I propose a simple account of how we generate intuitive opinions on complex matters. If a satisfactory answer to a hard question is not found quickly, [the intuitive capacity] will find a related question that is easier and will answer it. I call the operation of answering one question in place of another

substitution. -Daniel Kahneman,Thinking Fast and Slow

I have asserted that operations or concepts that cannot be defined briefly with result in significant struggles for our students. For example, the idea of “proportionality.” When we build the appropriate context and then define proportionality as “based on constant ratios,” our students are successful. When we barge in without the context and then say “proportionality means the graph has this appearance, the table of data has this property, and the equation uses these operations,” our students flounder.

Square roots are a huge problem. I cannot find a satisfyingly brief definition, and as the quote above suggests, when students can’t wrap their brains around “the square root of 100,” they *substitute* a problem they can understand: “what is 100 divided by 2.” Exponents. Same thing. How do we help students build the context they need for these operations? I suggest one element is creating the need for the operation. Most 8th graders have never had any reason to use a square root (recall Cain’s Arcade and the checksum on his free play cards). Perhaps if we created that need and then referred back to it. Suggestions?

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