At a conference I recently attended, the keynote speaker Dr. Randy Charles spoke of “two inversions”. The first I was very familiar with, that of changing
Mathematical tool -> Problem solved by tool
Problem solved by tool -> Mathematical tool
That is, motivating mathematics by presenting problems which force the students to develop the tools they are supposed to be learning in the first place.
He spoke of another inversion which I was not familiar with:
Standards -> Understandings
Understandings -> Standards
A “standard” might be “the student knows how to add two fractions” while an understanding might be “why do the denominators need to be the same when adding two fractions?”
This portion of the talk was somewhat foggy, and honestly the analogy with reversing problem solving and mathematical tools isn’t the best since the problem-solving approach leads to an obvious change in sequence while the understandings approach leads to a change in … well, I’m not sure, really. Perhaps it is a change in assessment. I have in the past included problems like “Why does the graph of tan(x) have asymptotes at and ?” rather than just “Graph this function” on tests. Does every understanding needed be an explicit question? I’m not sure how this is an “inversion” rather than just “we should be thinking about this more.”
However, I did want to give my honors students a fraction test that involved very few numbers, and I think it gives an idea of what Dr. Charles was talking about.
1. Why is the top of a fraction called a “numerator” and the bottom called a “denominator”?
2. Why can each fraction be represented in an infinite number of ways?
3. Explain in your own words a method for reducing a fraction.
4. When is reducing fractions important? When is reducing fractions not important?
5. Why do the denominators need to be the same when adding two fractions?
6. Describe the error made below, and explain why it is an error.
7. Why do the denominators not need to be the same when multiplying two fractions, and why does multiplication involve multiplying both the numerators and denominators?
8. How can an integer (like 6 or -2) be written as a fraction? Why does this work with any integer?
9. Why does a fraction equal zero when a zero is in the numerator?
10. Why can’t you have a zero in the denominator (division by zero)?
11. Explain, without referring to the rule “dividing by a fraction is the same as multiplying by the reciprocal,” why dividing by ½ is the same as multiplying by 2.