This week, we spent one day with 15 math teams (almost 50 teachers and administrators) from 15 elementary schools in Surrey. (I’ll blog about this project soon.) Part of this day was devoted to having teachers work together to solve problems. These problems help set the stage for some of the important themes schools will be exploring by participating in this project over three years. These include:
- conceptual understanding
- concrete, pictorial, and symbolic representations
- use of manipulatives
- communication
- connections between mathematical ideas
- learning and teaching through problem-solving
- multiple solutions
- reasoning
- attitudes and self-confidence
We gave the following problem, from Figure This!:
Take two identical sheets of paper (8½ inches by 11 inches). Roll one sheet into a short cylinder and the other into a tall cylinder. Does one hold more than the other?
A common misconception is that the two cylinders hold the same because the two pieces of paper are the same size. Teachers use a variety of strategies to explore the relationship between surface area and volume.
The first approach most teams take is to calculate the volume of each cylinder. They’ll ask for (or google!) formulas. Even after determining the volume of each cylinder, many will remain unconvinced. Prompted with “How might young children solve this?” teachers will fill each cylinder with manipulatives available at their tables and compare the results, similar to the third act of Dan Meyer’s Popcorn Picker or John Scammell’s Surface Area vs. Volume.
One of the things that I enjoy most about posing these problems to teachers is that each time someone will come up with a solution that I haven’t seen before. For example, this week one team solved this problem by solving a simpler problem. That is, they compared rectangular prisms. (Is ‘squarular prism’ a thing? It should be.)
They argued that the conclusion would be the same but the calculations would be easier. They found a way around the formulas V = πr²h and C = 2πr. True problem-solving!
Part of me geeks out at seeing innovative solutions. The other part of me kicks himself for not making this a bigger part of my own classroom. A lot of missed opportunities– maybe one day I’ll get a do-over.
Chris Hunter 
I wonder if it is because some people feel there is a “right” way to solve it and that it probably is hard and involves algebra that we reach for those formulas right away. When a problem is presented in “real” life I have seen more creative thinking.