How To Teach Mathematics
Posted: Wednesday, April 08, 2009
by Mark Sanford
AlphaLane.com
I once asked a math major what integration meant. The question must have caught her by surprise since she stopped, looked at me and said "I don't really know, but I did get all A's in Calculus."
Math teachers are notorious about introducing a concept with abstraction before explaining the relation it has with real life. Take the first day of a typical college Analytical Geometry class; the teacher introduces the class by drawing a graph on the board and asking, "Is that a relation or a function?" Relations and functions have different properties and the question is legitimate, but ‘you lost your audience professor' – except for the top 3%. This is because most of the students have no concrete concept of functional or non-functional graphs. First, explain what function vs. relation means and then let the class try and create both.
Mister ‘Math' could not solve this one and it was frustrating! I was getting a taste of what a lot of students dealt with every day. What really made an impression on me was what the teacher said at the next meet, "If you did not draw this problem, you can not solve it!" He, then, proceeded to draw each object in the problem and, visually, add and subtract material from each while writing the appropriate equations. I sat there thinking how simple this problem is when you can see it and how difficult it is when you cannot!
This is why we lose our audience when teaching Math, whether it is grade school, high school, college or advanced problems. If you do not first teach your class to draw and visualize concepts, they will never get it! The best Math students have the ability or have trained themselves to visualize what is being asked in real time and real space. As an example, if you asked even a third year Math college student what the third integral of a closed graph would be, they might be at a loss to answer unless they were trained in special visualization of concepts.
If teachers really want to jump start their classes in Math and get most of the class on board, the school should make it mandatory to take conceptual visualization classes before ever starting rote Mathematics. This teaching concept should be implemented at the earliest levels and carried on throughout college. If students were shown how to visualize abstraction at an early age, they would not only feel more competent in Math but they would eventually free up their cognitive abilities and synthesize new concepts.
After all, the main difference between Sir Isaac Newton and the average student is that Newton visualized physics and Mathematics as a natural practice while most students don't know how. If you truly want to teach Mathematics or any subject, properly, first train your audience in the art of visualization.
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Top-level comments on this article: (1 total)Thanks, Mark for the information. I am a visual learner and one of the "had no clue" kids in math class from junior high on. "New math" was introduced when I was in 7th grade and it wasn't handled well. My mind, and I, rebelled and I took as little math as possible to graduate. Funny this is, now, I handle our budget and do it quite well.Thanks for the feedback, Lorrie. Understanding concepts involves visualization whether we call it abstraction or not. All primary and secondary education should continually teach visualization techniques.Mark
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