Mathematical understanding and multiple representations

 

What convinces (or doesn't convince) you in the authors' argument?

I was convinced that mathematics can have internal and external representations. Internal representation is more abstract whereas external representations can be expressed through different means. Mathematic as a subject that host huge amount of abstract ideas, being able to represent or “externalizing an internal abstraction” is crucial. The author mentioned that the nature of an external representation can influence the nature of the internal one, and vice versa. I think this is true. For example, if the internal concept I have is “set”, then my external representation of a set can be a basket. We can extend the idea of a set to study open and close sets. Since a set can be open or close, a good representation would be a basket with lid. But what about a set that is both open and close? The external representation of using a basket can no longer demonstrate. We would need to change the way to represent. Hence the nature of external representation changes due to the evolving internal concept. 

·       What kinds of mathematical representations are included and excluded in this article? Can you think of an example of a mathematical representation of a particular math concept (from secondary or elementary school curricula) that is not included, but that might be helpful for students in developing understanding? Describe briefly how you might teach using this representation.

In this article, author presented mostly visual representations for mathematical concepts, as well as using patterns as representation (the Fibonacci sequence). One example of a mathematical representation of a math concept can be found in representing polynomials using algebra tiles. The following chart shows the tiles and what they represent.

We can represent a given polynomial expression. We need to determine which tiles we need and lay or draw them out side by side. For example, if we want to express 2x^2 – 3x + 3, we need 2 positive x^2 - tiles, 3 negative x-tiles, and 3 positive unit tiles.