Mathematical Understanding and Multiple Representations

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Representation is a strong tool to help understand mathematical concepts. Representation comes in a lot of different forms. For example, drawing a rough sketch to solve a word problem, or plotting points on a graph to see how a function looks. One of the author's main convincing point in his argument is the experiment Tchoshanov performed in Russia. To be honest, I do not find the results of this experiment surprising. I believe that a student who is able to put his thoughts(sketches) on paper combined with strong analytic skills will excel greatly compared to students who are only strong in one suit.

An example of a mathematical representation would be the blocks used for teaching kids how to count. Kids are able to count quickly if they learn to represent 10 blocks as the 1 long stick, or to represent 100 blocks as the big flat piece. Another example they use is the different ways of presenting a set of 20 dots.
On the left, the 20 dots are scattered and in no particular pattern. The image on the right groups them in 5, which shows kids how to count in groups, as opposed to counting one by one.

One mathematical representation that I find useful is that the integral of a function is equal to the area under the curve. For a lot of students, it's hard to visualize what an integral is exactly, and this link to the area helps them see what they are doing. Take for example if we asked two students to integrate 3x from x=0 to x=3. The 'pure analytic' student would follow the integration rules to reach their answer. The 'representational' student would be able to sketch a rough graph of the function and see that the area under the curve is simply a triangle, which may be easier to understand for most students. 

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