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. 

A great rule of thumb in getting your point across to people is to draw a picture of what you are talking about. I thought I knew this completely until I ran across an Algebra problem given to us by a good Math teacher in Freshman College.  I usually whipped through the Math homework with ease but I found myself stuck on this problem.  It was a simultaneous solution of only three equations but composing the last equation was not happening!  So, after an hour of getting nowhere, I said, "This is one for the teacher."

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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