«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
The frequency of C-coded responses was striking, considering the literature on student attitudes towards MI. A fairly common misconception uncovered by Ernest (1984), Harel (2002) and others, holds that the basis step is needed only to satisfy the instructor. In a study by Baker (1996), large numbers of secondary and university students failed to recognize a missing base case in a proof analysis task. A student who views the basis step in MI as an unnecessary formality should be less likely to find fault with one that is vacuously true. But the students offering C-coded critiques held the opposite view. What explains these students’ level of unease with the basis step in this argument?
Some C responses suggested an empirical proof scheme. Smith (2006) suggests that the use of IBL with this group of students may have been significant as well. In IBL environments, students grew more likely to expect mathematical proofs to explain why a given statement is true. A vacuously true statement, while perfectly valid, may raise more suspicion among IBL students than ones who place less emphasis on the explanatory role of a proof. IBL students were also observed frequently using examples to help them make sense of mathematical statements whereas their nonIBL peers appeared to view examples as unhelpful because they were not proofs. Smith’s findings suggest that the use of examples by IBL students can a highly effective way of working and may lend itself to a deductive (transformational) proof scheme, but it may raise a question for IBL practitioners. Does the role of examples as a sense-making tool have implications for the way IBL students understand the basis step of MI?If so, how should IBL users support students in the transition towards axiomatic reasoning?
15TH Annual Conference on Research in Undergraduate Mathematics Education 465 G and H Responses: Contextual and External Symbolic Proof Schemes 3.3
A response was coded G when the writer showed suspicion related to the non-mathematical context – that a mathematical argument, even if correct (!), simply cannot apply to people:
Humans are not numbers.
Nearing the opposite extreme were some H-coded responses which focused on the performance of algebraic procedures and suggested an external symbolic proof scheme:
When the substitution step is done … Also there’s a problem when they’re evaluating k + 1 … If the 1st human through the k th human was set to k [ …] we could substitute in for 1st human through the k th human with k − 1 but even then, k − 1 does not equal k + 1 and there is no way to prove that.
Although zero H-coded responses would surely be preferable to three, the literature on MI describes a significant reliance on procedural thinking. Perhaps the low frequency of H-coded responses should be encouraging.
1. How might these student responses differ from the responses of students in a non-IBL environment?
2. Do IBL students differ from non-IBL students in their ways of understanding (or misunderstanding) the basis step in MI?
3. how should IBL users support students in the transition towards axiomatic reasoning?
References Avital, S. and Libeskind, S. (1978). Mathematical induction in the classroom: Didactical and mathematical issues.
Educational Studies in Mathematics, 9(4):429–438.
Baker, J. (1996). Students’ difficulties with proof by mathematical induction. In Annual Meeting of the American Educational Research Association, pages 8–12.
Brumfiel, C. (1974). A Note on Mathematical Induction. Mathematics Teacher, 67(7):616–618.
Ernest, P. (1984). Mathematical induction: A pedagogical discussion. Educational Studies in Mathematics, 15(2):173–189.
Harel, G. (2002). The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction. In Learning and teaching number theory: research in cognition and instruction, page 185. Ablex Publishing Corporation.
Harel, G. and Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in collegiate mathematics education III, 7:234–282.
Marshall, D. C., Odell, E., and Starbird, M. (2007). Number theory through inquiry. MAA Textbooks. Mathematical Association of America, Washington, DC.
Pólya, G. (1954). Mathematics and plausible reasoning. Vol. I, Induction and analogy in mathematics, volume 1.
Princeton University Press.
Smith, J. C. (2006). A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses. The Journal of Mathematical Behavior, 25(1):73 – 90.
This work aims to establish a new theoretical construct, mathematical activity for teaching, and to investigate relationships between students’ mathematical activity, mathematical activity for teaching, and teacher’s instructional moves. Mathematical activity for teaching refers to the mathematical activity teachers engage in as they work to support students’ mathematical activity. This construct represents one component of the Mathematical Activity for Teaching Cycle, a conceptual framework that guided my analysis of classroom interactions between mathematicians and their abstract algebra students. Through this analysis I was able to exemplify each component of the Mathematical Activity for Teaching framework and begin to identify relationships between teachers’ mathematical activity and those of their students’.
Key Words: Teaching, Mathematical activity, Mathematical knowledge for teaching
As a way to address the challenges mathematicians face while implementing inquiry-based curriculum, researchers have looked to link mathematical knowledge to certain teaching demands (Johnson & Larsen, 2011; Speer &Wagner, 2009). While these studies begin to identify the process by which teachers’ knowledge influences their teaching, there remain questions about how teachers’ mathematical knowledge directly relates to the mathematical activity of their students. Presumably, it is not enough for teachers to simply have the mathematical knowledge that underlies their curriculum. Teachers also need to be able to use their mathematical knowledge in a way that supports their students’ mathematical activity. With this distinction in mind, one question that could be asked is: What types of mathematical activity for teaching do teachers engage in to support their students’ mathematical activity? This paper addresses this question in the context of an inquiry-oriented, abstract algebra course.
Conceptual Framework Guiding my work is a framework, the Mathematical Activity for Teaching Cycle, developed to investigate the relationships between students’ mathematical activity, mathematical activity for teaching, and teacher’s instructional moves (see figure 1). Because the abstract algebra curriculum for the course was heavily influenced by the Realistic Mathematics Education heuristic of guided reinvention (Freudenthal, 1991), as students work to reinvent group theory concepts, I expect to see instances in which students make conjectures, pose questions, and generalize ideas. Additional activities of interest include symbolizing, algorithmatizing, and defining. Such activities serve to exemplify the students’ mathematical activities component of the Mathematical Activity for Teaching Cycle.
15TH Annual Conference on Research in Undergraduate Mathematics Education 467 Figure 1. The Mathematical Activity for Teaching Cycle As students engage in such mathematical activity, one would expect that teachers would also need to engage in mathematical activity. For instance, faced with a novel proof, the teacher may need to evaluate a student’s proof to determine the validity of the argument and possible (dis)advantages of this new approach, both in terms of the current task and in terms of their students’ mathematical development. Such evaluation may include proof analysis (Lakatos, 1976; Larsen & Zandieh, 2007) and identifying connections between the student’s proof technique and other mathematical justifications the students would be likely to encounter during the course of the curriculum.
Additionally, within the last few years there has been research done to investigate mathematicians’ abilities to engage in specific skills related to the implementation of inquiryoriented curriculum. For instance, Speer and Wagner (2009) investigated a mathematician’s ability to provide analytic scaffolding during whole class discussions, where “analytic scaffolding is used to support progress toward the mathematical goals for the discussion” (p.
493); and Johnson and Larsen (2011) investigated a mathematician’s ability to interpretively and/or generatively listening to their students’ contributions, where interpretive listening involves a teacher’s intent of making sense of student contributions and generative listening reflects a readiness for using student contributions to generate new mathematical understanding or instructional activities (Davis, 1997; Yackel, Stephan, Rasmussen, & Underwood, 2003).
While such skills may not necessarily be mathematical in nature, I hypothesize that they may rely on a teacher’s ability to engage in certain mathematical activities. For instance, in order to engage in interpretive listening, a mathematician may need to interpret a student’s imprecise language, generalize the student’s statement into a testable conjecture, and then identify a counterexample (see Johnson & Larsen, 2011) Indeed, while both Speer and Wagner (2009) and Johnson & Larsen (2011) connected the mathematicians’ ability to successfully engage in these activities to the mathematicians’ mathematical knowledge for teaching, Ball et al. (2008) warn against a purely static view of mathematical knowledge for teaching. Instead stating that, their interest was not limited to the knowledge that teachers hold, but also in “how teachers reason about and deploy mathematical ideas in their work” including “skills, habits, sensibilities, and judgments as well as knowledge” (p. 403). The mathematical activity for teaching component of my framework consists of the mathematical activity that supports such skills and reasoning.
468 15TH Annual Conference on Research in Undergraduate Mathematics Education The last component of the Mathematical Activity for Teaching Cycle is instructional moves.
This component represents the mechanism through which the teacher’s mathematical activities influence the students’ mathematical activities. Such instructional moves could include providing counterexamples, restating student concerns for class discussion, exhibiting a proof for the class, or types of pedagogical content tools (Rasmussen & Marrongelle, 2006).
Research Method To understand ways that instructors engage with the inquiry-oriented, abstract algebra curriculum we have collected data from the classrooms of three mathematicians over the course of two years. During these two years, there have been four implementations of the curriculum.
For each implementation, every class session was videotaped and members of the larger research team took field notes. Additionally, mathematicians participated in interviews related to their experiences both in class and in using the curriculum materials.
The Mathematical Activity for Teaching Cycle guided the data analysis process. Initially, instances in which students would likely engaged in mathematical activity were hypothesized based on an analysis of the curriculum materials. For instance, during the group unit students are asked to prove some basic theorems related to the order of group elements. Given such a task, I would expect students’ mathematical activity to include proving. Such analysis of the instructor materials served to inform my first round of classroom videotape data analysis, in which I identified instances in which students’ mathematical activity of interest appeared. These episodes were reanalyzed to see if and how teachers were engaging in mathematical activity, using cases of listening and analytic scaffolding as a signal that teachers may be engaged in such activity. In the third round of data analysis instructional moves that bridged the teacher’s mathematical activity and that of the students’ were identified. Finally I looked for changes in students’ mathematical activity following the teacher’s instructional moves.
Results Here I will provide an example to illustrate my analytic process, the components of the Mathematical Activity Cycle for Teaching, and the relationships I am trying to investigate.
During the deductive phase of the group unit the students were asked to prove that, if the order of b is 4 and ab = b3a, then ab2 = b2a. After a chance to work alone, a student presented a proof by contradiction to the class. In this proof the student assumed that ab2≠ b2a and was able to deduce that ab ≠ b3a. However, this student’s steps relied on the fact that if you start with two things that are not equal (b3ab≠ b2a) and multiply both expressions on the left by the same element, then your resulting expressions are still not equal (bb3ab≠ bb2a). The creation of this proof represents an example of student mathematical activity.
Following this proof, some students questioned if was valid to assume bb3ab≠ bb2a based on the fact that b3ab≠ b2a. The teacher, Dr. Bond, generatively listened to these students’ concerns and used them as a way to guide the trajectory of the course, asking the students, “if we take two things that we know aren’t equal and we multiply, do we know that they are still not equal”?
Initially Dr. Bond stated that this question did not need to be resolved, instead she just wanted to make sure that the students were aware that “this is an important question to ask”. Indeed, during the debriefing meeting following this class, Dr. Bond admitted that, “ I hadn’t decided if it was valid or not … I really hadn’t thought it through yet”.