Mathematics Learning: Visualising for Success

[verty]

I've come to realize lately that I seem to learn differently from mathematicians, which seems odd because I did really well at math in school. Throughout my 12 years of school, I was top of my standard (grade) in mathematics.

But I've realized that I am unsuited to mathematics by proof. Looking back, I entered an olympiad in grade 7 and came first in my 1/10 or so of the country. But in subsequent years, I didn't do too well in olympiads. I hadn't thought too much about it, but I've realized that the reason for the difference is because that grade 7 olympiad was full of questions that one could visualise.

It had such questions as rolling a dice(die) or calculating the volume of water in a swimming pool, but the later olympiads had more theoretical questions like solving tricky equations.

Now recently I've been trying to learn mathematics but I seem to struggle when the discussion is very abstract. Especially where proofs are concerned, where something is proven and then used to prove something else, I have tremendously difficulty following it, or even staying interested.

When I build a puzzle, I pick up a piece and study the picture to locate where the piece goes; I then place the piece on the table in the correct spot. This is how I have always built puzzles and I seem to be pretty good at it.

So I'm thinking that the same should apply to how I learn, in that I study a piece with reference to the picture to see where it goes.

If I must build the sky in a puzzle or some such with little assistance from the picture, my usual method is to separate the pieces based on their shape. Some might be longer than wide and some wider than long, or some have cutouts offset similarly. Perhaps some portion of the sky is darker and some lighter.

Then having grouped the pieces, I choose a group and try the pieces one by one where I think they should go. As I place the piece, I reevaluate where the most likely place to find another fitting piece is. Perhaps by placing the piece I notice an unusual shape has formed on one of its edges, so I can then focus on that detail to place the another piece.

But nevertheless, this is very much slow going whereas I am much faster than that when I have the picture to aid me. Building in this shape-fitting way seems to me to be vastly inferior.

This shape-fitting method also presupposes that one can see all the pieces and group them so as to know where they might likely fit, so if one is not fortunate enough to have the picture or a suitable picture is not available, one can benefit greatly from seeing all the pieces and being able to group them.

Certainly if one was given one piece at a time, it would be very tough indeed because any piece might not fit yet. Perhaps if the pieces were organised such that they do fit in that order and the task is merely to figure where, the task is then one of mindless effort. All that can be learned then is dogged determination to exhaust all permutations. No insight goes along with it.

I supposed it might be hoped that after the puzzle is built, one could reflect on the pieces and notice that they can be categorised, so that any future assembly is much easier, but it does seem that this piece-wise learning is sure to be inefficient.

As for myself, I seem to get very distracted if I must do redundant work. Assembling the puzzle piece-wise by sole virtue of mindless effort and the pieces being furnished in the proper order is almost certainly bound to be highly frustrating and counterproductive.

Perhaps having paid vast amounts of money to enter a study program at a prestigious university, a student has enough motivation to battle through it, but again this can't be the best way to build a puzzle. If a picture is available, why hide it?

Is this analogy fitting? Is it that students are fed pieces in fitting-order by their lectures/professors/course designers, and expected to battle through the mindless process of fitting them, to reflect thereafter? And if so, is this not an inferior method to teach the subject?

If we were to teach puzzle-building, I imagine there would be three methods taught. One would be when one has a sufficiently detailed picture and a significant number of pieces can be placed piecewise by determining where they fit in the picture. Another would be to be used when no picture is available or the picture is insufficiently detailed to locate pieces piece-wise, and would emphasize grouping the pieces by similarity so as to reduce the number of permutations one need try. The third method would surely be when one has neither a sufficiently detailed picture, nor the ability to see a significant number of pieces. The third method would be mindless repetition whereby one simply tries one piece after the next.

I would think these methods would be applied in the order I have presented them, not the other way round. Surely the way to build the mathematical puzzle is to study the picture first of all, placing piece-wise as many pieces as one can, then to group the remaining pieces however one can to better optimise one's progress, and finally at the boundary of research to simply try whatever piece one fancies in whatever way.

Would this be a fair application of this puzzle-building analogy to mathematics? If so, are there recognised authors/books/academies that follow this approach?

Have there been mathematicians or similar folk who have not been adept an progressing by proof, but have yet played a pivotal role in the history of mathematics?

 

[matt grime]

I don't think that linking 'proof' with 'piecing things together piece by piece' is at all accurate. Or, more precisely, what makes you think that maths is special? Any other science is like this - physics, chemistry et al all have progressed by this means. Or perhaps you see, for some reason, that organic chemistry pertains to high energy physics in some way?There are certainly famous and great mathematicians for whom the modern method of proof would be very alien. Indeed all of them from before the late 1800s, such as Gauss, Newton, Leibnitz, Euler, Euclid... pretty much everyone, in fact.

The view you have of mathematics, and its learning, is definitely one rooted in the late 20th, early 21st century.

You are also suffering from the misapprehension that the way things are presented in a textbook is the natural way one would have discovered things. Books are written with hindsight - the material in them appears in reverse order of discovery, approximately. Of course, if you're basing this view on something like Finney's calculus book then you have an even less realistic idea of mathematics since books like that bear no relation to mathematics. Start with Polya, and Gowers (google is your friend), as well as Erdos's biography. Aigner and Ziegler have a nice book, though it is not as elementary as it seems to think it is, unless you happen to be Hungarian.

 

[verty]

You are also suffering from the misapprehension that the way things are presented in a textbook is the natural way one would have discovered things. Books are written with hindsight - the material in them appears in reverse order of discovery, approximately.

Thank you, Matt. I should perhaps try reading the latter parts first, or skim through a book before a more focussed second reading.

It is good to know that mathematicians don't all think like computers.

Thank you for the book suggestions, I'll look into them.

 

[verty]

By Polya, did you mean "Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving"? It's a bit expensive ($120). Hmm, I see there is a book by Lakatos, "Proofs and Refutations" for a third of the price. Perhaps that would be worthwhile.

 

[verty]

Interestingly, that "Aigner and Ziegler" book is one I had already ordered. I'm waiting for it to arrive. I'll also be buying that Lakatos book and "How to read a book" by MJ Adler, because obviously there is skill in reading difficult books well.