Mathematical understanding and multiple representations

 I do agree mostly on the ideas presented by the author of this article. I agree that both internal and external representations are both important elements in the learning journey of the students. This is especially true when it comes to learning Math. Being able to visualize math concepts and ideas make learning Math easier (and more fun!). But in our previous reading about instructional and relational learnings, we see that relational learning is equally important for the learning development of the students. So if a student can show clear and concise representation of a certain math topic (able to explain a graph, model correctly for instance), does that mean he/she really understand it? This can be done by another student who learn the topic instructionally without understanding the topic as well. So, as a teacher myself, I do think we have to dig deeper, we may have to present a variation of the same math problem or ask the student to present another way of solving the problem. In this way, we can make sure the student is learning relationally. 

One representation that is not mentioned in the article is analogy. I still remember when I first learnt limits in my Calculus class, my teacher taught us by telling us doing limits is like 2 trains (the left and the right hand limit) approaching their stations. The function itself is like the train tracks. I still remember this vividly because I can relate something I learn in Math to what I see in real life. The fact that my teacher makes something so abstract into something tangible helps us understand better the concept of limits. I think that this kind of representation is very helpful and I will definitely use this in my teaching in the future as well. To teach using this representation, we have to make connection between the math concept and a real life example. Another example that I have used in the past with my student is the idea of the sum of an infinite series. The question of my students ask is: How is it possible if you have infinite number of terms but there is a finite sum? I tried to explain to them by using this analogy: If you have 10 million dollars and I give you a penny, does it really make a difference to the overall amount at all?