Principal Investigator: Jennifer Christian Smith, Assistant Professor, Mathematics Education, The University of Texas at Austin
The development of an understanding of mathematical proof is one of the benchmarks of a major in mathematics. It is clearly vital for a student who intends to pursue graduate study in mathematics to be able to construct, understand, and validate formal mathematical arguments, but it is also important for those students who will be teachers of mathematics to develop a solid understanding of mathematical proof (CBMS, 2001; Selden & Selden, 2003). Many universities require mathematics majors to enroll in so-called transition courses, in which the students are introduced to mathematical formalism. It is in such courses that most students develop their understandings of formal mathematical proof; hence these courses provide an opportunity for researchers to study students' conceptions of proof in mathematics. At many universities, more than half of the mathematics majors are pre-service secondary teachers (CBMS, 2001), and so it is in these transition courses that future teachers begin to develop their understandings of proof. According to the National Council of Teachers of Mathematics (2000), opportunities to develop mathematical reasoning and construct informal proofs should be included in students' mathematical experiences throughout schooling. In order for teachers to provide children with environments in which this can occur, teachers themselves must have a solid understanding of proof (CBMS, 2001; Ma, 1999).
There has been a great deal of interest in the mathematics education community in research focusing on the difficulties undergraduate students have with proof. Researchers have found that students struggle to make sense of logical quantifiers (Dubinsky & Yiparaki, 2000; Tall, 1999) and of the structure of various kinds of proofs (Harel & Sowder, 1998; Healy & Hoyles, 2000), and find it challenging to determine for themselves the correctness of a mathematical argument (Selden & Selden, 2003; Segal, 2000). Weber (2002) argues having a solid understanding of logic and methods of proof is not enough; students must have a deep conceptual understanding of the subject matter in order to be able to construct proofs.
Most of the research described above has been based on in-depth interviews with individuals outside of the classroom context; to date, no research has explored the development of proof in an inquiry-based classroom setting at the undergraduate level. An example of an inquiry-based approach to teaching undergraduate mathematics is the "Moore Method," developed by R. L. Moore at the University of Texas during the mid-20th century. Instructors who employ a version of Moore's rather strict philosophy today refer to it as the "Modified Moore Method" (MMM). An instructor using the MMM typically presents students with a series of theorems to prove individually or collaboratively. The students then present their proofs in class and these are discussed by the whole group (Mahavier, 1998). The role of the instructor in a MMM course is that of a guide rather than a dispenser of information and knowledge (Renz, 1999).
The proponents of this style of teaching are enthusiastic and passionate about its impact on students. In particular, its practitioners believe that students who have taken MMM courses have a much greater ability to solve problems and construct proofs than those who have not (Mahavier, 1998; Renz, 1999). However, very little research has documented the effectiveness of the MMM, and almost no learning theory has influenced its development.
As a brand new mathematics education researcher at the University of Texas at Austin in the fall of 2002, I was invited by an experienced professor of mathematics to help design an evaluation of an MMM course. I interviewed students enrolled in several sections of number theory that were being taught either through lecture or using the MMM, and the results of that pilot study were quite intriguing. Though the sample was small, there were interesting differences between the students' approaches to proof and beliefs about the role of proof in mathematics. For example, students in the MMM sections tended to approach the construction of a proof much more holistically than did the students in the lecture sections, who tended to "throw" proof techniques at a problem without making an attempt to understand the statement first. Classroom observations of the MMM classes revealed a fascinating community of learners starkly different from a typical lecture-based undergraduate mathematics course in that the students were discussing mathematics and presenting their ideas at the board while the professor acted as facilitator, adding in comments occasionally and asking questions. This particular environment provides a unique opportunity to study the development of students' understanding of proof in and out of the classroom.
There is a growing body of research indicating that participation in a community of learners is a vital part of students' success in mathematics. Learning is regarded as a product of the reflexive relationship between communally developed and shared classroom practices and individual constructive activity (Cobb & Yackel, 1996). However, such communities are rarely developed in undergraduate mathematics courses, in which lecture is frequently the sole teaching technique. Research has demonstrated that lecture-based environments are not conducive to learning for most students, and that students at all levels benefit from participation in discourse about mathematics (Bransford, Brown, & Cocking, 2000).
I attempted to study the development of mathematical practices in my dissertation work (Smith, 2002), but found the limited discourse of a typical lecture-based undergraduate mathematics course did not lend itself easily to such analysis. One of the intriguing characteristics of a MMM class is that there is a great deal of discourse; students present their ideas, ask questions of each other, and discuss mathematics in a manner rarely seen in undergraduate mathematics courses. As a result, I believe that an MMM course provides a unique opportunity to study the development of students' understanding of proof both collectively and individually.
The research questions that we plan to address with this study are as follows:
In the fall (2003) semester, four sections of number theory will be taught at the University of Texas at Austin; all will be taught by experienced instructors, two of whom plan to use the MMM and two of whom plan to teach in a lecture-oriented style. Students enroll in the various sections fairly randomly, though they will be aware that the sections taught by two of the instructors will employ the MMM. This may have an impact on their choice of section. Four primary types of data will be collected:
The role of the principal investigator in this project will be to coordinate and participate in the collection of data as the leader of the research team. This team will consist of 2-3 mathematics education graduate students, one of whom will be supported by a mathematics department teaching assistantship for the course in question, and the other of whom would be supported by this grant. An additional graduate student may participate, though financial support for that individual is not currently available. Videotaping of one MMM course section will be done by the project's research assistant. All members of the research team will interview students and instructors and observe class sessions.
Analysis of the data will be ongoing and will inform the collection process, though the majority of the data analysis will take place during the spring 2004 semester. The team will employ qualitative analytical techniques and use a grounded theory approach (Strauss & Corbin, 1998).
It is somewhat rare to see inquiry-based teaching strategies such as the MMM employed in undergraduate mathematics courses. There are several factors that may be contributing to this rarity. There currently is little concrete evidence that the MMM is as effective as its proponents insist. In fact, there has been almost no research conducted to determine the effectiveness of this particular type of teaching method at the undergraduate level. The traditional separation of mathematicians and mathematics education researchers may have contributed to this gap in the literature. Second, instructors typically start using the MMM as a result of a personal experience with a MMM course that they either took as a student or observed firsthand as a more experienced colleague taught. There seems to be no systematic attempt to inform the mathematical community at large of this teaching method. It is clear that research such as this proposed project could help to inform the mathematics and mathematics education communities about this intriguing instructional approach.
This research will also contribute to the growing body of work on the nature of undergraduates' understanding of proof. It is rare to have an opportunity to observe undergraduate mathematics students in a highly interactive environment for an entire semester. The video-tape data gathered in this project will be a rich source of information not only for this study, but also for further investigation of the development of mathematical practices at the college level.
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Smith, J. C. (2002). An Investigation of Undergraduates' Understanding of Congruence of Integers. Doctoral dissertation, University of Arizona. Dissertation Abstracts International.
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