I recently gave a test to my Geometry Concepts students which included the following:
Write a counterexample for the following: There is no number between 2 and 3.
First off, the only right answer students put was:
No 2.1, no 2 1/2, no 2.67. Only 2.5, in decimal. I don’t know if there’s a conclusion to be made here.
Wrong answers the students put were
? (or a blank)
In some cases the students hadn’t been paying attention to the meaning of a counterexample so they forgot it. But in at least one case (based on what the student said) someone wasn’t able to think of a number between 2 and 3; the statement seemed to them to be true. Now, it isn’t that the students didn’t know of the existence of 2.5, it’s just they were mentally “closed” in the natural number system and assumed from context “number” meant something you can count.
Some students were getting tripped up by the idea of a counterexample here and instead made an example of a number not between 2 and 3. Other students may have made a double negative; converting “there is no number between 2 and 3” into “there is a number between 2 and 3” and forming an counterexample to that statement rather than the original.
Perhaps most interesting is the students who put a number not between 2 and 3, but still used decimal. It’s like they started with the correct idea of a counterexample and latched on to the part about being between two natural numbers, but made a mental error afterwards when it came to picking a number.
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