«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
Keywords: intellectual need, Realistic Mathematics Education, mathematizing, bridge course Introduction Harel’s (1998) Necessity Principle states, “Students are most likely to learn when they see a need for what we intend to teach them, where by ‘need’, is meant intellectual need, as opposed to social or economic need” (p. 501). The aforementioned Necessity Principle puts forth a conjecture about how students learn (Speer, Smith & Horvath, 2010) and has been used extensively by Harel as a component of a larger conceptual framework called Duality, Necessity, and RepeatedReasoning (DNR) (Harel, 2001). More recently, Harel, (2011) has refined and expanded these intellectual needs into five inextricably-linked categories: the need for certainty (to establish that a statement is true), the need for causality (to determine why a statement is true), the need for computation (to quantify and calculate), the need for communication (to persuade others of truth and to agree on conventions), and the need for structure (to re-organize knowledge into a logical system). Harel has illustrated each of these categories of intellectual need using both examples from his own research, as well as documented accounts from the history of mathematics, which suggests that intellectual need permeates throughout the discipline of mathematics.
Theoretical perspective Several theoretical perspectives—both in curriculum design and in the preliminary data analysis, influence this research study. First, the group theory curriculum (Larsen, 2004) that was used in this teaching experiment was inspired by the design theory of Realistic Mathematics Education (RME) (Freudenthal, 1991). One of the tenets of RME is the importance of guided reinvention, in which a student is encouraged to “invent something that is new to him, but wellknown to the guide” (Freudenthal, 1991, p. 48). In this particular setting, guided reinvention is the process by which students’ model of an abstract group first emerges as a model of the symmetries of 15TH Annual Conference on Research in Undergraduate Mathematics Education 603 an equilateral triangle and evolves into a model for the abstract dihedral group of order six (Larsen, 2004). A core component of Larsen’s curriculum is the mathematical activity of axiomatizing, which requires students not only to construct a system of rules for operating on their symmetries, but also to refine these rules to a minimal list of axioms that could be used independently of the objects from which they were abstracted (Larsen, 2009). Therefore, the researcher in this study considered students’ axiomatizing as a mathematical activity analogous to that of symbolizing (Gravemeijer, Cobb, Bowers, & Whitenack, 2000), and sought to examine some of the intellectual needs that the students articulated when developing their axiomatic system. Consequently, the DNR conceptual framework was used as an indispensable tool for analyzing the data. Specifically, the categories of intellectual need were used to code instances in which students referred to knowledge that they would need to construct to resolve a problematic situation (Harel, 2011).
Background and research methodology Over the course of nine weeks, the teacher-researcher and his students progressed through a subset of an RME-inspired curriculum (Larsen, 2004) for re-inventing the concept of group.
Extensive written and video data were collected from this teaching experiment, which occurred in an elective mathematics bridge course at a medium-sized, suburban community college. The teacher-researcher was a full-time community college instructor with more than ten years of experience teaching courses ranging from arithmetic through integral calculus. The participants were nine community college students (five female and four male) whose ages ranged from 17 to 35 years. Four of the students were math majors, two were engineering majors, one was a music major, and two students had not yet declared a major. The students’ mathematical experience varied greatly: four had taken courses through differential equations, one had completed calculus III, three had completed calculus I, and one student had only completed college algebra. None of the students had taken a junior-level collegiate math course, but two students were familiar with some group theory concepts from taking bridge course the previous year. A retrospective analysis (Cobb & Whitenack, 1996; Stylianides, 2005) was conducted on the classroom video data collected from this term-long teaching experiment. In the initial pass of the data, the researcher identified instances in which students may have had opportunities to address intellectual needs—specifically, where they were confronted with a problematic situation that was unsolvable by their current knowledge (Harel, 2011). The majority of these problems came directly from the instructional prompts that were part of Larsen’s curriculum, but other problematic situations originated either from the teacher or from students in the class. In a second pass of selected classroom episodes, students’ acts of axiomatizing were analyzed and correlated (when possible) with Harel’s existing categories of intellectual need.
Preliminary results of the research At this point in the analysis, a few themes have emerged. First, the data lends credence to one of Harel’s claims about the need for computation—that it is indeed a robust intellectual need.
Eight of the nine students in the teaching experiment seemed to be motivated to invent rules that aided in computation and for one student in particular, the associative axiom seemed completely unnecessary because he saw no computational need for it. Secondly, the data suggests that axiomatizing is a mathematical activity that could provide students with opportunities to address a variety of intellectual needs. For example, globally there existed a constant tension between the need to create rules that made students’ computations more efficient, while at the same time, keeping the list of rules as small as possible to avoid redundancy. This tension provided an opportunity to discuss the differences among mathematical terms such as definitions, axioms, theorems, and lemmas and to point out the advantages and disadvantages of lengthening or 604 15TH Annual Conference on Research in Undergraduate Mathematics Education shortening the list of rules. As the students’ model progressed from a model of toward a model for, decisions about how to state certain axioms appeared to be influenced by their needs to communicate, compute, and structure. In fact, throughout the term, the students formally axiomatized five different versions of their list of rules, which provides strong evidence for the existence of the need for structure. Finally, there is evidence in the data to support re-examining Harel’s initial category of the need for elegance, which he described as “what we associate with mathematical beauty, efficiency, and abstraction” (1998, p. 502). In making decisions about notational conventions and which rules to keep or discard, students’ choices may be motivated not only by the existing categories of intellectual need, but also by an intellectual need that is epistemic to the discipline of mathematics—the need for elegance. One of the students in the teaching experiment seemed to be periodically motivated by this need and used a powerful metaphor to
describe the need for elegance of an axiomatic system, as this excerpt illustrates:
Chris: It’s like you know, you got a hammer sitting at home…you get a blue hammer. You go out and get a blue hammer, so you hammer in nails with a blue hammer instead of a red hammer. Cuz we already got the red hammer and the red hammer works just as well to solve the problems as the blue hammer…and we already have it.
Later, Chris acknowledged that the creation of a new axiom would make certain computations “faster,” but he stated that such an axiom did not make the system “stronger.” Sinclair (2004) adds to the importance of this need by stating, “In terms of the aesthetic dimension of mathematical judgments, the emphasis placed on the aesthetic qualities of a result implies a belief that mathematics is not just about a search for truth, but also a search for beauty and elegance” (p. 269).
Questions to further future research In traditional mathematics curriculum, students are rarely given opportunities to develop their own notations, conventions, or axioms, so examining the role that students’ intellectual needs plays in designing and enacting RME-inspired curriculum may be very useful for the field. In particular, Harel (2011) claims that “DNR’s Necessity Principle is an analogue of the RME dictum that students must engage in mathematical activities that are real to them, for which they see a purpose” (p. 23). If that is the case, then how do other acts of mathematizing correlate with DNR’s categories of intellectual need?
Another area that might be worthy of future investigation concerns the function and role of bridge courses. If one of the primary functions of bridge courses is “to ease the transition from lower division, more computational [emphasis added], mathematics courses to upper division, more abstract, mathematics courses such as modern algebra and advanced calculus” (Selden & Selden, 1995, p. 135), then it seems reasonable that students in bridge courses should engage in mathematical activities that give them opportunities to address intellectual needs other than those necessitated by computation. Arguably, proof and the activities associated with it attend to this larger goal, so it is not surprising that much of the research on bridge courses has centered upon proof. However, in addition to proof, what other elements could or should be included in bridge courses to support student learning of more abstract mathematics?
Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30(3), 213-228.
Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Norwell, MA:
Kluwer Academic Publishers.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling, and instructional design. In P. Cobb, E. Yackel, & K. McCain (Eds.), Symbolizing and communication in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Lawrence Erlbaum Associates, Inc., 225-273.
Harel, G. (1998). Two dual assertions: The First on learning and the second on teaching (or vice versa). American Mathematical Monthly. 105(6), 497-507.
Harel, G. (2001). The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction. In S. Campbell & R. Zaskis (Eds.). Learning and Teaching Number Theory. New Jersey, Ablex Publishing Corporation, 185Harel, G. (2011). Intellectual need and epistemological justification: Historical and pedagogical considerations. In K. Leatham (Ed.), Vital Directions for Mathematics Education Research. Manuscript in preparation.
Larsen, S. (2004). Supporting the guided reinvention of the concepts of group and isomorphism: A developmental research project. Unpublished Dissertation, Arizona State University.
Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. The Journal of Mathematical Behavior, 28(2-3), 119-137.
Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.
Sinclair, N. (2004). The role of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6(3), 261-284.
Speer, N., Smith, J., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. Journal of Mathematical Behavior, 29, 99-114.
Stylianides, A. J. (2005). Proof and proving in school mathematics instruction: The elementary grades part of the equation. University of Michigan.
This theoretical report aligns itself with Arcavi’s (1994) work and the tradition of ontosemiotic research in mathematics education (Font, Godino, & D’Amore, 2007) and is situated in
the context of statistics education. This report will:
• articulate a notion of symbol sense in statistics
• explain the importance to student understanding of the development of symbol sense.
The goal of this work is to guide both research and curriculum design efforts for introductory undergraduate statistics courses. The paper begins by describing statistical analogs of Arcavi’s algebraic symbol sense, then furthers this by noting the importance of reading symbols generally, reading symbols through the context of the question, and the reading of symbols related to the visualization or selection of the display. Finally, the paper briefly explores how the understanding of symbols becomes more difficult and important in the use of the Central Limit Theorem and estimation of parameters.
Keywords: Statistics, symbols, symbol sense, semiotics
0. Introduction and Motivation While there have been investigations of students’ understanding of measures of center (Mayen, Diaz, Batanero, 2009; Watier, Lamontagne, & Chartier, 2011), variation (Peters, 2011;