Dora (intralimina) wrote in creativescieng,

Creative Math

I've been purposefully drowning myself in math for quite some time now, after having done no formal math at all for about 18 years. This has given me a rather strange, distanced point of view on how math appears to be presented, or taught.

There are two things that seem to be taught: proofs (sometimes, if you're lucky), and steps for solving a problem. There is an emphasis on the latter, and an expectation that the student will learn those steps and use them. "If you follow 1., 2., 3., you will have the solution." This isn't just how things are being presented in the college calculus classes I'm currently taking, but I also recall it rather poignantly from my grade school and high school experiences years ago (most of which were frightful, despite the fact that I love math and understand it easily, b/c I usually need to invent my own solution steps).

It is curious to me, especially now that I am older and have a clearer sense for what works for me and what does not, that a key concept recognized in other disciplines is not recognized in mathematics--at least it's not being recognized at the academic level I'm currently at. That key concept is this: if a person has deep, real understanding and ownership of an idea, they can use that idea creatively. Conversely, creative application of an idea leads to deeper understanding. One is expected to make creative use of learned concepts in the arts, right? And even the graduate science course I am taking expects (and grades on) creative problem exploration. Why is the expectation in math that one will simply plug in values to a memorized series of steps that were invented by someone else and be happy with that? That that's enough. Why isn't there an emphasis on using concepts creatively to find an individually optimized series of steps for solving a problem, or using concepts creativity to get at the core of what is wrong or right with methods of solution. If a concept is understood enough to be used creatively, then that implies a much deeper understanding of the concept than rote regurgitation.

I wonder if it is because we are taught in grade school that there is always "one right answer" to a math problem that we get bogged down into thinking that mathematics is purely formulaic. The closer I get to more advanced mathematical concepts, the less I find this paradigm useful (not that I've ever found it particularly useful). What would be different if from the start children were taught mathematical concepts instead of someone else's steps, and were told to find optimal methods of solution? What if from the start children were encouraged to play with their math, to discover for themselves how each jeweled facet of it works?

Eh, maybe things have changed in the way math is taught in primary education in the 20-30 years since I was a kid, but still, I've been thinking about this a good deal. I'm not entirely certain where I'm going with it yet, but I thought the ramble had relevance to this community :-)

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I had my own way of doing multiplication as a kid, say 24 x 33 = blah.
It really was 'backwards' i think but I can't remember anymore. I was in the math olympics and over time, I was assimilated to use the standard conventions.

My solution "worked" in the sense that I could answer the problems given to me. I am willing to accept that the solutions didn't 'scale well', considering it might well have required some memory which would break down when doing say 4 digits X 3 digits... Too much to compute in a mind without loosing some precision.

Oops, did you just find a flaw in the system?
I think you did!

Is it always appropriate to use a Btree?
How about storing 10 numbers out of sequence?

Without challenging the ways that structure is formed, we discourage more optimal solutions, even if those solutions are for the individual.

Nice Post.

From my time of trying to be an Electrical Engineering Major (one semester), and since, trying to find out why I failed, I have found out this - the problem is, for most professors, if you are good at teaching, you get fired. If you are good at research that brings in grants/theories that get the university on the map/controversial subjects (math and feminism, maybe... mathematical proof that women are better than men, etc...)that get the university acclaim, credit, and grants (mostly grants...), you are tenured. This being the case, the thing is that a lot of university professors have only enough time to do one of these well. Thus, the one-size-fits-nobody approach of most of the "successful" ones. Good teachers stay as lecturers - low pay, no tenure, transient, at-will positions. Good/Controversial/Financially adept ones proceed to the top of the heap (as long as the money keeps rolling in...). Sadly, teaching, like medicine, should be an art, but has to be run as a business to survive in the competitive, cut-throat world we are in.

There is an entire field of study devoted to creative math in the way you imagine -- Ethnomathematics. For a more general look at how math can be taught as discovery, you might look in to Mathematics Education Research.
nice post.
I was debating along the same lines.
Also growing up, I hated math. I never met a math teacher I liked until college. My thing is if math were taught differently maybe more people would excel in it. People who are good with words might like math more if they learned the history and concepts behind math to essentially help them connect better with it.

My 0.2 cents
nice post. I was always good at math but found it uninteresting to memorize facts for the sake of getting a good grade on the test. I had to take pre-calc two years to graduate and now that I see how it is applied to physics, I find it much more interesting. I think creating steps to a known answer or solving a physics problem are the most rewarding applications of math. Here's a link to a physics series that explains calculus concepts and includes the history and people that came up with them:
thank you!