I have written about understanding in math, and the education establishment’s view of it. With all this talk about how it is important for students to “know math” and not just “do math” the question arises: “How do we measure a student’s understanding as opposed to their ability to go through procedures?” That is, how do we differentiate someone who truly understands from that of a “math zombie”.
In my opinion, the most reliable tests for understanding are proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning. Here’s an example.
On a placement test for entering freshmen at California State University, asingle item on the exam correlated extremely well with passing the exam and subsequent success in non-remedial college math. The problem was to simplify the following expression. (Multiple choice test):
Without verbalizing anything or explaining one’s answer, simply recalling the arithmetic properties…
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