Pape and Tchoshanov define mathematical representation as both the internal and external manifestations of mathematical concepts in the "The Role of Representation(s) in Developing Mathematical Understanding" (2001). The Skemp article had me thinking about the different ways students understand and this article has me thinking about the different ways students get to their understanding.
I mostly agree with the opinions of the authors. I also believe that representation is a vehicle for discussion and that it is an inherently social activity. However, I think there are often cases where students have a wildly different internal manifestation of the concept in question. Differences in internalization may be influenced by the student's cultural and personal experiences. I think it is important for students to have the opportunity to explain how they understand concepts in order to further their understanding and add discussion to the classroom. Communication is an important step of the learning process where representations are co-constructed by teachers and their students.
The clearest example of representation that comes to mind is from balancing forces in the vectors unit of calculus in grade 12. We had problems with static objects experiencing or exerting three forces in total. It helped me to go from the word problem to two diagrams: one with the object and the forces and the other with the three vectors to form a triangle. In order to balance the forces, the triangle needed to be complete. There were trig equations to determine the answer but seeing the triangles and the angles boosted me through the problems.
I especially like the example of how the number 1123581321345589 is only able to be 'memorized' once the student understands that the Fibonacci sequence is utilized. This made me realize why I often preferred math classes to science classes on test days: When I understood how something works, through a representation or otherwise, there were seemingly less things to remember because on a test I could quickly rebuild the pieces to the puzzle I knew I could solve. I would like to find a book that outlines the best practices as instructors for which representations to utilize by concept in K-12 math education.
Lovely! Great examples of the forces vectors forming a triangle, and of the long number that is easy to reconstruct if you understand the Fibonacci sequence. I agree with you: math should be simple because there is much less to remember once you can reconstruct things from basic principles. Then the question, as you say, is how to represent things so that more students can do this reconstruction effectively. I don't think there exists a book of the kind you're looking for, but check out the work of John Mason and Dave Hewitt in the UK, especially their publications through the "ATM" (Association of Teachers of Mathematics). That's the closest I've seen to this!
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