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?
