If you noticed, a significant of aspect this blog is digression all over so I thought it is only natural for me to write about it. It turns out that one of the main reasons why math is ‘hard’ for those who feel that it is hard is that they learn math the hard way which is why it is hard. That doesn’t obviously sound like a thing, but the truth is that it is as good as any other reason. Let’s take an example.
Many of us take the most basic courses in college like calc, linear algebra, maybe discrete math as part of a ‘requirement’ for the degree that you intend to get in the future. ‘Requirement’ is the keyword. Imagine that you want to buy a car or bike, the first thing that one might consider doing is to learn how to drive or ride one of those. This seems to be somehow very obvious to all of us because, well because we all agree. In the same way, if one wants to get a degree in say biology, one should know how to make simple calculations like the growth of an organism, size etc.. But the problem is that the basic math courses are offered by the math department. In other words, you learn math from people who make their living out of math. So what? Now it puts an immense pressure on the teachers on how to teach these basic courses. Why, they’re after all experts duh? It turns out that in most cases, the courses are specifically designed so that if one takes these courses, they will be able to do specific aspects of computations in their fields, that is, biology or engineer or whatever. Hence as a mathematician, they are forced to dumb down a lot of things, that is, they skip lot of details, exams become hard because they don’t have enough time to cover everything in the class. The people who go to office hours and ask questions find this out very soon and manage to do well. If you do well, often you don’t hate it, it is only when you spend some time on math and you don’t well when the hate becomes real. One specific example is the Linear Programming course at UW Madison, it always excites me whenever I talk about this course. First of all, it is a requirement for any graduate student in Industrial Engineering whereas it is counted for credits for math, cs, stat and maybe many others. Well the course is an application of linear algebra basically which is a course that all the undergrads in math, stat, cs, engineering take. But the problem is that the cs, engineering don’t use linear algebra in the way that is expected of as a prerequisite for the Linear Programming course, for example, they might have never worked with words like null space, image space, span etc. in their entire life. Now this is a problem because the basic component of one of the basic algorithms called as the simplex algorithm is just entirely based on this. This is not the case with the other department students usually assuming they have taken few other math courses, because either they actually took some more linear algebra courses or because they got used to definitions. The second reason is the secret ingredient because the first reason is like any other skill, once we spend enough time on something we know about that thing in some detail (hopefully). The second reason is not a direct reason and this is how one can learn math. I can quote lot of mathematicians, for example, Neumann – ‘Young man, in mathematics you don’t understand things. You just get used to them’. This is not a really good statement in some sense and I wasn’t comfortable with this. But a more matured version of this statement (or what he really meant) can be seen from Alexander Grothendieck who said, ‘‘.
Probably I should have put that quote in the beginning. Let’s assume that we would like to crack a nut. One way to do this is take a hammer and try to break it and the other way is from Grothendieck. When you take a math course to apply or use it in your field, then mathematicians are forced to teach it in the hammer way, that is, they teach math as an array of tricks that can be used to solve problems of a certain type and this array seems so artificial which either makes one wonder how they came up with it (and the one starts liking it) or one starts to hate the subject. (Again, this is not their choice or how they want to do it, it is because they are forced to this.) Also, the nut becomes something special or in other words we don’t need a hammer to break a fruit whereas when we are trying to crack a nut, it becomes extremely important that we use a hammer to open or we will never be able to see what’s inside. What we are trying to do in this blog is the other way, that is, I deliberately digress, you can think of it as surrounding the nut with water. This process is slow but eventually it will pay off in the sense that we can think of hammer as as certain type of liquid that can penetrate much more. We can also do things in a much more optimal way since we can just use the hammer at places that are hard for the liquid to penetrate. This means that the idea of liquid is much more powerful than the idea of hammer which is what we are aiming for because then we learn about the nut in a much more detail way in one shot and as usual everything is nice and beautiful.