The Role of Representation Article Sep 23

 

There are numerous aspects of this article that I find very convincing. First, students should create and use representations to organize, record, and communicate math because representations are a fundamental way of understanding, modeling, solving, and communicating complex ideas. Furthermore, representations should be a vehicle for exploration and a tool for understanding, explaining and justifying. Representations should not be a goal or final product.

Second, representations improve mathematical learning, thinking, and ability because brains work better with representational patterns. The article’s example about memorizing the Fibonacci sequence visibly demonstrates this idea. It is almost impossible to memorize this sequence before recognizing the pattern but is very possible, even easy, afterwards.

Third, I agree that multiple representations make learning easier because in my work as a math tutor, I regularly have students that require alternative explanations in order to grasp a concept. Explaining an idea using a different or non-standard approach is often effective when the initial explanation has failed. Similarly, math is better understood with representations that are concrete, tangible manifestations. This is especially true when introducing a new, mathematical idea because this allows for building on top of what is already understood and makes connections explicit.

Fourth, I agree that mathematical learning benefits from social interaction with peers and teachers in order to solve problems. This matches the conclusions drawn from the classroom videos we saw contrasting passive math learning and interactive, assertive math learning.

 

One mathematical representation that was not mentioned in this article is the representation of perfect squares and their visualization or representation as surface areas. This representation can be both conceptual and concrete. It is a valuable tool for understanding the connection between perfect square numbers, which can be externally seen using tiles aligned to make a square, and surface area. This idea can be extended to multiplying a variety of lengths and widths to extend the application to surface areas of rectangles and then other more complicated shapes like circles and triangles and then possibly volumes as well.

I would teach this idea with tiny squares that can be put together to make bigger squares and extend this idea to the real world with concepts like the square footage of a room or house. Then, we could use more abstract mathematical ideas to determine the real world value of square footage in different neighborhoods. We could determine how the value of square footage varies from place to place. This exercise would also be valuable to students interested in real estate work, which can be very profitable. It also has value when comparing two homes for purchasing considerations.