How to think like a mathematician
REAL mathematics develops by stumbling forward, and occasionally backwards. Take a problem, try to solve it, reflect on what you’ve learned and then change the problem and try again. Heuristics. It’s not how the math textbooks present the subject. So, how to give kids a feel for thinking like a real mathematician? And gaining the confidence to think like a mathematician?
As this simple problem (taken from Reardon Problem Solving available from www.as.ysu.edu/~thomasr/pss.htm) seemed to have worked, it’s worth sharing.
As we started I emphasized the aim was for all students to feel how it feels to think like a mathematician: We were going to do a simple problem, focus on what it feels like and what we did… and then change the problem and try again.
The problem: Take the sequence 1, 2, 3, 4, 5 and five circles arranged in a cross — with one circle in the center and two circles top and bottom and two circles either side. Place each number in a circle such that the sum of the numbers in the circles across equals the sum of the numbers in the circles going down.
I asked my classes to try it out and emphasized I wanted them to remember their first thoughts on tackling the problem.
Before I had finished introducing the problem hands were going up with students proudly declaring “done!” and sitting back with an answer on their paper. I asked a student to put up their answer. Immediately another student announced they had a different answer. And then a third, yet another answer. I asked if there were a fourth? With no takers, I asked students to recall their first thoughts.
“I took 1, 2, 3, 4 and put them in the outer circles and then the 5 in the center, and went from there.”
“I knew the sums couldn’t be more than 10, so I opted for a sum of 8 and made it work.”
“I just put numbers in the circles till it worked.”
“I spotted 4 + 1 and 3 + 2 both equal five, so I put the 1 in the center.”
“I’m hungry… no, I mean, I put the biggest number in the center .”
So, I pointed out, there’s at least three answers. And different students seem to have had different thoughts about how to start. Remember, in group work different students will have different approaches. We agreed the first attempts were starts to guess and check.
Then we looked at the three answers on the board. Can you spot any patterns? I prompted. After an initial silence and a few stabs along the lines of the 1 and 5 always fall in the same line of circles, the focus was on the the number in the center circle. Students described the numbers variously as “odd”, “not-even”, or the “first, middle and last numbers”. Someone pointed out the totals were 8, 9 and 10. The sum of the top and bottom numbers must equal the sum of the two side numbers another pointed, leading a discussion that focusing on the outside numbers also gave clues.
Ok, now try 2, 3, 4, 5 and 6.
The response was quicker. And most students now carried on after finding one solution. Students put answers on the board. First thoughts included, “I went for the odd numbers in the center, but there were just two, so I put the even numbers in the center circle”.
“I looked for pairs of numbers that had the same sum.”
“I looked for the first, middle and last numbers and put them in the center circle.”
So, is there a fourth solution? I asked pointing back to our first list of 1, 2, 3, 4, and 5. Students seemed confident there wasn’t. Explanations focused on putting an even number in the center circle: there was then no way the four remaining numbers could be paired to give equal sums, or, the sum of the remaining numbers was odd and could not be divided by 2.
So, will it work for any set of five consecutive numbers (we played hangman to get the word “consecutive”)? Yes, was the consensus. Students tried with their own numbers. Examples gave sums of tens, hundreds and even thousands, but with all students able to experiment with their own sequence. By now some students were branching out: It works for 30, 45, 60, 75, 90 said one (do they have common factors I suggested). Can we use decimals? asked another. What about negative numbers? another ventured.
We summarized: If the first five numbers we looked at had been one of the latest student-generated examples, we’d have all got stuck and would have given up, I suggested. We went simple, guess and checked, learned from that, changed the problem and tried again using what we’d learned from our first try. And again.
OK, does it work for any consecutive five numbers? Yes most agreed. But, bearing in mind this is an algebra class, how can we prove it?
After a bit of prodding, x was suggested. Variables someone else explained.
Looking back to 1, 2, 3, 4, 5 what if x =1, how would we write 2? I asked. y? said one student. z? another.
Getting to (x + 1) didn’t come immediately. But (x + 2) came tumbling out once that obstacle was surmounted followed by (x + 3), each written below the appropriate digit 1 through 5. Now, the problem was to find three solutions for the consecutive sequence of x, (x + 1), (x + 2), (x + 3) and (x + 4). Use what you’ve learned I suggested.
For some students this was a struggle, though most seemed to be able to follow along and find a second solution once a member of their group cracked it and gave one of the three solutions.
Meanwhile, one student was changing the operation from addition to multiplication and another had expanded the sequence to nine numbers and circles. You can make it work he proudly announced.