Re: Mathematical understanding and multiple representations

 I totally agree with the author on how graphs are not just a static end result or an answer, it should be a vehicle for a better understanding and establish a relationship between your answer and a social context. My experience totally encoded with this statement. For me to fundamentally understand and apply mathematical phenomena, it's curial to visualize the question and have the ability of interpret, construct, and operate effectively with both external and internal representations. There is a lot of concept-based knowledge in math that is hard to "show". Our brain just works better when we can realize the pattern so that we can encoding and decoding the information. To have a thorough understanding of a math problem, students should be able to not only "solve" the questions but also "make-up" similar questions as well. 

The author mentioned multiple representations include graphs, symbols, diagrams, physical and virtual manipulatives, etc. One presentation I think would be helpful to students to develop their understanding of the concept of distance is sound. Humans have the ability to distinguish the loudness and frequency of the sound depending on how fat they are from the sound source. Doing an in-class activity of such a presentation will help students to notice the relationship between distance and math in our daily life by exploring the representation of sound and distance.