NCTM News Bulletin, March 2008
"Is 4 × 12 closer to 40 or 50? How many paper clips can you hold in your hand? If the restaurant bill is $119.23, how much should you leave for a tip? How long will it take to make the 50-mile drive to Washington? If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"OK. I've generally ignored it when people talk about "number sense", but here it is from the president of NCTM. He says: "Number sense is important and needed—right now." What is it, exactly? Is it estimation? No, it seems to be more than that. Does Mr. Fennell define it? No. He just gives the examples above. Let's look at each one.
"Is 4 × 12 closer to 40 or 50"Do it exactly in your head. It's part of the times table.
"How many paper clips can you hold in your hand?"How accurately do you have to make this estimate to show number sense? He doesn't say, but I don't think he is talking about plus or minus 25% accuracy. I don't think I could guess that closely.
"If the restaurant bill is $119.23, how much should you leave for a tip?"Does he think that those who have mastered the traditional method of multiplication are stuck doing this calculation right-to-left on paper? This is a straight estimation problem, and my traditionally-taught wife takes pride in calculating these things down to the penny in her head.
Is number sense more or less than estimation? It seems to be both more and less. Number sense means more than just estimation, but it doesn't require you to provide accurate estimations.
"How long will it take to make the 50-mile drive to Washington?"About an hour? What are the assumptions? Is number sense equal to estimation with common sense added in? Apparently the common sense level is not very high. How about the number sense to determine how long it will take to drive 400 miles on the highway, accounting for stops for gas, eating, and traffic? What if I gave you the exact times for stops and the lower speed in the traffic? Mastery of the basics leads to number sense, not the other way around.
He seems to be making the case that there is no linkage between mastery of the basics and number sense. But then he really isn't talking about mastery of estimation. Schools could hand out "Arithmetricks" and practice, practice, practice. No, he seems to be talking about some sort educational number sense osmosis. Low expectations.
"If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"He goes on to say:
"Students who have a good sense of number are able to provide a reasonable response to the examples above, including the driving example. And they know that there is no proportion-driven response for the final example."Sure there is. If a 10 year old child is 5 feet tall, then proportion tells you that there is some other effect going on when they get to 20. If you were talking about some unusual species of tree, then proportion or number sense is not going to help. You need content knowledge, and we all know what schools think of content. I think I'll coin a new term: Content Sense. That's what we need-right now! Just like with fractions on the real number line, kids need to be able to place major historic events on a timeline.
After all of this, I still don't know what he means by number sense or what performance level is required. Whatever it is, it seems pretty low. Mr Fennell can't define it, but he wants it fixed "right now".
There is another example:
"A sense of number emerges that is built on the foundations discussed above, which"the pieces were bigger in fourths"?
yield responses such as, “I knew 3/4 was more than 3/5 because the pieces were bigger in fourths.” This is what all math teachers want. Such “aha!” classroom moments remind us about the importance of understanding."
So number sense is something other than math; something other than mathematically knowing why 3/4 is greater than 3/5. And "understanding", according to him, is something other than mathematical understanding. What if the student said that 3/4 = .75 and 3/5 = .6? Is that number sense? Does that show understanding or is that just rote knowledge?
Math is all about tools and methods that you can rely on to give you correct results in spite of the fact that what you might be doing defies common sense. That's the power of math. As I've said before, let the math provide you with the understanding. If you're worried about estimation, teach it directly. Number sense, whatever it is, will take care of itself.