kitchen table math, the sequel: the number sense, redux

Wednesday, March 19, 2008

the number sense, redux

Unbeknownst to the NCTM, it seems, people have been looking into the question of what the number sense is or is not:

This led to Dehaene’s first encounter with what he came to characterize as “the number sense.” Dehaene’s work centered on an apparently simple question: How do we know whether numbers are bigger or smaller than one another? If you are asked to choose which of a pair of Arabic numerals—4 and 7, say—stands for the bigger number, you respond “seven” in a split second, and one might think that any two digits could be compared in the same very brief period of time. Yet in Dehaene’s experiments, while subjects answered quickly and accurately when the digits were far apart, like 2 and 9, they slowed down when the digits were closer together, like 5 and 6. Performance also got worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and 8. When Dehaene tested some of the best mathematics students at the École Normale, the students were amazed to find themselves slowing down and making errors when asked whether 8 or 9 was the larger number.

Dehaene conjectured that, when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this. “It is a basic structural property of how our brains represent number, not just a lack of facility,” he told me.

[snip]

These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas.

Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—“Ambition, Distraction, Uglification, and Derision,” as Lewis Carroll burlesqued them—a chore. It’s not so bad at first. Our number sense endows us with a crude feel for addition, so that, even before schooling, children can find simple recipes for adding numbers. If asked to compute 2 + 4, for example, a child might start with the first number and then count upward by the second number: “two, three is one, four is two, five is three, six is four, six.” But multiplication is another matter. It is an “unnatural practice,” Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.) The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 x 6, the reflex activation of 7 + 6 and 7 x 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 x 8, the error rate can exceed twenty-five per cent.
The Numbers Guy
by Jim Holt
The New Yorker


in a nutshell:
  • number sense--right now! is bunk. number sense--5 or 10 years from now! might be more like it, depending on what kind of numbers we're talking about. For instance, number sense for exponents--not in this lifetime! would capture a normal human being's ability to grasp intuitively the nature of exponential growth. [Is an exponent a "kind" of number? Probably not.]
  • the fact that I spent 30 years of my life believing that 7x6=43 is perfectly normal and nothing to be ashamed of.

Which reminds me: I never wrote part 2 of my post on cumulative practice.

15 comments:

ElizabethB said...

I worked with a young lady whose first language was Korean. When I complained to her about having to think for a while about which months were later than others and, for some months, which number they were (a few I know automatically,) she said that she didn't have that problem at all since the numbers of the months were their names in Korean, and that translated over to her English as well, she instantly thought of a number when she heard a month and knew which ones came first instinctively.

I also find myself having to sing the alphabet song occasionally (silently, of course!) when alphabetizing, I don't always have a sense of which letters are first. I have periodically tried to teach myself to associate numbers with each letter (A-1, B-2, C-3, etc.) but have never had enough motivation to memorize more than the first few, so I'm not sure if it would truly be helpful or not. (Although based on my Korean-speaking co-worker's experience, it probably would be.)

ElizabethB said...

On the all math looks alike, associative inference:

one of my husband's work phone number (the last four, that's all you have to dial on a military base) was the exact opposite of the last four numbers of another phone number I frequently dialed. It was really hard to remember, when it seemed like it should have been easier. I didn't memorize the number, just remembered that it was slightly different. Then, I would have to think of the other number, then remember just what the difference was, then apply that difference.

Catherine Johnson said...

I'm so glad I came across this passage (haven't read the article yet).

It's exactly what I need to "hold onto" this idea.

Catherine Johnson said...

btw, I now believe that this aspect of math makes massed practice even less effective than it is with other content - less effective and in some cases outright destructive.

I'm come to the conclusion that, with math, "mixed review" - or "shuffled" problems should be the norm.

Teachers should always assign problem sets with individual problems drawn from many different lessons in the book.

That's what C's teacher has done this year and he's learned a huge amount.

It's ridiculous the books aren't published with shuffled problem sets. It's obviously a huge amount of work for the teacher to put these assignments together.

Saxon Math is the only series I know that has shuffled problem sets.

Catherine Johnson said...

The fact that Glencoe doesn't come with shuffled problem sets makes things harder for students and parents, too.

Instructivist said...

[For the hardest problems, like 7 x 8, the error rate can exceed twenty-five per cent.]

Not if you learned this nifty trick:

56 = 7 x 8

(Numbers in consecutive order)

Independent George said...

Six times nine - Forty-two?

I always knew there was something deeply wrong with the universe.

SteveH said...

"How do we know whether numbers are bigger or smaller than one another?"

I don't think this is what they mean. I think their idea of number sense derives from their belief that traditional math just teaches rote procedures that require no thinking about whether a number seems right or wrong. However, this is more of a problem nowadays with an over-reliance on calculators. Kids punch in numbers adn give answers to 5 decimal places.

How many paper clips can you hold in your hand?

How long will it take to drive 50 miles?

If a 10-year-old is 5 feet tall, how tall will the child be at age 20?

How many pounds does a baby weigh?

This is not strictly an estimation skill. Their idea of number sense relies almost entirely on content knowledge and a little common sense.

My main point is that they seem happy enough even though they can't define it accurately. If they can't define it, then they won't know how to improve it.

Related to math, it seems that a lot of what people call number sense has to do with units, like lengths, areas, weights, volumes, and rates. They could directly improve this skill by studying and examining different types of units directly. Numbers are numbers, but units give meaning or sense to numbers.

How much does a loaf of bread weigh? How much does a car weigh in pounds or kilograms? How many square feet in a football field. How many square feet in a typical house? What is a typical house construction cost per square foot? This is all content knowledge. You could look these things up, but can you call it number "sense" if you have to look it up? You could also tie this approach in with a proper development of dimensional analysis.

Unfortunately, the best that Mr. Fennell can do is talk about some sort of magic number sense osmosis process that has to happen right now.

bky said...

I think there are two kinds of numbers sense. One is mathematical and one is really a quantitative understanding of some content area, which is what steveh was referring to.

Here is an an example of mathematical number sense. On some math/ed blog I recently read about middle schoolers who couldn't figure out some proportions involving 1.5 and 3. My reaction to that was that my fourth grade kids might not know what proportional means, but they know, from everyday math lessons and working with rulers, that half of 3 is 1.5 That's number sense of the mathematical sort. You might say, no, that's number facts, but since it's something that is absorbed rather than specifically taught, I think of it as number sense -- fluency with numbers.

Physicists talk about number sense and they mean knowing orders of magnitude -- being able to give the exponent in 10^n for various questions (how many miles from here to there, how many elementary particles in the universe, that sort of thing). So an example of that would be, for the question of how tall the 5-ft ten-year old will be when he is twenty, knowing not to answer "ten feet tall."

Here is a mixed example (in which I am the hero). I was reading a history book to my kids and we learned that North America has "thousands of square miles." I knew that the distance coast to coast is "thousands of miles" and therefore the area of the continent is like a thousand squared, so "millions of square miles." That requires knowing some physical facts plus what area is and how exponents behave under squaring. Nothing you can expect grade school kids to decisively master, in combination, at any specific grade.

Tex said...

. . . they know, from everyday math lessons and working with rulers, that half of 3 is 1.5 That's number sense of the mathematical sort. You might say, no, that's number facts, but since it's something that is absorbed rather than specifically taught, I think of it as number sense -- fluency with numbers.

I do not understand how you can say that fluency with numbers is absorbed rather than taught. It seems that the most efficient way to impart the fluency/number sense that is being referenced is to teach it directly, not require a student to absorb it in a constructivist-driven fashion. From my experience, this is directly related to the sad situation of so much of my child’s time being wasted in the classroom.

Tex said...

Their idea of number sense relies almost entirely on content knowledge and a little common sense.

That’s my experience. I recall some of the questions in the early elementary grades required more life experience than my daughter had at the time.

It also strikes me that this definition of number sense could work against some cultural groups in an unintentional way. Or maybe I’m missing the point, and this is all very intentional. The educrats’ goal may be to try to teach all our kids certain cultural content, and all I want is for my kid to learn math.

bky said...

Tex -- you can't directly teach everything. There is no Lesson #N that covers half of three. In the process of doing the work kids do on various tasks, which in my kids' case does involved direct instruction, they pick up a whole slew of understanding, some of which comprises random facts (or intuition about facts), e.g. what is half of small odd integers. It's like with reading. People who read pick up words that they understand, at least receptively, without having to have it on a vocabulary list. "Number sense" is common sense about numbers; it is not innate, but something some people have more talent for than others; and it is learned, but not explicitly taught. The same is true about fluency with language. It is learned, but not explictly taught. Fluency accretes around those things that can be explicitly taught.

SteveH said...

bky wrote:

"I think there are two kinds of numbers sense. One is mathematical and one is really a quantitative understanding of some content area, which is what steveh was referring to."

I will agree with that. Dan K. gave this example on the other thread.


"I guess the antithesis of number sense, to me, is when a student computes the third side of a triangle with sides of length 20 and 30—by the Pythagorean Theorem or the Law of Sines or however—and his calculator tells him that the third side is something nutty like 0.561 long."


This is not an estimation problem and it isn't a units or content problem.


Then Tex wrote:

"It seems that the most efficient way to impart the fluency/number sense that is being referenced is to teach it directly, not require a student to absorb it in a constructivist-driven fashion."

I agree with this. You can't teach everything, but you can have a plan. That was my main issue with Mr. Fennell's comments. His examples of number sense were poor and he didn't give any idea, other than by osmosis, on how to achieve the vague goal.


I made some suggestions in a previous post about direct instruction of units, and there are ways to directly teach kids about other numerical common sense rules. You can also directly teach estimation. Some reform math curricula try to do this, but they NEVER emphasize mastery, even with their alternate approaches.

For example, many of their alternate algorithms require you to do the addition, multiplication, etc. in a left to right fashion; add the hundreds column, then add the tens column, then add the ones column. This is how I do estimation in my head. I stop when I have enough significant digits. But the goal of reform math is only conceptual understanding, not mastery. I think they do these alternate algorithms because they place less emphasis on knowing your number facts. The kids do it on paper, not in their heads.

If they don't like the idea of doing things only one way, then teach the kids both the traditional (right to left) algorithms and their alternate ones (left to right). But the key ingredient is practice and mastery, not conceptual understanding. They just don't like hard work.

Having a detailed teaching plan is a good place to start and mastery should take care of the rest.

TerriW said...

Bear with me, because I'm not sure how related this is (heh) -- but to this day, as a nearly-35 year old, my brain has trouble processing facts that are "One or the other" types. Occasionally, I still have to put my index fingers and thumbs up of each hand to figure out which makes the "L" and is thus my left hand.

College-educated (a BS even), but left vs. right confounds me.

SteveH said...

I write with my right hand.

Actually, I learned right versus left in third grade. From where I sat, the windows were on the right. Imagine my problems in a modern day classroom where desks are arranged in circles.