In either case, kids call the triangle by name, tell you it has three sides, and/or draw a three-sided figure that you would call a triangle. So - good news - they got it!

The issue is that you checked to see if they understood what you had said, instead of checking to see if they DIDN'T understand what you said. You have fallen victim to your own

*confirmation bias*, which leads to you seek data to*confirm*what you already believe (in this case, that your kids have learned what a triangle is.)
Don't feel bad, though; this happens all the time. We look for evidence to confirm what we believe

*all the time,*so much that it affects our perception. In a classic study, linguists played tapes of people from northern Michigan and then asked Michiganders what they heard. The researchers told the control group that the tapes were of speakers from Michigan. The experimental group, however, was told that the speakers were Canadian. Shockingly, the groups actually heard the speakers differently. The control group clearly heard "out" where the experimental group heard "oot." The researchers even reported the experimental group mocking the speakers for the thickness of their 'Canadian' accent. If our hearing is affected by what we expect, it should come as no surprise that our professional judgment as teachers can also be negatively affected by our expectations.
My solution is to reframe

*checks for understanding*as*checks for misunderstanding*. This is what many great teachers already do instinctively, but I think this framing is helpful for those of us that don't always do this naturally.
In the triangle example above, think of a way to probe to see where kids have misunderstood you. Perhaps they think triangles and rectangles are just two different names for all closed figures. Perhaps rectangles have more than three sides? Perhaps triangles have three sides but need not be closed?

To address these possible misconceptions, I could imagine having all kids draw something that is almost a triangle but not quite, and to explain what would need to change for this to be a rectangle. I could also imagine giving kids an impossible task: Draw something that is both a rectangle and triangle at the same time.

In any case, these tasks seem more likely to get incorrect answers, which is really what you want. You shouldn't ask questions to confirm that every kid "gets it"; instead, you should ask questions to try to draw out misconceptions or misapplications of what has been learned.

Let's try another math example: You have taught the definition of prime numbers (common definition: a number that has exactly two factors) and have given examples.

**Checks for understanding:**So is 17 a prime number? Great; what about 18?

**Checks for misunderstanding:**Is 9 a prime number? (students could reasonably think 3 and 9 are the factors of 9.) Given that 1x5, 2.5x2, and 0.2x25 all yield a product of 5, what kind of number is 5, prime or composite?

Note that this is different from (though sometimes concurrent to) the concept of higher-order thinking or task complexity. The key, however, isn't necessarily to make things more complicated, but to give students a chance to misapply what they (reasonably) believe to be true.

I would love to collect more examples of

*checks for misunderstanding*, from any discipline. Leave me some goodies in the comments if you have a chance!
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