Intrigued by Matt Parker’s tweet yesterday, I decided to have a go at it.
Arrange the numbers 1-17 so that adjacent numbers sum to a perfect square.
It’s the kind of problem that makes students who struggle at math, or hate puzzles, shutter. I decided to take this problem on, in my lowest level remedial math class. This class has about 20 kids, 9th-12th grade, all who have failed a previous math class. These kids range in ability as there is a large variety of classes failed.
I started off by having them number little pieces of paper with the numbers 1-17. I think this helped in setting up the task in a low pressure way. Numbers 1-17, how hard could it be?
Then we talked about what numbers were square numbers. Those who thought they could begin were off.
Others who still couldn’t wrap their head around how the numbers could be arranged, together, came up with some examples of pairs of square numbers. Then, all students were able to make triples with adjacent square sums. They built on those smaller sets and began to come up with strings of 4 or 5 or 6 numbers. One student noticed that the highest square number that could be made from the numbers 1-17 was 25, so 16 and 17 needed to be on the ends of the arrangement since only one other number each would sum to 25 from the list.
Toward the middle of the task, I saw students getting frustrated that they only had a few left and they couldn’t seem to place them. We talked about how 1 could pair with multiple numbers on the list. That discovery seemed to re-energize them to rearranging more numbers and persevere in solving the problem.
The students had differentiated themselves at this point and some were working alone and some together. What I found very interesting is that when one of them solved it, they weren’t immediately drawn to that person to show them the answer. They wanted to figure it out on their own. They weren’t rushed either as students finished. Sometimes when students finish quickly, others become frustrated and just want the answer.
The students also developed some interesting strategies, like grouping pairs that totaled 16 and 25. By the end of the 30 minutes, every single student had arrived at the correct solution. I’m not sure if it was the physical manipulative or the puzzle-like feel of the task, but I was so proud of this group of kids. These are students who have already failed at math and have convinced themselves that they are inherently bad at it. Today they proved that not only is the latter completely false, but also that success is in math is achievable with perseverance and resilience.

# Aha for patient problem solving

I teach two different classes with a similar (if not identical) mix if students: college algebra and accelerated probability and statistics. I have been using problem solving in college algebra as a basis for our classroom discussions and I like the material I’ve chosen. However, it didn’t seem as though the college algebra students were developing those patient problem solving skills as much as I’d hoped. Most were working hard, but many of them were stopping when they hit a snag and then waiting for the “smart” kids to come up with the formula or equation.
The aha moment came when I gave a problem to my accelerated prob and stat students that was similar to that of the college algebra class: there were multiple entry points, many solving methods and a high ceiling. I noticed that as the problem progressed, more of the students in the stats class were still working on formulating a solution than would normally be doing the same in college algebra. The students in stats valued all of the methods as productive in some way, whereas in college algebra, many students reject ‘guess and check’ as it doesn’t seem like ‘real math’ to them. I realized why this was: they didn’t have a concrete formula they were searching for. They truly had to discover it on their own. Having accepted that there was no formula, they then trudged onward toward a solution that made sense. I made an assumption, that my college algebra students confirmed that in algebra, they have always been told that they need to set up an equation, find the right formula, or pick the right method. So when problems get hard, they know they can wait for the smart kid to figure out the formula and they can then apply it to a similar scenario next time. And advanced kids are very good at repeating a process, as long as it’s easy to figure out which specific process applies.
Don’t get me wrong, these are VERY smart, hard-working, awesome kids. I am just struggling with how to get them to, well, struggle, a little bit longer.