Entrance Slip #3: Mathematical understanding and multiple representations

Image source: https://www.guangyangpri.moe.edu.sg/wp-content/uploads/departments/math/math01_diagram.jpg

An interesting note from the article that I hadn't thought about fully before was how a small child first learns the names of the numbers but doesn't necessarily understand that they are counting up to a total. It made me think about my friends who introduced their son to the symbols of multiplication and division along with the names before he was even two years old. They said their intention was to get their son familiar with the symbols, even if he didn't understand their significance.

Although I agreed with the authors encouragement of teaching both analytic and visual representations, I'm not sure that the idealistic case of a student using both always works. For example, in my case, I felt more comfortable with analytic methods when I was growing up. Even if we were taught geometric or other representations, I tended to gravitate towards the method I liked best. I have seen the same in some of the students I have tutored or helped - they tended to choose a method (analytic, geometric, or other) and use that whenever they could. That being said, the author does suggest an exploration aspect be incorporated in the classroom and maybe that was what was missing in my education growing up and in the classrooms I have worked in. 

The main types of representations listed were analytic and visual, and the connection of internal and external representations. There were also three stages mentioned which were enactive, iconic, and symbolic. Enactive (manipulating concrete materials) and iconic (pictures and graphs) fall under the visual representation category while symbolic (numerals) is more analytic. The article also mentioned combining analytic and visual, and I believe real-world examples fall into this category; specifically, translating a real-world problem into a math problem, solving it, and communicating the results. I think the idea of internal and external representations was interesting. I particularly liked the breakdown explanation of how we use numerical or language equivalents for visualizing a number set, and how it is easier for us to take in visuals that are organized in a certain way. 

The types of representations that I noticed were excluded were oral motion, and musical. Sometimes students learn more when they are explaining concepts to others or voicing their thoughts aloud. Those who are more inclined to move might benefit from activities such as using their strides as measurements in solving problems (i.e. the units become "5 strides of Jim" instead of metres). In terms of musical representation, the first example that came to mind for me was the clapping exercise that I had to complete during piano exams. Both listening to and having to parrot a pattern of beats was a form of representing fractions. I also had music directly used as a memorization strategy when I was learning multiplication. By either following our teacher's tune as she played the piano or by using bells and drums, the whole class sang along to the multiplication "songs." I can still recall the beat that goes along with the words "6 x 6 is 36, 6 x 7 is 42, 6 x 8 is 48." To implement music in my classroom, I might devise a similar strategy to help memorize important facts or formulas by creating a catchy tune to play once a day until it stayed with them (because, knowing teenagers, the only ones who would be singing would be those who were doing so ironically). Alternatively, it may be a good project idea if a student wanted to create their own song full of useful math.