Advanced mathematical thinking: Implications of various perspectives on advanced mathematical thinking for mathematics educatio

Advanced mathematical thinking:

Implications of various perspectives

on advanced mathematical thinking for mathematics education reform.

M. Kathleen Heid

The Pennsylvania State University

ik8@psu.edu

Guershon Harel

University of California- San Diego

harel@gte.net

Joan Ferrini-Mundy

Michigan State University

jferrini@pilot.msu.edu

Karen Graham

University of New Hampshire

kjgraham@hopper.unh.edu

The Advanced Mathematical Thinking (AMT) Working group has taken on the task of constructing definitions that capture what appeared to be three different perspectives on the nature of ãadvanced mathematicalÊ thinking.äÊ Over the past year, three groups, led by Barbara Edwards, Chris Rasmussen, and Guershon Harel, have developed definitions of advanced mathematical thinking that capture the characteristics each group deemed salient to the issue.Ê Those papers will be made available to participants in the AMT Working Group at PME XXII via the Advanced Mathematical Thinking Working Group list-serve (contact HSUPAO@MAINE.MAINE.EDU for information on how to access the list serve), they will be referenced in this paper, and they (and one other paper) will be a central focus of discussion at the meetings of the AMT Working Group at PME-NA XXII in Tucson.

Three Perspectives on ãAdvanced Mathematical Thinkingä

One can think about advanced mathematical thinking as characterizing the thinking that occurs primarily in the study of advanced mathematics at the collegiate or graduate level.Ê The Edwards team took the perspective that advanced mathematical thinking requires two simultaneous conditions: 1) advanced mathematical thinking requires precise reasoning about mathematical ideas, and 2) these mathematical ideas are not entirely accessible to the five senses (Edwards et al., 2000).Ê The authors point out several examples of ways in which it is insufficient evidence of advanced mathematical thinking that one and not both of these conditions are fulfilled.Ê For example, the authors point out that although ãlimitsä is a mathematical idea that is not entirely accessible to the five senses, ãevaluating limitsä is probably not advanced mathematical thinking since it may involve only the implementation of an automated routine and not precise reasoning about a mathematical idea.Ê The authors develop several other examples that fit their definition of advanced mathematical thinking.

A second perspective on advanced mathematical thinking offered by Rasmussen and his colleagues focuses on ãadvanced mathematical activityä since the authors, as supported by Sfard (1998), conceive of mathematical learning as participating in doing mathematics (Rasmussen et al., 2000).Ê These authors specific ally do not limit advanced mathematical thinking to undergraduate and graduate mathematics, although the primary examples they develop in their paper are drawn from courses in differential equations and college geometry.Ê They center their conversation about advanced mathematical thinking on the phenomena of horizontal and vertical mathematizing (Treffers, 1987), whose definitions they expand to allow for horizontal mathematizing in pure mathematics settings.Ê The Rasmussen team characterizes horizontal mathematizing as transforming a mathematical or real world problem setting in such a way that it lends itself to further mathematical analysis.Ê The group conceives of vertical mathematization as activities that are grounded in or build on horizontal mathematizing.Ê The authors clarify their stance on advanced mathematical thinking by illustrating horizontal and vertical mathematizing through the activities of symbolizing, algorithmatizing, and defining.Ê For example, the authors describe horizontal mathematizing as using symbols to record and communicate mathematical thinking and vertical mathematizing as using the symbolizations so developed as inputs for further mathematical reasoning.

The third perspective on advanced mathematical thinking, developed by Harel, characterizes mathematical thinking as advanced if mathematics education research can substantiate that ãits development necessarily involves epistemological obstacles.äÊ Harel holds that advanced mathematical thinking develops over long periods of intensive effort.Ê He gives examples of ãways of mathematical thinking (a) that are essential to the learning of advanced mathematical content and (b) whose development must start in an early age when elementary mathematical content s are taught.ä (Harel, 2000).

Questions Raised by these Three Perspectives on ãAdvanced Mathematical Thinkingä

Each of the three perspectives on advanced mathematical thinking generates its own list of questions that could be investigated through further refinement of the theories or through empirical research.Ê The Edwards team raises the issue of mathematics that is not entirely accessible to the five senses.Ê To what extent are accounts of instances of mathematical thinking classifiable as ãnot entirely accessible to the five sensesä?Ê To what extent does this characterization capture the mathematical thinking in which research mathematicians engage?Ê Is the viability of the definition largely a function of the type of mathematics being considered?Ê If, as the Edwards team posits, this definition of advanced mathematical thinking lies at one end of a ãmathematical thinkingä spectrum, what characterizes the role of ãaccessibility to the five sensesä in the intermediate stages between advanced mathematical thinking and elementary mathematical thinking (the thinking at the other end of the spectrum).Ê If one of the goals of secondary mathematics is to prepare students for later advanced mathematical thinking, what will prepare students to conduct mathematical thinking about mathematical ideas that are less accessible to the senses?

The Rasmussen team centers its discussion of advanced mathematical thinking on horizontal and vertical mathematization.Ê Further development of this definition will lead to additional refinement of ways to characterize and identify vertical mathematization.Ê Of interest would be an investigation into the relationship between vertical mathematization and problem solving.Ê Is one a subset of the other?Ê Is vertical mathematization a necessary component of successful problem solving.Ê If so, what are the other components?Ê The authors claim that advanced mathematical thinking is not confined to collegiate mathematics.Ê Are examples of vertical mathematization at the secondary level fundamentally different from those that typify the collegiate level?

The definition of advanced mathematical thinking offered by Harel (2000) is intimately connected to research questions.Ê To qualify as ãadvanced mathematical thinking,ä mathematics education research needs to substantiate that ãits development necessarily involves epistemological obstacles.äÊ Methodological questions arise about how one might investigate whether the development of particular ways of thinking involve epistemological obstacles.Ê To what extent are these epistemological obstacles individual?Ê To what extent are they generalizable?Ê What characterizes growth in mathematic al thinking that evidences the successful maneuvering of epistemological obstacles?

Implications for Teaching and Learning in the Context of Reform-Oriented Teaching

The past two decades in mathematics education might be characterized as decades in which mathematics education reform was conceptualized.Ê The past two decades have witnessed a major effort to reform the teaching of calculus and several major thrusts to reform mathematics teaching at the school level.Ê The most recent document to characterize the nature of that reform is NCTMâs Principles and Standards for School Mathematics (2000).Ê Principles and standards takes a strident stand on behalf of mathematical thinking.Ê While half of the documentâs ten standards concern the content of school mathematics, the other half speak to the processes of school mathematics.Ê As such, these later standards help to characterize mathematical thinking at the school level.

Discussions of the nature of advanced mathematical thinking can help to illustrate and illuminate the standards in Principles and Standards.Ê Instead of providing those illustrations in this paper, we will simply raise a few questions about the impact of the three perspectives on how we might interpret the standards.Ê The definition of advanced mathematical thinking offered by the Edwards team emphasizes the need for precise reasoning about mathematical ideas.Ê The Reasoning and Proof Standard provides some illustrations of how students at the school level might reason about mathematics.Ê The goal of reasoning about mathematical objects that are not entirely accessible to the five senses, advanced by the Edwards team, suggests that teachers of school mathematics must learn how students develop their capacity to reason in the absence of concrete examples.

The definition of advanced mathematical thinking offered by the Rasmussen team speaks to an expanded notion of horizontal mathematization that includes the communication of purely mathematical relationships.Ê In the context of this definition, implementation of the Communication Standard may require that special attention be paid to studentsâ capacities to symbolize their mathematical ideas.Ê How can students in school mathematics learn not just to symbolize their ideas but also to reason from those symbolizations?Ê What combination of emphasis on reasoning and mathematical connections is needed for students to develop their capacity for vertical mathematization.Ê

Finally, one possible, and very interesting, exercise would be to analyze specific expectations in the Principles and Standards from each of these three perspectives; namely, which categories of goals stated in Principles and Standards constitute advanced mathematical thinking or are seeds that will form a foundation for advanced mathematical thinking.Ê For example, which of the following mathematical activities can be characterized as advanced mathematical thinking or as a seed for advanced mathematical thinking according to the criteria of: precise reasoning about mathematical ideas not entirely accessible through the five senses; epistemological obstacles; or vertical mathematization?

1. Use representations to model and interpret physical, social, and mathematical phenomena;

2.Ê Make and investigate mathematical conjectures;

3.Ê Organize and consolidate their mathematical thinking through communication; and

4.Ê Understand how mathematical ideas connect and build on one another to produce a coherent whole.Ê

Conclusion

In this paper, we have summarized the perspectives on advanced mathematical thinking offered by three teams of mathematics educators.Ê We have suggested theoretical and empirical research that might be conducted to further understand each of these theories, and we have raised issues about implications for teaching and learning in the context of reform-oriented teaching.

References

Edwards, B., Dubinsky, E., Krussel, L., and McDonald, M. (October, 2000).Ê Advanced mathematical thinking.Ê Paper presented at the 22^nd annual meeting of the International Group for the Psychology of Mathematics Education ö North American Chapter. Tucson, AZ.

Harel, G. (October, 2000).Ê Advanced mathematical thinking across the grades.Ê Paper presented at the 22^nd annual meeting of the International Group for the Psychology of Mathematics Education ö North American Chapter. Tucson, AZ.

National Council of Teachers of Mathematics (2000).Ê Principles and standards for school mathematics.Ê Reston, VA: The Council.

Rasmussen, C. L., Zandieh, M., King, K. D., and Teppo, A. (October, 2000).Ê Advanced mathematical thinking: Aspects of studentsâ mathematical activity.Ê Paper presented at the 22^nd annual meeting of the International Group for the Psychology of Mathematics Education ö North American Chapter. Tucson, AZ.

Sfard, A. (1998).Ê On two metaphors for learning and the dangers of choosing just one.Ê Educational Researcher, 27(2), 4-13.

Treffers, A. (1987).Ê Three dimensions: A model of goal and theory description in mathematics education: The Wiskobas project.Ê Dordrecht: Kluwer Academic Publishers.