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Sorry, there hasn’t been much activity on this blog lately – which is an understatement, I acknowledge! But, for now, here’s a small article by Terry Tao on CNNOpinion.

In mathematics you don’t understand things. You just get used to them.”

— John von Neumann

I had been wanting to write on this topic – no, I am not referring to the above quote by von Neumann – for quite some time but I wasn’t too sure if doing so would contribute anything “useful” to the ongoing discussion on the pedagogical roles of concrete and abstract examples in mathematics, a discussion that’s been going on on various blogs for some time now. In part coaxed by Todd, let me share some of my own observations for whatever they are worth.

First, some background. A few months ago, Scientific American published an article titled In Abstract: Avoid Concrete Example When Teaching Math (by Nikhil Swaminathan). Some excerpts from that article can be read below:

New research published in Science suggests that attempts by math teachers to make the subject easier to grasp by providing such practical examples may actually have made it tougher to learn.

For their study, Kaminski and her colleagues taught 80 undergraduate students—split into four 20-person groups—a new mathematical system (based on several simple arithmetic concepts) in different ways.

One group was taught using generic symbols such as circles and diamonds. The other groups were taught using practical scenarios such as combining liquids in measuring cups.

The researchers then tested their grasp of the concept by seeing how well they could apply it to an unrelated situation, in this case a children’s game. The results: students who learned using symbols on average scored 80 percent; the others scored between 40 and 50 percent, according to Kaminski.

One may read the entire article online to learn a bit more about the study done. Let me add that I do agree with the overall conclusion of the study cited: in mathematics concrete examples (in contradistinction to abstract ones) more often than not obfuscate the underlying concepts behind those examples, thus hindering “real” or complete understanding of those concepts. However, I also feel that such a claim must be somewhat qualified because there is more to it than meets the eye.

Sometimes the line between abstract examples and concrete ones can be quite blurry. What is more, some concrete examples may even be more abstract than other concrete ones. In this post, I will assume (and hope others do too) that the distinction between an abstract example and a concrete one (that I have chosen for this post) is sharp enough for our discussion. Of course, my aim is not to highlight such a distinction but to emphasize the importance of both abstract and concrete examples in mathematical education, for I firmly believe that a “concrete” understanding of concepts isn’t necessarily subsumed under an “abstract” one, even though a concrete example may just be a special case of a more general and abstract one. What is more, and this may sound surprising, abstract examples may sometimes not reveal certain useful principles which, on the other hand, may be clearly revealed by concrete ones!

Let me illustrate what I wrote above by discussing a somewhat well-known problem and its two related solutions, one of which employs an abstract approach and the other a concrete one, if you will. Some time ago, Isabel at God Plays Dice pointed to an online article titled An Intuitive Explanation of Bayesian Reasoning by Eliezer Yudkowsky, and I borrow the problem I am about to discuss in this post from that article.

PROBLEM: 1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

How may one proceed to solve this problem? Well, first, let us look at an “abstract” solution.

“ABSTRACT” SOLUTION: Here we employ the machinery of set-theoretic probability theory to arrive at our answer. We first note that what we really want to compute is the probability of a woman having breast cancer given that she has tested positive. That is, we want to compute the conditional probability P(A/B), where event A corresponds to that of a woman having breast cancer and event B corresponds to that of a woman testing positive for breast cancer. Now, from Bayes’ theorem, we have

\displaystyle P(A/B) = \frac{P(B/A) P(A)}{P(B/A) P(A) + P(B/A^{c}) P(A^{c})}.

Also, we note that P(A) = 0.01, P(B/A) = 0.8, P(A^{c}) = 0.99 and P(B/A^{c}) = 0.096. Plugging these values into the above formula immediately yields P(A/B) = 7.76%. And, we are done.

A couple of observations.

1. It is not hard to observe that the derivation of Bayes’ formula follows from the definition of conditional probability, viz. P(A/B) = P(AB)/P(B), where P(B) > 0, and the usual set-theoretic rules involving the union and intersection of sets (events). And, this derivation can be carried out through sheer manipulation of symbols under those rules. By that I mean, if a student knows enough set theory as well as the “laws” of set-theoretic probability theory, then the derivation of Bayes’ theorem makes absolutely no (or, almost no) use of the “intuitive” faculty of a student.

2. The abstract method presented above also subsumes the concrete method, as we shall see shortly. What is more, Bayes’ formula can be generalized even further. This means that once we have this particularly useful “abstract” tool at our disposal, we can solve any number of similar problems by repeatedly using this tool in concrete (and even abstract) cases. In addition, Bayes’ theorem can also be thought of as a “black box” to which we apply certain inputs in order to get our output. This should not surprise us, for in mathematics the use of theorems as black boxes is a common one.

Now, the above two observations may lead one to believe that indeed there is almost no need to find a “concrete” solution to the above problem. After all, the abstract case takes care of the concrete cases completely.

However, let us see if we can come up with a concrete (that is, a far less abstract) solution and examine it more closely to see if we can extract some useful ideas/techniques from the same.

“CONCRETE” SOLUTION: Suppose we choose a random sample of 100,000 women of age forty. (We choose that figure for reasons that will be clear soon.) Then, we have two groups of women.

1st group: 1,000 (1%) women who have breast cancer.

2nd group: 99,000 (99%) women who don’t have breast cancer.

Now, in the 1st group, 800 (80% of 1,000) women will test positive, and, in the 2nd group, 9,504 (9.6% of 99,000) women will test positive. So, it is clear that if a woman tests positive, then the probability that she belongs to the 1st group (that is, she really has cancer) is 800/(800+9504) = 7.76 %. And, we are done.

Let me quickly point out a very important advantage the above solution has over the abstract one we saw earlier.

Indeed, we finally “see” what’s really going on. That is, from an intuitive standpoint, we observe in the above solution that there is a “tree structure” involved in our reasoning. The sample of 1,00,000 women bifurcates into two distinct samples, one of which has 1,000 women who have breast cancer and the other that has 99,000 women who don’t. Next, we observe that each of these two samples in turn bifurcates into two samples, one of which comprises women who test positive and the other that comprises women who don’t. This clearly reveals to the student the “tree structure” in the above reasoning. This makes the concrete solution much more appealing and “satisfying” to the average student. In fact, the generalization we talked about earlier in regard to Bayes’ theorem can even be carried out in this particular method: we will only need to increase the depth and/or breadth of our “tree” by extending more nodes from existing ones!

Moreover, one may recall that the use of such “trees” in reasoning is quite common in mathematics. For instance, the two most basic rules of Combinatorial Principles, viz. the Rule of Sum and the Rule of Product are proved using such “trees”. So, this is one instance in which a concrete solution reveals much more clearly a quite fundamental principle/technique (use of “trees” in reasoning) in mathematics that isn’t clearly revealed at all in the abstract solution we examined earlier.

In other words, much thought needs to be put in deciding if abstract examples should necessarily be “favored” over concrete ones in mathematics education. From a pedagogical standpoint, sometimes concrete examples are simply much better than abstract ones!

I have just begun reading Herbert Meschkowski’s Evolution of Mathematical Thought, and I have finished the first few chapters. The book is certainly an interesting read. The second chapter titled “Foundations of Greek Mathematics” provides some good insights into the way mathematics was understood by the Greeks during the classical Greek period. Based on that second chapter, I will write a little on why Plato laid so much stress on the study of geometry.

Tradition has it that the inscription above Plato’s Academy read, “Let no one enter who is ignorant of geometry.” We are, of course, not sure if that was really true, but we wouldn’t veer too far away from the truth if we do ascribe the above quote to Plato. And the reasons are quite compelling.

Platonic idealism (according to Wikipedia) is the theory that the substantive reality around us is only a reflection of a higher truth. To elaborate, in layman’s terms, there is a higher world (independent of us) out there and this world, according to Plato, consists of “ideal forms” or Ideals. These ideal forms are blueprints of all the tangible objects around us. So, in a sense, tangible objects are instances or copies of these ideal forms. For example, there exists the ideal circle (or the perfect circle) out there and all the circles we draw are copies that are close approximations of the ideal circle. In some sense, the approximations can only “aspire” to be similar to that ideal circle. Now, according to Plato, these ideal forms are more real than the instances of those forms, and he carries this idea over to the discipline of mathematics, thus providing a justification for the study of geometry.

In the Republic, Plato states:

And do you not also know that they (mathematicians) further make use of the visible forms and talk about them, though they are not thinking of them but of those things of which they are a likeness, pursuing their inquiry for the sake of the square as such and the diagonal as such, and not for the sake of the image of it which they draw?… The very things which they mold and draw, … they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind.

In other words, geometers (or mathematicians) draw points, lines, squares, circles and so on for explaining theorems (or doing proofs), and the figures they draw are quite “inaccurate” ones, in the sense that no one can ever claim to have drawn a perfect round circle or a perfect straight line. However, this doesn’t prevent the mathematician from looking deeply into the “reality” behind those figures to obtain useful insights from the same. Such insights guide the mathematician in proving geometric results, which otherwise cannot be “extracted” from the (crude) figures.

So, for instance, no matter how one draws a triangle, if one measures all the three interior angles with, say, a protractor, the sum of the three angles will never equal two right angles (180^{\circ}); the sum will always be a tad less or a tad more than two right angles. But, this doesn’t prevent the mathematician from proving that the sum is indeed equal to two right angles! (Of course, this is true only in Euclidean geometry.)

Thus, for Plato, knowledge of mathematics amounts to insights in the realm of ideas, and the pursuit of mathematics was a road to insights of a universal nature.

Thus, education through mathematical thinking frees man’s mind to “see” the world of ideal forms, which was more real to Plato than the tangible. This is further illustrated in Plato’s Seventh Letter in which he distinguishes among the different ways of viewing a geometric concept. For instance, consider a circle and let’s look at it’s different “interpretations.” First, a circle is something, “the name of which I just now uttered”; second, the concept defined verbally; third, the physical image of the circle as it is produced by the draftsman or the lathe-turner; and finally, the ideal circle that “approaches nearest in affinity and likeness to” the “real circle”, which alone is the object of scientific perception.

In conclusion, for Plato, the study of geometry (mathematics) was an intellectual exercise in training the mind in overcoming the “illusions” of the tangible world by learning to see the “ideal forms” or Ideals, which to him was more real than the physical reality around us.

Here is an interesting article (by Terry Tao) and the accompanying discussion on the relation between high IQ and doing good mathematics. Moral of the story: there is hope! 🙂

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May 2023