When we first selected the following problem, Dale and I knew it would be a challenging one for the Grade 3s – some of them were still struggling with rounding, estimation and one-digit addition and subtraction – but we wanted to see what they would do with the challenge.
I was amazed at the thinking that emerged with some students in my class this morning!
Mr. Trinder has 23 model trucks and cars. He has 5 more trucks than cars. How many cars does Mr. Trinder have?
At first, many students tried to add 23 and 5. Some added the resulting number to 23, and came up with a number in the 60s. Predictably, others subtracted 5 from 23. Although a few of them thought they had solved the problem with these strategies, many realised that something was not quite right.
But they weren’t sure what it was.
Today, I realised the importance of letting them wrestle with these incomplete solutions for a while, before stepping in to guide their thinking. And when I did step in, it was to encourage them to do more thinking, rather than to provide them with the “correct” path: After about 15 minutes of work time, several students had begun to realise that either their solutions didn’t make sense, or that they didn’t even really understand the problem itself, whereas others were well on their way to either solving the problem, or perhaps even already done solving it correctly. At this point, I asked students to be honest with themselves, and self-select a partner who was clearly on a different learning path than they were, then meet to discuss their thinking to this point. I encouraged them to use “ask questions” and “ask for clarification” (from our “Rich Talk” chart), rather than “disagreeing constructively”. It was the first time I had explicitly referred to this chart during math class (we’ve used it extensively while dialoguing, in groups, about rich mentor texts in other subject areas).
After this short, 5-minute peer conversation, I sent students back to work on the problem, and after 3 minutes, I interrupted them again, as I could sense that – although many of them now realised they had done something wrong in attacking the problem, and why their original answers didn't make sense – they still really didn’t know where to go next. Now I presented the problem visually through hand and arm movements, "drawing" two groups of “vehicles” in the air, cars and trucks, and encompassing them with my hands to show that altogether, there are 23 vehicles, but that we do not know how many of each there are.
“But what do we know?” I wondered out loud.
Several students were able to correctly communicate that we knew that one number, the one representing the trucks, was 5 more than the other number. I confirmed this, and reiterated that both numbers, when added together, must total 23.
The students went back to work.
By the time the period was over, not only had many students solved the problem correctly (and using a variety of strategies), but one of them had even figured out that his strategy would work for all such problems, regardless of the numbers. He had discovered that if you deduct the “more than” amount from the total, than divide the resulting number in half, and add the number you originally subtracted to one of the halves, you would get the two numbers you needed to make the total correctly – it was very exciting; even I had not realised the general application of his solution strategy until we explored it together, after an “I wonder if...” on my part!
The morals of the story?
Offer a single, rich problem, provide lots of opportunity for independent thinking and then oral language to share thinking, and, most of all, ensure limited and very intentional teacher interference!