«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
Alcock & Inglis, 2008; Michener, 1978). Examples, and non-examples, of a theorem can aid in the process of proving the theorem and understanding the conditions involved. Example spaces are similarly needed for definitions, because they can demonstrate the importance and use of particular aspects of the definition. To achieve clarity, the examples should also differ along a narrow set of parameters (Fukawa-Connelly, Newton, & Shrey, 2011; Goldberg & Mason, 2008). Interestingly, the knowledge gleaned from being presented with examples does not seem to be as great as when students generate examples on their own (Dahlberg & Housman, 1997).
Zazkis and Leiken (2008) emphasize the importance of students creating their own examples, both to the students and the instructor who is trying to evaluate student comprehension.
Textbooks obviously present the reader with examples, but are the example spaces appropriate?
Do they texts include essentially the same examples, leading to a conventional example space that teachers then expect their students to become familiar with (Watson & Mason, 2005)? The reader should be given a range of illuminating examples, but also should be led to generate personal examples through the text or exercises. The combination of the two ways to enhance an example space seems to be the best way to increase initial understanding of a concept.
Methods: In this study, over a dozen abstract algebra textbooks were considered, some of which were later editions of another text in the collection. The years of publication ranged from the 1960s to 2010. Many popular texts were used, such as Fraleigh’s A first course in abstract algebra (2003, 1976), Gallian’s Contemporary abstract algebra (1994, 2010), Herstein’s Abstract algebra (1986) and Topics in algebra (1964), and the classic textbook, A survey of modern algebra, by Birkhoff and Mac Lane (1965). Sometimes, specific content areas like rings and groups, which could be found in all the textbooks, were examined. Other questions led to a consideration of the book as a whole.
One method of analysis that was used in this study involved reader-oriented theory.
Within this framework, I tried to find characteristics of the intended, implied, and empirical readers. Many times information given in the preface of the book served as an indicator of the intended reader. Other factors under scrutiny were the language used by the author, the example spaces, the style of proof, and the level of detail given in explanations. For instance, when the author uses the pronoun “we” or imperatives such as “suppose”, then he or she indicates that the reader is part of the mathematical community and a peer of the author (Rotman, 2006). On the other hand, when over a dozen examples are given for one definition an author implies that the reader requires more guidance and need not develop their own examples. This indicates a discrepancy between the implied reader and the intended reader, and could lead to a limited level of discovery and understanding by the student. The style of proof can be revealing as well.
Differences such as paragraph style versus list style, or more details versus fewer, give evidence of what knowledge the reader is expected or needs to possess in order to comprehend the proof.
Of particular interest in this textbook analysis were the example spaces of the textbooks.
The examples for rings, groups, and equivalency classes have been examined. Some assessments under consideration include: number of examples, types of examples, and difficulty of examples.
Also taken into account were the examples that were given or asked for in the exercises.
Preliminary Results: Results thus far point to some discrepancies between the intended, implied, and empirical reader of abstract algebra textbooks in terms of maturity of language and 15TH Annual Conference on Research in Undergraduate Mathematics Education 365 style. However, the type of examples seems to match nicely with the intended reader. The prefaces indicate that the authors are well aware that their reader is a student, but the language and level of detail are often appropriate for an experienced person from the mathematical community. For example, most authors seem to use a paragraph style of proof with the minimum number of steps or details. Some of the proofs refer to previous theorems or lemmas by number without any description, even though it is likely that most students do not memorize theorem numbers and may not look them up. Although the intended reader is usually a student in their first abstract algebra class, the implied reader is a mathematician comfortable with sophisticated proving techniques.
One aspect about example spaces that stood out, coming from sections on equivalence classes, was the contrast in how many real world examples were used, both within the section and in homework exercises. Gallian, who wants students to see that the “concepts and methodologies are being used by working mathematicians, computer scientists, physicists, and chemists,” listed many applied exercises and motivated the topic with examples in a physics setting (2010). Birkhoff & Mac Lanes's motivating example was the classic modulo 12 description of how we measure time, which corresponds to their desire to use “as many familiar examples as possible” (1965). Herstein, who aimed for a “chatty” presentation and to “put the readers at their ease,” has the first example set in a grocery store (1986). The most recently published textbook that I examined, by Bergen, included sixty-six exercises with no applied problems. Bergen, in the preface, explains that abstract algebra can especially help those who plan to teach mathematics at the high school level by clarifying the concepts encountered in high school (2010). It seems that the number of applied examples correlates with the goals and objectives of the authors in terms of their intended reader.
Despite the differences in real-world examples of equivalence classes, after comparing the other examples for equivalence classes as well as groups and rings, preliminary results indicate that the example spaces of abstract algebra textbooks are remarkably similar. Often, as new editions or new books are published more examples are added to the texts, but even those examples have distinct parallels. This indicates that the authors tend to agree on which examples best demonstrate a definition or theorem, creating the conventional example space that Watson & Mason describe (2005). The large number of examples and exercises in the texts, however, may not be beneficial to students. There is little to no motivation for the reader to generate their own examples and hypotheses. For instance, the definition of a ring may be followed by examples that are commutative, non-commutative, with unity, without unity, fields, or not fields. The reader has no need to think deeply about the definition or theorem to create such examples since they are immediately given.
1. The quantity of textbooks that I have available make the study time-intensive and it is hard to succinctly describe the differences and similarities. Would the benefits outweigh the disadvantages of considering every textbook for every question?
2. Should I narrow the focus to look only at one example space, such as equivalence classes?
3. I wanted to consider the changes that abstract textbooks have taken over time (1960s to 2010), and did find some interesting patterns. How can I figure out why certain examples began to take precedence over others, and why certain theorems became more or less important?
366 15TH Annual Conference on Research in Undergraduate Mathematics Education
Alcock, L. & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111-129.
Bergen, J. (2010). A concrete approach to abstract algebra: From the integers to the insolvability of the Quintic. Burlington, MA: Elsevier.
Bills, L. & Watson, A. (2008) Editorial introduction. Special Issue: Role and use of
exemplification in mathematics education, Educational Studies in Mathematics. 69(2):
77-79 Birkhoff, G. & Mac Lane, S. (1965). A survey of modern algebra. New York: Macmillan.
Borasi, R. & Seigel, M. (1990). Reading to learn mathematics: New connections, new questions, new challenges. For the Learning of Mathematics, 10(3), 9-16.
Crouch, R. & Walker, E. (1962). Introduction to modern algebra and analysis. New York: Holt, Rinehart, and Winston.
Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33(3), 283–299.
Eco, U. (1970). The role of the reader: Explorations in the semiotics of texts. Bloomington, IN:
Indiana University Press.
Fraleigh, J.B. (1976). A first course in abstract algebra. Reading, MA: Addison-Wesley.
Fraleigh, J.B. (2003). A first course in abstract algebra. Boston, MA: Addison-Wesley.
Gallian, J. (1994). Contemporary abstract algebra. Lexington, MA: D.C. Heath and Co.
Gallian, J. (2010). Contemporary abstract algebra. Belmont, CA: Brooks/Cole.
Gilbert, L. & Gilbert, J. (2009). Elements of modern algebra. Belmont, CA: Brooks/Cole.
Goldenberg, P. & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183–194.
Herstein, I. N. (1986). Abstract algebra. New York: Macmillan.
Herstein, I. N. (1964). Topics in Algebra. New York: Blaisdell.
Hungerford, T. W. (1990). Abstract algebra: An introduction. Philadelphia, PA: Saunders College Publishing.
Jacobson, N. (1974). Basic algebra I. San Francisco, CA: W. H. Freeman and Co.
K-12 Mathematics Curriculum Center. (2005). The changing mathematics curriculum: An annotated bibliography. Newton, MA: Education Development center.
Kang, W., & Kilpatrick, J. (1992). Didactic transposition in mathematics textbooks. For the Learning of Mathematics, 12(1), 2-7.
Lithner, J. (2003). Students’ mathematical reasoning in university textbook exercises.
Educational Studies in Mathematics, 52(1), 29-55.
Mason, J. & Watson, A. (2008). Mathematics as a Constructive Activity: exploiting dimensions
of possible variation. In M. Carlson & C. Rasmussen (Eds.) Making the Connection:
Research and Practice in Undergraduate Mathematics. (pp189-202) Washington: MAA.
McNamara, D. S., Kintsch, E., Songer, N. B., & Kintsch, W. (1996). Are good texts always better? Interactions of text coherence, background knowledge, and levels of understanding in learning from text. Cognition and Instruction, 14(1), 1-43.
Michener, E. (1978). Understanding Understanding Mathematics. Cognitive Science, 2(4), 361TH Annual Conference on Research in Undergraduate Mathematics Education 367 Morgan, C. (1996). “The language of mathematics”: Towards a critical analysis of mathematics texts. For the Learning of Mathematics, 16(3), 2-10.
Fukawa-Connelly, T., Newton, C., & Shrey, M. (2011) Analyzing the teaching of advanced mathematics courses via the enacted example space. In (Eds.) S. Brown, S. Larsen, K.
Marrongelle, and M. Oehrtman, Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education, Vol. 1, pg 1-15. Portland, Oregon.
Reys, B. J., Reys, R. E., & Chaves, O. (2004). Why mathematics textbooks matter. Educational Leadership, 61(5), 61-66.
Robitaille, D. F., & Travers, K. J, (1992). International studies of achievement in mathematics.
In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.
687-709), New York: Macmillan.
Rotman, B. (2006). Toward a semiotics of mathematics. In R. Hersh (Ed.), 18 unconventional essays on the nature of mathematics (pp. 97-127). New York: Springer.
Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.
Weinberg, A. & Wiesner, E. (2011). Understanding mathematics textbooks through readeroriented theory. Educational Studies in Mathematics, 76(1), 49-63.
Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69(2), 131-148.
This paper will take a close look at the construction of a graphical image for reasoning with approximation in the context of Taylor series. In particular, it is a comprehensive case study of the genesis and evolution of an image created by one student, who draws extensively on other images and knowledge from calculus and physics to supplement gaps in his understanding of Taylor series and reason with Taylor series approximation tasks. His process resulted in a graphical representation that was leveraged to build knowledge and reason with the situation, even while lacking key considerations that are central to an understanding of Taylor series. The preliminary report sets the stage for a paper that will speak to considerations both of students’ understanding of particular content, as well as a detailed examination of the processes of constructing a visual image used for problem solving and obtaining and utilizing evidence to amend that visual image.
Keywords: Taylor series, graphical representation, calculus