A Developing Approach to Studying Students’ Learning through Their Mathematical Activity

Abstract

We discuss an emerging program of research on a particular aspect of mathematics learning, students’ learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which students pass in the context of particular instructional approaches or learning environments. Generally missing from the literature is research that examines the process by which students progress from one of these conceptual steps to a subsequent one. We provide a conceptualization of a program of research designed to elucidate students’ learning processes and describe an emerging methodology for this work. We present data and analysis from an initial teaching experiment that illustrates the methodology and demonstrates the learning that can be fostered using the approach, the data that can be generated, and the analyses that can be done. The approach involves the use of a carefully designed sequence of mathematical tasks intended to promote particular activity that is expected to result in a new concept. Through analysis of students’ activity in the context of the task sequence, accounts of students’ learning processes are developed. Ultimately a large set of such accounts would allow for a cross-account analysis aimed at articulating mechanisms of learning.

ACKNOWLEDGMENTS

This research was supported by the National Science Foundation (REC-0450663). The opinions expressed do not necessarily reflect the views of the foundation. We acknowledge Drs. Karen Koellner, Kay McClain, and Laura Van Zoest for helpful comments on this article.

Notes

We consider the classroom teacher to be the most important determinant of the quality of classroom mathematics learning. However, unlike most of the first author's prior work, this program of research is not focused primarily on the teacher.

diSessa and Cobb (2004, p. 90) refer to cutting the “fabric of in-class learning” to focus on particular phenomena that are not easily studied in that context.

Simon (2006) proposed the construct of empirical learning processes to create a category that is more encompassing than Piaget's construct of empirical abstraction, the latter referring to abstraction of properties of physical objects or the material aspects of a physical action. Empirical learning processes include all abstractions limited to input-output patterns, for example those generated through repeating an arithmetic operation.

Understanding the logical necessity is meant to indicate a qualitative distinction. We suggest that within the category of understanding logical necessity, there is likely a spectrum of understandings possible.

This description is compatible with Davydov's (1990) claim that scientific concepts are not the result of empirical thinking.

Steffe's collaborators have included John Olive, Ernst von Glasersfeld, and John Richards, among others, and a number of talented doctoral students. For efficiency of writing, we will refer to “Steffe” and not “Steffe and colleagues” in the remainder of this section.

Jason was using a computer microworld, TIMA: BARS, that has a function that partitions a unit bar into a desired number of equal size parts.

Understanding can be thought of as a progression rather than as something that is present or absent. We use the language of “understand” and “not understand” throughout to suggest two different states of understanding, not the complete absence of understanding or total understanding.

For example, JJ was asked to give the rabbit a speed that would make it go over and back (200 feet) in 7 seconds.

These are attempts to describe the predominant characteristics of recent microgenetic work in developmental psychology. As such, there are, of course, exceptions to these generalizations about the work that has been done.

Here is a brief example of a description of the nature of the change process. Siegler and Svetina (2006, p. 1007) reported, “Thus, acquisition of understanding of class inclusion appeared to include three periods: a period of consistently incorrect responding, a period of chance responding, and a period of consistently correct responding.”

This is not to say that coordination class theory cannot make a contribution to the development of mathematical objects, only to point out the particular goal for which our program of research was designed.

diSessa and Cobb (2004, p. 88) pointed out, “We note that teacher interventions constitute a potential competitor to ‘student knowledge’ as an answer to the question ‘how did students manage this?’”

Of course, within a pair of learners, the active and passive roles can switch many times during the solving of a mathematical task.

Steffe (2003, pp. 249–250) also encountered the disadvantage of working with two students,

• Fortunately, Jason was absent from the teaching episode held on 10 January 1994 because it permitted Laura to be the primary actor in solving tasks posed by the teacher. Even though Laura assimilated Jason's language and actions in Protocols I and II and thereafter acted as if she had constructed a commensurate fractional scheme, she did not solve commensurate fractional tasks independently of Jason's solutions.

And Hershkowitz, Schwartz, and Dreyfus (2001, p. 204) provided a similar rationale to ours: “Our reason for choosing a single student in an interview situation as a first paradigmatic case is that in this setup the epistemic actions are relatively easy to identify.”

We understand that performance on parallel tasks is neither a necessary nor sufficient basis for a claim that a conceptual advance was made. However, we see it as a way to strengthen the credibility of such a claim.

At the same time, we seek to answer the question of whether there are different categories of learning that might necessitate the postulation of distinct mechanisms of learning. For example, might the development of a concept of division, or ratio, or integral be different from reinvention of an algorithm for multiplication of whole numbers or for solving of quadratic equations?

Simon had designed a sequence of tasks and conducted exploratory teaching, and Zembat (2004) had done a dissertation study adapting Simon's original tasks. Zembat's study did not produce useful data for our purposes, but did add to our ability to generate a more useful task sequence.

Much of what one would want students to learn about division of fractions and about fractions more generally was neither assessed nor taught in this study.

The rationale for this part was the following. Prior study had revealed that many elementary teachers are unable to write a division of fractions word problem. One theory (Simon, 1993) is they respond to a demand for a division problem by spontaneously selecting a partitive model. They are unable to fit the anticipated division into the partitive model, and because they are unaware that they have selected a particular (and not the only) model, they are not able to amend their selection when unsuccessful. Thus, we hypothesized that explicit knowledge of the two different models and knowledge of the relationship of the quotitive model to division of fractions word problems would contribute to their being able to envision division of fractions word problems.

We use past tense when referring to data that have already been presented or a summary of data and present tense when referring to a data excerpt that follows.

In our transcript data, we use the symbolic form of whole numbers (e.g., “5”) or numerators and the written form of fractions or denominators (e.g., “2 thirds”). The purpose here is to avoid assuming what the student means or how the student interprets an expression such as 2 thirds. For the student, it might be a single fraction or it might be 2 copies of the unit fraction, 1/3. In English, we have a built in ambiguity.

As mentioned earlier, inability to anticipate the answer (without doing the mental run) further supports the conclusion from the pre-assessment that Erin did not have the target understanding prior to the teaching experiment.

The reason for Erin's “of a whole” is unclear. Our tentative interpretation is that, because at one level the parts have no fixed size, all she can say about them is that they are equal parts of a whole.

This was accomplished through the researcher's leading question and therefore was not consistent with the teaching experiment approach that we use. This was an a priori decision, because, for our research purposes, we were not interested in that aspect of her learning. The uncommon denominator problems were brought up to assess whether she would over-generalize and to show her the greater power of what she had accomplished.

For composite units, we only have evidence of her reasoning with powers-of-ten multiples.

Young children who want to add two amounts count out the first, then the second, and then count all the objects together. At a certain point, the children no longer need to count all the objects to get the combined amount. They start from the amount of one of the sets of objects and count on using the second set of objects to reach the total (e.g., “3, 4, 5, 6, 7”).

The conditions for reflection (i.e., seeing commonality) were present. Erin had the same goal in the two problems and called on the same activity sequence, which caused her to focus on the same quantities in both cases. Fundamental to a claim of commonality is that the person's goals and activity predispose them to focus on that which is common.

Although Erin had no difficulty generating the solution, some might argue that the first task that Erin had to solve using a diagram was a problem. However, considering that first task as a problem does not change our claim, because the conceptual advance took place at a later point in the task sequence.

This claim contrasts with von Glasersfeld's (1995, p. 68) description of Piaget's theory of learning, “Cognitive change and learning in a specific direction take place when a scheme, instead of producing the expected result, leads to perturbation, and perturbation, in turn, to an accommodation.”

See note 22.