DEFINING AS A MATHEMATICAL ACTIVITY: A REALISTIC

MATHEMATICS ANALYSIS

Chris L. Rasmussen	Michelle Zandieh
Purdue University Calumet	Arizona State University
raz@calumet.purdue.edu	zandieh@math.asu.edu

Abstract: The purpose of this paper is to report on a college level classroom teaching experiment in geometry and to discuss one aspect of students' activities with mathematical definitions. The instructional design theory of Realistic Mathematics Education is used as a framework for understanding students' mathematical activity involving the modification of new definitions out of familiar ones. Our analysis extends the work of Gravemeijer (1999) and suggests that the creation of new mathematical objects and their definitions involves four interrelated types of activities: situational, referential, general, and formal.

Introduction and Theoretical Perspective

The purpose of this paper is to further the notion of definition as a mathematical activity (Mariotti & Fischbein, 1997; Freudenthal, 1973) and to elaborate a theoretical means to make sense of students' defining activities where new mathematical objects and their definitions are created out of familiar ones. From a mathematical point of view, definitions should be useful in that they serve to single out a concept with certainty, they should be minimal and elegant (Vinner, 1991), they should capture or synthesize the mathematical essence of the concept (Borasi, 1992), and they are links in deductive chains of organization (Freudenthal, 1973).

Freudenthal (1983) described mathematical concepts, structures, and ideas as our inventions, created to organize the phenomena of the physical, social, and mental world. In the research reported here, we view mathematical definitions as one aspect of the mathematical structures and ideas that learners' create. Drawing on the work of Freudenthal, we view mathematics itself as essentially a "human activity" in which one engages for the purposes of generality, certainty, brevity, and exactness, where defining is one aspect of this activity (Gravemeijer, 1994). Freudenthal (1973) distinguishes between two different types of defining activities in mathematics, descriptive and constructive. Descriptive defining "outlines a known object by singling out a few characteristic properties," whereas in constructive defining one "models new objects out of familiar ones" (p. 457). Focusing on descriptive defining, DeVilliers (1998) argued that students should be actively engaged in the defining of mathematical concepts and elaborated the notion of descriptive defining, framing his analysis within the cognitive theory of Van Hiele levels. In this report, we focus on what Freudenthal refers to as constructive defining, framing our analysis within the theory of Realistic Mathematics Education (RME).

Previous work within the theory of RME has mainly centered on the learning and teaching of K-12 mathematics (for some exceptions, see Gravemeijer & Doorman, in press; Rasmussen & King, 2000) and has not elaborated the role of definitions in the learning and teaching of mathematics. The theoretical significance of this report is in its elaboration of definitional activities in the theory of RME. In this paper we draw parallels between the types of activities elaborated by Gravemeijer (1994, 1999) in his analysis of the role of emergent models and the role they play in fostering the constitution of formal mathematics. In Gravemeijer's analysis, there is a "global transition in which 'the model' initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning" (p. 155). In a similar way, our analysis suggests that definitions first come to the fore as a definition of students' previous activity and later these definitions serve as tools for further mathematical reasoning.

Methodology

We conducted a classroom teaching experiment (Cobb, 2000) during a 5-week summer session with 25 students at a large Southwestern university in the United States. The course used the textbook, Experiencing Geometry on Plane and Sphere (Henderson, 1996) and instruction generally followed an inquiry-oriented approach. The course was taught by one of the research team members. A second researcher attended every class session. Data consisted of videotape recordings of each class session, copies of students' written work, bi-weekly student journal entries to specific questions developed by the research team, researcher and instructor field notes, audio-recorded debriefing sessions between the instructor/researcher and the other researcher, and videotape recordings of individual interviews conducted with 22 of the 25 students at the beginning and end of the course.

Typical class sessions consisted of a brief introduction of the problem by the instructor followed by small group work on the problem and whole class discussion of students' reasoning and interpretations. Students' activities that form the crux of the data for this paper occurred during the second and third weeks of the course. During this time period, students sorted through for themselves the definition of a triangle on the plane, explored the world of triangles on plastic spheres, and investigated several questions about spherical triangles.

Results and Discussion

We posit that students' defining activities in which new mathematical concepts and their definitions are created out of familiar ones can be understood in terms of four interrelated types of activity: situational, referential, general, and formal. Consistent with the manner in which Gravemeijer (1999) described these types of activity in the modeling process, we do not view these levels as a strict hierarchy or as a strict developmental progression, but rather as a general trend in students' ways of reasoning, acting, symbolizing, etc. Students' activities at one level often fold back (Pirie & Kieren, 1994) to another level. Next we clarify what we mean by these four levels of activities.

Situational Activity

As described by Gravemeijer (1994), situational activity involves interpretations and solutions that depend on understanding how to act in the problem setting. The hallmark of situational activity is the fact that students deal with experientially real settings. This may include imagined experiences like packaging candies in a factory or people getting on and off a bus. In such settings, students' interpretations and solutions are very much grounded in the real world context, using their understandings of the experientially real setting serves as a means to solve the problem. From the perspective of RME, however, such experientially real situations need not be limited to real world settings. Depending on the level of the student, experientially real settings can also refer to more abstract mathematical contexts. This latter interpretation is the one taken in this paper. In our example, the problem setting is the plane where straight lines have zero extrinsic curvature and the task posed was the formulation of a definition for a planar triangle. In this case, students have a way to proceed because they have had many prior experiences (both in school and out of school) with straight lines and planar triangles and thus have built up a rich concept image (Vinner, 1991; Edwards, 1999) about triangles.

Referential Activity

Referential activity involves interpretations and solutions that refer to activity at the situational level. For example, as students explored the world of triangles on the sphere there was considerable amazement at how "different" these triangles are from planar triangles. Considerable discussion occurred as to whether they really were triangles (since, for example, the interior angles didn't sum to 180 or a triangle had two obtuse angles). Students' decisions as to whether such objects were actually triangles ultimately referred back to their definition of a triangle (where straight line segments were taken to be great circle segments), rather than to their imagery of planar triangles. It is in this respect that students' activities are referential, both to their imagery of planar triangles and to their definition of triangle. The following journal excerpt, which begins by referring to a figure that a group of students were sharing with the whole class, illustrates what we are calling students' referential activity.

From the figure [they had] drawn, it didn't seem like it was a figure at all, but in close observation it was a triangle! Yes, a triangle. It was a triangle based on the definition we chose in class. The definition of a triangle matched up with the figure. Though this was true, the figure did not look like a triangle. I did not see the triangle until someone brought up that it was a triangle by definition. Better yet, there were two triangles! Yes, the inside AND the outside were both triangles."

General Activity

Students activities with triangles on the sphere begin to take on a life of their own, independent of the situation specific imagery. That is, independent of planar specific imagery. Evidencing this independence, students began to formulate conjectures (without prompting from the instructor) about the range for the sum of the interior angles. Illustrating this point is the following quote from a student's journal. "Another surprising observation was when Group 1 gathered information about triangles on a sphere and concluded that the maximum number of degrees that a triangle can have with respect to its angles is 1080 degrees. These observations [have] changed my view on spheres. All along I was thinking and limited to a 2-dimensional perspective." Furthermore, this excerpt illustrates that students are working within a new mathematical reality, characterized by new mathematical objects and relationships on the sphere that previously did not exist (for students, that is). From our perspective, the creation of such a space of possibilities is reflexively related to students defining activities.

Formal Activity

At the formal level, triangles on the sphere become an object in their own right that can be modified to suite particular purposes and needs. For example, when investigating whether two spherical triangles for which two sides and the included angle are congruent necessarily mean the triangles are congruent, students created a new class of spherical triangles for which this theorem is true. They defined this new mathematical object as a "small triangle" and these definitions varied from group to group. Furthermore, students considered the equivalence of these various definitions of small triangles and used small triangles as links in chains of deductive reasoning.

In this report we furthered the notion of defining as a mathematical activity from an RME perspective. Our analysis suggests that the RME constructs of four different levels of activity, first elaborated in the context of modeling, are a viable framework for making sense of students' activities where new mathematical objects and their definitions are created out of familiar ones. One direction for future research would be to investigate the extent to which the framework elaborated here might guide the design of instructional sequences where the defining is a prominent aspect of students' activities.

References

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