How does a student learn to 'dig behind the scenes' or 'feel' math like a Fields Medallist?

Question

_{Source 1: Siobhan Roberts, Mathematical Beauty: A Q&A with Fields Medalist Michael Atiyah, Quanta Magazine, 2016/3/9.}

Is there one big question that has always guided you?

I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.

People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.

_{Source 2: Sir Michael Atiyah on math, physics and fun.}

  [...] You've got to have a lot of input and material from somewhere, you've got to have ideas coming from physics, concepts from geometry. You've got to have imagination, you're going to use intuition, guesswork, vision, like a creative artist has. In fact, proofs are usually only the last bit of the story, when you come to tie up the... dot the i's and cross the T's. Sometimes the proof is needed to hold the whole thing together like the steel structure of a building, but sometimes you've stopped putting it together, and the proof is just the last little bit of polish on the surface.   So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.
  $\color{#009900}{\Large{[2.]}}$ And I think that's one thing the field can get wrong when they're being taught to students. They can see a very formal proof, and they can see, this is what mathematics is. My story I can tell. When I was a student I went to some lectures on analysis where people gave some very formal proofs about this being less than epsilon and this is bigger than that. Then I had private supervision from a Russian mathematician called Bessikovich, a good analyst, and he'd draw a little picture and say, this -- this is small, this -- this is very small. Now that's the way an analyst thinks. None of this nonsense about precision. Small, very small. You get an idea what is going on. And then you can work it out afterwards. And people can be misled, if you read books, textbooks or go to lectures, and you see this very formal approach and you think, gosh that's the way I gotta think, and they can be turned off by that because that's not an interesting thing, mathematics, you see. You aren't thinking at that point imaginatively.

1. Is there a term that describes for what I bolded above (about ex ante presentiment)? 'Intuition' can be ex ante or ex poste, and so is ambiguous.

2. Assumption per 2 above: Too many textbooks and lecturers state only theorems and proofs, and fail to explain the presentiment.

Then how can a maths student who is not a maths genius, learn presentiment:

3. for a specific (proven) theorem and proof?

4. for solving unseen problems or proving unseen theorems?

Answer 1

In addition to other insightful answers/comments/remarks, apart from the issue of "degree" and "what kind of genius", I think a large part of the problem is exactly that mathematics is mostly portrayed as a school subject, in which a large part of "success" is measured by a certain conformity to "the teacher"'s expectations, and conformity with the worldview manifest in the textbook.

I claim that this problem exists at all levels of schooling and mathematics. :)

That is, kids and other people are mostly led to believe that they will/must learn mathematics in some sort of school, whether accelerated or whatever, and that their success will be measured by some form of external approval. This is not toooo ridiculous, but given the tendency to commodify, the goals for the teachers also become externalized and commodified (and thereby corrupted), to meet "no-one left behind", "graduation rate", and other administratively imposed mis-measures.

That is, to really understand mathematics or anything else, probably people should first "be allowed" to trust their own judgement. Few school-math scenarios even remotely come close to encouraging this viewpoint. It's not that teachers try to persuade students, but "command", from a position of authority. Authority is not the same as expert-ness. An expert can persuade by showing how wonderfully a thing can be done.

So, sure, if one is genuinely curious, and not worried about invalidation or failure, one behaves very differently than the opposite.

Now, if the goal is to make this happen in the framework of "school", then there is a fundamental conflict, I do seriously think, since the measure of success in "school" has almost nothing to do with one's curiosity or standards. Oop.

Answer 2

This question makes me think of the James Gleick quote on two kinds of genius:

"There are two kinds of geniuses: the 'ordinary' and the 'magicians'. An ordinary genius is a fellow whom you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what they've done, we feel certain that we, too, could have done it. It is different with the magicians. Even after we understand what they have done it is completely dark. Richard Feynman is a magician of the highest calibre."

I don't know how to learn to think like a Fields's Medalist. Those are the magicians. But helping students get plain old mathematical insight might be something us ordinary mortals can shoot for.

In How Children Fail, John Holt describes (as anecdotes) his experiences where he has seen children make the switch from rote processing to having insight into the meaning of the problem.

The text of the book is available online at http://www.schoolofeducators.com/wp-content/uploads/2011/12/HOW-CHILDREN-FAIL-JOHN-HOLT.pdf.

Unfortunately, the online PDF does not include page numbers. (!!!) But some sections do have dates. If you run a search for "March 11, 1961" you will find the story of Dorothy who, through sheer repetition "got a feel" for determining whether a number was evenly divisible by 2, 3, 4 etc:

"People to whom I have described this child's work have found it all but impossible to believe. They could not imagine that even the most wildly unsuccessful student could have so little mathematical insight, or would use such laborious and inefficient methods to solve so simple a problem. The fact remains that this is what the child did...."

Also search for the date "October 1, 1959" to ready the story of Dr. Gattegno's demonstration of children with learning disabilities. Holt describes the demonstration:

"I thought, what must it be like to have so little idea of the way the world works. So little feeling for the regularity, the sensibleness of things? . . . Then as I watched, the dark haird boy saw! Something went "click" inside his head. And for the first time, his hand visibly shaking with excitement, he reached without trial and error for the right rod..."

I first read that story close to 30 years ago, and it still moves me. Unfortunately, these are mere anecdotes. Holt isn't able to definitively answer how to catalyze the transformation from trial and error to genuine insight. But I find his accounts incredibly moving. Maybe you can determine from his stories what the "essential ingredients" are.

Answer 3

Take a look at this article:

The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra (Tracy Goodson-Espy, 1998) http://link.springer.com/article/10.1023/A:1003473509628

So much has been written about the construction of a mathematical "edifice," by the building of abstraction upon abstraction, that I'm not even going to try to cite a source.

The basic premise of the article, as I understand it, is this: Students need to work (e.g., solve problem with) with an abstract idea until it begins to feel concrete to them. But once the abstract notion has been reified, students (or mathematicians, for that matter) can use it in the construction of even more abstract ideas. ("Reify," by the way, comes from Latin res, meaning "thing.")

For example, some time ago I read John Derbyshire's Prime Obsession on the Riemann Hypothesis. The hypothesis is easily stated (from Wikipedia):

"The Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2."

The problem is, to understand the Riemann Hypothesis requires understanding the Riemann Zeta Function. And understanding the Riemann Zeta Function requires layer upon layer of abstraction. The problem is I read the book like a novel, instead of textbook. Once I thought I understood a particular abstraction, I moved on to the next chapter. But understanding the words in a sentence is a far cry from understanding at a gut level. In order to truly understand the Riemann Hypothesis, I would probably have to spend years reading and digesting Derbyshire's book, along with a textbook on number theory and plenty of practice problems.

So I think that is part of the answer to your question: Don't go too fast. Don't move on to a more abstract notion until the ones that it builds upon are deeply ingrained in you (or your students). That takes time, practice, trial and error, and a certain capacity for tolerating abstract thought.

EDIT (3/11/2017): Here's a related quote, taken from Feynman's What Do You Care What Other People Think, that also refers to the need to base new ideas on firmly rooted old ideas:

It's natural to explain an idea in terms of what you already have in your head. Concepts are piled on top of each other: this idea is taught in terms of that idea, and that idea is taught in terms of another idea, which comes from counting, which can be so different for different people! I often think about that when I'm teaching some esoteric technique, such as integrating Bessel functions.

Answer 4

I suggest taking a look at Polya's two volume set on induction:

https://en.wikipedia.org/wiki/Mathematics_and_plausible_reasoning

In particular, the preface and first chapter have some good comments on learning by guessing, guessing well. And on the differences between formal proof of ideas as opposed to derivation of them.

In Polya's spirit, I would actually try a few problems in the first chapter! (He said even induction as a CONCEPT should be studied VIA INDUCTION. Brilliant!) But you really won't need to study the whole two volumes to get Oolya's main points.

I think there are some questions one could ask to chip around this question. First do all Fields medalists think similarly? What are the major commonalities and differences. [The format of the question and even the Siobhan/Atiyah remarks seems to assume that they mostly think similarly and hey let's learn about it. Could even be right. But I wouldn't assume it from outset. Even trying to look for differences amongtst the population maybe reveals similarities, even similarities that are different from the general population.]

Second, perhaps writings of more famous "magicians" could be revealing. Ramujan is an iconic example. Of interest, it is how much he enjoyed and was inspired by math contest-y, manipulation style things like Edwards Calculus or that famous Tripos crib book. [Things that Hardy disliked!]

Third and this is a deeper (harder) area to evaluate but one to consider: what is the difference between intelligence and creativity?

I guess other questions to ask the Fields guys are "did you always think this way or did something change over time". Consider Greg LeMond who said "it never gets easier, you just go faster". Is it like that? Or different.

Another thing to ask the Fieldsers could be "do you see differences in how you think" (versus math students, versus lesser professional mathematicians).

Polya has some interesting comments in Solve It about problem solving. Even for very simple problems, it is possible for students (non Fields caliber!) to make some jump of intuition. He gives the example of cube diagonal formula derivation. And even for students that don't figure it out on own (hard), you can get them SOME of the feeling and process of mental "jumping" by using hints and gradual ones, rather than going straight to explaining it or to giving too much of a hint.

I wonder if this in some ways is similar to the self discovery process involved in teaching by Socratic dialog (something I tend to do almost subconsiously...rather than just explaining how to do the car tire problem:

"is there anything in this week's chapter that applies?"

Uh...maybe the ideal gas law? But...blabla...don't know how to apply it.

"Well what does the ideal gas law say"

(Recites it.)

"WEll from the problem statement, do we have ANY of those quantities listed"

Uh...OK. I got this and that but missing X and Y.

"So is there anything similar to X? Maybe not exactly same but transformable to X?"

Hmm...well this thingie (more help given if needed)

(more questions to push towards standard molar volume if that is the needed conversion factor)

"Ok plug it in and see how it goes"

(Does)

"Now check your answer in the back"

(Does. Happiness.)

Answer 5

I don't know how to explain it. I sometimes have a natural passion to do some of my own thinking from scratch independently of what I thought before. I kind of have an idea of how I could get a deep understanding from scratch about why the associative law of natural number addition is true and like it. The study of what I have the potential to do is kind of a topic in and of itself to see and notice things in and like the feeling of when my mind is on it. Maybe it really is for the things I see and notice right then.