How do you get pupils to think like mathematicians? The challenge I set them is: Add up the numbers to 100. It’s simple to explain and yet the responses vary in depth and subtlety, and in the process of tackling it many pupils get a nascent understanding of the ways in which mathematicians think.
The class consists of about 30 pupils who are at our high school for an induction day, just a few days before they leave primary.
They are typically 12 years old, the transfer to secondary school being a little later here in Scotland than in England and Wales. And with our catchment area being among the largest in Britain, they come from 14 different primary schools, many of them small.
There’s a full range of abilities in the room, with support staff to help individuals. Some pupils are full of trepidation, others are full of themselves. It’s a real mixture.
I begin by asking pupils what they think a mathematician is, what makes them different from other people? Not surprisingly, this brings the response that mathematicians are people who are good with numbers. I suggest that a mathematician is just a person who thinks clearly - exactly what I hope they’ll be doing when they tackle the problem.
Often, the whole group appears clear that what’s required is to add all the counting numbers to 100. But should we include 112 in the list? Should the problem be stated with greater care? Add up all the whole numbers from 1 to 100.
A mathematician is someone who states what they want to do as precisely as possible.
There is a collective reaching for calculators, but I explain that they are not to be used. Some pupils start adding: 1 plus 2 is 3, plus 3 is 6, plus 4 is 10 - probably wondering if this is a new form of cruelty.
A mathematician is someone who looks for short cuts, some efficient way of reaching an answer.
If the problem still seems too daunting, a mathematician reduces a complex problem to a simpler one and from tackling that simpler problem learns something about how to tackle the original question.
Some pupils are now up and running; others need a guiding hand. One approach that I often see (or hint at as I tour the room) is adding the first 10 whole numbers to get 55.
A hand goes up and I go over to a pupil who is having problems. “I’ve got it now. The answer is 550,” the youngster says.
This is obviously wrong, so we get around to discussing the actual connection between the partial sums (1 to 10 gives 55, 11 to 20 gives 155, 21 to 30 gives 255). The correct answer of 5,050 is now reached quickly by a few pupils.
Meanwhile, other approaches are being used, including summing linked pairs, 1+100, 2+99, 3+98, and so on, a method fancifully attributed to Gauss, the German mathematician and scientist, by his first biographer. There are connections here to the calculation of the mean and of the median that pupils will need later, so tease out the ideas.
Actually, it’s simpler to pair 1 with 99, 2 with 98 and then add 100 (or 0+100).
Here’s an extension for those who finish early. A mathematician looks for methods that answer not just the original question but all questions of a similar type. The patterns in the sums to 10, 100, 1,000 and 1,000,000 are easily discerned by some. A mathematician is someone who looks for patterns.
One pupil leaves having done something amazing.
“What did you do on your induction day, dear?”
“I added up all the numbers to a million.”
Chris Pritchard is principal teacher of mathematics at McLaren High School in Callendar, Falkirk.
