ON BEING EMBODIED IN THE BODY OF MATHEMATICS:

MATHEMATICS KNOWING IN ACTION

Elaine Simmt	Tom Kieren
University of Alberta	University of Alberta
elaine.simmt@ualberta.ca	tom.kieren@ualberta.ca

Abstract: Mathematics knowing is explored in this interpretive study of a parent and child’s actions and interactions as they are occasioned by a variable-entry prompt, each other and the artifacts of their own thinking. We explain their knowing in interaction as embodied in the body of mathematics in so far as it is observed to fit within and contribute to the local, contemporary and historical mathematics communities.

Introduction

There is considerable interest in mathematics education in how a person’s mathematics knowing arises in interaction with others and with elements of his or her environment (e.g. Cobb and Bauersfeld, 1995; Davis, 1996). Over the past several years, we have contributed to this conversation by exploring the consequences of thinking of mathematical knowing in action not as the matching of pre-given models or answers to mathematical queries (although a knower might provide answers as part of their actions), nor even as problem solving (although some of the data discussed here could be viewed in those terms), but as a fully embodied phenomenon which brings forth of a world of significance (including mathematics) with others in a sphere of behavioural possibilities (Maturana and Varela, 1992; Varela, Thompson and Rosch, 1991). Observing mathematical knowing in the actions and interactions through which such bringing forth occurs, involves observing the personal/structural, social/interactional and cultural/mathematical dynamics of the situation all at once. In previous papers, we have focused on the interplay between personal constructive activity and social interaction (Simmt, Kieren, 1999; Kieren, Simmt, Mgombelo, 1997).

Research Focus

In this paper, we turn our attention to the ways in which a person’s embodiment in the body of mathematics co-emerges with his or her mathematical actions and interactions and the potential that arises out of mathematics knowing in action. In other words, we interpret a person’s mathematics (actions, utterances and forms of reasoning, in part) and demonstrate how it can be viewed as occasioned by the mathematical practices and artifacts of his or her local community as well as by the practices and artifacts of the broader contemporary and historical mathematics communities. This perspective allows us to ask, “What occasions such embodied actions and what are the possibilities for embodiment in the body of mathematics that arise when, let’s say, adults and children engage in a particular mathematical setting differently?”

Research Setting and Methods

In our work we have been considering such questions by discussing and interpreting the work of parent-child pairs (the children are between the ages of 8 and 13) as they respond to variable-entry prompts offered in an extra-curricular mathematics program (Simmt, 1997). It is important for us to note that we understand the “starters” used in the program as prompts rather than problems in that we observe it is the participants, in their various actions, who specify the problems with which they engage rather than the starter itself. Such a characterization of the prompts also points to the feature that a person can enter into, or engage with these prompts through a variety of appropriate mathematical actions. Thus the parent’s or child’s mathematical actions (mathematics) should not be thought of as caused by the prompt in the sense that these are simply an effect of it; nor should one think of the parent’s and child’s actions and mathematics independent of the prompt. We find it useful to say that the mathematical actions (the mathematics) of the parent and the child are occasioned (see Kieren, Simmt, Mgombelo, 1997) by the prompt and co-emerge with it and their environment.

Our method of study is interpretational in nature; we invoked the second-order cybernetic method of observing observers (von Foerster, 1981). It involves the gathering of a variety of data (e.g. audio or video tapes; mathematical artifacts created by the participants; field notes) and the creation of other “data” (such as transcripts of the tapes; mathematical activity traces; sequences of still pictures from video with related annotations). All data are studied recursively to build up the interpretations which we use in our paper and are studied by teams of researchers, each of whom brings their own interests to the study. This overall research philosophy is consistent with our view that the statements we make in our interpretations are radically contingent ones; they are co-dependent on the structures of the participants, the conditions and interactions in the settings, the nature of the observations, and the lived histories and structures of the observers.

In our paper, we elaborate on our conception of being embodied in the body of mathematics. We show that while the inter-action between the parent and child affected the mathematical actions of each, both their lived histories and their embodiment in the body of mathematics (the ways in which they engaged in mathematical activity) were unique; hence different spheres of possibilities were created for each person. While we focus on only one parent-child pair, we believe their case is illustrative of the many pairs we have observed working with variable-entry prompts in the parent-child mathematics program.

Interpreting the Mathematics of a Father-Daughter Pair

Imagine posing the following prompt to a father (Jake) and his 9 year old daughter (Cathy) (Figure 1).

Figure 1. Variable-entry prompt offered to Cathy and Jake

Assuming this is a variable-entry prompt (extensive data on this support our assumption), we anticipated that it would be approached in a variety of appropriate ways by the participants in our study and that different individuals and parent-child pairs would be occasioned to consider very different problematics. This certainly proved to be true for Jake and Cathy. They entered into mathematical activity by specializing; that is, they considered the particular case of nxn squares. Cathy drew a sequence of three squares beginning with a 2 x 2 square and then superimposed a 3 x 3 square on the 2 x 2 square and finally a 4 x 4 square on the 3 x 3 square. (I have drawn what her image looked like over time in figure 2.) Although the diagonals drawn were not accurate, both the father and his daughter, in their own ways, were able to imagine how the diagonals ‘should’ look and they quickly saw a generalization for squares. Jake explained, a “four by four is four ’cause you just have to go to the middle one—cross diagonally each one.”

Figure 2. Replication of the image Cathy drew with its changes over time

Cathy then turned her attention to (non-square) rectangles. “Can I just show you something?” she said to her father as she drew a 3 x 5 rectangle. Jake responded by commenting, “Let’s do this step-by-step.” As Cathy started to draw a 4 x 6 rectangle in a different part of the paper Jake stopped her and said, “Don’t go all over the place. What we want is to go over the top of the three by five ... and just keep expanding ... otherwise we will not have enough room.”

At one level, we might observe the exchange above as a feature of the interaction between a parent and a child— in this case an interaction whereby the parent exerts control over his child. Viewed in another and not necessary disjoint way we might note how this interaction demonstrates Jake’s desire for or value of order and neatness. In action, he “knows” the value of step-by-step activity, or the value of ordering his activity in particular ways. Jake’s statements and consequent actions (his ordering) could be observed as an embodiment in the body of mathematics in that the actions of ordering and progressing in a systematic manner are common mathematical actions and ones which are promoted in school mathematics.

Although Cathy followed Jake’s instructions not to go all over the place, she found her father’s strategy of drawing one rectangle over another one too confusing. She whispered to herself about how “messy” and “complicating” she found it to work in this way. On one hand, it may be that Cathy’s structure and her lived history prevented her from seeing the value in her father’s strategy of systematically generating cases. On the other hand, from previous work, we know that Cathy can envision geometric patterns and is oriented to representing and exploring her thinking geometrically, with diagrams. Thus, even though we might conclude that her father is directing her in a mathematically appropriate direction, Cathy starts to work divergently and soon she and her father are working quite independently of each other, each keeping their own records (figure 3).

Figure 3. Extract from Cathy’s (left) and Jake’s (right) working papers

It is not surprising that the nature of Cathy’s mathematical practice is different from her father’s but it is notable for a number of reasons. Most importantly (for this paper), we are given the opportunity to observe and consider the mathematical potential in what Cathy and Jake are doing given that their utterances, forms and reasoning differ from each other’s. While their records do not look much alike, it is clear that both Jake and Cathy, in their own way, are searching for patterns. Notice that such a search can be observed as part in parcel of Jake’s mathematics and his engagement in the body of mathematics, certainly at the level of the local practice of the parent-child mathematics program. In fact, this practice appears to lead him to powerful mathematical patterning involving the dimensions of the rectangles in terms of factors and relatively prime factors. From his work with the rectangles, Jake created a series of tables in which he noted the dimensions of the rectangles and the number of unit squares the diagonal passed through. Eventually he stated a relationship between the number of unit squares and the dimensions of the rectangle, if the dimensions were relatively prime—length plus width minus one (m + n - 1).

In contrast, Cathy’s forms are quite distinct from Jake’s and her “work” lends itself to different mathematics. Cathy’s approach allows her to organize her actions and keep her records in her own fashion. However, she was influenced by the tables her father kept and the utterances he made about the relationship he noted for relatively prime rectangles. After hearing her father explain his reasoning to the researcher, Cathy made a table of values based on her drawings and tried to find a relationship among the numbers in her table. But more than that, a careful study of her work shows that her mathematical actions opened her to mathematical ideas not available to her father in his work. In her drawings one can find small marks she made on the 4 x 6 and the 4 x 8 rectangles where she has distinguished (visually) patterns along the diagonal and hence the simplest ratio between the two sides. We might observe that by marking the “prime” rectangles in each of the “composite” rectangles her sphere of behavioural possibilities holds different potential from her father’s. From her mathematics one could begin to develop notions of equivalent ratios, for example—something that is not easily developed from her father’s diagrams. Further, an observant teacher might notice the potential in this prompt for promoting a geometric basis for thinking about equivalent and non-equivalent ratios.

Consequences of Embodiment in the Body of Mathematics

It is clear that Jake's and Cathy’s mathematics and actions were determined by their own lived histories (personal, structural and dynamic) and hence the worlds they brought forth were at once different but co-emergent with the prompt. At the same time, their mathematics was also occasioned by their interaction. Jake, occasioned by Cathy’s method of presentation for squares, developed his sequential overlapping method which, in turn, occasioned him to contrast various cases and make generalizations based on the relationship of the factors. While Cathy did not follow her father’s work, she was occasioned by it to engage in pattern making and generalizing. Although Cathy and Jake engaged the prompt in distinct ways, their approaches were mathematically appropriate; further, both Cathy and Jake were, at once, participating in the practices of contemporary and historical mathematics and contributing to that culture.

So what might these observations of a very unique mathematical activity session offer to the mathematics education community? In our paper we explored the divergences between Cathy’s and Jake’s actions and interpreted them within a framework of mathematics as fully embodied. In particular, we explored the cultural dimensions of their embodiment by considering the mathematical potential that was triggered from Cathy’s and Jake’s different forms, utterances, and reasoning. We assert that engaging differently in mathematical practices or being embodied differently in the body of mathematics by one’s actions, opens one to different mathematical forms and practices within the local, contemporary and historical mathematics communities. Finally, we believe that thinking about personal mathematical knowing in terms of its embodiment in the body of mathematics provides both researchers and teachers with new insights into pedagogical mathematics knowing and into ways in which mathematics knowing in action affects the curriculum in action. It suggests to us that by offering even a slightly rich prompt, the teacher is providing herself multiple opportunities to look not simply for expected behaviour or results but to be tuned into the actions, utterances and reasoning which reveal various embodiments in the body of mathematics. Although the teacher may offer certain practices in a particular setting, she would be wise to take up the mathematics and mathematical practices of her students as an opportunity to observe just how students belong to and are part of the body of mathematics through their actions.

References

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Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland Publishing, Inc.

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Maturana, H., Varela, F. (1992). The tree of knowledge: the biological roots of human understanding (rev. ed.). Boston: Shambhala Publications Inc.

Simmt, E. (1997). Reflections on a Parent Child Mathematics Program. Delta-K, 35(1), 56 - 60.

Simmt, E. and Kieren, T. (1999). Expanding the cognitive domain: the roles and consequences of interaction in mathematics knowing. In F. Hitt and M. Santos (Eds.) Proceedings of 21st annual meeting of Psychology of Mathematics Education Vol. 1 (pp. 298 - 305). Columbus, OH: Eric Clearing House for Mathematics Science and Environmental Education.

Simmt, E., Kieren, T. and Gordon Calvert, L (1996). All-at-once: Understanding and inter-action in mathematics. Paper to be presented at the Psychology of Mathematics Education North America, Panama City, FL.

Varela, F., Thompson, E., and Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge: MIT Press.

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