I am not very acquainted with the philosophy and psychology of the learning and doing of math. However, it gets more and more apparent to me that mathematics is built on three fundamental brain activities. Each of them has nothing in common with the other, and the links between them are very loose.
- Visual – where I include all internal or external views and images with mathematical content. The most obvious examples are graphs of functions, and geometric figures. But you may also include Venn diagrams in set theory, commutative diagrams in algebra, or any arrangement of mathematical objects in diagrams, matrices, graphs etc. In calculus, include vector fields, tangent planes, gradient vectors, or the derivative of curves depicted as vectors.
- Arithmetic – including symbolic algebra, and calculus. There is not much to explain here, since we all have been tortured with pages and pages of computations during our school days. Arithmetic is, at least on a lower level, what a clever program can do for us.
- Logic – as the general question, why mathematical things are as they are, and why they have to be that way. Logic is necessary as soon as we ask for the general result, instead of a specific one. The answer is derived by showing that certain assumptions necessarily lead to certain consequences, in each and every model example one could come up with. Logic is also used and abused to construct things, which go beyond intuition.
It seems to me that each of the above techniques have to be learned separately. Doing math only by images and pictures does not help in computing things, and it can only be a vague guide in proving. Logic alone is bound to fail, if there is no imagination of the general picture. E.g. in synthetic geometry, a geometric picture helps to understand the situation, but does not automatically yield a mathematical proof. In topics like set theory or set theoretical topology, you are left with logic, and there is only little visual guidance.
It is obvious that logic and visual mathematics are related to different brain activities. But I think that arithmetic is also something different. Since we have now computers which can do a lot of arithmetic, it seems to be an activity on a lower level. It is guided by logic, or at least it should be guided by logic. For we want to know why we are allowed to perform a specific arithmetical step. Of course, arithmetic can also be done on a very high level. Then it is an artful juggling with patterns rather than a blind schematic activity. In any case, it is connected to pattern recognition, with patterns in formulas of non-geometric content.
If that is so, it is no surprise that mathematics is such a difficult object. The need to learn three different and distinct techniques, and to switch between them in a masterful way is clearly a challenge. Good mathematicians are good in all three areas in a balanced way. Students concetrate too much on one aspect. Not only is mathematics based on these three different pillars, but it is also a very huge amount of content, a big house to live in, split in areas inhabited only by experts. Each semester I am surprised how much information I pass to the students. It cannot be expected that they know and master each definition and theorem, not even the best students can.
School mathematics was long built on arithmetical skills too much, which is what the not so brilliant students seem to like. Things are changing, however. With the explosion of the computer power available to students, a simple schematic problem solving strategy is no longer adequate. At least, we should pose more open problems, where there is a choice of solutions, even if each of them is still schematic. Moreover, open problems force the student to make a visual sketch of the situation. If the problem is such, that it also forces a logical decision about the right or wrong way, even better.
And finally, all arithmetical activity in schools should be linked to logic. If possible, force students to reflect about each step of a computation. Let them add the correct implication during the computations. Contemplate with them, if a verification by insertion of the result into the question is absolutely necessary, or if it is just a way to make sure the result is correct. This is far more important than double underlining the result.