Talk:What makes Mathematics hard to learn?


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experiental learning

What you are describing here is experiental learning versus "normal" performance based learning. There is however one small problem with experiental learning : it does not work.

It sounds very well. Just get students interested. Well, unfortunately, every single test it again turns out that the performance of students in actual problems ... worsens. This is logical if you consider development psychology. It's the same divide as between capitalism and communism really. The basic problem is this : learning is work, (hard) work, and yes some people do the work with very little direct pressure (these people are obviously overrepresented in academic settings, so studying college kids radically schews the data).

However people do *not* want to learn. Unless direct pressure is present, they will not become capable of solving problems. Unless there is a "threat" involved, people do not become problem solvers. Mostly these threats come from parents ("if you don't learn you'll spend your life flipping burgers and crying yourself to sleep every night"), and this works very well. Given this pressure and a teacher that can answer direct questions, basically any book will do.

Given the absence of this pressure, no book, no laptop, nothing at all will make a significant population learn. (ie. except the 5% or so "geeks" who will do it anyway).

This is the same as what happens in "social justice" (communism) : a few people, who get truly fascinated by a subject, will still achieve moderate results (and yes, sometimes even exceptional results). The large majority, however, will not, and will start using social pressure on the few achievers to hide that they excel at these subjects. This pressure becomes larger over time and may spill over into physical violence.

Books are not the problem. 95% of the problem is parental attitudes, another 4.99% is teachers forcing "equality" (forcing people to be equal ... equally bad that is) on people and 0.01% is dumb teachers (less knowledgeable about the subject they're teaching than the students before them).

This is one of those things that seems like a good idea, and is a VERY bad idea in practice. Trying this can literally destroy the learning potential of thousands of people, making absolutely sure they will never succeed in academia, because (a) they don't understand the basics of their own subjects at 18 years (b) they never experienced the hard work->understanding and better marks thrill, therefore they see it as nonexistent and thus are not prepared to fix (a).

The potential of the olpc project is to enable geeks to learn in any population, thereby (potentially) creating a set of people that can function as teachers for their peers. But olpc cannot replace teachers, and will just be thrown away unless people are warned IN ADVANCE of the consequences of not learning (the world will make sure that there are consequences, so you cannot fix that), but a 2 year old does not understand the use of letters. A 10 year old does not understand the use of mathematics.

high school anecdote

I had a friend who, one year behind me in high school, was already very talented at mathematics, excelled in math competitions. after he had taken first-year high school math, he was taking geometry. he didn't like it. Every year we would have the same conversation, as he tried to find reasons to drop or stop studying for his math courses.

10th grade : why should I learn this geometry? I already know everything I need to about algebra and trigonometry and mathematics - everything my calculator can do, very useful stuff. why should I learn this? it's a waste of time.

11th grade : why should I learn this calculus? I mean sure, algebra and geometry are useful everywhere, and this is an abstraction above those -- and taking derivatives and integrals to evaluate some basic physics. But a few weeks in I know the basics, most of it is mechanical -- why should I learn this just to be 'educated'?

12th grade : why should I learn abstract algebra? How could this ever have meaning in my life? I already know algebra and geometry, which use abstractions, and calculus and function analysis is useful in lots of my science work, but this is just manipulating sets -- I already know how to do anything that might come up in a real world problem. Why should I spend time studying this?

By the end of each year, he was the strongest proponent of how interesting and useful and enlightening a math course was, and could rattle off a dozen ways in which it helped improve on or influence work in some of his other courses (he went on to study cognitive science).

On the one hand, he is right -- today he doesn't use any of the above math on even a weekly basis. On the other hand, it took him about 6 months into the year before he was loving what he was doing and seeing relevance in everything else he studied. The teachers and books never inspired that in him, however, he had to internalize it and start seeing the world differently before he found inspiration to carry on. --Sj talk 22:06, 26 February 2008 (EST)

College Anecdote

In college, to fulfill a degree requirement, and depending upon your entrance test scores, you had to either take 3 courses of 'general math', which duplicated subjects like 'long division' and 'balancing a checkbook', or complete an arc of a higher level set of classes. I had tested into the Algebra program. We spent a major part of the semester reducing complex polynomial equations to their simplest form, first with two terms, then with three, and so on. After weeks of this, one student raised a hand in class and asked "Dr. Wu, What is this FOR? ...I mean; when will we use this, and for what purpose?" The professor paused, cocked his head in thought for a moment, and said "You don't need to worry about that now."

This is, again and again, the error in Mathematics education: poor pedagogy. Either you have math teachers teaching directly from the book at the limit of their skills, or professors who find the subject beneath. Or some mix between. Only rarely have I had a teacher who would answer such a question properly: to explain the next step until the questioner and class were struggling, then say "and this part that's confusing now, it relies on what we'll teach in two weeks. Then this will make sense." It's those teachers, the ones who give you a goal, or a narrative structure to the problem at hand--even if it's a challenging one--that should be teaching.

Great essay!

I love it! I felt parts of my brain coming to life after years of dormancy just reading through it once! We still have to figure out what this means in terms of what software is built for the XO and how teachers and students use it.

I believe that praxis of software development and of education has to come from a close relationship between the teachers, students and developers. The hardest part so far seems to be finding a meaningful way to communicate between those groups. Maybe a linguist (Chomsky?) or someone versed in educational training (preferably Freierian style) can help find ways to put these ideas in to software and practice in schools.

In any case, we're strongly engaged in the deployment of XOs and building relationships with teachers We'll keep these ideas in mind when we get to discussing the ways teachers work and students learn on a daily basis.

I can't wait to read the other essays (right after I get through my next couple of hours of work drudgery :-)


Gregorio 09:51, 2 May 2008 (EDT)

I just read an Alan Kay article that says essentially the same thing including making the analogy between learning and Topology!

As it happens I'm reading the book "Poincaré's Conjecture" about topology.

Going way out on a limb here's a new conjecture.
Consider a child's understanding of an idea as a "learning manifold". The child's cognitive maps are "charts" of her own learning manifold on to some other manifold which is being learned (call it the knowledge manifold). All the charts making up the atlas of the child's learning manifold represent the child's full understanding of the knowledge manifold. My conjecture is that the manifolds of all children are homeomorphic to each other.

If we could find that homeomorphism between children's learning manifolds we would have a deeper understanding of what learning means.

For knowledge manifolds, if we could find a simply connected compact manifold without boundary to which they are all homeomorphic, we would have a deeper understanding of what knowledge means!

That obviously needs a lot more detail starting with a definition of learning concepts that can be used mathematically (e.g. what and how many are the dimensions of the manifolds?).

Also, I think a manifold is defined by its relationship to Euclidean space. So the problem of defining the "knowledge manifold" for an idea may be the same as finding the Euclidean version of the idea (like its Platonic ideal). That's not very satisfying as I believe all cognitive maps are meaningful and there is no ideal one. Please correct me if I'm wrong about the relationship between the concept of "manifold" and Euclidean space. At a minimum the relationships seems obscure in dimensions greater than 3.

Just thought I would share a Friday afternoon idea generated by all those neurons and axons which the article brought back to life :-)

Gregorio 13:58, 2 May 2008 (EDT)

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