# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

Leron & Dubinsky, 1995). The course is often the first encounter with higher mathematics for many students; in particular, they are exposed to algebraic structures which form unifying threads throughout the rest of mathematics (Edwards & Brenton, 1999). Unfortunately, many students struggle with this transition to higher mathematics and fail to understand even the subject’s most basic and fundamental concepts (Dubinsky et al, 1994). As a result, many students who are initially interested in mathematics experience a complete reversal of opinion and become indifferent or disengaged. Leron and Dubinsky (1995) even go so far as to state that “[the] teaching of abstract algebra is a disaster, and this remains true almost independently of the quality of the lectures” (p. 227). To this end, alternative approaches to teaching abstract algebra must be explored.

In response to this need, two notable innovative approaches have been developed in recent years. Leron and Dubinsky (1995) developed an instructional method using the programming software ISETL to allow students a more interactive experience with basic algebraic concepts, such as group, subgroup, normal subgroup, coset, and quotient group. Larsen (2004, 2009) used the theory of Realistic Mathematics Education (RME) to develop local instructional theories supporting the guided reinvention of group, group isomorphism, and quotient group. Both of these methods emphasize example-driven approaches which serve to highlight and elucidate the foundational concepts of group theory. Similar research in the area of ring and field theory, however, is exceptionally scarce. Based upon the literature regarding student difficulty in comprehending the definition of a group (Dubinsky et al, 1994), it is reasonable to suspect that many students are just as unsure of the importance of the ring axioms and the subtle differences among such ring-theoretic structures as ring, integral domain, and field. Thus, this research project seeks to address this need by developing an original approach towards increasing student proficiency with the definitions of ring, integral domain, and field.

Literature. As mentioned previously, Larsen (2004) developed an innovative method of group theory instruction by testing and revising an instructional theory which supports the guided reinvention of group and group isomorphism. He explicated three iterations of the constructivist teaching experiment (Cobb, 2000) as part of a developmental research design (Gravemeijer, 1998). Larsen’s instructional activities employed symmetries of regular polygons as a means by which students are able to interact with the group structure. Gradually, the students harnessed their informal experience with the symmetries of a triangle and square and ultimately were able to reinvent the concepts of group and isomorphism by way of stating a precise mathematical definition. Similarly, Larsen, Johnson, Rutherford, and Bartlo (2009) 380 15TH Annual Conference on Research in Undergraduate Mathematics Education developed an instructional theory for the reinvention of the quotient group concept and included results for how such a theory might be implemented in a classroom setting.

The only reference in the literature which directly addresses student learning in ring theory is a case-study of one student’s work with the commutative ring Z99 (Simpson & Stehlikova, 2006). In particular, the student’s self-guided explorations of the structure by such devices as equation solving enabled her to recognize and address several fundamental properties of rings with little external prompting, confirming the ideas of Filloy and Rojano (1989) who asserted that equations are a means by which students transition from arithmetic thinking to algebraic thinking. Kleiner (1999) echoed the importance of equation solving by stating that “in the solving of the linear equation ax+b=0, the four algebraic operations come into play and hence implicitly so does the notion of a field” (p. 677). Indeed, the informal act of solving basic equations seems to provide a nice context for motivating both the ring and field axioms and other concepts central to ring and field theory, but no research exists which analyzes this claim.

Furthermore, Larsen has laid the groundwork for a novel, reinvention-minded approach in group theory, yet no such research exists in ring theory. This research project addresses these gaps in the literature.

Research questions. The overarching questions which guide this research project pertain to the reinvention of the definitions of ring, integral domain, and field, as well as how they might be motivated and distinguished from one another: How might students reinvent the definition of a ring? How are students able to motivate the need for the subsequent ideas of integral domain and field? What models or activities will enable students to clearly differentiate between these structures? Supporting research questions include: What activities, models, processes, or ideas are involved in developing these concepts when the students start with their own activity and knowledge? With what types of informal knowledge are the students able to begin the process of reinventing the definition of a ring? What kinds of activity can support the transition of the students’ informal knowledge into more robust methods of thinking?

Theoretical Framework This study utilizes the ideas of developmental research as a means to evaluate and revise a local instructional theory (Gravemeijer, 1998). The initial local instructional theory and the subsequent instructional activities were guided by the heuristics of Realistic Mathematics Education (RME). The notion of initial local instructional theory can be likened to Simon’s (1995) hypothetical learning trajectory, which he defined as a “prediction as to the path by which learning might proceed” (p. 135). Moreover, Gravemeijer (1998) recommended that the initial instructional theory be designed with regards to “informal knowledge and strategies of the students on which the instruction can be built” and “instructional activities that can foster reflective processes which support curtailment, schematization, and abstraction” (p. 280). The RME heuristic which largely guided the design of the initial local instructional theory was that of guided reinvention, the main idea of which is to allow students to discover the desired mathematics for themselves (Gravemeijer, 1998).

The Initial Local Instructional Theory Due to the familiarity of most students with solving basic equations, the historical importance of equation solving (Kleiner, 1999), and its potential for motivating the structure of a ring (Simpson & Stehlikova, 2006), I designed instructional activities and the overarching initial local instructional theory with the idea that the ring structure would emerge as a result of solving equations. In particular, I am viewing the general structure of a ring as an emergent model (Gravemeijer, 1998) brought about by the activity of solving equations. Using the modelTH Annual Conference on Research in Undergraduate Mathematics Education 381 of/model-for transition as detailed in Gravemeijer (1999), I anticipated that the ring structure would initially emerge as a model-of the students’ informal knowledge of solving equations and would gradually evolve into a model-for more formal mathematical activity to motivate the distinctions between the definitions of ring, integral domain, and field. At the crux of this hypothesized emergence of the ring structure is the students’ mental transition from thinking about properties simply as the properties used to solve equations into those properties which explicitly characterize a mathematical structure. Also of significance is the subsequent identification of those properties which make certain equations solvable on some structures but not others; these will be exactly those properties which distinguish general rings from integral domains and fields.

Research Design The research design is comprised of three iterations of the constructivist teaching experiment (Cobb, 2000) that I conducted myself with pairs of undergraduate students. Each iteration consisted of up to 12 sessions of 1.5-2 hour sessions each. The participant pool included students who had recently taken a course in discrete mathematical structures and had not yet had a course in abstract algebra. The multiple iterations of the teaching experiments allowed for the instructional theory to be in a constant state of revision. The data, which consists of both transcribed video data and written work, was analyzed both between sessions within teaching experiments and also between the teaching experiments themselves. The data was analyzed and the instructional theory revised by means of multiple iterative analyses similar to that of Larsen (2004). Other theoretical constructs employed to support the reinvention process and enhance data analysis include Larsen and Zandieh’s (2008) Proofs and Refutations framework and Zandieh and Rasmussen’s (2010) defining as a mathematical activity framework.

Results and Implications As of the submission of this proposal, data collection and analysis is still ongoing, so any statement of conclusive results may be premature. However, based on the literature and my experience with the participants in the teaching sessions, I expect to be able to present preliminary results regarding the revision and evaluation of an instruction theory which supports the guided reinvention of ring, integral domain, and field. These initial results and implications based on the data from the teaching experiments (and the corresponding analysis) will be complete in time for the conference. I hope to engage in conversation with other researchers interested in both my content area (teaching and learning abstract algebra) as well as my research method (RME and guided reinvention) to help me refine the conclusions I am able to harvest from my data.

Questions

**I will ask the following questions:**

In your experience, what are some other problematic concepts for students in an introductory course on ring and field theory?

(Continuation of previous question:) How might this study be able to address those problematic concepts given its current design?

If you were teaching a course in ring and field theory, how might you modify this instructional theory for your classroom?

What other frameworks, pieces of literature, or research contacts might be relevant to or helpful for my work?

Do you have any suggestions for future research which could further the work done here?

Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. Kelly & R. Lesh (eds.), Handbook of Research Design in Mathematics and Science Education (pp. 307-334). Mahwah, NJ: Erlbaum.

Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267-305.

Edwards, T.G., & Brenton, L. (1999). An attempt to foster students’ construction of knowledge during a semester course in abstract algebra. The College Mathematics Journal, 30 (2), 120-128.

Filloy, E. & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2). 19-25.

Gravemeijer, K. (1998). Developmental research as a research method. In A. Sierpinska, & J.

Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 277–296). Dordrecht, The Netherlands: Kluwer.

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2). 155-177.

Hazzan, O., & Leron, U. (1996). Students’ use and misuse of mathematical theorems: The case of Lagrange’s theorem. For the Learning of Mathematics, 16(1), 23-26.

Kleiner, I. (1999). Field theory: From equations to axiomatization. American Mathematical Monthly, 106(7). 677-684.

**Larsen, S. (2004). Supporting the guided reinvention of the concepts of group and isomorphism:**

A developmental research project (Doctoral dissertation, Arizona State University, 2004) Dissertation Abstracts International, B 65/02, 781.

Larsen, S., Johnson, E., Rutherford, F., Bartlo, J. (2009). A local instructional theory for the guided reinvention of the quotient group concept. Conference on research in undergraduate mathematics education. Raleigh, NC. Retrieved April 15, 2011 from http://sigmaa.maa.org/rume/crume2009/Larsen_LONG.pdf.

Larsen, S. & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics, 67, 205-216.

Leron, U. & Dubinsky, E. (1995). An abstract algebra story. American Mathematical Monthly, 102(3), 247-272.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.

Journal for Research in Mathematics Education, 26, 114-145.

15TH Annual Conference on Research in Undergraduate Mathematics Education 383 Simpson, S. & Stehlikova, N. (2006). Apprehending mathematical structure: a case study of coming to understand a commutative ring. Educational Studies in Mathematics, 61, 347Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning. Journal of Mathematical Behavior, 29(2), 57-75.

384 15TH Annual Conference on Research in Undergraduate Mathematics Education The Use of Dynamic Visualizations Following Reinvention

Preliminary Research Report This research is a part of a larger project to gain insights into how calculus students might come to understand formal limit definitions. For this study, a pair of students participated in an eightday guided reinvention teaching experiment in which they created a formal definition for sequence convergence even though they had not previously received instruction on formal limit definitions. During the reinvention process they identified and coordinated relevant graphical attributes of sequences as they recognized and resolved problems with their emerging definition.