THE OPERATION TABLE AS METAPHOR IN LEARNING ABSTRACT ALGEBRA

number: 983-97-554. Abstract This paper discusses seemingly idiosyncratic ways that students make distinctions among mathematical concepts. The findings are taken from the author’s dissertation on student learning and understanding in an undergraduate abstract algebra class. Analysis of interviews and written work from five students revealed that the students often made nonstandard linguistic and notational distinctions. For example, Carla used the term coset to describe not only individual cosets of a subgroup but also the set of all cosets. This kind of understanding was characterized as being immersed in the process of generating all of the cosets of a subgroup. Similar characterizations described and explained instances of the phenomena of failing to make standard distinctions between a set and its elements, between quantifiers such as “any” and “all,” and between the specific and the general and their notation in mathematical arguments. Implications for teaching and for the process/object distinction are discussed.This paper discusses seemingly idiosyncratic ways that students make distinctions among mathematical concepts. The findings are taken from the author’s dissertation on student learning and understanding in an undergraduate abstract algebra class. Analysis of interviews and written work from five students revealed that the students often made nonstandard linguistic and notational distinctions. For example, Carla used the term coset to describe not only individual cosets of a subgroup but also the set of all cosets. This kind of understanding was characterized as being immersed in the process of generating all of the cosets of a subgroup. Similar characterizations described and explained instances of the phenomena of failing to make standard distinctions between a set and its elements, between quantifiers such as “any” and “all,” and between the specific and the general and their notation in mathematical arguments. Implications for teaching and for the process/object distinction are discussed. Detail (from Findell, 2001) Given the function f from U8 to Z4, given by f(1) = 0, f(3) = 0, f(5) = 2, and f(7) = 2, Carla had identified the kernel of the homomorphism as the set {1, 3}. When I suggested that she call that set K and that she investigate the sets aK, she responded, “Left cosets. All right. So to do aK, we see what happens . ” (line 63). She struggled momentarily to recall the definition of coset and reconstruct the process for computing them, but then completed the calculations quickly, yielding {1, 3} and {5, 7}, and concluded, “So we have two elements in the left coset” (line 70). Her nonstandard syntax suggested unusual thinking. She explained, “{1, 3} and {5, 7} are each of, are the two elements that result from aK” (line 74). I ask her whether aK refers to all of the cosets or one specific one. 82 Carla: aK is the general formula that gives you all of the cosets. Then you choose specific as for whatever K you are given, and you will find all the specific cosets. Thus, Carla had an efficient and reliable procedure for computing cosets, and the procedure was supported by the notation aK. By calling the cosets elements, she seems to have had little difficulty seeing cosets as objects, suggesting that she had encapsulated the process of coset formation. Furthermore, her language “the left coset” for the collection of cosets suggests that she had further encapsulated that collection as an object, a set of sets, a point of view that is helpful in order to see the set of cosets as itself a group under the appropriate coset arithmetic. Nevertheless, Carla did not see the relevance of the distinction I was trying to make, and thus did not distinguish between the particular coset aK and the set of all cosets of K. Instead, a varied as part of a procedure specified by the formula aK, which gave all the cosets. Carla’s language suggests that her thinking was in the transitional process between two objects: the particular coset aK = {ak | k ∈ K} and the set of all cosets of K, which might be written {aK | a ∈ G}. In the midst of the process, a is varying, so aK denotes neither a particular coset nor all of them. For Carla, aK denoted and specified the process. Being immersed in the process allowed Carla some flexibility in her thinking. On the one hand, she could consider a particular coset by stopping the process for a moment. On the other hand, she could consider all the cosets by completing or imagining she had completed the process. From within the process she could broaden her viewpoint slightly and see both a particular coset and the set of all of them as two aspects of the concept of “coset. ” Thus, “aK tells you how to find the cosets” (line 80) and together there are “two elements in the left coset” (line 70). Carla maintained this dual role of the term coset through the fourth interview, and even maintained her process orientation, saying, for example, “The left coset gives the set of (23) and (132)” (line 239, emphasis added) rather than the left coset is that set. Other students in the study used the term coset in similar ways. The paper generalizes the characterization of this phenomenon to include similar difficulties that the students had distinguishing, for example, between a particular multiple of 4 and the set of all multiples of 4. Together, these findings provide explanations for the common phenomenon of failing to make notational and linguistic distinctions between a set and its elements. The findings also provide nuance in the relationships between process and object conceptions of mathematical ideas.