The Seven Sticks and what mathematics is

This week I provided games and puzzles at a welcome lunch for new students in the Mathematical Sciences degree programs. I had big logic puzzles and maths toys and also a list of some of my eight most favourite puzzles on tables with paper tablecloths to write on.

One of the puzzles is the Seven Sticks puzzle, which I invented:

Seven Sticks
I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them.
Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them.
How long is the shortest stick?

I sat down at one of the tables and I could see the students there were working on the Seven Sticks, so I asked them to explain what they had done. (WARNING! I will need to talk about their approach to the problem, so SPOILER ALERT!)

They told me about their reasoning concerning lengths that don’t form triangles, and what that will mean if you put the sticks in a list in order of size. They had used this reasoning to write out a couple of lists of sticks on the tablecloth which showed that a certain length of shortest stick was possible but that a longer shorter stick wasn’t possible. And so they knew how long the shortest stick actually was.

Only they said to me they hadn’t done it right.

I was surprised. “What?” I said. “All of your reasoning was correct and completely logical and you explained it all very clearly. Why don’t you think you’ve done it right?”

Their response was that what they had done wasn’t maths. They pointed across the table to what some other students were doing, which had all sorts of scribbling with calculations and algebra, and said, “See? That’s maths there and we didn’t do any of that.”

I told them that actually what they did was exactly what maths is — reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.

Only later did I realise the implication of what had happened: these students are coming into a maths degree, which means they have done the highest level of maths at school, and they perceived that something was only maths if it had symbols in it. A plain ordinary logical argument wasn’t maths to them. And a scribbling of symbols was better than a logical argument, even if it didn’t actually produce a solution.

This made me really sad.

I wonder what sort of experiences they had that led them to this belief. I wonder what sort of experiences they missed out on that led them to this belief. I wonder how many other students have the same belief and I will never know. I wonder how I can help all the new students to see more in maths than calculations and symbol manipulation, and allow themselves to be proud of their work when they do it another way.

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One Response

    Hi David, Every week we send to primary schools around Australia Maths worded problem sheets. The problems on the sheets reflect current events — pop stars, sport, movies, TV programs, special seasons (Christmas, Easter, ANZAC DAY, etc.), but are written in such a way that teachers love them, students love them, and they combine reading comprehension with building Maths problem-solving skills.

    I’d be delighted to email you copies of our recent Aussie Maths 4-Level Packs, and to receive from you a crit of them. In particular, we’d be keen to know how, from your perspective, we can improve the content of our sheets. We see them as mainly for student use, but with a definite “professional development” component, particularly for younger primary teachers, many of whom struggle with some areas of the Maths curriculum. You can call me on (03) 8746 8332, or email me on

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