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

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Within the context of an advanced calculus instructional design teaching experiment, four students encountered interesting difficulties with sigma notation. This report tells the story of those students’ progress; it describes the nature of the difficulties encountered and the ways

**these difficulties were resolved. Specifically, we wish to answer the questions:**

1) How do post-calculus students talk about and use sigma notation?

2) How do they handle the transition from discrete to continuous cases in their use of sigma notation? In particular,

a) What challenges do students encounter when transitioning from sums involving the terms of a sequence to sums involving approximate area under a function?

b) What skills or tools do students use to meet these challenges?

Key words: Sigma notation, calculus, analysis, concept image/definition, mathematical discovery, semiotics Literature Review Mathematics education research that focuses specifically on students‟ understandings and interpretations of sigma notation is scarce. Studies (Alcock & Simpson, 2004, 2005; Ferrari,

2002) have attributed student difficulties with mathematical topics that rely on sigma notation to a lack of semiotic control (Ferrari, 2002)--students‟ ability to properly interpret and manipulate symbolic expressions involved in tackling a mathematical task. However, the nature of these difficulties, and students‟ conceptions of sigma notation in general, have not been well documented. Some researchers (Arcavi, 1994, 2005; Hiebert, 1988; Pimm, 1995) have focused on notation/symbol use as a whole. This corpus of work, as well as general work on the connection between symbols and concepts, informs our work on students‟ understanding of sigma notation.

Much of the work on symbols highlights the importance between a symbol and its referent (e.g. Arcavi, 1994, 2005; Hiebert, 1988; Pimm, 1995; Tall & Gray,1994). Tall and Gray in particular highlight the connection between processes, objects, and the symbols used to represent them. They refer to this triad as procepts. For example, sigma notation is used to represent both the process of adding together the terms of a specific sequence and the resultant sum. Moving between process, object, and symbol with relative ease is an important part of fluency. Arcavi (1994, 2005) coined the term „symbol sense,‟ to describe such fluency with symbols and their referents. The term is used analogously to how the term „number sense‟ is used in relation to numerical reasoning (see, Sowder, 1992 for a review). He describes several categories of reasoning which exemplify various forms of symbol sense as they relate to algebra. This includes flexible strategic choices of symbolic referents, being able to smoothly transition from algebraic symbols to their referents when it is prudent, and noticing higher-order structure within algebraic expressions. Arcavi is not intended to be a complete catalog of types of symbol sense but instead paints a picture of the types of reasoning that symbol sense can encompass.

550 15TH Annual Conference on Research in Undergraduate Mathematics Education In the development of fluency with concepts and the mathematical-symbols and notation practices that are used to work with them, many things can go astray. Tall and Vinner (1981) studied the disconnect between mathematical formulations of concepts and students‟ uses and interpretations of these concepts in action. They refer to these student notions as concept images.

Concept images are often inconsistent, context dependent, and removed from formal mathematical notions.

While Arcavi‟s (1994, 2005) work highlights the things that can go right with students‟ understandings of mathematical notations, Tall and Vinner‟s work helps illuminate the varied nuances that occur when students‟ conceptions are misaligned with formal mathematical notions.

Both of these bodies of work provide useful tools with which we can describe students‟ interactions with and understandings of sigma notation. In our presentation we will give examples of what symbol sense looks like in relation to sigma notation and we will present a picture of what students‟ concept images look like in relation to sums.

Background In the Spring of 2011, we began an Advanced Calculus teaching experiment. The purpose of this experiment was to investigate the efficacy of a new instructional sequence for Advanced Calculus (Real Analysis). This sequence was designed to provide the students with tasks that would leverage their knowledge of calculus to motivate further investigation into its theoretical underpinnings. The first sequence of tasks had the students investigate notions of area, with increasing formality and rigor, in order to motivate the study of sequential limits. Along the way our students demonstrated some of the challenges they faced in using sigma notation to talk about area. Specifically, while the students were able to use sigma notation to denote the sum of odd integers without difficulty, they were unable to use it to accurately to reflect a rectangleapproximation to the area under a curve, at least initially. These challenges led naturally to the

**following questions:**

1) How do post-calculus students talk about and use sigma notation?

2) How do they handle the transition from discrete to continuous cases in their use of sigma notation? In particular,

a) What challenges do students encounter when transitioning from sums involving the terms of a sequence to sums involving approximate area under a function?

b) What skills or tools do students use to meet these challenges?

This report will add to the body of knowledge of how students think about and use math symbols in general and sigma notation in particular, with potential application to improved instruction in calculus, statistics, and analysis.

Method For this initial investigation four students who had completed an introductory calculus (up through Sequences & Series) with high marks were recruited to work in pairs on the prescribed sequence of tasks. Both pairs of students worked in sessions of 60-90 minutes, with the first group participating in fourteen sessions and the second group in nine. Two researchers ran the interview/experiment sessions, with each session being video and audio recorded. The video and audio data were reviewed after each session by the researchers to facilitate ongoing modifications to the instructional sequence and to plan the next session.

Analyzing the students as individuals, we will use a grounded theory approach (Strauss & Corbin, 1990) to explore and explain the challenges faced by students in attempting to use sigma 15TH Annual Conference on Research in Undergraduate Mathematics Education 551 notation in the context of area under a function, in addition to what eventually helped them overcome those challenges.

Preliminary Findings Below is an excerpt from the teaching experiment. Early in the third session of the first teaching experiment, Betty and Kathy had worked out the sum of the first ten odd integers using sigma notation (Figure 2). We then returned the context to area. After drawing an arbitrary function on an arbitrary interval, they successfully wrote out a long-hand approximation for the area with eight rectangles using the left-hand rule (See Figure 1). When asked to do the same with sigma notation, they encountered difficulty.

Though there is not room to present further excerpts, these two students also experienced difficulty when dealing with the varying sizes of „the change in x‟ and its relation to the sigma notation representation of the approximate area. This was surprising given that the students were able to use sigma notation to deal with non- integration related sums during other portions of the teaching experiment. These difficulties led to an important insight into the behavior of the index variable in sigma notation, namely that the rule to increment that index by 1 each time is not an explicit part of the notation.

Results and Applications Sigma notation is a useful and widespread standard for describing finite and infinite sums.

This research makes inroads into mapping common student understandings (and misunderstandings) related to its use. Of particular interest was the difficulty that students experienced (which they demonstrated multiple times in the first teaching experiment) in making the transition from using sigma notation in discrete situations to continuous ones. This research is a worthwhile endeavor for two reasons: as a field, it adds to our knowledge base on this topic, and has the potential to inform improved instruction.

Discussion As this is a preliminary report, we hope to receive constructive feedback and suggestions for future research. In particular,

- What are some steps the research team should take in developing this theory?

- How can this theory be used to inform instruction?

- Are there other theoretical frameworks that might be useful in analyzing this data?

552 15TH Annual Conference on Research in Undergraduate Mathematics Education References Alcock, L. & Simpson, A. (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner‟s beliefs about their own role. Educational studies in mathematics, 57(1), 1-32.

Alcock, L. & Simpson, A. (2005). Convergence of sequences and series 2: Interactions between nonvisual reasoning and the learner‟s beliefs about their own role. Educational studies in mathematics, 58(1), 77-100.

Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24-35.

Arcavi, A. (2005).Developing and using symbol sense in mathematics. For the Learning of Mathematics, 25(2), 42-47.

Ferrari, P. (2002). Understanding elementary number theory at the undergraduate level: A semiotic approach, In S. Campbell, & R. Zazkis (Eds), Learning and teaching number theory: Research in cognition and instruction (pp. 97-115). Ablex Publishing: Westport, Conneticut.

Hiebert, J. (1988). A Theory of developing competence with written mathematical symbols.

Educational Studies in Mathematics, 19(3), 333-355.

Pimm, D. (1995). Symbols and Meanings in School Mathematic, Routledge: New York.

Sowder, J. (1992). Estimation and number sense. In Grouws, D.A. (Ed.), Handbook for research

**on mathematics teaching and learning (pp. 371-389). Macmillan Publishing Company:**

New York.

Strauss, A. & Corbin, J. (1990), Basics of Qualitative Research: Grounded Theory Procedures and Techniques. Sage Publications, Inc.

Gray, E., & Tall, D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116-140.

Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity, Educational Studies in Mathematics, 12(2), 151-169.

Vinner, S. (2002). The Role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 65-81). Dordrecht: Springer Netherlands.

15TH Annual Conference on Research in Undergraduate Mathematics Education 553 FIGURE 1 - Sigma Notation and Area FIGURE 2 - Sigma notation for a simple sum

**Abstract:**

Because proof-writing involves both understanding mathematical ideas related to the theorem, as well as structural norms of formal proofs, we hypothesized that students could improve both content and structure of their proofs using the drafting techniques common to English Composition research. Our research question is, “Does proof revision lead to improved proofwriting skills?” The intervention group revised their proofs and turned in up to three drafts of each formal proof. This pilot led to the development of a coding tool to categorize the types of student individual errors. In this proposal, we share the coding tool as well as the ongoing analysis of two sets of Linear Algebra student proofs. Preliminary results suggest the drafting group engaged with the work more often than the control group during the semester and on the final, while the control students were more likely to skip proofs rather than attempt them.

**Key Words:**

teaching experiment proof-writing transition to proof undergraduate mathematics

Introduction Teaching undergraduates to write proofs involves much pulling of hair for both students and professors. We know quite a bit about the issues students typically face while proving such as whether a proof is convincing, understanding quantifiers, etc (e.g. Weber (2007); Dubinsky & Yiparaki (2000)). However, proof-writing techniques of instruction for the undergraduate student is under researched.

In contrast, the field of English Composition pedagogy has long considered the question of how to teach essay writing. One school of thought, offered by John Bean (2001), is that students learn to think critically by revising their own writing. Often, a student struggling with difficult ideas will make basic grammatical mistakes with surprising frequency, due to a sort of cognitive overload. But, though drafting—reorganizing the essay multiple times— the student develops a more sophisticated mastery of the content, and automatically corrects the grammar and spelling errors on their own. (Bean, (2001)). Instructors are advised to help the student clarify her ideas, but not to focus on the mechanics of the paper.