27 September - Mathematical understanding and multiple representations

What struck me was the following quote: "It is now well accepted that the use of particu-lar modes of representations (e.g., visual or concrete) leads to improvement of students' mathematical abilities and development of their advanced problem
solving and reasoning skills " (p.120). I do believe that citing sources to experiments is a good thing, however, just because someone quotes a source doesn't make the argument valid. While I don't disagree with the author's intention, I will play a bit the devil's advocate and say that, while the use of representation definitely helps, if it's done too much, we can be hurting the child's ability to reason abstractly. Representation could be the equivalent of watching a movie, versus reading the book. When we talk about concepts, for example "a set", sometimes it's best to simply make it up in our own mind, whether we imagine boxes or circles. This very picture-making exercise helps with visualizing concepts. Sometimes the way we visualize things in mathematics makes or breaks whether we'll discover something. Having the right pictures in mind definitely helps. However, if everyone has the same forced picture in their mind, it could become a problem since we are not leaving room for imagination.

The author includes a lot of set representations, especially a grouping of the elements. A representation that is not included is the "tiles" representation of a quadratic equation. I'd use this representation to show that X^2 is an actual square of different size, than let's say 2x.