In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Let me describe a one-liner that I’ll use when I want my class to figure out a pattern, thus developing a theorem by inductive logic rather than deductive logic.

Today’s one-liner is easily stated: “Gosh, I’ve seen that somewhere before.”

For example, In my statistics class, here’s the very first illustration that I show to demonstrate how to compute a standard deviation:

Find the standard deviation of the following data set: 1, 4, 6, 7, 8, 10.

The first step is finding the average:

.

We then find the deviations from average by subtracting 6 from all of the original data values:

Deviations from average = -5, -2, 0, 1, 2, 4

With these numbers, we can compute the standard deviation:

.

After asking if there are any questions of clarification about the nuts and bolts of this calculation, I’ll proceed to the next example:

Find the standard deviation of the following data set: 5, 8, 10, 11, 12, 14.

The first step is finding the average:

.

We then find the deviations from average by subtracting 10 from all of the original data values and then constructing the square root as before:

.

Then, in a loud obvious voice, I’ll declare, “Gosh, I’ve seen that somewhere before” and then wait a few seconds for the answer. Students can obviously see that the two answers are the same — which gets them thinking about why that happened.

Obviously, the two answers are the same. The real conceptual question that I want my students to figure out is why the two answers are same. Eventually, someone will come up with the correct answer — the second data set was made by adding 4 to all the values in the first data set, which may change the average but does not change how spread out the numbers are… so the standard deviation should be unchanged.

I love the “Gosh, I’ve seen that somewhere before” line after a couple of carefully chosen examples, as it cues my class that they really need to think a little harder than the dull and mechanical operations toward a deeper conceptual understanding of what’s really happening.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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