A recurring question which keeps coming up in discussions is how mathematics should be taught. There is a strong view that given the computing power which is available to us, we should relook the basic maths curriculum. So i went looking, and found this video which i feel looks at the problem in a logical way.
Conrad Wolfram is giving some compelling reasoning for why maths education should change, and gives a description of how it should change, too.
Hand-culcating the mundane way should not be the focus on the curriculum. Rather, the focus should be on building and enforcing the concepts, and leave the calculating to computers. In other words, as he says, students should be taught the procedures which define fundamental concepts, but the implementation of those should be left to computer programs. For example, students should know what a square root it, how it is calculated, but they shouldnt have to calculate, beyond illustrations. And here is the cool part he says … focus on teaching students how to write programs to calculate square roots, rather than having them mechanically execute the procedure for calculating. This will immensely help students clarify their concepts (how can one write a program without understanding the underlying principles to a very large extent?), while at the same time help them become more comfortable with the concept of application of these concepts. In other words, our mathematics curriculum should stress understanding and application (application to real world problems is a very good way of teaching these concepts) rather than stress on the mundane calculations which stress out students as well as parents alike. After all, why should a child lose marks in an exam (thats what happens) if he or she takes the square of 5 to be 10 when all the conceptual aspects of the solution are correct, and the only mistake is a calculation mistake?
Connect this with the post i had written earlier, and a rather innovative picture of mathematics teaching emerges.
Thats a somewhat crazy question. I came across this video, and i would say its quite a nice watch.
This video makes a compelling case for the why and the how of teaching maths in a visual way. After all, maths is not about words or languages, is it? If so, why should mathematics education be so language heavy? It should be about the concepts of mathematics. And i have found that the visual impact of mathematics is quite powerful. I tried this trick with my then 10-year-old … I introduced him to the concepts of integers, and multiplication with negative numbers without bringing in negative numbers to begin with.
We started with directions. Left and right. So, theres 3 to the left and 2 to the right. Or, 4 to the right and 5 to the right. Now, adding the 4 to the right and the 5 to the right is easy, but how does one add 3 to the left and 2 to the right? This is where the concept of the negative sign came into the picture. And once this was done, then it was a simple extension of this to see how multiplication with negative numbers simply changed the direction and nothing else.
What this did is help him build a mental picture of these concepts. In my experience as a trainer, i have found that these mental pictures are far more enduring than theoretical concepts. To take an example, i used to teach the concept of min-max planning, or the sawtooth curve, using the analogy of mom planning to go out and buy rice. Now, a few days after the class, the students, even if they had forgotten the min-max planning theory, or the sawtooth curve, still remembered the rice story, and this helped them to relate to the concept. The rice story here helped build a mental picture which is more enduring than theoretical concepts. So this isnt just about children learning mathematics, but also about adults learning.
Now, knowing him, he would have built a picture of Ben10 doing something which saves the world from the wildest aliens imaginable while multiplying two negative numbers, but hey, thats a picture i can live with!
Oh by the way, where there are integers, can vectors be far behind?