Developing Mathematical Reasoning: Avoiding the Trap of Algorithms
Chapter 6: Not Functional Reasoning 2
Transformation: A function represents a relationship between x and y as well as a relationship between symbolic and graphical representations. Functional reasoning in the context of transforming a function involves recognizing how changes to the parameters in the equation affect the graph of the function. In this video, the student is unable to visualize or represent what the graph of y = x2 looks like, not yet able to connect how changes in x affect y and what that would look like. When asked to consider y = 2x2 in terms of y = x2, she concludes (incorrectly) that y = 2x2 needs to be “double the size.” and motions a wider graph. With y = ½ x2, she incorrectly shows a skinnier graph and calls it “half of it”. Another indication that this student is not yet reasoning functionally is that she describes that one graph might be “longer” than the other, when in reality both graphs are infinitely tall.
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Developing Mathematical Reasoning: Avoiding the Trap of Algorithms emphasizes the importance of teaching students increasingly sophisticated mathematical reasoning and understanding underlying concepts rather than relying on a set rule for solving problems. This book illuminates a hierarchy of mathematical reasoning to help teachers guide students through various domains of math development, from basic counting and adding to more complex proportional and functional reasoning.
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