppoxt the contention that cognition is necessarily contextually bounded The discussion will also illustrate the intimate relationship between contexts, goals, and beliefs

Contemporary philosophy of science provides an initial example of the crucial role that context plays in problem solving There is a growing consensus that the scientist's actions are guided by a largely implicit contextual framework, be it called a world view or paradigm [Kuhn, 1970], a research program [Lakatos, 1970], or a research tradition [Laudan, 1977] Kuhn hinted at the essential role played by paradigms when he observed that "in the absence of a paradigm or some candidate f01' a paradigm, all the facts that could possible pertain to the development of a given science are likely to seem equally relevant" [p. 151 In other words, a paradigm or world view restricts the phenomenological field accessible to scientific investigation. The general context within which the scientist operates therefore constrains what can count as a problem and as a solution [Barnes, 19821. Analogously, the general contexts within which children do mathematics delimits both what can be problematic and how problems can be resolved [Cobb, 1985a]

The second example, cited by Schoenféld [1985], demonstrates that the psychological context within which one gives a situation meaning can radically affect subsequent behavior Kahneman and Tversky [1982] presented subjects with the following personality sketch

Linda is 31 years old, single, outspoken, and very bright.. She majored in philosophy As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations

For the Izarning of' Mathematics 6, 2 (June 1986) FL M Publishing Association, Montreal, Quebec, Canada |

Respondents were then asked which of the fOllowing two statements about Linda was more probable: (A) Linda is a bank teller, (B) Linda is a bank teller who is active in the feminist movement. It was found that 86% of undergraduate students and 50% of psychology graduate students judged the second statement to be more probable.

Kahneman and Tversky argued that these judgements ignore baseline statistics and are therefOre not perfectly rational (i.e. the set of all bank tellers who are active in the feminist movement is a subset of the set of all bank tellers). They suggested the subjects who chose (B) used what they called the representativeness heuristic by which judgements are based on the degree to which the characteristics of X (e g. Linda) match archetypal characteristics of Class A. A considelation of the subjects' intentions and purposes extends the analysis and allows us to understand why they might have used this heuristic. As Kahneman and Tversky noted; the subjects who chose (B) were not attempting to play the statistics game at all—they were operating in a different context To play the statistics game successfully, one must learn to view individuals as members of classes.. Save for the properties that specify their class membership, theil individuality is of no interest However, the subjects who chose statement (B) seemed to re-present Linda's individuality—their goal was to build a detailed representation of Linda And the process of modeling, be the product a scientific theory or a re-presentation of another person, involves searching for that which fits while adapting to the incongruous Knowing that Linda is a bank tellez is, at best, neutral with respect to their re-presentations of Lindæ Although it is possible that she could be a bank teller given what they know about her , they would not have predicted it In contrast, knowing that she is active in the feminist movement fits their re-presentations— this is what they would expect or, in other words, it is probable.

Ihis analysis indicates that to merely say that the subjects' behavior is not completely rational misses an impoltant point. Within the constraints of the modeling context, their behavior was perfectly rational—they know from past experience that it works. It is the process by which we each construct models of others. The students' fåifure was not necessarily due to a lack of understanding of elementary probability.. Instead, it reflected the goals they established and the contexts within which they interpreted the term "probable '

The next example drawn from research on everyday reasoning was also cited by Schoenfeld [19851 Again, I believe that more can be said. Perkins, Allen, and Hefner [1983] gave adult subjects two of the following four questions and asked them to reach a position,

Would restoring the military draft significantly increase American's ability to influence world events?

Does violence on television significantly increase the likelihood of violence in real lifé?

Would a proposed law in Massachusetts requiring a five-cent deposit on bottles significantly reduce litter?

Is a controversial modern sculpture, the stack of bricks in the Tate Gallery, London, really a work of Subjects were aksed to state and justify their position and follow up questions were asked„ It was fOund that approximately three fourths of the flawed arguments did not reflect faulty reasoning. Instead, they resulted from a fåilure to conduct a sufficiently elaborate analysis by, say, considering various counterexamples Further, most subjects could produce counterexamples to theil own arguments when asked to do so. Perkins et al concluded that what they call naive reasoners

act as though the test of truth is that a proposition makes intuitive sense, sounds light, lings true They see no need to criticize 01 revise accounts that do not make sense—the intuitive feel of it suffices. [p. 186]

Ihe above characterization of the evewday reasoning context can be compared with what might be called the context of academic reasoning. When we, as academic reasoners, write to produce publications, to share our ideas informally, or to clarify our own thinking we anticipate possible counterarguments because we know from experience that they will almost celtainly be offered. We also buttress our al guments against potential criticisms when we have discussions with colleagues A crucial feature of these situations is that critical scrutiny is the rule rather than the exception Further, we usually have time to reflect and conduct an internal dialogue before translating thought into action

In contrast, the overall goal of eve-I yday reasoning is not to construct compelling arguments in the face of potential scrutiny. Instead, it is to act so that the individual can achieve his or her particular goals in a specific situation

In many cases, the more systematic and precise approach would result in less effective practical action since it would take more effOrt to develop and would be less flexible in the fåce of unanticipated opportunities or constraints. Effective practical problem-solving may proceed by using tacit knowledge available in the relevant setting rather than by lelying on explicit propositions [Rogoff, 1984, pp 7-8]

Consequently,

what is regarded as logical problem-solving in academic settings may not fit with problem-solving in everyday situations, not because people are "illogical" but because practical problem-solving requires efficiency rather than a full and systematic consideration of alternatives [p 1 7]

The error made by the subjects of the Perkins et al study was not that they failed to construct carefully reasoned arguments per se Instead it was that they failed to think of producing such arguments—they did not anticipate that their arguments would be scrutinized and, consequently, they failed to play the academic reasoning game.

The examples given thus far illustrate that behaviours that might initially be dismissed as irrational begin to make sense when the contexts within which the subjects operated and the goals they attempted to achieve are considered The fbcus on the subject's rationality is compatible with

.3

Smedslund's [1977] contention that

the only defensible position is always to presuppose logicality in the other person and always to treat his understanding of given situations as a matter of empirical study From this point of view, people are always seen as logical (rational) given their own premises, and hence behavior can, in principle, always be understood This also applies to small children. [p 3]

The work of Wilker and Milbrath [1972] also illustrates that seemingly irrational behaviors have an underlying rationality within the goal-directed contexts that frame them, In contrast to Kahneman and Iversky and Perkins et al Wilker and Milbrath attempted to account for behavior that is produced spontaneously by subjects in the course of their evewday lives The phenomenon of interest to these political scientists was that "large numbers of people show by questioning on policy issues that they cannot link the policy outcome that they desire with the stands of public officials that they support" [p 41]. Such people will explain why they disagree with the policies of a

particular political candidate and yet vote for that candidate From the perspective of the "informed citizen" who engages in political activity with the overall goal of pursuing such tangible things as tax reform, improving U S military power, 01 increasing environmental controls, this behavior seems illogical.

Wilker and Milbrath suggested that many citizens' political activity is not carried out within the context defined by the instrumental pursuit of desired policy goals. These individuals are playing a different game—their political action is expressive rather than instrumentaL In par ticular , election campaigns

give people a chance to express discontents and enthusiasms, to enjoy a sense of involvement (in the political process) This is palticipation in a ritual act, however; only in a minor degree is it participation in policy formation. [Edelman, 1964, p. 3]

"By voting an individual "proves" the veracity of the civic myths that give him a sense of well-being" [Wilker & Milbrath, p.. 53]. Their action corroborates the belief "that the order is a rational one, that we all control our destiny, and that the world is indeed a friendly place" [p. 531 In other words, their overall goal is to be good citizens And by voting, these individuals "mobilize the body of myth that lies behind society's definition of what it means to be a good citizen" [pp. 56-67].

The work of Wilker and Milbrath and the two preceding examples demonstrate that an analysis of an individual's goals, intentions, and purposes is a crucial feature of contextual analysis, To infer the context within which an individual is operating is to infer the overall goals that specify the framework within which action and thought is carried out, This conclusion is reminiscent of Lewin's [1951] field theory, a field being a close relation of Kuhn's world view.

Field theoreticians contend that a field can only be desclibed by to the goals, pulposes and needs that are involved These purposive factors are the prime motivating forces and provide the impetus for structuring the field. [Wilker & Milbrath, p. 51]

The relationship between goals or purposes and beliefS has been clarified by the pragmatist philosophers Peirce and James. They argued that a "willingness to act, and, in the case of James especially, the assumption of some risk and responsibiity for action in relation to a belief, represent essential indices of actual believing" [Smith, 1978, p. 24] The general goals established and the activity carried out in an attempt to achieve those goals can therefOre be viewed as expressions of beliefs In other words, beliefS can be thought of as assumptions about the nature of reality that underlie goal-oriented activity. With regard to Wilker and Milbrath 's analysis, for example, to say that an individual's overall goal when voting is to be a good citizen clarifies the individual 's beliefS about the political process. 'Ihe individual believes, at least implicitly, that the democratic process is a rational one and that he or she can substantiate this rationality by voting. The alternative possibility of pursuing policy goals does not seem to arise—it is a separate context

To summarize the discussion thus far , firmly held beliefS constitute, for the believer, current knowledge about the world They are a ucial part of the assimilatory structures used to create meaning and to establish overall goals that specify general contexts. ^{F}lhe act of formulating a goal immediately delimits possible actions; the goal, as an expression of beliefs, embodies implicit anticipations and expectations about how a situation will unfOld. For example, students who have constructed instrumental beliefs about mathematics [Skemp, 1976] anticipate that future classroom mathematical experiences will "fit" these beliefs They intend to rely on an authority as a source of knowledge, they expect to solve tasks by employing procedures that have been explicitly taught, they expect to identify superficial cues when they read problem statements, and so forth.. Alternative ways of operating do not occur to them.. Consequently, an examination of the situations in which a student's expectations are corroborated and contradicted by experience plovides valuable information about his 01 her beliefs.. If, for example, we are interested in a student's beliefs about the role of the mathematics teacher , we might focus on teacher-student interactions, Observations of the student asking the teacher to verify his 01 her work as soon as an answer is produced indicate that the student regards the teacher as an auth01ity— the student expects the teacher to say whether or not the answer is correct. This inference would be further substantiated if the teacher responds by attempting to initiate a Socratic dialogue and the student shows irritation or frustration.

To conclude this discussion of context, I will consider two fU1ther examples that deal with mathematical behavior, The first is drawn from the work of Lave, Murtaugh, and de la Rocha [1984]. These researchers compared adults' ability to solve arithmetical problems that arose while they were shopping for groceries in a supermarket with their perf01mance on a paper-and-pencil arithmetic test.. The subjects' "scores averaged 59% on the arithmetic

test, compared with a startling 98% — virtually error free— arithmetic in the supermarket" 82]. Lave et al conducted a correlational analysis between these scores and certain demographic variables and concluded that

arithmetic problem-solving in test and grocery shopping situations appears quite different, or at least bears different relations with shoppers' demographic characteristics. Analysis of the specific procedures utilized in doing arithmetic in the supermarket lends substance to this conclusion [p. 8.3]

It would seem that the test and grocery shopping situations were separate contexts for most of the subjects. One can speculate that they approached the arithmetic test with the intention of solving tasks by attempting to recall and use procedures they had been taught in school. In the shopping situation, alithmetical problems arose when the subjects could not immediately make a practical decision such as which of two items is the better value And in these problematic situations, they did not try to recall a general method. Instead, they used self2generated methods that were tailored to the concrete decisions that had to be made In other words, the shoppers' practical arithmetic procedures were constructed within the constraints of specific contexts narrowly defined by their on-going practical activity.. Their overall goal was simply to select the items they wanted by making appropriate practical decisions As Scribner [1984] observed, skilled practical thinking is goal-directed and varies adaptively with the changing properties of problems and changing conditions in the task environment In this respect, practical thinking contrasts with the kind of academic thinking exemplified in the use of a single algorithm to solve all problems of a given type [p. 39]

Ihe final example comes from Schoenféld's [1985] investigations of the beliefs held by high school geometry students His data indicate that high school students, almost without exception, take an empirical approach to geometry tasks that involve making a construction. Their sole criterion f01 accepting or rejecting a solution to problems of this type is the accuracy of the construction—whether or not it looks light. Schoenfeld presented a wealth of evidence to demonstrate that many of these students could make deductive arguments that would have allowed them to deduce an appropriate construction without difficulty It appears that the construction and al gumentation 01 proof situations were separate contexts for the students From theil perspective, proof had little if anything to do with either discovery—the generation of conjectures—Ol verification. However, they could invoke proof either when the teachel explicitly demanded it or when they believed that such demands had been made implicitly This suggests that their overall goal in the proof context was simply to satisfy the expectation of an authority—they attempted to do what they thought they were supposed to do, In contrast, their overall goal in the construction context did not seem to involve a strong reference to others They wanted to produce constructions that satisfied their criterion of acceptability—they must "look right. "

The examples discussed in this section of the paper lend weight to Sigel's [1981] admonition that "decontextualizing the child's cognitive development is just as much in error as denying the role of internal processing by the individual" [p 216] As Rogoff [1984] put it, "context is an integral aspect of cognitive events, not a nuisance variable" [p 3]

Meaning-making in context

To say that cognition is context-bounded is to say that beliefs are intimately involved in the meaning-making process In Othel words, the elaboration and coordination of contexts is essential to the achievement of the most general of goals, the constiuction of a world that makes sense Weizenbaum's [1968] analysis of the process of conducting a conversation and Minsky's [1975] reflection on problem solving processes illustrate this aspect of meaning-making

In real conversation, global context assigns meaning to what is being said in only the most general way The conversation proceeds by establishing subcontexts, sub-subcontexts within these and so on. [Weizenbaum, p. 18]

At each moment one must work within a reasonably simple framework. I contend that any problem that a person can solve at all is worked out at each moment in a small context and that the key operations in problem solving are concerned with finding or constructing these working environments. [Minsky, p

It would be misleading to imply that the meaning-making process proceeds in an orderly top-down manner with the establishment of less and less general contexts culminating with the specific context of on-going activity. On purely intuitive grounds, one can argue that activity is always on-going—the individual is always in a specific context.. Further, the ability to change perspective (i e. context) when problem solving illustrates that contextually bound experiences can precipitate contextual reorganizations This phenomenon is also evidenced by the manner in which most children modify their beliefS about mathematics as they proceed through the elementary school grades

The view that meaning-making is a top-down process also leads to a contradiction. A problem arises when one asks how general contexts are established. The only alternative to the conclusion that the construction of viable contexts is a trial and error process is that the establishment of contexts is triggered by the perception of cues in an external environment But the act of perceiving such cues is itself contextually bounded, for otherwise all possible observations would seem equally relevant [cf. Kuhn, 1970] In short, it is necessary to assume the prior establishment of a specific context in order to account for the establishment of a more general one

A viable alternative is to regard more general contexts associated with overall goals and the specific contexts of on-going activity as being mutually interdependent As Lave et al [1984] put it, "neither setting [or general context] nor activity exists in realized form, except in relation with each other; this principle is general, applying to all levels of activity-setting [i activity-context] relations" [p 741 "In sh01t, activity is dialectically constituted in relation with the setting" [p. 73] This interdependence is exemplified by the dialectical relation between theory and observation. Theory constrains observation but unanticipated observations (i.e.. anomalies) can precipitate theoretical refinements which in turn modify what will be observed A similar analysis holds f01 the relation between a scientist's general world view and specific theories Thus, in line with Kuhn's [1970] thesis, the mutual interdependencies between observations, theory, and world view suggest that the scientist's conversion from one world view to another can be precipitated by anomalies that are either empirical or conceptual in origin [Laudan, 1977]. Current debates in the philosophy and sociology of science indicate that the task of accounting for the specifics of this process of conversion is itself extremely problematic Nonetheless, the general emphasis placed on anomalies indicates that the scientist changes allegiance from one world view to anothel in an attempt to resolve problematic experiences Similarly, children's modification of their general beliefs about mathematics, the analogues of the scientist's world view, can be viewed as attempts to deal with problematic situations This contention extends Vergnaud's [1984] argument that concepts are solutions to problematic situations from what might be called the cognitive to the metacognitive domain, Thus, a child's reolganization of his or her beliefS represents an attempt to restore a temporary stability to experiential reality—to make sense of seemingly incoherent situations This sought-for stability is itself indicated by the ability to achieve one's goals Since beliefs constrain what can be problematic within a context, the child who reorganizes his or her beliefs and changes more general goals deals with problematic situations by modifying what can count as a problem rather than by striving to find ways to solve the original problem. This implies that the task of understanding why children reorganize their beliefs involves inferring what might have been problematic for therm In the next section, will suggest that many of the problematic situations that precipitate children's reorganization of their beliefS about mathematics are social rather than mathematical in origin

Learning in interactive situations

Thus far, the discussion has considered the construction of meaning by the individual without reference to inter actions with others. Sociologists of science have alerted us to the limitations of this approach [e.g Brannigan, 1981; Barnes, 1977, 1982; 1976; Knoll, 1980; Cetina, 19811 Barnes [1977] argued, for example, that the gener ation of scientific knowledge "must be accounted f01 by reference to the social and cultural context in which it arises" [p. 2] Similar comments apply to the development of the individual's knowledge. As Sigel [1981] put it, "to understand the source and course of cognitive growth, the detailed analysis of social experience is necessary— it is the interaction that is crucial" [p 216]

'Ihe importance of social interactions as sources of constraints that delimit possible avenues of development is readily apparent when one notes that schools in general and mathematics classrooms in particular are socializing institutions [Bishop, 1985a; Stake & Easley, 1978]. Saxe, Guberman, and Gearhalt [1985] offered an analysis of the process by which interactions with others influence the child's development As their analysis focuses on goals, it is highly relevant to the problem of explaining how and why children reorganize their beliefs. Both Saxe et al and Scribner [1984] noted that the socialization process involves engaging the child in socially and culturally organized activities.. For preschoolers these typically include counting collections of items and setting a table for a meal and, for first graders, joining and partitioning collections and completing addition and subtraction worksheet By engaging the child in these activities, the more mature member of the culture influences the goals that the child generates This is not to say, however , that others set the child's goals. Instead, "children transform the goal stiucture of cultulal activities into fOrms that they understand and these forms may differ over the course of early development" [Saxe et al, p.. 5] The authors' analysis of one-on-one interactions between mothers and their children as they engaged in numerical activities indicates that there is typically a negotiation of the activity, The mothers' overall goal seemed to be the achievement of a coherent interaction or dialogue To achieve and maintain this, they monitored their child's behavior and frequently intervened by giving implicit and explicit cues that included temporarily restricting the activity to a sub-goal of the original activity. They might, for example, initially ask the child to count just one IOW of an array of items. In general, the goal structule that the child constructed was dynamic and emer ged in the course of the interaction as he or she attempted to give meaning to the mothers' actions. More specifically the mother and child each continually adapted to the behavior of the other The mothers' interventions were adaptive, purposefUl responses to observed but unexpected 01 undesired behaviors. The child, for his or her part, adapted to these interventions by modifying goals, giving the mother oppoltunities to make fU1ther observations These interactions exemplify the general process that occurs in communicative situations when individuals attempt to construct an understanding of each other and thus establish what Maturana [1980] calls a consensual domaim With regard to school mathematics, the question that arises is how do social interactions in the classroom influence the overall goals that students establish and thus their beliefS? An adequate answer to this question will involve analyzing what, in the classroom setting, might be problematic for students.

D'Ambrosio [1985] called the cultural activities that typically constitute school mathematics academic mathematics. "The mechanism of schooling replaces these [self: generated] practices by other equivalent practices which have acquired the status of [academic] mathematics, which have been expropriated in their original f01m and returned in codified version" [p. 47]. Two aspects of academic, codified mathematics can be distinguished [Hiebert, 1984]. The first concerns the conventional mathematical symbol systems per se and the second refers to the interrelated mathematical objects that are expressed both by the symbols themselves and by procedures for manipulating symbols The academic mathematics of high school geometry, for example, involves both abstract, geometrical objects (e g. points, planes) and deductive argumentation as expressed in formal proofs, Schoenfeld's [1985] study indicated that the mechanism of schooling has limited success in replacing students' empirical geometry with academic geometry.. At the first yade level, academic arithmetic includes but is not limited to numerals, number words, addition and subtraction sentences as well as the conceptualization of addition and subtraction in terms of the uniting and partitioning of sets, the commutative ploperty of addition, and the inverse relationship between addition and subtraction A major goal of first grade arithmetic instruction as it is typically practiced is to replace children 's self-generated, counting arithmetic with academic, settheoretical arithmetic.

Hiebert [1984] observed that "many children experience difficulty in learning school mathematics because its abstract and formal nature is much different from the intuitive and informal mathematics the children acquire" [p. 498]. Two potential sources of problems can be identified that stand apart from the abstract nature of academic mathematics per se (i e. difficulties arising from a mismatch between the child's concepts and those implicit in symbolic fbrms.) The first concerns the general contexts of self#nerated and academic mathematics, D'Ambrosio [1985] contrasted academic mathematics with what he called ethnomathematics. ms is the "mathematics which is practiced among identifiable cultural groups Its identity depends largely on focuses of interest, on motivation, and on certain codes and jargons which do not belong to the realm of academic mathematics" 45] In palticular , a clucial feature of ethnomathematics is that the ideas 01 concepts are put to use for practical purposes.. The analogy between ethnomathematics and children's informal, selfgenerated alithmetic is thought-provoking, although it stretches credibility to talk of, say, first graders as an identifiable cultural group. Like ethnomathematics, child-generated mathematics is constructed in the general context of pragmatic problem solving. The criterion of acceptance for self-generated methods is that they work, that they allow the child to attain his 01 her practical goals. Ideally, academic mathematics is also constructed within a problem solving context However, the pragmatic criterion that a method must work is not sufficient. In addition, concepts and procedures must be expressed in terms of conventional symbol systems that sociocultural history has provided as tools for cognitive activity A primary motivation f01 doing so is to facilitate the communication of mathematical thought. There is thus an important diffel ence between the l'ole and intent of selfæenerated and academic mathematics that is analogous to that between everyday and academic reasoning as discussed by Lave et al [1984] and Rogoff

Self-generated mathematics is essentially individualistic. It is constructed either by a single child or a small group of children as they attempt to achieve panicular goals It is, in a sense, anarchistic mathematics.. In contrast, academic mathematics embodies solutions to problems that arose in the history of the culture, Consequently, the young child has to learn to play the academic mathematics game when he or she is introduced to standard fOrmalisms, typically in first grade. Unless the child intuitively realizes that standard f01malisms are an agreed-upon means of expressing and communicating mathematical thought, they can only be construed as arbitrary dictates of an authority. Academic mathematics is then totalitarian mathematics The child's overall goal might then become to satisfy the demands of the authority rather than to learn academic mathematics per se.. This goal can be achieved, at least in the short term, by either covertly constructing and using selfæenerated methods or by attempting to memorize superficial aspects of formal, codified procedures.. If the latter approach is adopted, mathematics becomes an activity in which one applies superficial, instrumental rules

The second, related source of difficulty in learning academic mathematics concerns the nature of the interactions between teacher and student The socialization process usually involves engaging the child in joint activity with more mature members of the culture who attempt to regulate the activity in accordance with sociocultural patterns, As Bishop [1985b] noted, there is a necessary power imbalance in the learning-teaching relationship.. The central question is then how the teacher translates his or her power into action.. Bishop [1985a] discussed two general means of doing so, negotiation and imposition, that represent endpoints of a continuum. The analysis in Saxe et al of the interactions between a mother and her child exemplifies negotiation The mother's overall goal was to maintain a coherent interaction 01 dialogue and, to this end, she continually adjusted her interventions in light of the child's responses. The mother's activity was characterized by an attempt to communicate, to construct shared meanin& In gener al, teaching by negotiation is "more concerned with the initiation, control, organization, and exploitation of pupils activity There is a dynamic, organic-growth, feeling in the classroom" [Bishop, 1985b, p 26] It was also noted that the goals established by the mother and child were dynamic and emerged in the course of the interaction Similarly,

the teacher has certain goals and intentions for the pupils and these will be different from the pupils' goals and intentions in the classroom Negotiation is a goal-directed interaction, in which the participants seek to [modify and] attain their respective goals [p 27]

Negotiation therefore involves concerted attempts by the participants to develop their understandings of each other

This type of interaction can be contrasted with that in which the teacher's overall goal is to regulate students' activity by attempting to impose his or her knowledge of academic mathematics on them, The teacher's primary focus is on "a compartmentalized list of specific knowledge or skills to be taught from nothing, and to be finished in a set time" [pp. 26-27] Such "mathematics classrooms are places where you do mathematics not where you communicate or discuss mathematical meaning" [p. 27]. Here, teachers

exert their power through their contr 01 ofmathematical knowledge, which can be extremely abstract and opaque to pupils. In that sense they would be like teachers of a fOreign language in whose presence the pupils would feel relatively powerless and dependent on the teacher.. [Bishop, 1985a, pp.. 11-12]

As we have seen, negotiation is characterized by the manner in which teacher and students mutually adjust their goals and activity as they interact. Consequently, the teacher 's continually changing expectations are unlikely to be construed as demands by students, Imposition, in contrast, places a greater onus on students The teacher's expectations are far more ligid—it is expected that the students will solve certain sets of tasks in prescribed ways. In this situation, it is the students' responsibility to adjust their goals and activity to fit the teacher's expectations. Expectations can then be construed as demands and the students' primary goal might well become to find a means —any means—of satisfying them.. If this occurs, the students' overall goal becomes to solve problems that derive from socia! interactions Their activity is directed towards the goal of either blinging about 01 avoiding certain responses from the teacher.. Well-meant interventions made with the intention of facilitating students' construction of mathematical knowledge then become, in and of themselves, a source of majol problems for students In the absence of dialogue, there is a gross mismatch between the goals that the teacher thinks he or she is getting for students and the goals that students actually seek to achieve. In other words, the teacher believes that the students are operating in a mathematical context when their overall goals are primarily social rather than mathematical in nature

The dangers of this situation become apparent when it is observed that, to satisfy the teacher's demands, students have to behave as though they know how to play the academic mathematics game.. 10 succeed, the students have to produce evidence that they have perfOrmed appropriate symbolic manipulations As Schoenfeld's [1985] analysis of high school geomet1Y students' problem solving activity indicates, it is quite possible that this can become an end in itself. He observed that the students attempted to write formal proofS only when they perceived explicit or implicit demands to do so. However, their goal was not to construct and express a deductive argument in terms of conventional fOrmalisms. Instead, it was to produce an acceptable f01m per se The conceptual aspects of geometry that underlie the form were not a major focus of the students' activity Cobb [1985b] reported a similar finding when he investigated filSt graders' beliefs about arithmetic As the school year pi ogressed, the children increasingly produced appropriate addition and subtraction forms (i numeral, operator sign, numeral, equals sign, numeral) that did not make sense when interpreted in terms of sensory-motor activity (e.g.. 7 + 4 = 3, 4 - 7 = 3)

In general, students who come to believe that school mathematics is an activity in which one attempts to produce appropriate forms and thus satisfy the perceived demands of the teacher thereby increase their dependence on the teacher. This is because they have lost three valuable sources of feedback that could guide their construction of mathematical knowledge These are the inconsistencies that arise when individuals attempt to share mathematical meanings, when concepts and procedures are expressed in terms of actual or re-presented sensory-motor actions, and when the results of related procedures are compared In their absence, students have no option but to appeal to an authority in order to know whether theil solutions are appropriate. Unfonunately, this phenomenon is only too well known to mathematics educators [e.g. Confrey, 1984; Hoyles, 1982; Peck, 1984; Wheatley, 1984; Wheatley & Wheatley, 1982], With regard to the primary grades, many children

follow what they remember to be an appropriate rule and, on that basis, believe the answer must be correct regardless of what the symbols say, The result is that unreasonable responses such as 8 +21 =13, become quite common in the primary grades [De Corte & Verschaffel, 19811 This is especially disconcerting, since beginning first graders do not make these kinds of enors. [Hiebert, 1984, p. 503]

It should be stressed that imposition is but one end of a continuum that captures the various ways in which teachers exert theil power. However , Bishop [1985a] suggested and observations made by Goodlad [1983] and Stake and Easley [1978] indicate that most teachers' activity is far closer to the imposition than the negotiation pole. Further, the manner in which most students reorganize their beliefS as they PI ogress thr ough the elementary school grades is compatible with the suggested influence of teaching by imposition on students' goals and thus on the contexts within which they do academic mathematics The analysis therefore constitutes an initial framework within which to relate children's beliefs about mathematics to their classroom experiences of doing mathematics In particular, the analysis suggests that it is necessary to consider the social as well as the purely cognitive aspects of classroom mathematics in order to account for the meanings students give to the formalisms of academic mathematics This, it should be clear, does not mean that meanings and beliefS are communicated directly from the teacher to students Instead, I have argued that the beliefs students construct, the overall goals they establish, and the contexts in which they do mathematics are their attempts to find a viable way of operating in the classroom.. The are expressions of students' underlying rationalities, of the ways they try to make sense of classroom life. In short, students' beliefS about mathematics are their attempted solutions to problems that arise as they interact with the teacher and their peers

References

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/267685032

The crucial role of narrative thought in understanding story problems

**Article** · November 2017

DOI: 10.33683/ddm.17.2.3.1

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**THE CRUCIAL ROLE OF NARRATIVE THOUGHT **

**IN UNDERSTANDING STORY PROBLEMS**

Rosetta Zan

Department of Mathematics, University of Pisa, ITALY

*Abstract: Within the solution of a mathematical word problem, the preliminary understanding process is viewed as crucial. This process shows critical points both in the case of a rather poor text of the problem and in the case of a very rich text, as shown by many answers given by students and reported in the literature, characterized by an apparent ‘suspension of sense-making’. In this communication we propose an interpretation of this phenomenon, based upon the interplay between what Bruner calls the ‘ narrative’ and ‘logical’ modes of thought. *

**1. INTRODUCTION **

Mathematical word problems deserve an important role in mathematics teaching, mainly at primary school.

Even though ‘word problem’ literarily means only ‘problem expressed through a text’, in mathematics education the term usually stands for:

a text (typically containing quantitative information) that describes a situation assumed familiar to the reader and poses a quantitative question, an answer to which can be derived by mathematical operations performed on the data provided in the text, or otherwise inferred. (Greer, Verschaffel and De Corte, 2002, p. 271)

The definition given above, with its reference to a ‘situation assumed familiar to the reader’ seems to refer to *story problems* as well: and in fact the expressions *word problem* and *story problem* are generally used as synonyms^{1}.

The International literature on word problems offers several examples of students’ behaviours which suggest an apparent ‘suspension of sense-making’ (Schoenfeld, 1991). Trying to interpret these behaviours research moved on along different lines. The studies carried out so far (for a survey see Verschaffel et al., 2000) on the one hand led to identify in the suspension of sense-making the responsibity of the scarce realism of word problems, of the stereotypical nature of the text and of the implicit and explicit norms which govern the problem solving activity. On the other hand, they led to highlight the fact that many of the difficulties met by students lie in the preliminary phase of the construction of an adequate representation of the problem situation, and this makes it difficult to detect possible difficulties in the solution phase.

~~ ~~

^{1} For example Verschaffel et al. write (2000, p. ix): 'word problems (or "verbal problems" or "story problems")', e Gerofsky (1996, p.36): ‘Mathematical word problems, or “story problems”, (...)’.

**2. UNDERSTANDING (STORY) PROBLEMS **

In the literature, studies on the nature of problems highlight in particular that:

Although the numerical tasks are embedded in a context, the sterotyped nature of these contexts, **the lack of lively and interesting information about the contexts, and the nature of the questions asked at the end of the word problems** jointly contribute to children not being motivated and stimulated to pay attention to, and reflect upon, (the specific aspects of) that context. (Verschaffel et al., 2000, pp. 68-69, emphasis added) Nesher (1980) underlines:

The student who receives the well-defined SCH-PROB [school problem] does not have to be engaged in qualitative and quantitative decisions (...). In most cases, however, he is not able, because of the condensed style, to reconstruct the context from which the data was taken. In short, he **is not able to imagine the domain of objects and transformations that the author had in mind**. Instead he develops another strategy. He tries from the verbal formulation of the SCH-PROB text to infer directly the needed mathematical operation. (Nesher, 1980, p. 46, emphasis added)

The inference of the mathematical operation(s) from the verbal formulation of the problem may occur in a variety of ways, among which Sowder (1989, p.104) identifies the following: look at the numbers (they will ‘tell’ you which operation to use); try all the operations and choose the most reasonable answer; look for isolated ‘key’ words to tell which operation to use; decide whether the answer should be larger or smaller than the given numbers (if larger, try both addition and multiplication and choose the more reasonable answer; if smaller, try both subtraction and division and choose the more reasonable). Few students, even capable ones, give evidence of using the ‘mature’ strategy ‘Choose the operation whose meaning fits the story’. In other words, few students base their problem solving processes upon a representation of the problem.

Sowder notes that students who use the strategies listed above will be successful on many, if not most, one-step story problems in the whole-number curriculum. Also Gerofsky (1996) claims that often the process of representation of the problem is not essential to get to the correct solution of a word problem. Looking at word problems as a linguistic and literary genre, she identifyes a three-component structure: 1) a “set up” component, establishing the characters and location of the putative story; 2) an “information” component, which gives the information needed to solve the problem; 3) a question. Gerofsky writes:

...component 1 of a typical word problem is simply an alibi, the only nod toward “story” in a story problem. It sets up a situation for a group of characters, places and objects that is generally irrelevant to the writing and solving of the arithmetic or algebraic problem embedded in the later component. In fact, **too much attention to the story will distract students from the translation task at hand,** **leading them to consider “extraneous” factors from the story **rather than concentrating on extracting variables and operations from the more mathematically-salient components 2 and 3. (Gerofsky, 1996, p.37, emphasis added)

Verschaffel et al. (2000) observe that the description made by Gerofsky of word problem solving is ‘a description of bad word problem solving – that (...) bypasses the situation model and goes directly through some superficial cues from the text to the mathematical model’ (Verschaffel et al., 2000, p.147).

De Corte and Verschaffel (1985), referring to the huge amount of data collected in their work with beginning first graders, claim that the lacking construction of an appropriate mental representation of the word problem is actually an obstacle to correct solution processes, and enables the researcher to understand some apparently absurd answers given by students. According to the American psychologist Richard Mayer^{ }(1982) one of the major contributions of cognitive psychology has exactly been the distinction between two stages in problem solving: *representation* (understanding the problem) and *solution* (searching the problem space). The distinction between these two phases, Mayer observes, is not always possible, although it suggests that difficulties noticed within problem solving activities may come from an inadequate representation.

In the end, the phase of representation of the problem is recognized as essential and, at the same time, critical moment of the solution process. Researchers on the one hand underline that the representation process may be hindered by an excessively concise context (which rather favours the enactment of cognitive shortcuts, like those described by Sowder), on the other hand, claim that a story which is too rich can possibly ‘distract’ the student (Gerofsky, 1996). The following examples, drawn from Italian research studies, seem to confirm this ‘distracting’ effect:

- Within a study carried out by researchers of the University of Modena on the probabilistic intuitions of 2-3 grade children the following problem was used (Zan, 2007):

*Every time she goes to visit her grandchildren Elisa and Matteo, granny Adele carried a bag of fruit-candies with her and offers them to the kids, asking them to take the candies without looking into the bag. Today she arrived with a bag containing 3 orange jellies and 2 lemon jellies. If Matteo is the first to take a jelly, is it easier that he gets an orange or a lemon jelly? Why? *

Some children answer ‘orange’, with the justifications: *‘Because he likes them better;* ‘*Because he looked inside’;* ‘*If Matteo took the lemon jelly, only one was left and instead it is better to take the orange jelly’.*

- The following problem was assigned to students from 7 to 13 grade (Ferrari, 2003):

*In a house a Chinese pot was broken. In that moment 4 guys are in the house: Angelo, Bruna, Chiara and Daniele. When she gets back, the landlady wants to know who broke the pot and interrogates all four, one at a time. These are the single statements: *

*Angelo: "It was not Bruna"; Bruna: "It was a boy"; Chiara: "It was not Daniele"; Daniele: "It was not me". *

*Can you find out who is guilty? Careful: out of the 4 statements, 3 are true while 1 is false. *

*Who broke the Chinese pot? Explain how you found the answer. *

These are some of the answers collected by Ferrari :

"*It was Angelo, because he was not cleared by anyone *"; "*It was Chiara: nobody names her because they want to cover her *"; "*It was Daniele: he clears himself, therefore it was probably him *".

This and other studies about word problems suggest that when the context is very poor (as it generally is the case for those quoted in the literature) students tend to forget about the story and infer the operations to be done straight from the text. Rather, when the context is very rich (as it is the case for granny Adele and the Chinese pot problems), they get confused with ‘«extraneous» factors’ (Gerofsky, 1996), and get ‘lost’ in the story. In this concern, Toom (1999, p. 38) remarks that the text of a word problem must be ‘purged of all irrelevant data’.

**3. STORY PROBLEMS AS PROBLEMS FORMULATED BY OTHERS **

The points made above about the role of the representation of the problem situation in the solution process lead us to underline a characteristic of word problem solving, that in our opinion has a great relevance to understand students’ behavior.

The presence of a text that characterizes a word problem is connected to a feature of word problems that makes them really different to *real life problems*: the one who is to solve the problem (the student) is other from the one who proposes it (either teacher or textbook). In other words, problems students work on at school are ‘proposed, and formulated, by another person’ (Kilpatrick, 1987), and the (written) text is the usual way they are *posed*.

Some important implications for the process of understanding a problem follow from the fact that school problems are *hetero-posed* (i.e. posed by others).

The first implication is the presence of an explicit question, with the function of communicating to the one who solves it what his/her goal is: when the problem is self-posed the one who solves it does not need to make his/her own goal explicit to him/herself.

The second implication concerns the particular goal that characterizes those who pose a word problem. Nesher (1980) stressing the stereotypical nature of arithmetic word problems, underlines the role played by the intentions of the author of the problem text:

for the sake of simplicity, the qualitative and quantitative considerations for a given REAL-PROB [real problem] have already been made by the author of the text. (...) He has in mind a mathematical operation, or a mathematical structure with whose applications in real life he would like the students to become acquainted. The author then chooses one of the real life contexts and imagines a situation (...) which will call for the application of the given mathematical structure (...). In order to simplify it for the student he then adds, in the most concise manner, all the qualitative and quantitative information needed for solving the problem and arrives at a kind of SCH-PROB [school problem] which has all the stereotyped characteristics already described. (Nesher, 1980, p. 45) Thus the goal of either the teacher or the author of the text is *internal* to mathematics. Nevertheless, as Cobb (1986) remarks:

(...) there is a gross mismatch between the goals that the teacher thinks he or she is getting for students and the goals that students actually seek to achieve. In other words, the teacher believes that the students are operating in a mathematical context when their overall goals are primarily social rather than mathematical in nature. (Cobb, 1986, p. 8) Cobb explicitly points to a ‘social’ context, meaning that students’ activity ‘is directed toward the goal of either bringing about or avoiding certain responses from the teacher’ (ibidem, p. 8). More in general he claims that ‘the psychological context within which one gives a situation meaning can radically affect subsequent behavior’ (ibidem, p.2). In the case of a story problem the psychological context involved in the phase of representation requires ‘penetrating to the pragmatic deep structure’ (Nesher, 1980, p.46) of the story. Understanding the stories of people, of their reasons, intentions, feelings is linked to a form of thought that Bruner (1986) defines as ‘narrative’, and that the scholar juxtaposes to ‘paradigmatic’ or ‘logico-scientific’ thought:

There are two modes of cognitive functioning, two modes of thought, each providing distinctive ways of ordering experience, of constructing reality. The two (though complementary) are irreducible to one another. (...)_{ }One mode, the paradigmatic or logico-scientific one, attempts to fulfill the ideal of a formal, mathematical system of description and explanation. It employs categorization or conceptualization and the operations by which categories are established, instantiated, idealized, and related one to the other to form a system... (...)_{ }The imaginative application of the narrative mode leads instead to good stories, gripping drama, believable (though not necessarily "true") historical accounts. It deals in human or human-like intention and action and the vicissitudes and consequences that mark their course. It strives to put its timeless miracles into the particulars of experience, and to locate the experience in time and place. (Bruner, 1986, p. 11-13)

The representation of the situation described in the word problem – the ‘story’ – thus requires the student to get into a context (in the sense of Cobb) that we might call *narrative*. Then, on the representation of the situation, the solution process (and the answer with it) should be built, and logical thought plays a crucial role in this.

Distinguishing between the two phases of *representation* and *solution* as well as the role played by narrative and logical thought in these phases leads us to distinguish between information relevant to representation (that we might call ‘narratively relevant’), and information relevant to the solution, that is to answer the question** **(‘logically relevant’). In the interplay between the two processes (representation / solution) a crucial role is played by the question, which we have already mentioned to be consequence of the fact that the problem is hetero-posed: in order to solve the problem, a child must represent the situation but *also* understand the sense of the question. Therefore, the more the representation of the described situation evokes the question to the child, the more that representation will promote an understanding of the question, needed to get to the solution. Particularly meaningful from a narrative standpoint is therefore that information that enables the child to grasp the problematic nature of the story and point out the link existing between the story itself and the posed question. _{ }

The point here is that the data a child needs to represent the problem are not necessarily those he/she will need to use in the solution.** **

For example, in a study carried out by d’Amore et al. (1996) children from 4 to 8 grade had to re-formulate the following word problem: “Three workers take 6 hours to complete a certain job. How long will 2 workers take to complete the same job?”. All the children added information about *the reason why* the workers reduced from 3 to 2 (for example: “one got ill *and so* only 2 were left”). This information was relevant to grasp the story, in particular its problematic nature and (therefore) its relationship with the question. In a later study I supervised for a first degree, some children explicitly commented upon the original text as follows: *“I can’t imagine the scene because I don’t know what their job is”, “I can’t understand how to answer the question because the workers are initially three and then they become two, it is not explained very well”. *

It might also happen that the pieces of information needed to solve the problem are not necessarily consistent from a narrative viewpoint, and if they are inconsistent, they will probably be ignored by those who read in a narrative mode.** **

For instance, in the ‘Granny Adele’ problem the final question (‘Is it easier to get an orange or a lemon jelly?’) is not a realistic one from a narrative standpoint: it is an artificial question, with no meaningful links to the narrated story. Actually the reported answers suggest that the children either answered a different question, or simply completed the story. Similarly, in the ‘Chinese pot’ problem – which follows the narrative plot of a ‘thriller’– the piece of information ‘*Careful, though: out of the 4 statements, 3 are true whereas 1 is false*’ is not consistent from a narrative viewpoint (who may know?) and it is nevertheless a fundamental one to answer the question.

In the end, in order for narrative thinking to *support*, through the process of representation of the story, logical thinking, which is necessary for the solution process, it is important that information needed for the solution be consistent from a narrative viewpoint, and that information needed for the representation be consistent from the logical viewpoint, in particular consistent with the posed question.

In actual fact the standard formulation of story problems standard generally pays little attention to these aspects.

When the context is extremely poor, there is not enough information to represent the story: we might even say that sometimes a *story* as such is missing, given that a

crucial dimension of stories is that of time (Bruner, 1986). In particular the question does not follow in a *narrative* way from the context, it is rather an artificial question about the context. The role of the context is reduced to that of container of necessary (and generally sufficient) data to be able to answer the question. Thus, not surprisingly, the student focuses on the question, while the context is read with relation to that question (in particular by selecting key-words and numerical data).

On the contrary, when the context is pretty rich, the richness of the story promotes the enactment of narrative thinking, needed to understand it. It is not trivial that this understanding should support the solution process, and particularly the understanding the sense of the question. If the posed question does not narratively fit with the narrated story, or the story itself embeds essential information from the logical point of view but inconsistent from the narrative point of view, narrative thinking enacted by the story will not support the student in solving the problem. It should even be an obstacle in the solution process, leading the child to answer a question that better fits with the narration, or rather to get lost in the ‘fictional wood’^{6} we have built for him/her.

In our hypothesis this is what happens in the ‘Granny Adele’ problem as well as in the ‘Chinese pot’ one. In fact the answers we reported suggest that children have completed the story *narratively*, regardless of either the posed question or the given constraints: the story of a grandmother with her grandchildren, the ‘thriller’ of the Chinese pot. In these cases a phenomenon described by Cobb seems to take place: the child works in a setting - the narrative one in this case - that differs to the logical mathematical one, expected by the teacher._{ }In this setting the answers reported are fully legitimated, and it does not make sense to talk about mistake or even ‘lack of logical thinking’.

**4. CONCLUSIONS **

The phase of understanding the problem is acknowledged to be at the same time a crucial and a critical moment of the solution process. Research on word problems, in particular, highlights that on the one hand the process of understanding may be hindered by a context that is too concise, on the other hand that an excessively rich story may ‘distract’ children. Our remarks suggest a possible interpretation of this phenomenon, which requires further investigation: the difficulties highlighted in the two cases are due to a lack of consistency between the narratively relevant and logically relevant information, more than to the features of the context taken as such (concise / rich). In the case of a rich context lacking of this consistency, narrative thinking enacted by the context does not support logical thinking, needed to give the answer: a consequence of this might be that narrative thinking prevails and the child gets lost in a *fictional wood*. Conversely, in a ‘well formulated’ problem the story described in the context supports, and does not hinder, the solution of the problem itself.

Because of the purposes of the activities in the classroom with word problems, the teacher (or the author of the problem) pays most attention to the solution process, and therefore to the question and to the information needed to answer. The same attention though, is not paid to the phase of representation, and therefore to the information a child needs to represent the problem to him/herself: information concerning the ‘story’ are often viewed as ‘irrelevant’ details, source of confusion rather than help. In other words, it is the logical structure of the problem that deserves the attention of those who pose the problem, whereas the narrative structure is not considered enough. Hence, what generally happens is that there is a ‘narrative rupture’ in the text of the problem, i.e. the question and the information needed for the solution are not consistent from the point of view of the narrated story.

The analysis proposed in this paper refers to word problems characterized by a *story*. Our remarks suggest that research about word problems should appropriately consider *story problems* as ‘particular’ word problems, whose specific features deserve researchers’ attention. Further investigation about story problems is needed: in particular, it is possible that the process of understanding may be favoured by the presence of a story. For instance, the time-dimension which characterizes a story is also the main difference between *static* and *dynamic* problems, where a change happens, and research pointed out that dynamic problems turn out to be easier for children than static ones (Nesher, 1980).

Another implication of our remarks is that two logically equivalent problems might be very different from a narrative viewpoint. Being able to recognize the same mathematical structure in different story problems is an important skill in mathematics, which involves logical thinking and which cannot be viewed as a prerequisite. It is rather an end-point of mathematical education, requiring time and attention to the critical points we stressed. The link between contexts and modes of thought (logical / narrative) on the one hand, and between contexts and goals, on the other, as underlined by Cobb (1986), points out another important objective for mathematics education, at a meta cognitive level: educating students to recognize contexts in which logical thinking better fits with their purposes. But, again with Cobb, an individual’s goals are in turn linked to his/her beliefs, that ‘can be thought of as assumptions about the nature of reality that underlie goal-oriented activity’ (Cobb, 1986, p. 4). This link highlights the role of the beliefs students build up by interpreting their own experience.

One last remark. Even though a mathematics teacher’s task is to develop logical thinking, in my opinion narrative thinking should not be viewed as an obstacle to logical thinking, or anyway as a lack of rational thinking. It might work in contexts where logical thinking fails. For example, narrative thinking may let a logically absurd problem make sense. In the problem known as ‘the age of the captain’ (‘*There are 26 sheep and 10 goats on a ship. How old is the captain?) *some children answer after summing up the numbers found in the text and justify this with arguments like *‘Perhaps the captain got an animal as a gift for each birthday’* (IREM de Grenoble, 1980), i.e. they build up a *story* thanks to which the answer (and thus the question) makes sense. Hence, given the problem “*In a meadow there are 20 sheep, 7 goats and 2 dogs. How old is the shepherd?*” (logically, but not narratively equivalent to the previous one) a child answers:

My particular reasoning path was: if the shepherd has two dogs for few animals, perhaps he needs one of the two dogs because he is blind. Hence I deduce that he might be about 70-76 years old.

Does this answer really show a ‘suspension of sense making’?** **

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