Throughout reading this article, there is an overwhelming amount of support for the use of representation(s) in aiding a student through their journey in understanding mathematics. The author makes many arguments within the paper, but the strongest claim that representation is an important asset in math is with the experiment conducted by Tchoshanov. The findings of this experiment was that the group of students taught using different representations scored significantly higher than those taught using a pure approach in either an analytic or visual approach. The students in the representational group were able to switch on the fly between different modes, depending on which was relevant to the question on hand, which ultimately lead them to have a higher understanding of the material.
As mentioned above, the kinds of mathematical representations included in this article are analytic, visual, and in a more broader sense, internal and external. In addition, there are three levels of engagement with representation, namely enactive, iconic, and symbolic. Some examples of representations excluded from this article include videos and simulations to name a few. A coding simulation of particular algorithms to find the derivative of a graph or the area under the curve in calculus could be a huge boon in helping students develop understanding in said topics. In high school calculus classes, at least from my own experience, calculating derivatives and integrals was relatively simple as we only used simple functions with a well-defined rule. Riemann sums is a topic which eludes most students, and showing a simulation of how they help in defining the integral could prove useful. Students could then, in theory, apply this knowledge to more advanced functions.
Thanks for this excellent blog post on the article, Kyle! You've given many excellent examples of representations of mathematical relationships that weren't considered by the authors -- great ideas.
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