Our education research group here at University of Colorado had a visit and a very interesting talk by Sanjoy Mahajan, director of the teaching and learning laboratory at MIT and former physics professor, last semester. He focuses on understanding and improving students number sense, mostly through use of approximations and estimations. He’s a very provocative fellow. Here are some highlights from his message to us.
There are 26 sheep and 10 goats on a ship. How old is the captain?
That was a question given to 2nd and 3rd graders in France back in the late 1970’s. The answer, of course, is 36. Or at least so stated most of the children who answered it. Here’s an interesting writeup from a researcher who reports on several variations on that original experiment, with odd and disturbing results. Children argued that the number of the flock determined the age of the shepherd, and if members of the flock ran away, then that affected the shepherd’s age! Do click on that link above, it’s very interesting. One of the researchers whose work he discusses said:
The students he interviewed not only failed to note the meaninglessness of the problems as stated but went ahead blithely to combine the numbers given in the problems and produce answers. They could only do so by engaging in what might be called suspension of sense-making – suspending the requirement that the way in which the problems are stated makes sense … There is reason to believe that such suspension of sense-making develops in school, as a result of schooling.
Here is another example, from Sanjoy Mahajan, about a lack of number sense. In a national assessment of mathematics ability, students 13 and 17 years old were asked:
Estimate 3.04 x 5.3
It’s even easier than you think. They were given a set of answer choices:
E) No answer
Here are the responses of the students, 13 year olds and 17 year olds
A) 1.6 28% 21%
B) 16 21% 37%
C) 160 18% 17%
D) 1600 28% 11%
E) No answer 9% 12%
The conclusion I draw from this? We’re doomed. I mean, the 17-year olds did a little better than the 13 year olds, but not that much. And get this! 70% of those students could correctly do the algorithmical multiplication problem. This isn’t a problem of multiplication. It’s a deeper problem of not understanding our number system.
Students just wander around in a random walk in solution space, he says, until they get something that looks like an answer. And they put a box around it. We’d rather, of course, that they have a guide, a sort of nose for where they should go in solving the problem. Then a path to the solution will be more direct. But that requires understanding, rather than rote learning. Rote learning, believes Sanjay, is an evil thing to be eradicated from our learning system. Not everyone in our group agreed.
In another example he gave, he demonstrated how much more comfortable students are with algorithmical numerical calculations than with other solution methods. Even when a graph was right in front of them, demonstrating the answer to the question, they ground through the calculation. Sanjay argues this shows a lack of reasoning and understanding. Students have been taugh that numbers are a more valid way of reasoning, and that this is what teachers are looking for in answers, rather than pictures and graphs.
Or, how about this one. You drop a ball on the table and it bounces. What are the forces on the ball at the moment that it’s stationary on the table? Think about it a moment.
Did you answer “mg”? That’s what most students answer. We’re so used to the normal force being equal to mg when items are stationary. So, Sanjay has his students put out their hand on the table. He places a rock on their hand.
Sanjay: What’s the force of the rock on your hand?
OK, no problem. Now he holds the rock above their hand and makes as if to drop it.
Sanjay: Hey, why are you moving your hand?
He places the rock again on their hand. “That’s what mg feels like. Why are you afraid of mg?”
OK, so they decide it must not be mg. It must be, maybe, 2mg. That seems plausible, given all those momentum conservation problems they’ve done. So he puts two rocks on their hand. That’s 2mg. That still feels OK. So it must be more than 2 mg.
Now that they have that physical intuition, he says, they’re ready to see the symbolic manipulation.
Here’s the answer as he described it. Acceleration goes as the velocity of impact divided by the time of contact. What is the time of contact? The bottom of the ball hits the ground, but the top keeps going until it gets the signal that the bottom has hit, that there’s no more room to move down, and it’s time to start moving up. That happens at the speed of sound.
Sanjay argued that doing this kind of qualitative reasoning is both a diagnostic tool (to see if students have understood you), a treatment (to get students thinking qualitatively) and fun. This gives students a tool to understand and estimate numbers in any problem, not just physics. He wants them to have a feel for what’s going on before they start plugging in numbers.