In 3rd grade Math, we were working on problem solving strategies, and looking at the problem 720 ÷ 360. This was embedded in a word problem where a bus could drive 360 miles in one day and needed to go 720 miles to get from DC to Alabama. The question we were trying to answer was how many days it would take. One student suggested solving a simpler problem as our strategy. I asked what a simpler problem would be, and N said to take off the zero on both numbers, leaving 72÷36. Darren immediately said the answer would be 2 and when I asked how he knew, he responded that 36 + 36 = 72. The other students agreed, and I suggested we move back to the more difficult problem.
Being the TZST Teacher, and also being a gifted resource teacher whose job it is to help students think at high levels, I often do so by deliberately trying to confuse them with extraneous information, faulty or illogical reasoning, or outright mistakes. This was one of those days.
So I then said, “Oh, I get it, we took the zeroes off of the hard problem and worked a simpler problem and got the answer 2. Now, when we go back to the hard problem, all we need to do to get the answer is put the zero on our 2 to get 20 and we have the right answer!” Students were nodding all over the room as I wrote 20 as the answer of the problem 720÷360. Some looked confused, so I asked if everybody got that. C said I went too fast, so I offered to explain it again.
“See,” I said, “when we looked at the hard problem, we took off the zeroes on the end of the numbers. 720 became 72 and 360 became 36. That’s like dividing these numbers by 10. Then we worked the simpler problem 72 ÷ 36 and got 2. Since we divided by 10 to get the simpler numbers, we need to multiply by 10 to go back to the harder problem and get the answer. Make sense now?” (I was also drawing arrows from the zeroes on the first problem to the 2 as the answer in the second problem.) All the kids were now nodding their heads to say they understood my explanation.
My next question–“So, if you agree that 20 is the answer, stand up.” Some stood, some didn’t. I didn’t ask for alternative answers–I instead looked at one still sitting and said, “You don’t understand this? Do you see where we took the zeroes off and then added them back on?” and went through the illogical explanation again. At that point he agreed, and said, “Oh, I get it now” and stood up. So did the others.They sat down as I asked my next question.
I then repeated my request with a slight twist–If you understand why we took off and then added back the zero, please stand up.” Everyone stood. Then: “If you agree the answer to our hard problem is 20, stay up. if you think it’s something else, sit down.” Darren sat down.
Now, Darren is a kid with GREAT number sense, who can do problems quickly and intuitively, using the deep mathematical knowledge he has. He sees patterns and relationships in numbers, and clearly knew 20 was not a reasonable answer. However, who wants to be the ONLY child in a classroom who is sticking their neck out for a different answer than the crowd? I decided to push Darren today and help him see his strengths!
I looked at him and said, “Oh, come on, Darren–don’t you understand what we did with the zeroes?” and went through the faulty reasoning yet again! I pushed him–“Now, do you agree? do you understand?” (deliberately asking two questions to make him focus on the second.) ” If you do, stand up, if you don’t sit down.” Again, I was leaving it vague so he was having a hard time showing his thoughts. He was sort of in between sitting and standing, WANTING to sit down, but not sure that was where he needed to be to disagree with our answer.
Watching EVERY OTHER child in that room agree with what I thought was a really ridiculous answer to our original problem, I decided it was time to let Darren share his thinking and cause cognitive dissonance for the others. I could plainly see in his face he knew something was, as the saying goes, “rotten in the state of Denmark.”
So, I asked everyone to sit down. Then I looked at Darren and said, “I can tell you still don’t buy that it would take 20 days to go from DC to Alabama, do you?” He hemmed and hawed, and several other children were now offering to give him the same faulty logic I had shared, and so he said, “Yes, I do.” However, having deliberately phrased the question that way, I could see several other students begin to question that answer as well. I KNEW Darren didn’t believe that was reasonable–he has too much number sense–and so I said, “No, you don’t. You have a different answer in your head, don’t you?”
Now, this child knows me–he was suspecting at this point he was being played, but not quite sure where to trust himself and where to stay in the pack. My 3rd graders aren’t quite sure enough of their own talents and knowledge to stand up alone at times–but they will be much stronger at it by the time they leave my class!
So, I kept asking things like, “are you sure you don’t have another answer? I can see you’re thinking something. I can see on your face you don’t agree. Trust yourself and share your thinking so we can understand YOUR idea.” He was still wavering, and finally Sammie said, “Oh, come on, Darren–just tell her. You KNOW she can read minds! She sees what you’re thinking, so just get it over with and tell her!”
After laughing out loud, I asked the class to stand up again if they believed 20 was our answer. N (another child with extraordinarily good number sense) stayed down this time, everyone else stood (Darren somewhat reluctantly). I then asked yet another child, S, who had been across country for vacation last summer, how long it took him to get clear to the other side of the United States, asking if it took 20 days. When he said no, I then said, “But you believe it takes 20 days to go from DC to Birmingham, Alabama?” He said no, and promptly sat down. Another child, T, sat down at this point, saying. “It didn’t even take me 20 days to get to Mexico when we went !”
Then, Darren began looking very much more confident and less puzzled and he confidently sat down. I went through the work once more–“Look, guys, we had a hard problem and decided to use the strategy of working a simpler problem, 72÷ 36. We know 36 + 36 is 72. If that’s true, then is 720 ÷ 360 equal to 20? Think about a reasonable answer and write it down on your paper.” I gave them a few minutes, then asked Darren to share what he was thinking, and what he thought the answer was. He clearly explained that if 36 + 36 is 72, then 360 + 360 = 720, so the answer would stay 2.
We then discussed their answers (all of which were correct, by the way, despite me!) and talked about trusting their instincts, their number sense, their knowledge. We talked about looking for a reasonable answer and not getting caught up in following an algorithm or listening to faulty reasoning to the detriment of meaning making.
Do I read minds? Well, teachers who pay attention to their students CAN see a lot of the thinking in their faces. . .and Sammie has faith that I DO know what she’s thinking. She says I’m her favorite teacher cause I ask her questions to help her figure out the answer without telling her, and that makes her feel smart. What a great description of scaffolding!
Maybe I don’t read minds. . but I do try to grow ’em smarter!