In the article, Pape and Tchoshanov (2001) argue that internal and external representations should be utilized to develop student's understanding of mathematics (p.118). They define internal representation as "abstractions of mathematical ideas or cognitive schemata that are developed by a learner through experience." (p.119). These are the mental images that learners have in mind, and draw relation to personal experiences. External manifestations of mathematical concepts can be represented by visuals such as graphs, tables, diagrams and charts. External representations also include numerals, and algebraic equation. I believe visual representations of mathematical concepts would be helpful for students in understanding the mathematical concepts. They give a clearer sense as to what the content is, and they make math problems less dull. Visual representations also make concepts notable and deepen the memory in the learners minds because I often find students pay more attention to visuals than just plain words.
Verbal representation was shown in Figure 1 on page 119 in the article, but the authors do not provide an example for how it works. In verbal representation, students use words to relate math symbols and ideas. This type of representation can be useful in word problems by deciphering the given information and what is being asked to find.
Example of how the verbal model works can be found in this link.
https://www.target.k12.mt.us/cms/lib7/MT01000812/Centricity/Domain/68/math_iwthout_numbers.pdf
Good!
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