Prototypes, Metaphors, Metonymies And Imaginative Rationality In High School Mathematics By Norma C. Presmeg

Prototypes, Metaphors, Metonymies And Imaginative Rationality In High School Mathematics By Norma C. Presmeg

Norma starts her article with an interesting question “Why is it that there are such vast individual differences in students' learning of mathematics, even within one classroom?” I believe there are a lot of reasons behind this variation such as student’s mathematical knowledge, students' intelligence, student’s willingness to learn, student’s personal’s issues, etc. However, I assume, Norma’s question has neglected these reasons and only focuses on the same classroom and teacher’s issues. In fact, she asks if everything is the same, how is the student’s learning process?

From Norma’s point of view, student’s understanding of the question or topic directly affects students’ learning. By “understanding the problem”, she means imagining the mathematical subject which can be varied from a shape to an algebraic equation. She highlights that students visualize the problem to comprehend its meaning. A significant portion of the article is allocated to explaining and elaborating of the visualization and imagination. The authors include the view of several researchers to clarify how students might imagine mathematical subjects. From my perspective, Metaphors and Metonymies are the two important kinds of visualization. The author describes the Metaphor using the example of a student who visualizes a Sin graph as the sea waves that goes up and down; it helps her to learn and memorize the subject. In describing the Metonymies, the author uses the Lakoff's (1987, p. 79) definition as “Metonymic usage occurs because of a diagram or image, by its nature, depicts one concrete case. “She also uses the Johnson’s (1987, p. 171) categorization of "synecdoche" and "metonymy proper”, as two kinds of metonymy.

After an excellent discussion of imagination’s types, Norma states that type of visualization isn't mattered and as long as one type helps students to understand a topic, it is good and applicable. However, she had not neglected the teacher’s role in drawing or creating an image in student’s mind. In fact, the author believes that the view of a teacher regarding a mathematical topic affects student’s view and might even change it. I agree with Norma and think that an image that is drawn by a teacher (even an imaginary not a real one) sometimes remains in student's mind forever. For instance, if an equilateral triangle, for the first time, has been introduced to a student as A triangle, he will visualize a triangle in the form of equilateral. However, the role of a teacher in creating an image in student’s mind is entirely related to teacher’s ability to teach, a great teacher who is good at transferring knowledge has this kind of influence on his/her students.

This article reminds me of Poly’s suggestion in solving a new mathematical question in his book “how to solve it.” He proposes to imagine or draw a picture of the questions to have a better understanding of a mathematical problem. Although I think, it is more applicable in solving geometric problems; Norma highlights that it is even practical for an algebraic problem. This part of Norma’s article is interesting for me as I never thought it is possible to imagine an algebraic subject. Now that I am writing this reflex, I am also thinking how it is possible to imagine numbers or operations? And I remember a film that we have watched in our last class, the teacher knock on the wall to show the increasing or decreasing the numbers. Although he did not draw a number line, I think, his action will remain in his student’s mind as some hidden number line.

How about you, do you ever think of imagining an algebraic subject? Or did you ever draw a picture (imaginary picture) in your student's mind to help them memorizing or learning an algebraic topic?