The author argued that "representations must be thought of as tools for cognitive activity rather than products or the end result of a task"(p.124), and I agree with this statement. When we represent mathematical concepts, we are making sense of the internal and external meaning of them, and representation is used as a tool to reduce the cognitive load for this activity. If we treat representation as an end goal, then it is easy to produce it according to sample templates without actual understandings. One example is to find the slope of a graph, where most students are taught to connect two points on the graph. The resulting "line" representation of the slope is rather pointless if the students are ignorant about the physical meaning of the x-axis and y-axis, and students will not be able to use the idea of slope in a wide range of applications.
This article discusses a variety of representations:
- Internal representation: abstractions of mathematical ideas, cognitive schemata
- External representation: numerals, algebraic equations, graphs, tables, diagrams, charts, symbols and etc.
I also want to add phonetic representation to the above list, where music, math and patterns can be taught to enhance the understanding of one another. Reading musical pieces are similar to reading math symbols, and the construction of music sometimes also follows mathematical patterns. Teachers can show students some beats constructed following mathematical patterns and analyze the aesthetics in those sequences of beats.
OK, good. I watched much of the Marcus Miller TedX talk and didn't yet find the representation of math with the saxophone -- but maybe I haven't seen that part yet?
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