Our journey “Solve & Share” will take us to exploring math tasks and sharing some reasoning out loud. I thought the following word problem would be a good way to start as it illustrates, I hope, how much you can learn about your child’s understanding in math by just listening to him/her.
The problem was from the South Dakota Booket I discussed previously (here). As always, my child Rosie, 8, could solve the problem in “any way that makes sense to her” (Carpenter et al, 2014). She could model the problem, with manipulative, a drawing or a trial error approach, she could use counting strategies, or number facts. As always, it was up to her.
The problem was:
Kenata has 167 coins in her jar. 50 of them are dimes, and the rest are pennies. How many pennies does she have?
I had seen Rosie solve problems that looked similar to me. Using counting strategies and number facts. And confidence.
With this one, she froze.
At some point, she drew the picture shown on the right, with not much conviction though. Used to the Base Ten Blocks and their representation on paper, she drew the 167 coins as 1 Hundred (i.e. the “gridded square” on the top ), 6 Tens (i.e. the tallies), and 7 Ones (the little squares). She wrote an equation with the unknown number : I was kind of expecting Rosie to finish up.
But she froze again.
She tried with smaller numbers, but it did not seem to help.
I suggested another strategy she had been using successfully in the past when she is stuck : change the story. We talked about pets, dogs and cats, instead of coins, dimes and pennies. Rosie did not have any issue to solve it.
But when she went back to the initial problem, she… froze again.
We went back to her drawing. At this point, however, I noticed through her explanation that the Tens in her jar no longer represented 10 coins but… 10 cents i.e. … 1 dime. No wonder why she was confused. Dimes and pennies are so often associated to cents in word problems, that she could not see them as just coins anymore.
I could have helped her, and said “Rosie! Your Tens represent Tens of coins, not Tens of cents!”.
But I did not.
Because I rarely do. Following the steps of Cognitively Guided Instruction (Carpenter et al, 2014), I prefer waiting that it comes from her, even if it requires an additional 5-10 minutes. Or more. But little step by little step, going back and forth from her drawing to the problem, from the problem to the drawing, she saw it. At some point, she saw where her confusion came from. And provided an answer of 117 pennies almost instantly. With a priceless expression on her face.
It may take time to let a child fully make sense of a problem, or a math concept. But, to me, as a parent as much as an educator, it seems so worth it.
- Carpenter, T., Fennema, E., Franke, M., Levi, L. and Empson S. (2014). Children’s Mathematics, Second Edition: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. ISBN-13:978-0325052878.