I have to admit that I have a somewhat dismissive nature sometimes, and have been known to make critical remarks for non-public consumption; “Programmers cannot do math at ALL” (hi Dale!) Of course, this is more out of shock than a belief in any natural order. I firmly believe that anyone of everyday intelligence can learn math. I am of the opinion that I could teach calculus* to a dead twig if the twig where sufficiently motivated.

Which is why I was happy to see that someone had written a blog entry on learning math being what you make of it. The only thing I would criticize is that he is completely wrong.

Ok..just kidding, but I do have to say that I would not follow his councel on exercises. When I read a GTM on a new subject (papers don’t usually have many exercises :), I don’t really view the problem sets as seperate from the explanitory text – I do every single problem. This is because the author of the text did not view the problem sets as seperate either. It goes beyond “having been shown the idea, cement it in your mind with the excercises”. Most of the time, realizations that the author wants you to have are set up in the problem sets because they would be TOO padantic and verbose in the main text.

*This isn’t restricted to calculus of course.

You say, “I firmly believe that anyone of everyday intelligence can learn math.” It would seem that there is no room to argue with this statement. We all have pretty much the same basic brain and the same basic cognitive tools at our disposal. However, the vast “non-math” public doesn’t believe that they can learn math and there appears to be a certain amount of pride attached to the idea of being bad at math. So tell me how you’re going to convince someone that they can learn math and that the learning is valuable.

I have no magic answer to that. It is a chasm that is hard to bridge, and given todays climate of attack on science and rational enquiry by fundamentalist in this country, one that may be unbridgable.

What is it a person is learning when he or she “learns math”? I think there are two aspects they learn: One is an ability to solve certain types of problems in a certain way. The other is a certain way of looking at the world (or that part of it being studied). In any particular mathematical domain, both aspects are needed, although one may be more to the fore than the other. Although both are needed, students may be better at one aspect than the other in any particular domain. And some domains (eg, category theory) focus on understanding more than on problem-solving.

I believe, from my personal experience as a student, that it is possible to learn how to solve problems in the approved manner, yet still not view the world in the supposedly-correct way. I was good enough (or hard-working enough) at solving probability problems to end up with an honours degree in mathematical statistics, yet I still could not get my head around the standard account of probability theory. It was only 20 years later that I discovered that lots of other people had had similar problems with the standard account, going all the way back to Leibniz. Statistics departments are usually not very good at informing their students of this dissident thread in the history of their discipline.

I reject the standard probability axioms (due to Kolmogorov) for models of uncertainty applied to real-world decision domains. Does this mean I haven’t yet learnt probability theory?

At least in the US, learning math seems to be only that aspect that covers the ability to solve problems in the accepted ways. In order to “pass” the boatload of standardized tests now used to determine if a student has acquired knowledge (and thus, determine the quality of the school), the only available course of instruction involves problem solving. Learning to think analytically and rationally takes time and is not easly tested for. The climate in the US today with its attacks on science (as pointed out by Michael) does not encourage this type of learning either.

I wonder how frustrating this is for the college math professor who is faced with a classroom full of students who have learned not to think.

I barely managed to pass an undergraduate course in probability with a C, so it’s clear that I haven’t learnt and don’t understand the subject. No way am I touching the question of whether someone else has.

Want to say something more on that? I’m a great believer in using whatever mathematical tool does the job so I’m curious to know what tools you use.

Sigfpe â€“

Let me try posting this again, as my inequalities were unfortunately interpreted as HTML tags.

A key problem with probability theory in real-world decision-making domains is as follows: Suppose you have some principled method of assigning probabilities to a statement S on the basis of evidence, E. For example, S could be the statement that a patient has a particular disease, and E is the medical evidence which supports such a diagnosis. So, having this principled method, we can assign a probability Pr(S) to S on the basis of E. We may also have some other evidence Eâ€™ for the negation of S, not-S. So we can again use our principled method to assign a probabiltiy Pr(not-S) to the statement not-S on the basis of Eâ€™.

Now, most of the time in real-world domains, our evidence is neither complete nor even consistent. In the case when it is not complete, we may have:

( Pr(S) + Pr(not-S) ) less than 1.

In the case when it is not consistent, we may have:

( Pr(S) + Pr(not-S) ) greater than 1.

In other words, we often donâ€™t have the standard Kolmogorov axiom:

Pr(S) + Pr(not-S) = 1.

The problem arises because our information about the world is messy and partial. A common probabilist approach to this problem is to force Pr(not-S) to be equal to ( 1 – Pr(S) ), i.e., not to use the principled method to determine the value of Pr(not-S). But this strikes me as incoherent and arbitrary. For if one had instead started with not-S then we would likely have different values for these two probabilities.

What alternatives are there to probability theory? Well the two main ones are Possibility Theory and Dempster-Shafer (or Belief Functions) Theory. You can find an excellent general mathematical treatment of these, alongside probability theory, here:

@BOOK{klir:wierman:book98,

author = â€œG. J. Klir and M. J. Wiermanâ€,

title = â€œUncertainty-Based Information: Elements of Generalized Information Theoryâ€,

publisher = â€œPhysicaâ€,

year = â€œ1998â€³,

volume = â€œ15â€³,

series = â€œStudies in Fuzziness and Soft Computingâ€,

address = â€œHeidelberg, Germanyâ€}

Alternative formalisms for representing uncertainty is a very active area of reseach, driven now mostly by people in AI, following various contributions from people in law, medicine, and economics.

Pingback: Technomadic » Adding Up Numbers is Very Fun

When I read a GTM on a new subject (papers donâ€™t usually have many exercises , I donâ€™t really view the problem sets as seperate from the explanitory text – I do every single problem.

I’m curious … how long does this take you for a typical text? Especially a springer “Graduate Text in Mathematics”, which is what I assume GTM means.

When reading a math book, I’m always torn between a) wanting to do as many problems as possible and thus learn the material very well, and b) wanting to cover more material. In theory I want to learn everything in depth, but in practice there is only so much time available.

It depends on the subject, and on how much free time I have to devote to it. When I worked my way through this book it was only a month or so, maybe two. When I worked through Jack Lee’s Smooth Manifolds text, he was writing the book as we where taking the course from him, so I was essentially proofreading as well, and that took something like 9 months. Hey…I had two other classes.

My point wasn’t that YOU must do every single problem in every single chapter. My point was that the excercises are often more than half the books content, and you should not ignore that fact. You should do as much as you can possibly stand.

I absolutely agree that the exercises are a critical part of books, and it is important to do them. But it takes time! Have you found that as you learned more math, it becomes easier (or at least less time consumgin) to learn more math? I have found in other subjects that concepts and approaches to solving problems transfer from one area to another.

I think that what I probably need to do is work hard to really master some basic material (in particular analysis and abstract algebra), to give me a solid foundation to build on. My foundation now is really pretty weak.

Again, it depends on the subject. In my experience, you gain understanding by surrounding the new knowledge with other aquired knowledge, sorta like capturing stones in Go. But this only reasonably applies to knowledge at the same “level” in the pyramid, and there are ALWAYS new abstractions adding levels. So the more you know, the faster you can build out, and even down, but up always seems time consuming.