This morning we had mini-pancakes for breakfast. I gave my youngest son (who is two) three mini-pancakes. When he was still hungry, I gave him two more.
Half-way through his final pancake, he said:
There is interesting research that suggests, through an ingenious experiment, that we are literally born with some knowledge of numbers. Not symbols, like the word “five” my son is using above, but numbers. It seems to me that while my son is developing his language, it is an excellent time to develop his innate understanding of number and connect it to the language he is learning.
So I pay attention to what my son says and even at two years old, when he is first developing language, we count together, we group objects when we play, and I help him give language to the thoughts he is already having.
My son, “Daddy, what’s 192 plus 192?”
Me, “What’s 200 plus 200?”
My son, “400.”
Me, “What’s 192 plus 192 then?”
My son, “I don’t know.”
Me, “Try breaking it down like this. 100 plus 100 plus 90 plus 90 plus 2 plus 2. Does that help?”
My son, “Hrmm. Okay, it’s 284. No, I mean 384. Do you want to know why I want to know?”
Me, “Absolutely. Why do you want to know what 192 plus 192 is?”
My son, “I started with 1 plus 1, and now I’m at 8192, and I want to know what 8192 plus 8192 is.”
Me, “Do you mean you added 1 plus 1 to get 2, and then 2 plus 2 to get 4, and so on?”
My son, “Yes.”
What makes this exchange especially interesting to me is that I remember attempting to do exactly the same calculations my son is doing, at about the same age.
This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.
The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.
A related, and more generic, question is:
How do I change this diagram/image/equation/formula so that it represents <something else>?
Use the numbers 2 and 3 only once (but any combination of mathematical symbols you know), make the largest possible number you can. Make the smallest possible number you can. Make a number as close to zero as you can.
Bonus, use any numbers you like, and then look for patterns between the numbers. In general, which collection of operations will produce the largest number for any two given initial numbers?
Thanks to Michael Pershan for the idea.
Chris Hunter writes on his blog about a student explaining how they would express 0.500 using ten-frames:
One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor.
Showing 500/1000 or 0.500 using ten-frames
Have you seen examples where students come up with innovative ways of representing numbers?
This is a great video of some students sharing their thinking while solving a problem posed by their teacher.
See this blog post where Deirdre Bailey and Amy Park frame the problem posed for the students (as well to get a copy of the text of the problem itself).
This post was shared to me by Jennifer Silverman. What do you think of this student’s inventive use of a pair of calculators?
In a ninth grade Algebra 1 class, we were studying systems of equations. The problem for the day was about drag races, so we started with: http://youtu.be/LGEWCsP1bR4 & http://youtu.be/ggxsrVWEf9E.
One of my students, Ben, was tackling this problem with a partner, and they weren’t making a lot of progress. Suddenly, Ben jumps up and says, urgently, “Mrs. Silverman, do you have another calculator?” The “loaners” were gone so I gave him mine. Pretty soon, he looks up with a big grin and says, “The Mustang wins!” He had built two sequences, one in each calculator, and was carefully monitoring each of them as he simultaneously hit ENTER on each calculator. He knew that the Mustang won because that calculator got to 500 first. When I asked him how long the race took, he asked me to hang on a second. He repeated the process, this time counting the number of times he hit ENTER. “A little more than 11 seconds,” he said, “because at the eleventh hit, the first calculator was at 495.”
Eventually Ben was able to model and solve these problems algebraically, but on the day of the final exam, he had two calculators with him, in case he needed to go back to his own intuitive approach.
Pretty cool, huh?