Entrance Slip Sep 27th– Mathematical understanding and multiple representations

          No doubt that internal and external representations are not separable, and representations help with understanding the abstract concepts in mathematical learning also with any other field of learning. The internal mind can be studied by learning the structure of external representations. The internal representation can be beneficial if the benefit of using external representations can offset the cost associated with the externalization process. This article reminds me of when I first interact with Canadian elementary mathematic textbooks regarding using visual representation symbols or concrete materials of base-ten blocks and rods in place value and addition. I felt the representation was unnecessary since I was so used to the way I was taught which didn’t involve so many visual representations other than using the place value of the positioning method in addition concept. I didn’t get the pedagogical theory behind the Canadian textbook taking large space by drawing many blocks to demonstrate the idea of addition and regrouping. I was concerning kids would feel boring and tired by counting or drawing those blocks. After I start tutoring, I start to understand that every student is different in terms of accepting concepts and build relations between internal and external representations. In my opinion, as educators, we don’t have to necessarily agree with every way of representation for teaching, but we need to have them on our teaching toolbox to help with students’ learning.

          

         This article includes many ways of external representations including graphs, pictures, concrete materials, models. I feel one element I want to add is Venn diagrams because  I think it is a useful and easy visual representation for problem-solving in finding possible logical relations between a finite collection of different sets. For example, it can be used for finding the relation between a set of numbers such as whole numbers, natural numbers, integers, rational numbers, irrational numbers, imaginary numbers, and so on.