Missing a Test Question: A Case Study

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:
2.5

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.

250

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.

4.5

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.

3 Responses

  1. I think there is a conclusion to be made for some of the students who answered 2.5. It is probably just a lucky guess based on the fact that they might not remember what a counterexample is, and just chose to ignore the word. They might then read the statement (not very carefully) as …

    Write the following: … number between 2 and 3.

    I skim writing these days … and I wouldn’t be surprised if students are doing it too.

    Another explanation is simply that the concept of “half” is an innate number understanding. Even very young children understand 1, 2, more than 2, and half. So there might also just be a natural gravitation towards “half” as part of human nature.

    Perhaps in the next iteration, you should ask for two counterexamples! Then you might see some even more interesting results! Thanks for sharing. I love trying to figure out what students are collectively thinking!

  2. I’d put the question part and the statement part on separate lines, maybe italics or quotes on the second part…

    But i like the question.

    No idea about that 4.5 though.

    Jonathan

  3. In the actual test it was something like

    Write a counterexample for each of the following statements.

    1. …
    2. …

    I do think the double negation was the main cause of confusion (since you want something that’s not not between 2 and 3).

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