Author: Presmeg, N. C.
Source: Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23(6), 595-610.
In this article, Presemeg firstly invites the readers some examples
of prototypical mathematical images that can be necessary for reasoning in
solving mathematical problems in high school classroom. Her study shows the
prototypical images acted as metaphors which lead the students to the
mathematical reasoning process, and there were many images that intend to
metonymic purpose in the students’ thought process. The students used
“concrete” images to solve the mathematical problems. For instance, one student
idiosyncratically thought of a ship on a water level as the metaphor to remind
her to utilize 180 and 360 degrees to solve the acute angles in the trigonometry
problem. The author acknowledges the difficulties in the problems of generality
that can be observed in metonymies, although she points out the strengths of
visual processing (as following).
The
author describes the category of imagery that pure relationships are depicted
in a visuospatial scheme was designated as pattern imagery. She explains
imagery can be used to illustrate abstract situations by concretizing the
referent or by using pattern imagery. Additionally, “the mediating link between
specific concrete cases and generalized mathematical concept lies in pattern
imagery which shares some of the imagistic nature of concrete images but goes
beyond them to distinguish regularities and commonalities and to abstract these
in a different kind of image which provides a basis for the formations of the
concept.” (p.605) In other words, pattern imagery gives us the generalization
of concrete images and invites us different concepts, but they are based on
individual leaners’ mathematical experience which can be shared with others.
This article made me rediscover the multiple meanings of image in mathematics.
When I introduce the new concepts or contents in the classroom, I usually bring
or explain the objects that the student might hold as a commonality. However,
through the lessons, the students might channel the mathematical concepts into
the image that they are (more) familiar with. Although it is difficult to
estimate the all images each student has, I wonder the introduction with the
image that the students hold can help the students to understand the
mathematical concepts earlier than using other images. Also, how can
mathematics teachers utilizes the different types of imagery in mathematical
reasoning?
The
other point this article reminds me is the power of images in mathematics
classroom. When I explained the meaning of surface area of solid figure to my
student who is a Japanese language learner, I firstly use the way of expression
that my other Japanese students could understand. (Think about the situation
that you soak this solid figure in a fish tank with filled water, the wet areas
equal surface areas of this solid) He understood what I said in Japanese, but
he still could not understand well what surface area is. After that, I considered
more universal objects in the world as much as I can think up and explained him
the same situation with a glass of water and a dice, and it finally worked for
him. Although this article mentions the images which are embedded in the
students’ mind, the images (including metaphors etc.) in math textbooks might
be common only for certain groups such as majorities.
Questions:
Do you think the description without images
in solving mathematical problem can develop students’ mathematics ability? (For
instance, the questions ask some angles in geometry but no images attached)
What do you think about the images in
mathematics textbooks? Do you think those images help students to understand
mathematics?
Images and other visualizing aid always help students to connect abstract mathematical concepts with concrete examples. In geometry, it is even more helpful since our symbolic system doesn’t do a great job in this area. To describe a spatial object or construction purely using words or symbols will sometimes cause confusions. I agree with you that multi-presentation should be considered to facilitate mathematical learning such as
ReplyDeletegraphs and functions and to interact with these objects. I have some experience with integration of Geogebra, a dynamic software which offers not only presentations but continuous interactions between students and the objects. What I notice recently is that some teachers have made arguments to insert those dynamic java presentation to ebooks, which may bring huge impact to the perception of static textbooks.
The images accompanied with clear analytical presentation helps learners in solving mathematics problems. There are various procedures to be followed ,such as reading the question and understand then identify the problem, paraphrase the problem , draw ,look for patterns and then review. Images shall always be in mathematics learning and therefore has to be well placed in the rising technological teaching and learning aids.
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