"The standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply, and divide. But this does not mean that you have to solve every problem in multiple ways. Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn't mean you have to go to school by car, then by bus, then walk, then bike--every single day!"
Last month I stopped by a second grade classroom where the teacher was administering an end-of-year math assessment. I paused by the desk of a student who, with a look of frustration on her face, was puzzling over this question:
"What's the matter?" I asked.
"I forgot how to use an open number line," she responded, head down, staring at the blank page.
"Do you know another way to solve the problem?"
She looked up at me. "Partial sums?"
"Could you show me how you would do that?"
Here's what she produced on a piece of scrap paper:
Question: Do you mark this wrong because she couldn't show her thinking on an open number line?
Continuing to make my way around the room, I came upon this response:
"Tell me what happened here," I asked.
"I got confused about using the number line."
"Do you know another way to solve the problem?"
"I could draw base-10 blocks."
"Could you show me how you would do that?"
Question: Do you mark this wrong because he couldn't show his thinking on an open number line?
I spent the next several days looking through other end-of-year assessments for examples of questions where students were being commanded to solve problems using specific representations and methods. Here's a sample from grades 1-4:
Bill McCallum
Last month I stopped by a second grade classroom where the teacher was administering an end-of-year math assessment. I paused by the desk of a student who, with a look of frustration on her face, was puzzling over this question:
"What's the matter?" I asked.
"I forgot how to use an open number line," she responded, head down, staring at the blank page.
"Do you know another way to solve the problem?"
She looked up at me. "Partial sums?"
"Could you show me how you would do that?"
Here's what she produced on a piece of scrap paper:
 |
| And wrote the answer, 79, in the space provided. |
Continuing to make my way around the room, I came upon this response:
"Tell me what happened here," I asked.
"I got confused about using the number line."
"Do you know another way to solve the problem?"
"I could draw base-10 blocks."
"Could you show me how you would do that?"
Question: Do you mark this wrong because he couldn't show his thinking on an open number line?
I spent the next several days looking through other end-of-year assessments for examples of questions where students were being commanded to solve problems using specific representations and methods. Here's a sample from grades 1-4:
- Use the break-apart strategy to solve each problem.
- Use the turn-around rule to solve.
- Explain two different ways you could use doubling to solve 6 x 8.
- Explain how you can think addition to solve 14-7.
- How can you find the sum using a number grid?
- Explain how you can use the near-doubles strategy to find the answer.
- Use base-10 shorthand.
- Use an open number line.
- Solve using partial-sums addition.
- Solve using U.S. traditional addition.
- Use partial products or the lattice method to solve.
- Use U.S. traditional subtraction (this for 38,000 - 23,177.)
As I recall, in my math classes growing up there were no multiple methods or representations. You memorized your facts and used the traditional, standard algorithm. I'm sure I had classmates clever enough to devise alternate strategies on their own. As for me, I was out of luck. That's too bad. I wish I had the exposure to the multiple methods and representations that are now considered essential components of math education today. If I had, maybe this wouldn't have happened.
But in the leap from standards, to curriculum, to assessment (especially assessment), something has gone awry. We want to expose kids to multiple representations and methods, and encourage them to experiment with, explore, connect, and analyze them. But do we want to force kids to use them on summative assessments? For a grade? The two students wrestling with question 16 above each had their own way of thinking about 43 + 36. But the directions to the problem, which instructed them to show their thinking on an open number line, only served to shut their thinking down. How did it make them feel? And how will they feel when they get their test back and see that a problem that they can find the answer to is marked wrong because the way they want to show their thinking is not what the test maker wants?
Providing access to and connecting different models, methods, and representations for students as they find their way to computational fluency is very important. But I think that in forcing the issue we run the risk of doing more harm than good. How kids get to school is dependent on many variables, none which are under their control. The ultimate decision rests with us adults. How about we let the kids decide for a change?
But in the leap from standards, to curriculum, to assessment (especially assessment), something has gone awry. We want to expose kids to multiple representations and methods, and encourage them to experiment with, explore, connect, and analyze them. But do we want to force kids to use them on summative assessments? For a grade? The two students wrestling with question 16 above each had their own way of thinking about 43 + 36. But the directions to the problem, which instructed them to show their thinking on an open number line, only served to shut their thinking down. How did it make them feel? And how will they feel when they get their test back and see that a problem that they can find the answer to is marked wrong because the way they want to show their thinking is not what the test maker wants?
Providing access to and connecting different models, methods, and representations for students as they find their way to computational fluency is very important. But I think that in forcing the issue we run the risk of doing more harm than good. How kids get to school is dependent on many variables, none which are under their control. The ultimate decision rests with us adults. How about we let the kids decide for a change?
