Learning Through Teaching Mathematics

"What Changes in Teachers’ Knowledge Occur Through Teaching? (p. 13-14)

What Changed in Einat’s Knowledge?

The unpredicted turn that the lesson took in relation to the solution of Problem 2 nurtured Einat’s learning of mathematics. According to her plan, Problem 2 was aimed at performing calculations using the Pythagorean theorem. But when a student raised an unforeseen (general) question related to the length of the two paths (Fig. 1d), new mathematical connections were constructed: the paths within the rectangle could be compared using the properties of triangles with equal areas and a constant basis or using the properties of the ellipse.

When Einat moved the internal rectangle from the center of the external one, it became clear to her that the length of the two paths will be different “because the position of the internal rectangle is asymmetric.” This intuitive assumption appeared to be correct for the concrete situation presented in Problem 2, but it was incorrect as a general statement. Einat discovered that not all asymmetrical positions of the internal rectangle resulted in paths of different lengths.

When points E and F are on the ellipse, the paths are equal in length. Thus, an incorrect intuitive assumption was refuted, and incorrect intuitions were replaced with correct mathematical knowledge. The second critical point for Einat’s learning was her intuitive agreement with Opher’s conjecture, which was also refuted. Our additional observation is related to the interrelationship between Einat’s mathematical and pedagogical knowledge. It was her pedagogical sensitivity that encouraged her to formulate new problems that led to mathematical discoveries. At the same time her mathematical knowledge allowed her to evaluate the difficulty of the refutations she had produced, and (again) being attentive to her students she designed a new instructional activity using the Dynamic Geometry software. In sum, in this example, we recognize the development of knowledge in the transformation of intuition into formal knowledge and in the mutual support between pedagogical and subject matter knowledge.

Example 2: Learning from a Student’s Mistake: The Case of Lora
Lora, an experienced instructor in a course for pre-service elementary school teachers, taught a lesson on elementary number theory. The following interaction took place:
Lora: Is number 7 a divisor of K, where K = 34×56? Student: It will be, once you divide by it.
Lora: What do you mean, once you divide? Do you have to divide?
Student: When you go this [points to K] divided by 7 you have 7 as a divisor, this one the dividend, and what you get also has a name, like a product but not a product. . .

Lora’s intention in choosing this example was to alert students to the unique factorization of a composite number to its prime factors, as promised by the Fundamental Theorem of Arithmetic, and to the resulting fact, that no calculation is needed to determine the answer to her question. This later intention is evident in her probing question.

What Did Lora Learn from the Above Interaction? First, she learned that the term “divisor” is ambiguous, and that a distinction is essential between divisor of a number, as a relationship in a number-theoretic sense, and divisor in a number sentence, as a role played in a division situation. She learned further that the student assigned meaning based on his prior schooling and not on his recent classroom experience, in which the definition for “divisor” used in Number Theory was given and its usage illustrated. Before this incident, Lora used the term properly in both cases, but was not alert to a possible misinterpretation by learners. The student’s confusion helped her make the distinction, increased her awareness of the polysemy (different but related meanings) of the term divisor and of the definitions that can be conflicting. This resulted in developing a set of instructional activities in which the terminology is practiced (Zazkis, 1998)."