Hilbert's Dream & Can AI replace Math teachers?

Can we get a computer program to solve Mathematics problems? Can we get an AI to check Mathematical proofs? This problem is very similar to a very old problem in Mathematics, which was thought of by the grand old man of Mathematics David Hilbert. However, the problem statement was changed multiple times as works of several great Mathematicians like Alan Turing, Alonzo Church and Kurt Godel. The answer to the question, till today, remains to be "no". So the world is probably going to need Mathematicians and no machine can replace them! Sir Timothy Gowers, in an LMS Popular Mathematics Lecture explains the details and subtleties of this problem and how he thinks the creation of such a programme would affect Mathematics, especially pure Mathematics.

Mathematicians are anyways a minority, a never understood minority. This kind of reminds me of an social media meme which claims that Albert Einstein had apparently told Charlie Chaplin that people appreciate his art although he doesn't say a him, to which Charlie Chaplin responds by saying that Einstein's art is even greater as people applaud him although they don't understand anything. Research in AI will probably have to get past various level before we can even think of replacing Mathematicians with AI. But how do we contrast Mathematicians with Mathematics teachers? Does an average school teacher even understand how a Mathematician thinks? Does an average school teacher have atleast basic research experience in Mathematics? Is a Mathematics teacher's role expendable, can we get an AI to take over?

I'm afraid that a lot of venture capitalists do think that the answer is a yes and you see that in the form of millions of dollars being invested in tech based personalized mathematics learning products which deliver better than what several under-qualified, under-trained Mathematics teachers do. So the question which I'm asking now, is what should a Mathematics teacher which an AI cannot do, which will keep AI as a tool and not as a substitute or replacement for teachers? I think this question is closely related to the question- why can't we get a computer to think like a Mathematician? Once we get an answer to this question, we ask- how can we get a Mathematics teacher to impart these capabilities to children. Let us explore this question with a middle school question- 32 x 64 (#LMS).

A computer would solve this question by multiplying mechanically, using the standard algorithm and it will in all likelihood beat human beings if they use the same method. However, a student who understands exponents wouldn't solve this problem that way, he'd recognize 32 as the 5th power of 2 and 64 as the 6th power of 2 and he'd respond that the answer is 11th power of 2, as 1024 is equal to 10th power of 2, the answer is 2048. How does a student develop these little hacks which help in solving a problem creatively and quickly? A question to ask here- did the student solve this problem creatively or mechanically? As Socrates observed, it is hard to say that, the student could have been taught this method in school. You can find it out through a dialogue involving good questions. A devil's advocate might ask, so why not just teach the student a lot of mechanical methods to solve problems? That's not a bad question, most of our current Mathematics programs work that way- Vedic Maths, speed Maths etc.

Let us ask another question, which will probably explain why Mathematics has to be taught as a creative discipline. There is a large 8x8 square which contains lots of 1x1 squares. Two diagonally opposite corner squares are removed. You are given 31 1x2 dominoes, can you find a way of covering all the 62 squares using these 31 dominoes? You can think about this puzzle, but from a Education policy reformer's perspective- how do we teach students so they could solve this problem? A businessman might ask- what mechanical methods can be taught so that a kid could crack this down? A businessman is probably lot more "shrewd" and "pragmatic" in comparison to a policy researcher, as he understands that we don't have those many passionate Mathematics teachers anyway, let alone qualified teachers.

It turns out that there is a very elegant and simple way to prove that this is not possible. Imagine a chess board and remove the corner squares. You have 2x1 dominoes- if one of the squares of the domino covers a black square, the other ought to cover a white square. So if the 31 dominoes cover all the 62 squares, you must have equal number of black squares and white squares. However both the corner squares are of the same color, so we cannot have equal number of black squares and white squares in this 62 square board. A brilliant and elegant proof isn't it? This probably explains what is beauty/aesthetics/elegance in the Mathematics context. But does one need genius to conceive such hacks/heuristics/ideas? Is it genius or mathematical process? So do we teach this so called genius or do we teach mechanical methods? Given that we don't have good teachers anyway, do we have answers within the mechanical problem solving paradigm?

Even before we try and check if we have solutions within the mechanical problem solving paradigm, let us ask if such problems are even important and if such elegance is even required? Well think about it- a software developer with better problem solving skills debugs a code lot faster, thinks of a neater/elegant way of writing a code. Isn't this kind of non-linear problem solving visible in tech/business situations. Why else do we celebrate Steve Jobs, Dennis Ritchee as geniuses (such an irony to name both in the same sentence)? So we do recognize that such problem solving skills are extremely important and they stay with the student all through his/her life even if he/she doesn't choose a STEM career. These skills help the child in drawing lateral connections between Mathematical experiences and real life situations - much like how Steve Jobs and Dennis Ritchee did!

So we do need a way of teaching the creative aspects of Mathematics- something which a computer cannot do. I shall not answer the "how" question, I have already answered the "why" question, I shall conclude by respond to the "what" question. What is the difference between a computer and a Mathematician? Knowing the answer to this question can probably help in populating a student/parent/school's wish-list of what they'd like from a Mathematics teacher. I am drawing some points from Sir Timothy Gowers LMS Popular Lecture to answer this "what question" and this cheat-sheet concludes this article.

Mathematician vs Computer cheat-sheet

1. Can you a Computer learn from mathematical experiences? Can it carry forward experiences of solving 10 problems, into a 11th problem?
2. Can the computer step back and look at what it is doing? If the computer is struck in a problem, can it look back and ask "why am I struck" and then take a recourse from the answers to this why question?
3. Can computers handle vague statements and vague ideas? Do computers always have to be precise? Do the "behind the scenes" work of Mathematics have to be "vague"?
4. Can a computer get a flash of a genius? (Usually we break an idea into small ideas and that's how we conceive these ideas. But then at times, we get ideas which can't broken down and we see them as a "flash of a genius"- impressive original idea. An example is the 8x8 square and 2x1 dominoes problem)
5. Can computer be taught Mathematics like how Mathematicians are taught? (Sir Timothy Gowers explains why Mathematician appear to be geniuses, although it is the mathematical thought processes which help them in solving problem. He gives three ways to appear like a genius. One- solve twenty problems, don't tell anyone about the other failed questions. Two- Make an accidental observation, make it sound deliberate. Three- Do a lot of work in private, don't reveal your thought process. If people look at your output and not the process, it will appear like a work of a genius, it'd be magic)
6. Can computers perform vagification (or crudely speaking-generalization or abstraction)? Can computers vagify two problems and make them so general that they can see similarities and then use the analogy to solve problems? (For example, we compare polynomial factorization with prime factorization of numbers then solve)
7. Can the computer bring in those little details which can be used to solve a problem? Can the computer figure out areas where interesting ideas/definitions could be applied? For instance, we apply the definition of a prime, that the factors are only 1 and itself, since ab=n, either a=n or b=n and so on, to solve a problem. Can a computer decide when is the right moment to bring in a definition to solve a problem?
8. Can the computer think of what we can do with the mathematical results? Can the computer think of what it can do with probability or algebra?
9. A computer can know a lot of facts, but can it link up different facts?
10. Can the computer teach itself to become a better Mathematician (self learning)?

LMS Talk of Prof. Timothy Gowers: https://www.youtube.com/watch?v=k_ordDFw588

PS: The author Tarun, is a Mathematician by training, is the Founder of sciensation.tv, an edumedia organization which is working towards scaling the MIT/Harvard/IISER approach to learning. Sciensation tries to get students to understand the thought process of scientists, economists and researchers in general. Checkout www.sciensation.tv to view Tarun's TEDx talk about research based learning and how Sciensation gets kids to think like researchers!