Representation in Mathematics

The one statement in this week's reading that I strongly concur with is the claim that mathematical representations must serve as a cognitive tool rather than the end product of a task. I would also like to add that, in my view, representations are most effective when acting as an accumulation of all the cognitive efforts that helped students to arrive at an answer.

For example, an analogy I have is how we use the GPS to guide us from point A to point B. The representation in this case is not only a series of lines are connected to the two points, but also why each step was taken knowing where point B is on the map, and if one of the "traffic intersections" was blockaded, what alternatives could we take to arrive at the same destination. In most cases, there are multiple ways to get to the final answer (both in terms of geographical location and mathematical problems).

There are two examples mentioned in the reading that I found interesting: 1) the usage of instructional materials like the base 10 blocks to help students with addition; and 2) using a combination of algebraic and pictorial approach to teach trigonometry. The first example, although mostly applicable for students at a early age, requires a high level of instruction (as mentioned in the article, which I agree with) that can successfully map the material representation with the internal math procedure. For the second example, it's conceivable that the referenced study was able to show success when both internal and external representations were used, because trigonometry requires visualizations more than other concepts in mathematics. These examples show that mathematics is both internal and external in its representations, as the author suggests. But in practice, what, when and how we use external representations are context based. They can vary among different teachers and disciplines, and the question whether students prefer external or internal representations should be asked. Teachers also need to be more reflective, as some could conform to using a 'correct' way of representation instead of coming up with ones that are more helpful and creative for students to experiment with. In addition, there could be weaknesses in using too many external representations, as some argue that having more than two different visual representations may be confusing and counterproductive.

One example of external representation that I immediately thought of when I started reading the article was the tree diagram used in teaching probability. This was how I learned probability effectively as a student, so it's something I can relate to. But the tree diagram is only one of thousands of ways to teach probability using external representations. As stated above, as teachers it's important to think about how we use representations to not only help students understand the abstract ideas, but also obtain a blueprint to apply the ideas in a flexible manner.