Sunday, February 24, 2008

7. Rafael Nunez’s article

Mathematical Idea Analysis: What embodied cognitive science can say about the human nature of mathematics---Rafael E. Nunez
Assignment requirement: Write a short (imaginary) letter from Nunez to a skeptical math teacher who believes that math has nothing to do with the body or the physical world. In the letter, Nunez should be trying to convince the skeptical teacher of his ideas.

Dear Mr. Wang:

I still remember the issues about mathematics education that we discussed last time when we first met each other. Your comments on what the nature of mathematics is got me think further. From discussing with you, I know that you, as a secondary school math teacher, have been suspecting for quite a long time about a problem that is whether or not math has anything to do with the body or the physical world. In your view, not all of the math concepts / ideas can be explained and understood explicitly, for instance, you illustrated an example “Why the empty set is a subset of all sets?”, the question posed by your students. Your reply was that “The empty set is a subset of all sets” is merely a definition and asked your students to memorize it rather than understand its true meaning. As to this point, I totally disagree with you. So, this is the main reason why I’m writing to you now.

From my perspective, mathematics is neither transcendentally objective nor arbitrary. The nature of mathematics is about human ideas, not just about formal proofs, axioms, and definitions. These human ideas are grounded in species-specific everyday cognitive and bodily mechanisms. In the last century, many influential mathematicians viewed human intuitions (which are not theorems, axioms, and definitions) as basis for helping explain and understand mathematics. These intuitions, although they were not proved scientifically, demonstrated implicitly that mathematics is based on aspects of the human mind. In recent years, there are some empirical findings about the nature of mind discovered based on the contemporary embodied cognitive science. These empirical findings show that we are able to understand the real meanings of mathematical ideas through bodily mechanisms or people’s physical world.

In this letter, I will draw on Mathematical Idea Analysis which comes out of the embodied cognitive science as a theoretical tool to further address why we can understand mathematical ideas through bodily, physical mechanisms and people’s everyday experience embodied by image schemas, conceptual metaphors (this one is very important, constituting the very fabric of mathematics), and etc in the following:

First, by using Mathematical Idea Analysis, I have found that mathematical ideas are grounded in bodily-based mechanisms and everyday experience. For example, we can use the everyday concept of a collection of objects in a bounded region of space to conceptualize the math concept of a set; use the everyday concept of a repeated action to conceptualize the math concept of recursion; use the everyday concept of rotation to conceptualize the math concept of complex arithmetic; the everyday concepts as motion, approaching a boundary, etc to conceptualize derivatives in calculus…

Second, conceptual metaphors are fundamental cognitive mechanisms which project the inferential structure of a source domain onto a target domain, allowing the use of effortless species-specific body-based inference to structure abstract inference. They are not arbitrary, because they are structured by species-specific constrains underlying our everyday experience- especially bodily experience. For example, Affection Is Warmth, here, Affection is conceptualized in terms of thermic experience. The same happens in mathematics. For instance, Classes (Sets) Are Container Schemas—Grounding metaphors; use ‘more than’ and ‘as many as’ for infinite sets—Redefinitional metaphors; Functions Are Sets of Points, the Sets Are Graphs—Linking metaphors (see the details of three types of conceptual metaphors in Lakoff & Nunez (2000)).

Third, image schemas are basic dynamic topological and orientation structures that characterize spatial inferences and link language to visual-motor experience. Their inferential structure is preserved under metaphorical mappings. So, if we go back to the instance ‘Sets Are Container Schemas’, we can see that the Container Schema(a common image schema in mathematics) presents the link between language and spatial perception, involving three parts: an Interior, a Boundary, and an Exterior. Image schemas can fit visual perception, for example, Venn Diagrams are visual instantiations of Container Schemas.

Of course, conceptual metaphors are closely related to image schemas. Now, please allow me to still focus on the example of “Sets Are Container Schemas”, here, sets initially metaphorically conceptualized as containers, and then Venn Diagrams work as symbolizations of sets to show their bounded regions in space. Therefore, for this mathematical idea ‘Sets’, we can realize the understanding through metaphorically relating them to containers and then using Venn Diagram as the image schema for containment. It means that we can conceptualize ‘Sets’ through our everyday life experience and body-based perception.

Furthermore, the ‘Laws of Container Schemas’ called by Lakoff and I are conceptual in nature and are reflections at the cognitive level based on the analysis of neural structures. So, I sincerely invite you to read my paper which includes more regarding this issue.

Yours sincerely,

3 comments:

Charles Wells said...

Your post is an excellent summary of Núñez' ideas. Here are some added comments -- not criticisms of what you said. (1) All metaphors in mathematics suggest good insights as well as misleading insights. The container metaphor for math is good, particularly in the Venn Diagram form. It is not good if you think of the containers as purses: 2 is in the intersection of {1,2,3} and {2,3,5}, but it causes cognitive dissonance to think of something being in three different purses. Students should be taught metaphors, and every time you should warn them about the misleading aspects of the metaphor. (2) You should encourage the students to expect to use several different metaphors or images or mental representations for the same mathematical concept. Another insight which I have pushed in my abstractmath.org website is to think of a set as a pointer to its elements. Pointer in the sense it is used in computer science. This is a strikingly accurate way of thinking about sets IF you know about pointers! When you are teaching a discrete math course, for example, typically most of the students will be computer science majors and to them it makes a difference.

Charles Wells

julialandai said...

Hi,Mr. Wells:
Thank you for your comments.I'm really appreciated.I come from Shanghai,China.I previously was a high school math teacher in Shanghai.In Shanghai school math curriculum, set theories are the first math concepts that Shanghai students need to learn when they start their high school academic learning.However,as we all know, all concepts regarding sets are quite abstract.Normally I use a class/group as a metaphor to describe sets and its students as another metaphor to explain the elements of sets.Venn's Diagram provides students with visual images to see the relationships of sets. This is also my first time to know about the concept of container metaphors. Your comment also give me another new perspective to view the concept of a set--to think of a set as a pointer to its elements.
I'm not so familiar with the idea of pointers used in computer science, but I will go to your website and learn about that.
Thanks the internet and your comments.
Best,
Julia

Susan Gerofsky said...

Julia --
It's great to see Prof. Wells and you engaging in an online discussion of these key issues in math education! I'm also unfamiliar with the computer science sense of a "pointer", and I'm also intrigued -- I will look into this further.

Speaking as your teacher, I agree that you've done a good job of summarizing Nuñez's article. I have some difficulty with the way you've done it though.

What I see in your blog is that you have quoted chunks of Nuñez's own writing in the article, rather than putting the ideas into your own words. This worries me because:

1) it gives me no indication whether or not you have understood or made sense of Nuñez's ideas. (It's quite possible to lift sentences and phrases wholesale from the original without having made sense of them, and I do wonder what sense you make of phrases like "image schemas are basic dynamic topological and orientation structures that characterize spatial inferences and link language to visual-motor experience. Their inferential structure is preserved under metaphorical mappings." Could you explain this verbally in ordinary language?

2) You have not actually engaged with the assignment, which was to try to convince a skeptical math teacher of Nuñez's ideas. Instead, you have produced a technically perfect shorter version of Nuñez's own words. This leaves me not knowing what you think about the article -- I only know that you have read it, and can quote from it.

My concerns may be partly cultural ones. There are different norms for academic writing in Chinese and Canadian cultures. From what I know of Chinese academic writing, there is a greater emphasis on deference to established authorities while Canadian academic culture emphasizes originality and building an individual argument. China has a 1400-year tradition of national high-stakes examinations where memorization and quotation of classics is emphasized -- Canada is closer to the American and British models which admire the feisty, original, and generally plain-talking individual who might well break from tradition.

In addition to all this, I know that you are doing graduate work in your second or third language, which is a difficult undertaking! I can see that you might be understanding the reading very well, but that Nuñez's own words might seem much more elegant than your own paraphrase. This is possible, even likely -- but I have no way of knowing simply by reading your blog posting!

Let's talk more about this in class this evening.

Thanks for all your hard work.

Susan