Archive for August 8, 2009
At Alan Kay’s suggestion (in the comments to the Geek Gene post on spatial and visual intelligence), I picked up a copy of The psychology of invention in the mathematical field by Jacques Hadamard. Hadamard was a mathematician who was also interested (inspired by a lecture by Henri Poincare’) on how mathematicians did what they did. From where does invention come?
Hadamard addresses interesting points about the role of imagery in mathematical thinking, of subconscious, and of logic vs. intuition. Hadamard spends much of a chapter considering mathematical thought without words, and those (few) who argue that the words are how they think (Max Mueller). Hadamard solicits input from several prominent mathematicians (including Albert Einstein) to support his theories. For me, one of the major take-away points was that mathematical invention seems to be a process of synthesizing a large number of possible solutions (perhaps subconsciously, perhaps through logic or intuition) then dispensing of most of them (again perhaps subconsciously, or perhaps through a sense of aesthetic). For Hadamard the role of mathematical symbol or words is to provide specificity — he is already completely convinced of the solution when he tries to specify it in symbol and word.
Hadamard describes invention, the process of a person coming to understand and solve a problem whose solution is unknown. In education (at least, school-based), the goal of the teacher is to get students to understand solutions to problems (concepts, skills, facts) that are known. Occam’s Razor would have us start with the assumption that these are the same cognitive processes in the mind of the inventor or learner. The goal for the learner is the same in both cases, to come to understand something, and it seems a stretch to imagine that the cognitive process would change depending on whether someone else already knows the answer.
If Hadamard is right, the challenge of supporting student learning by the teacher is to provide students (a) with the opportunity to understand competing solutions for the same problem and (b) the tools and techniques to choose between them. I asked Nancy Nersessian, a cognitive scientist who studies how scientists, engineers, and mathematicians come to know, and she says that Hadamard is well-thought of in that community — so there’s reason to work from the assumption that he is right. What’s involved in getting students to understand competing concepts, skills, and facts, then giving them the tools to choose one?
- Alan asked in the comments to my last blog post, “What is the equivalent of molecular basis of life, how and why chemistry works, and why evolution should be plausible — that cannot be omitted from a first course?” Students need to know enough that they can recognize a potentially correct answer as a potentially correct answer. They have to be able to see that the Web didn’t have to come to be as it is, and that there are several ways to design a language or make an operating system.
- Students have to understand the practices of “coming to know” used by scientists, mathematicians, engineers, historians, social scientists, and practitioners of other fields that we want students to learn about. This is exactly what Science for All Americans suggested 20 years ago, that students should learn science by doing science and come to understand how scientists make decisions.
- History plays a really useful role here. Through history, many competing ideas have been generated, and those have been filtered out to the best competing ideas today. Teaching the history of an idea might be a great way to teach it.
- The effort to understand is a large one. We have to pay attention to how we sustain motivation and keep students engaged through understanding alternatives and choosing between them.
This process demands a lot of effort on the teacher, and will take a lot more time than simply lecturing facts at students. We should only do this process for the things that we really want students to learn. Since this process will take so much more time, we will need to find some time — we better start removing those things that aren’t so important for students to learn from our curriculum. That’s a good thing. National Research Council and TIMSS reports for over 20 years have argued that the US curriculum is “a mile wide and an inch deep” (see this reference to this complaint in 1989). If Hadamard’s process of invention really is how people come to understand (not just parrot back on a test) anything, then we have to spend this much time on students’ learning. Spending less time doesn’t really lead to learning.
Hadamard published this book in 1945. I find it amazingly prescient that the implications of his findings are echoed in the work of cognitive and learning scientists up to today. I find it amazingly disappointing that these same recommendations have been made for over 20 years, and yet we still cram so much into our CS1 courses that any real learning is due to outstanding personal effort by the students. If we aren’t spending this much effort on facilitating student learning, we’re not really teaching.