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

This difference is one of many that indicates not only a mathematical but a dispositional difference between Gymnasium and university students. Whereas university students left open their understandings of these concepts for future sophistication, and most suspected that with this additional knowledge, they could then generate such an example, Gymnasium students did not consider mathematical possibilities or representations beyond their own experience.

It should be noted that in the course of an interview, the students were occasionally asked to generate an example that could not and did not exist (e.g., three linearly independent vectors in R2) and they often correctly identified these situations as not possible. As such, both the Gymnasium and university students knew that “no example” was a possible valid solution.

The students’ disposition towards applying their knowledge in a new unfamiliar mathematical context correlates with what one might expect given the different institutional norms in which the students had recently encountered linear algebra. At the Gymnasium, students were asked to solve tasks that mimicked or varied slightly the examples previously worked by the teacher or students in front of the entire class. Emphasis was placed on mastery of solving specific mathematical problems, ones that would be later encountered on student exams in the classroom and for the end degree, the Abitur. Successful completion of these exams is essential for students to “graduate” from the Gymnasium and go on to a university.

Compare this with the university setting, where courses are often given in a large traditional lecture format – providing students with the same information that might be available in a textbook (note that textbooks are not commonly used in German universities), complimented by smaller homework practice sessions and weekly problem sheets. These problem sheets often asked students to work through novel problems using the concepts introduced in the lecture or had students unpack concepts that were introduced in the lecture but had been previously unfamiliar.

Many students used a common space made available to them to collaborate with their peers, an approach to learning that is widely expected and encouraged by the faculty and students alike. As such, small groups of students would often work together to solve these novel problems, explain solutions and understandings they had, or question other students about their solutions and understandings. This corresponds well with the active, undeterred way in which the university students reacted to the novel example generation tasks in the interview.

What still remains to be explored how do we better use this data, particularly in conjunction with a theoretical framework, to account for these differences in mathematical experiences, individual disposition and institutional environments – and more importantly, how to relate these differences. This aspect of the research provides the main impetus for discussion.

15TH Annual Conference on Research in Undergraduate Mathematics Education 545

**Questions for Discussion:**

• How can the present literature like that for the emergent perspective be best used to unpack the variety of data presented? Or, are there other frameworks (e.g., learning progressions or trajectories) that can better flush out these changes in the individual students mathematical conceptions, beliefs and dispositions, the institutional changes between the Gymnasium and university, and how these two relate or blend together?

• How do we best obtain data for and analyze data to understand student changes in disposition? Recommendations for literature or frameworks and new ideas encouraged.

• Germany’s situation with linear algebra taught at the secondary school and university differs from the US, where linear algebra is primarily an undergraduate course. How does research coming out of different cultures contribute to research here – or rather, how can Germany and the US best learn from each other’s experiences despite major differences in the mathematics covered (and how it is learned) in secondary schools and universities?

**References:**

Cobb, P. & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31 (3/4), 175-190.

Dahlberg, R., & Housman, D. (1997). Facilitating learning events through example generation.

Educational Studies in Mathematics, 33 (pp. 283-299).

De Guzmán, M., Hodgson, B.R., Robert, A., & Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. In G. Fischer and U. Rehmann (Eds.), Proceedings of the International Congress of Mathematicians, Volume III. (pp. 747 - 762). Berlin.

Dorier, J.-L. (2000). On the teaching of linear algebra. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L.

**Dorier (Ed.), On the teaching of linear algebra, pp. 191-207). Dordrecht, Netherlands:**

Kluwer Academic Publishers.

Larson, C., Zandieh, M., Rasmussen, C., & Henderson, F. (2009, February). Student interpretations of the equals sign in matrix equations: The case of Ax=2x. Paper presented at the Twelfth Conference on Research in Undergraduate Mathematics Education. Raleigh, NC.

Selinski, N. E. (2010). Student reasoning in linear algebra: Student reasoning and justification for the invertible matrix theorem (Master’s thesis). San Diego, CA: Montezuma Publishing.

Tall, D. (1991). Advanced mathematical thinking. Dordrecht, The Netherlands: Kluwer Academic Publishers.

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

151-169).

546 15TH Annual Conference on Research in Undergraduate Mathematics Education A First Look at How Mathematicians Read Mathematics for Understanding

Abstract As students progress through the college mathematics curriculum, enter graduate school and eventually become practicing mathematicians, reading mathematics textbooks and journal articles appears to comes easier and these readers appear to gain quite a bit from reading mathematics. Previous research has focused on what early college students do as they read and the difficulties they encounter that interfere with understanding what has been read. This preliminary study was designed to help us begin to understand how more advanced readers of mathematics read for understanding. Four faculty members and four graduate students participated in this study and read from a first year graduate textbook in an area of mathematics unfamiliar to each of them. The reading methods of the faculty level mathematicians were all quite similar and were markedly different from all the students the researcher has encountered so far, including the more advanced students in this study.

Introduction Many would agree that reading is critical for gaining understanding within a discipline.

Yet, most teachers of first-year college level mathematics courses are well aware that even if they ask or require their students to read from their textbooks, that few students do so with understanding. Students complain about how hard it is to read their mathematics textbooks, and it appears that even good readers in general do not read their mathematics textbooks well (Shepherd, Selden & Selden, in press). But as students continue in mathematics courses through undergraduate and graduate work, and eventually become mathematicians, somehow they “learn” to read mathematics textbooks and similar writings in journals with deep understanding.

Is there some “thing” or combination of things that mathematicians “do” as they read that helps them understand better? Maybe mathematicians are better at monitoring their own personal understanding and have confidence that they can “fix” any misunderstanding. And the questions that motivates this study: (1) Are there obvious differences in the reading strategies of mathematicians versus first year undergraduate students, and (2) If there are differences, which differences appear to be significant in learning from reading mathematical text in this situation?

Literature & Theoretical Perspective Reading involves both decoding and comprehension. On the comprehension side of the coin, research has identified several strategies that good readers employ as they engage with text (Flood & Lapp, 1990; Palincsar & Brown, 1984; Pressley & Afflerbach, 1995). These strategies depend on the individual reader, the reader’s goals and the material being read.

The theoretical perspective used herein is aligned with the view that reading is an active process of meaning-making in which knowledge of language and the world are used to construct and negotiate interpretations of texts (Flood & Lapp, 1990; Palincsar & Brown, 1984;

Rosenblatt, 1994). Yet, it appears that for many students, a major factor in their ineffective reading is a lack of sensitivity to their own confusion and errors and an inappropriate response to them (Shepherd, Selden & Selden, in press).

15TH Annual Conference on Research in Undergraduate Mathematics Education 547 Research Questions There is considerable reason to believe that most mathematicians can read mathematics textbooks and other mathematical writing effectively. This must be done, not only to teach new courses, but to support a mathematician’s mathematical research. However few mathematicians seem to have received any instruction in reading mathematics and seem to have tacitly learned effective reading. Although we would like to eventually know why mathematicians appear to be effective readers and first year college students are not, we limit our research question for this preliminary study to attempting to understand some differences that mathematicians have in approaches to reading mathematics versus both first-year and advanced mathematics students and whether any observed differences seem to contribute to mathematicians’ apparent ability to learn from reading mathematical text.

Research Methods The participants were four students and four faculty members at a large southwestern university. Each participant attended a single interview/reading session. One student was a masters level mathematics student, the other three were all pursuing PhD level work in mathematics education. The four faculty members were all experienced teachers and researchers. All participants read from Lectures on Differential Geometry (Chern, Chen & Lam,

2000) starting at the beginning of the book and were given instructions that they were to read to learn the material. Two of the faculty members had taken coursework in Differential Geometry (because it was required), but none had done research in the area. The students reading sessions were done first as a pilot. All the reading sessions were video recorded and initial questionnaires were given to assess background and teaching/research experience of each participant. Each reading session lasted about 45-60 minutes. At the end of each reading session, the faculty members were asked to create a homework set over the material they had read.

Very Preliminary Results The advanced mathematics students used techniques and strategies similar to the firstyear undergraduate students, although they were more sensitive to monitoring of their own comprehension. These students essentially read the material word for word as undergraduate students appear to do, and worked through the problems or examples on their own which undergraduate students appear to do only when encouraged to do. The mathematicians rarely read word for word. They frequently read “meanings” instead of the words or symbols that appeared on the page. They were very cautious about their own understanding and frequently adjusted their interpretation to match more closely that of the authors of the textbook. More results will be obtained as analysis of the data continues.

Implications for Further Research and Teaching This research project is a preliminary step in understanding the broad scope what it means to read mathematical text for understanding. This is an initial pilot research project to understand the “expert” side of reading mathematical text. Previous research has focused on the “novice” or first-year undergraduate course student. As researching into reading mathematics textbooks continues, there are opportunities to understand not only what experts “do” differently, but how they learn to do this and what steps or phases of learning to read occur between novice 548 15TH Annual Conference on Research in Undergraduate Mathematics Education and expert. We can also anticipate the integration of reading for understanding with learning theories.

This current research has strong implications for teaching as we design tasks and textbooks, paper and online, what can we do to help our students move toward the “expert” end of the reading mathematics for understanding scale.

Discussion Questions

1. The text chosen was one on a topic unfamiliar to the readers. There were no theorems in the portion read. Would the reading strategies be similar for more familiar topics?

2. What would one expect a mathematician to do when “stuck” on understanding some topic or example while reading? Can we test this?

3. If one of the reasons mathematicians read more effectively is because they have had positive reinforcement that they can learn from reading, how do we achieve similar positive reinforcement with lower level students?

** References**

Chern, S., Chen, W., & Lam, K. (2000). Lectures on differential geometry. World Scientific.

**Flood, J., & Lapp, D. (1990). Reading comprehension instruction for at-risk students:**

Researchbased practices that can make a difference. Journal of Reading, 33, 490-496.

Palincsar, A. S., & Brown, A. L. (1984). Reciprocal teaching of comprehension-fostering and comprehension-monitoring activities. Cognition and Instruction, 1, 117-175.

Pressley, M., & Afflerbach, P. (1995). Verbal protocols of reading: The nature of constructively responsive reading. Hillsdale, NJ: Lawrence Erlbaum Assoc.