Reversing is the act of taking a problem and reversing a given and an unknown. It’s simple, but critical in building intuition and intellectual need for algebraic structure. One trap I’ve fallen into is teaching opposite directions of the same concept on separate days. While that can lead to superficial, short term success, it doesn’t promote understanding. Instead, introducing the Pythagorean Theorem and having kids solve for both the hypotenuse and the side the same day promotes productive struggle and more flexible understanding.

For me, reversing falls under MP.2: Reason abstractly and quantitiatively. It’s critical in building students’ connection between arithmetic problems and the value and power students gain in using variables and algebraic structure to give math meaning.

Some examples:

Find the area of a rectangle given two sides; find a second side of a rectangle given the area and one side

Find the mean of a data set; find the missing data point given the other data points and the mean

Simplify an algebraic expression; identify a missing coefficient given the simplified expression

Find an interior angle measure of a regular polygon; identify a regular polygon given an interior angle measure

Simplify an exponential expression; identify a missing exponent given the simplified expression

Use an equation to generate a table; generate an equation from a table

Manipulate logarithms; manipulate exponents

Calculate percentages; apply percentages to identify parts and wholes

I think the biggest pitfall to building abstract reasoning through reversing is dividing these objectives into separate parts. It’s tempting for teachers to try to isolate skills. One day, teach finding the area of a rectangle, have kids practice a bunch, they do well with it. The next day, have kids find the missing side length of a rectangle, have kids practice a bunch, they do well with it. It’s gratifying as a teacher for lessons to go well, but the conceptual understanding is quickly revealed as illusory, and students struggle to think flexibly about the big idea — the multiplicative structure of a rectangle.

While students will struggle with questions reversing a concept, their struggle is a struggle with rigorous, broadly applicable algebraic concepts. Cutting out those concepts reduces math to a set of discrete series of steps that must be memorized and consist largely of arithmetic, which is the opposite of MP.2.

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