Prior work has shown that metaphors are useful for bridging students’ prior knowledge about situations in ordinary life and mathematical ideas (Presmeg, 1998), and also to help students learn how to communicate mathematical ideas (Lakoff & Núñez, 2000; Pimm, 1981, 1988; Presmeg, 1997, 1998). The paper asks: How does a teacher use a metaphor to make students remember specific mathematical content?  This question is examined by analyzing videos of instruction of a teacher who taught a problem-based lesson in two geometry classes with the purpose of installing (Herbst, Nachlieli, & Chazan, 2011) the secant-tangent product theorem.  The analysis uses Duval’s work on apprehension of diagrams and Presmeg’s (1992) regarding prototypical images to investigate how a metaphor provided specific heuristics for students to remember a set of theorems when solving a problem. 

The notion of “apprehensions of diagrams” involves specific ways of looking at a geometric diagram and making sense of it while working on a solution to a problem.   According to Duval (1995), there are four types of apprehensions of diagrams: perceptual, sequential, discursive, and operative.  An interaction with a diagram always involves the perceptual apprehension. A sequential apprehension entails following a sequence of steps for making a diagram. The discursive apprehension involves the use of propositions or concepts that justify different operations with the diagram.  Finally, the operative apprehension involves performing specific operations to the diagram such as adding auxiliary lines or reorienting the diagram.  These four types of apprehensions can be used separately or simultaneously. The combination of Duval’s (1995) framework for analyzing interactions with diagrams and Presmeg’s (1992) work regarding prototypical images as metaphors is useful for understanding how a teacher triggers students’ memories of theorems that have already been taught in class, because the teacher’s endorsement of a prototypical image may necessitate students’ application of a specific type of apprehension.

The first step of the analysis was to create a timeline of selected episodes in each class.  The creation of the timeline included parsing the episodes into segments according to changes to the mathematical task (Doyle, 1988; Herbst, 2006).  The segments were coded according to the apprehensions of diagrams applied and the reliance on a prototypical orientation of a diagram. The next step of the analysis involved examining whether the use of prototypical images was connected with specific apprehensions of diagrams.  The last step in the analysis was to contrast the apprehensions of the diagram used in the episodes to examine whether the use of specific apprehension during the discussion of the solution of the problem carried on to the installation of the theorem.

In the second period class, there were three episodes: the discussion of the circle problem, the re-installation of theorems about the geometric mean, and the installation of the secant-tangent product theorem.  In contrast, in the third period class the discussion of the circle problem was followed by the installation of the secant-tangent product theorem.  While the second period class solved the problem by using the theorems about the geometric mean, the third period class applied similarity. The proof of the secant-tangent product theorem required using similarity.  The re-installation of the theorems about the geometric mean enabled the teacher to make similarity explicit in the discussion.

The teacher used a metaphor to trigger students’ memories about the theorems about the geometric mean.  The metaphor named, “the parachute guy,” involved a prototypical diagram: a right triangle with the right angle at the top, the hypotenuse aligned horizontally, and the altitude to the hypotenuse.  When installing the theorems about the geometric mean, the teacher had made a story about a man in a parachute descending along the medians of the right triangle.  The story was the metaphor that provided students the heuristics to remember how to relate different segments when applying the theorems.

The perceptual apprehension of diagrams was used in most of the segments, which means that most of the tasks involved using a diagram.  The discursive apprehension was the second type most used followed by the operative apprehension.  This means that most of the segments centered on making statements about geometric objects. The sequential apprehension was not used.

The examination of the sub-types of operative apprehensions shows that most of the operations performed were of the mereologic subtype (e.g., the diagram was modified by adding or deleting elements or by separating or recombining parts).  In addition, there were changes in the position of the diagram.  In some cases, the teacher combined the two sub-types.  For example, the teacher redrew one part of the diagram separately and changed its orientation for two purposes: for illustrating the application of the theorems about the geometric mean (e.g., the parachute guy) and for illustrating triangle similarity.

The two solutions of the problem (using the theorems about the geometric mean and using similarity) required students to apply the discursive and the operative apprehension of diagrams in tandem (in addition to the perceptual apprehension). Even though the application of the theorems about similarity required the use of the operative and the discursive apprehensions of diagram, most students said that they did not remember similarity.  The teacher used the metaphor for helping students remember the apprehensions of diagrams involved when applying a set of theorems regarding the geometric mean in right triangles to the solution of the problem.  When the teacher mentioned the metaphor, the discursive apprehension supported the operative apprehension. 

The case illustrates that teachers’ use of metaphors as a strategy to make students remember specific content can put teachers in a double-bind (Christiansen, 1997; Herbst, 2002; Mellin-Olsen, 1987, 1991).  Specifically, a metaphor can enable a teacher to make students remember how to read a diagram when applying specific theorems, because the metaphor can trigger students’ memory of a prototypical orientation of the diagram. At the same, the metaphor can make it difficult for the teacher to make students remember the justifications of those theorems, because by remembering what the theorem is about students do not need to remember how the theorem became known. 

References

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