I do believe students learn better when exposed to different techniques since they will be more flexible. It also opens up learning to students who learn better by visual/graphical processes which makes concepts accessible to other types of learners. That being said, I appreciated that the author used some visuals in his article to help make some of the points/examples clear. I would have benefited from some more examples and explicit definitions.
If we are viewing it as a process rather than an end product we should provide the students with the materials during an assessment. I think the author assumes that all students are able to connect the external and internal abstractions quickly and easily. Since we know that all students are different and have different strengths, they should all be allowed to have the materials that can help them succeed.
What kinds of mathematical representations are included and excluded in this article? Can you think of an example of a mathematical representation of a particular math concept (from secondary or elementary school curricula) that is not included, but that might be helpful for students in developing understanding? Describe briefly how you might teach using this representation.
The article did not include technology based representations. If the school has access to a 3D printer students will be able to explore 3 dimensional shapes and can get a better understanding of surface area and volume by figuring out how much material is needed to create a hollow vs solid shape.
It also did not mention using students as a representation. Recalling my history of mathematics course, I remember one group created a video where they performed proofs via dance. I think this is a great way to get students to physically understand what is happening since they are physically performing it themselves.
Furthermore, as mentioned in class, this type of physical movement could be used to graph parabolas and other types of graphs to get the roots and understand the shape and what is going on in the graph. They can examine the roots, areas of increase/decrease etc.
Yes! Very nice, Concetta! I'm so glad you mentioned both technological and embodied ways of representing mathematical relationships, as these are so often left out in much of the math ed literature. I agree too that the article could have been more explicit in its examples, and in discussing the core problem of how learners internalize and connect multiple representations to develop more flexible thinking and problem-solving.
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