Divisibility Tests and the Place for Logic in the Maths Classroom

Original tweet: https://twitter.com/nathanday314/status/1379853529708384259

What’s the problem?

This question/answer was shared by @Miss_M_Maths earlier today and provoked some really interesting discussions about whether this pupil has given a valid reason for their answer.

Despite being a question about multiples and divisibility on the surface, the real issue here is one of logic and inference.

The pupil has given ‘all multiples of five end in either a 5 or a 0’ as a reason why 575,435 (a number ending in either 5 or 0) is a multiple of five. And this would appear to be a sensible enough reason, but it is actually a perfect example of a logical fallacy known as the converse error(among other names).

In particular, while it is true that ‘all multiples of five end in either 5 or 0’, the reason why we know 575,435 is a multiple of five is because ‘all numbers ending in 5 or 0 are multiples of five’.

These two statements are converses of each other. That means that they express the same relationship, but reversed. In general terms ‘All X are Y’ and ‘All Y are X’ are converses of one another, as are ‘If P, then Q’ and ‘If Q, then P’.

And the difference between a statement and its converse is really important, especially when one is true but the other isn’t. ‘All cats are animals’ is a very different statement to ‘all animals are cats’.

The confusion in this case is caused by the fact that both statements are true, ‘a number is a multiple of 5 if and only if it ends in either 0 or 5’. But this does not mean the two statements are equivalent, and you cannot use one as a reason in the place of the other.

Why is this something worth discussing?

The discussion that followed the tweet raised a very interesting question – is this all just pedantic nonsense that makes people feel stupid and distracts from the mathematics?

I’d say no. In fact, I think this discussion highlighted how important it is to discuss these sorts of questions, both with each other as maths educators and with our pupils. Logic underpins all we do in mathematics and studying maths gives our pupils the best opportunity they’ll get to develop their ability to think logically and carefully through a problem, a skill that is vital well beyond the realms of factors and multiples. It’s also an aspect of maths that becomes increasingly important as pupils develop as mathematicians, with the need to understand and form coherent arguments such a significant part of what it means to do mathematics.

So, I think these are exactly the sorts of issues we should be explicitly considering in the classroom, across all ages. But we must be careful in how we go about this, as there are a few ways it could go wrong.

Keeping it real

It would be easy when teaching these ideas for it to become too abstract too early, and for things to get lost in a haze of Ps and Qs (and not the polite kind).

Instead, we should do as has been done going all the way back to Ancient Greece (and possibly beyond) and make heavy use of examples. These can either be non-mathematical (‘All cats are animals, Garfield is a cat, so Garfield is an animal’) or mathematical (‘All rectangles have four sides, a rhombus has four sides, so a rhombus is a rectangle?’), provided pupils are already secure in the mathematical topic under consideration.

I think it would also be worthwhile, curriculum time permitting, to introduce vocabulary like converse, contrapositive, inverse and negation, as it is always helpful to be able to give names to things when you are trying to understand them. I think questions like the original prompt make for excellent discussion points – ‘Is the reason valid? What would a valid reason be? How else could that reason be stated? For what question would the original question be valid?’, and it’s a topic that can easily stretch pupils of all abilities and ages.

Keeping it positive

One of the real dangers in maths, and especially ideas that are as fundamental as these, is that it is very natural to go from feeling confused to feeling stupid. It is an area where, above any other, knowledge cannot substitute for understanding and, like a magic eye image, the gap between those who understand and those who can’t seems impenetrably large. This came up in the Twitter discussion and it is a feeling we can all probably relate to. That is something we must be always aware of as maths teachers, but especially when dealing with topics like these. We should be prepared to explicitly confront any idea that not understanding something yet or not getting things instantly makes one stupid or incapable of ever understanding.

But that shouldn’t put us off from discussing these questions in a positive and constructive environment, as all pupils (and teachers) have a lot to gain from thinking about logic and understanding it more explicitly and deeply. The prevalence of deduction puzzles like sudoku demonstrate how interesting and satisfying it can be. And how wonderful would it be for the next generation to be more prepared to spot fallacious reasoning when it crops up in politics and other areas of day-to-day life?