In high school I failed terribly at Algebra. I failed 3 out of 3 years. The only way I got through it was summer school and my godmother tutoring me (she taught 4th grade math at the time I believe.) All three years I had the same Algebra teacher. The way I remember it being taught was; here’s the problem, here’s how the equation is supposed to go, you try it once or twice with me, now do your homework and there will be a test at the end of the week.
I always felt left behind and I never reached comfortable understanding with the work before moving on to the next thing. Why we couldn’t just slow down so I could catch up on what was already covered was never explained to me. What did I actually learn in Algebra class? I just didn’t have what it takes and Algebra was just something I did not have the ability to learn. I know school systems are driven by unfair expectations and that curriculum are not always as malleable as some teachers would like in order to actually ‘teach’ to the result of an educated student. But I feel I was a victim of this described situation (as are I suspect most people reading this are, in one capacity or another.)
I did great with Geometry in my senior year though. A lot of special reasoning and a different teacher; apparently I do good with that sort of thing.
So where am I going with this? I was recently discussing with my sister that my near future was going to be full of math of all kinds as I pursue an Engineering Technologies degree. I was forced to think about how those High School algebra classes were presented to me and compare my experience then to my next 15 years as an ‘adult’ and how I have grown in confidence in my ability to learn and perform hard tasks. I am at peace with math and the hard work ahead of me.
So I was reading Stephen Pinker’s, How The Mind Works today and am on the chapter Good Ideas, specifically taking about the evolution of our understanding of logic and consequently mathematics. I want to quote some lines that stood out to me in light of the above mentioned ruminations.
Speaking about our consistency of making certain errors and the difficulty of conceptual ideas framed in math and how best to overcome these shortcomings: “With enough time and patience, we discover why our own logical errors are erroneous. We come to agree with one another on which truths are necessary. And we teach others not by force of authority but socratically, by causing the pupils to recognize truths by their own standards.”
“Evolutionary psychology has implications for pedagogy which are particularly clear in the teaching of mathematics. American children are among the worst performers in the industrialized world on tests of mathematical achievement. They are not born dunces; the problem is that the educational establishment is ignorant of evolution. The ascendant philosophy of mathematical education in the United States is constructivism, a mixture of Piaget’s psychology with counterculture and postmodernist ideology. Children must actively construct mathematical knowledge for themselves in a social enterprise driven by disagreements about the meaning of concepts. The teacher provides the materials and the social milieu but does not lecture or guide the discussion. Drill and practice, the routes to automaticity, are called “mechanistic” and seen as detrimental to understanding.”
IUPUI Math labs seem to be organized in this manner. That’s why I think my sister was so successful in her math classes. That and her husband is an Engineer… Now he’s my new tutor. :P
I have long thought that was the case. I was lucky with math in that, as a child, my father would play algebra games with me, so when we finally got to algebra, I never found it to be that foreign. And I remembered how fun he had made it. If I hadn't had that background, though, I would have run aground, as the way it was taught was certainly not conducive to learning. In retrospect, it was that way with most of science. My dad loved physics, so we were always playing with physics and I never had any problems with physics class. But chemistry was really hard on me, as I'd never played in that field before. I grasped the concepts quickly, but by test time, I'd forgotten the rules and had to re-create them from scratch. And my school wasn't that bad of a school.
ReplyDeleteAnd I think a lot of it has to do with long summer breaks. Germany doesn't do that. And, therefore, they can build off the previous year's knowledge, rather than having to go over details that were covered previously. No refreshing necessary. And, because they can just keep building, they can actually go a bit more slowly. Mind you, their system has a number of problems. But there's a lot to be learned from it.
I would agree that with regard to most learning, and definately math in particular, the process of learning is much more about learning 'why things are' rather than 'what things are'. Specifically, learning how different mathematical idea connect and work is far more important, and useful, than learning how to 'do' any one or set of ideas. Aside from being more interesting, it is also more useful since if you truly understand the idea you can apply the concept in many situations, rather than only being albe to use it in the limited sense you were first taught it.
ReplyDeleteAny idea is more powerful when you internalize it -- when YOU understand it, rather than when you are taught it. So, i would wholeheartedly agree that learning math socratically is far more fun and useful than with rote memorization.