Mathematical understanding and multiple representations

I was largely convinced by the argument that the authors made. I could identify with many of the examples of cognitive representation that they gave, having had experiences which I would describe as either internalization or externalization. I was particularly motivated by their inclusion of language as an external representation. Many of my own "internal" thoughts occur in English; I know that only some people have an internal dialogue, but I certainly do. I makes me wonder in what ways my supposedly "internal" understanding is influenced by this external representation. 

As an example, I think about emotional intelligence. Many people struggle to understand and process their own emotions because they do not have the representational tools to grapple with them. Therapists can help people by providing language and other representations (people speaking to you, different states of being) that help manage the more ambiguous actual internal experience. This seems to me like a powerful tool, but also something to bear in mind when communicating with others. It's extremely difficult (perhaps impossible) to communicate our own internal thoughts without modifying them slightly to fit a shared representation. There's some interesting connections to the philosophy of Wittgenstein here, but perhaps I'll save that for discussion so as to keep focused.

I was also reminded of some research that I did on honeybee numerical understanding wherein the bees could be trained to understand symbolic representations of numbers. In the process of doing research on this, I learned about number-sensitive and number-selective neurons, which help to understand the actual neurological grounding of "twoness" as opposed to "threeness." 

Something else that I found persuasive about the article was how it spoke to my own experience teaching students who struggle to navigate between representations. In particular, there is great difficulty navigating between language and symbols. Students do almost all of their mathematical learning in symbols, not language, and thus are left stumped when asked what "13% of 54" is. If you were to ask them to calculate "0.13x54" I am certain they would be entirely capable of figuring it out (perhaps with a calculator, but still).

Finally, I agreed strongly with the assertion that students are not given enough time with new forms of representation before being instructed in their conventional use. To me, so much of the process of mathematics involves developing new ways to represent ideas that you are struggling to communicate. inventing language and frameworks is a crucial creative aspect of math. It is much more empowering, in my experience, to create such a framework as a student and then learn that one already exists. It makes the student part of a community of creative individuals, as opposed to an external observer.

As for things to use in a classroom, I am reminded of a game I learned at summer camp, a "Never-can-tell." Essentially, the purpose of the game is to determine what the hidden form of representation is. The game works by rolling some number of dice and declaring how many roses are showing, with how many total petals. As an example, a roll of 5 and 3 (on two dice) would be two roses and 6 petals. This is less a game, really, and more a puzzle for students to work out. The trick (and it's supposed to be a "Never-can-tell" so you can't tell anyone that I gave it away!) is that any central pip on the die is a rose and any pip surrounding a central one is a petal. The "roses" are an alternative visual understanding of the face of the die. Thus, a 1 is a rose with no petals, a 2 is no rose at all, a 3 is a rose with two petals, and so on.

I think that this could be a useful activity with younger students, as it demonstrates an alternative interpretation of a visual representation and also demonstrates how we can use groupings to count in different ways, much like the example of visually representing 20 that was given in the reading.