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Where are the assumptions used?

For many years I have given maths talks in schools and I have regularly asked the pupils to state Pythagoras’ Theorem. The usual answer is a confident “c^2=a^2+b^2“. Their faces turn to surprise and puzzlement when my reply indicates “not quite”. The assumptions (right-angled triangle, definitions of a, b and c) are usually only given with prompting from me.

Years of examining also leads me to believe that focussing on the conclusion is not particular to Pythagoras’ Theorem. For example, asking for a statement of Cauchy’s Theorem I’ll get answers baldly stating

\int_{\gamma } f(z) \, dz =0.

There is no mention of the crucial assumptions such as closed curve, the function is holomorphic on a domain and so on.

It is obvious then that students do not always realise the importance of assumptions. Furthermore, I think they don’t understand precisely what they do but I’ll leave that discussion for another day.

My solution to students overlooking assumptions has been to use one of the questions from the Halmos quote that started this series: “Where does the proof use the hypothesis?”

This is done in the simplest way possible: Give students a proof and ask them to find where the assumptions are used. This has been done with proofs they have just met and with proofs there have been expected to have read before class.

To begin with I try to use proofs where the assumptions are used at an obvious point: “using the assumption that the group is abelian, we see that…”. Next, the occurrences are more subtle, e.g., where the abelian condition is used in the middle of a calculation without comment. The final stage is using examples where the assumption is used in reference form. For example “by Theorem 3.12 we have that…” where the assumption needed is actually an assumption in Theorem 3.12. The students then begin to see that in order to apply an earlier theorem we require its assumptions to hold.

A further benefit is that the students begin to understand the proof by seeing part of its overall structure rather than seeing it as a sequential one thing after another. In my experience many students attempt to understand a proof by reading it linearly again and again until it goes in.
This activity shows them that they can understand the proof by looking at it from a different angle.

Hence, this activity helps students notice assumptions and learn their importance.

An activity that needs work
I attempted to go further with this and demonstrate the importance of assumptions by including an extra, unnecessary assumption in the statement of a theorem. One year I told my students to find the assumptions without telling them that there was one not used in the proof. My idea was that they would learn that they could eliminate the assumption from the statement. This, perhaps predictably, led to some students being annoyed later — after all, I was setting an exercise that could not be done.

The next year I learned my lesson and warned them that one assumption was unnecessary — this time hoping to add an extra dimension to the assumptions hunt. (In both cases these were exercises set at the end of a lecture to be returned to in the next and so they had plenty of time to consider their answer.) This seemed to work better in that I had fewer annoyed complainants.

Unfortunately, in both years the extra assumption caused later problems. At the end of the course a larger than expected number of students were confused: “Do we need this assumption or not?” and “In the exam do we have to prove it with that assumption or without it?”. Clearly, the lesson was not working as intended!

Do you have any thoughts or experiences on students overlooking assumptions? What can I do to solve the problem above? Feel free to leave comments below.

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