On learning mathematics as an interplay of mathematizing and identifying

"Oh, so you understand him but not me?" 

The view of learning as a particular type of the activity of communicating with others and with oneself has already been shown to unify the once separate lines of study devoted to cognitive and to social aspects of learning. This approach is now extended so as to include the study of affect. In this article, emotional expressions are treated as an aspect of communicational activity and are thus metaphorically described as emotional hue of utterances. Accordingly, the study of emotions becomes grounded in special types of discourse analysis. Our empirical example, featuring a small group of 7th graders grappling with an unfamiliar mathematical problem related to fractions, illustrates our basic assumption, according to which learning can be seen as an interplay between the activities of mathematizing (talking about mathematical objects) and of identifying (talking about participants of the discourse). As will be shown, mathematics classroom may become an arena of very intensive identifying, made conspicuous through its prominent emotional hues. In the present case, this highly emotional identifying activity revolves around issues of leadership in discourse. As this one example seems to suggest, the activity of identifying can prevent the student from taking advantage of what appears as a particularly promising opportunity for learning.

What is it that makes classroom learning effective and what may hinder its progress? Not long ago, those looking for answers focused almost exclusively on cognitive factors. Influenced by Piaget's powerful vision of human development, they believed that the quality of learning depends almost exclusively on the responsiveness of instruction to cognitive needs and capacities of the students. Sometime later, researchers' attention was drawn by aspects of learners' behavior described with adjectives affective or emotional. The study of affect and its impact on learning began running in parallel to cognitive research, but remained distinct in its epistemology and methods. More recently, not in the least due to the Vygotskian turn in research on learning, social issues came to the fore. Recognizing their importance, many researchers shifted their glance to interpersonal aspects of learning processes. Special conceptual systems and dedicated methodologies have been developed for this last type of research2. Most of these designated techniques have been imported from current social research. With all this tremendous work already done, just one thing seems to be urgently needed: If those who study human learning are to be able to communicate with one another and arrive at new insights, a unifying framework is necessary, where cognitive and affective, as well as intra-personal and inter-personal (or individual and social) aspects of learning processes would all be seen as members of the same ontological category, to be studied with an integrated system of tools, grounded in a single set of foundational assumptions.

In this article has been designed with this need in mind. Our goal was to contribute to the effort of creating a unified discourse for dealing with all three aspects of learning processes. In what follows we report on the progress made so far. In the nutshell, the proposed perspective, which continues the work done by Wood (2009; this issue), originates in the recognition of the centrality of communication in all our activities, including the uniquely human forms of learning. As presented in general lines in the introduction to this special issue, the resulting approach may go so far as to equate mathematics with a particular form of communication. In this article we propose to view cognition, affect and social matters as aspects of the discourse that takes place when people learn mathematics. More specifically, mathematics learning is to be seen as interplay between two concomitant activities: that of mathematizing – communicating about mathematical objects; and that of subjectifying, that is, communicating about participants of mathematical discourse. Of all subjectifying activities, the most consequential for learning seems to be that of identifying – the activity of talking about properties of persons rather than about what the persons do. Scrutinizing the activity of mathematizing is the commognitive counterpart of cognitive analysis, whereas studying the activity of identifying means attending to all those phenomena that other researchers label with the adjectives affective, interpersonal or social. Indeed, these are identity concerns, among others, that introduce the emotional elements to mathematical conversations and that shape interactions. The empirical material presented in this article demonstrates how the activity of identifying may interfere with the activity of mathematizing, and thus with the learning of mathematics.

In the introduction of the unifying discourse is accompanied by an example of its application. The example comes from a study of extracurricular mathematical activity of a group of 7th graders. In it, we are taking a close look at one 30 minutes-long classroom scene in which the students attempted to solve an unfamiliar mathematical problem related to fractions. What happened in this episode surprised us and left us perplexed: whereas the participants seemed impervious to one student's seemingly lucid and cogent explanations, they claimed to have undergone an ‘a-ha’ experience while listening to an apparently quite incoherent argumentation of another student. Our scrutiny of the flow of the "mathematical content" of the conversation proved of little help. Students’ utterances carried little hints about the mathematical sources of their confusion. It was this incident, among others, that intensified our quest after a unified set of conceptual tools for dealing with all aspects of mathematical learning, including those that, although not related to mathematics in any obvious way, may nevertheless influence the course and effectiveness of mathematical communication.