The role of representations in developing mathematical understanding

According to the author both internal and external representations are interconnected with each other. The examples given by the author to explain this theory convinced me in his argument. In case of internal representations we imagine the numbers and figures in our mind, and when we give them the shape on the paper they become external representations. On the other hand, when we have external representations, and we study and examine them in mind, then these representations become internal representations. Therefore, our mind analyse things internally and then visualize externally. The representational thinking depends on the individual's ability to interrupt, construct and operate effectively with both form of representations.

The article includes the base 10 blocks for multidigit addition, two different ways of presenting the same set of things, and seeing, imagining and analysing patterns of numerical values like Fibonacci series. It is explained in this article through example, one particular mode of representation does not improve student's conceptual understanding and representational thinking.  Students who use both analytical and visual representations are able to solve multiple types of problems.

 Mathematical representation of fractions is not included in this article. Fractions are very easy to understand through pictures. I will ask students what they can see in this picture. I will give them time to observe the picture. There must be different answers. Some will count the whole parts of the circle, some will count the shaded parts, whereas the others will count the unshaded parts of the circle. After that, to calculate the fractions I will ask them to observe how many parts are shaded out of total number of parts. In first case the answer would be 4 out of 5 is shaded, which is the fraction. We can do the same for unshaded parts. I will also explain the concept of numerator and denominator,