What Is Research?

March 21, 2008

The mind’s eye

Recently, when talking to an Olympiad aspirant Ashwath Rabindranath about how to prepare effectively for the International Mathematical Olympiad, I came up with a formulation that I realized I’d implicitly been using for some time. After I discussed it with him, he said that he’s been trying it a lot and it’s been fairly helpful to him.

The concept is called the Mind’s Eye.

The idea is simple: everything should be in the mind’s eye. In mathematics, it is not enough to know that something can be proved. Truth is there only when you know how it has been proved. But even knowing, in the abstract, how it can be proved, isn’t enough. To really feel that a proof is correct, one should be able to behold it in the mind’s eye. Thus, if I tell you that every nilpotent group is solvable, you shouldn’t be satisfied withknowing that there’s some proof in a dusty book somewhere. You should see why the statement is true, and you should see it immediately, in your mind’s eye. By that I mean you should behold the proof conceptually, or pictorially, in a way that you can magnify any component of the proof at will. You should be able to tell me what the related facts are, what the applications and lemmas used are, and what the possible generalizations could be.

The mind’s eye is particularly important for Olympiad preparation because of the format of Olympiads: students are expected to solve a few challenging problems in a short time-frame, and they cannot refer any existing texts. A lot of Olympiad students waste precious in-exam time going down wrong alleys. If the student has in her/his mind’s eye all the possible things that could be done with the problem, what the consequences of each path would be, and what the likelihood of success on each path would be, then the time and effort spent online (during the exam) reduces proportionately.

But the importance of the mind’s eye is not merely limited to closed-book examinations, or time-crunched examinations. The importance extends to the more general scenario of learning and teaching. I can look up in a book a proof that not every normal subgroup is characteristic, but having the counterexample in my mind’s eye means that I can explore variations more easily, go forward, generalize. Books and online references are useful to supplement the mind’s eye in storing information — they cannot supplant the mind’s eye. The greatest research, insights and breakthrough come by immersing oneself in a problem, which means one can see it in the mind’s eye.

The idea is so breathtakingly simple that it amazes me why people do not use it more often. The mind’s eye can begin right in high school, in fact, when students are studying physics, chemistry, mathematics, history, economics, geography or just about anything. For those who’re more tuned to sound, they could use the mind’s ear, and for those more tuned to touch, they could use the mind’s touch. And it can begin simply. You look at a long-winding text or explanation. It’s too big for the mind’s eye. You look at it again. You break it down, you think about it. You mull over it. You sleep over it, and your subconscious reorganizes the ideas and the next day, it fits into the mind’s eye. Now, anybody can ask you about that idea and you can explain it offhand. More importantly, though, you can see (or hear or feel) it.

The mind’s eye could do well if supplemented by other resources that specifically prod it on. One of these techniques I’ve been exploring is an idea for math-related wikis, that I’ve started implementing. The first mathematics wiki I started is a wiki in group theory. The ”one thing, one page” paradigm on this wiki, as well as the diverse ways in which pages are linked together, using different relational paradigms, supplements the mind’s eye pretty well. I’ve often got new insights simply by surfing the wiki — and that’s saying something considering I’ve written almost all of it. The wiki has pages on things that might get short shrift in a textbook; for instance, normality is not transitive.

This isn’t the only way to supplement the mind’s eye. I came up with some ideas long ago about the use of a method of properties to organize information. That didn’t take off too well, though some of its features have been incorporated pretty effectively in the group theory wiki. Then, of course, we can learn from the way advertisers work: they tie in the core idea using a number of different paradigms. In his series on the Palace of Possibilities, Gary Craig talks of two tools to reinforce concepts: repetition and emotion. Instructional design texts emphasize the importance of reiterating the same basic point from a number of different perspective,s appealing to audio, visual and kinesthetic sense in the students over and above their cognitive abilities.

It is important to distinguish between the mind’s eye, and rote (or memorization). In fact, rote is a very special case of the mind’s eye; basically where you juts memorize the text as text, or in a specific form. The mind’s eye, in its more general form, encourages a complete immediate grasp of the material, but not from a specific angle, but rather from a large number of angles. The mind’s eye works best by building redundancy: by having not just an eye, but several eyes, several ears, and several hands to touch and feel.

Another important point is that the specific methods one uses to build the mind’s eye could vary widely, which is why I’m not listing here how to do this. The core idea is to increase the number of ways ideas linked together in the mind, and this could be done through random association attempts, by using systematic paradigms, or just by exposing oneself to a lot of material and letting the subconscious do the organizing. The key is to get the mind’s eye in action.


1 Comment »

  1. I had the similar idea , but it was not complete . The article portrays the idea verbally very well .

    Comment by Krishna — October 29, 2008 @ 1:09 am

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