Beyond True/False: Always, Sometimes, Never

Posted by Kevin Houston on Mar 27, 2019 in How To Think Like A Mathematician, Mathematical thinking, Mathematics education | 0 comments

A classic method of assessment is to set true or false questions. These are easy to mark and students can even mark them without difficulty. There is another method called Always, Sometimes, Never that goes beyond straightforward true/false.*

For example, take the statement The sum of any two integers is odd. We could clearly frame this as a true or false statement in an assessment. But now look at it from the perspective of whether it is true always, sometimes true or never true. In this case it is sometimes true and then we can ask what assumptions are necessary for the statement to be true. This particular statement is true when exactly one of the numbers is odd.

Another classic statement in this area: The square root of a number is less than or equal to that number. Again, this is sometimes true and we can ask for a condition to make it true and also, in this case, when it is false.

This format helps with stressing the importance of assumptions. For example consider For any integer , we have . Students should recognise this as part of Fermat’s Little Theorem. However, the crucial assumption that is prime is missing. Hence, the correct answer is not Always but Sometimes.

In this example we have a problem if we ask students to determine precisely when the statement is true. Certainly, the statement is true for prime. But it is not the case that it is false for not prime. When is not prime the precise conditions for the statement are difficult to state. Pseudo-primes exist and we need to start talking about and being coprime and so on.

Hence, when setting a question with a Sometimes answer we need to be careful about the follow up question. It may be suitable to ask for some extra assumptions to make it true and some assumptions for which easy counterexamples exist.

Another advantage of Always, Sometimes, Never questions (if students are not required to give explanations) is that the chances of getting the answer right by guessing is lower than guessing in True/False questions.

* Top fashion tip: The phrase can be used to remember which of the three buttons on a suit you should button to look particularly smart. From the top button it is: always, sometimes, never!

Get the newsletter!

Fancy a newsletter keeping you up-to-date with maths news, articles, videos and events you might otherwise miss? Then sign up below. (No spam and I will never share your details with anyone else.)