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r2 is a multiple of 24 and 10. What is the smallest value?Here, from a few weeks ago, is Stanislaus Dehaene on multiplication:
[O]ur intuition of quantity is of very little use when trying to learn multiplication. Approximate addition can implemented by juxtaposition of magnitudes on the internal number line, but no such algorithm seems to be readily available for multiplication. The organization of our mental number line may therefore make it difficult, if not impossible, for us to acquire a systematic intuition of quantities that enter in a multiplicative relation. (This hypothesis is supported by the fact that patients can have severe deficits of multiplication while leaving number sense relatively intact; in particular, patient NAU (Dehaene & Cohen, 1991), who could still understand approximate quantities despite aphasia and acalculia, was totally unable to approximate multiplication problems).This passage precisely captures my experience learning arithmetic.
Addition and subtraction make intuitive sense to me; multiplication and division do not. It's really that simple. For me, "math" - math as opposed to simple counting - begins with multiplication and division.
This observation brings me to a corollary: people always say kids fall off the math cliff when it's time to learn fractions, but I think the math cliff comes sooner. I think the math cliff is multiplication, only nobody knows it.
Nobody knows it because falling off a math cliff isn't like falling off a real cliff; with a math cliff, you can walk right over the edge and just hang there for awhile, suspended in mid air, like Wile E. Coyote.
The real drama comes after the fall, which is when kids finally get to fractions. Fractions aren't the cliff, and they aren't the fall. Fractions are the crash-landing at the bottom.
At least, that's my guess for the moment.
Setting metaphor aside, though, if it's true that we are not equipped with an intuitive understanding of multiplication and division, and I believe it is true, why don't more people know this?
Everyone knows fractions are hard; why doesn't everyone know multiplication and division are hard?
It's true most people perceive that certain aspects of multiplication and division are hard. Namely: memorizing the times tables is hard (for many children) and learning to do long division is hard (for many children). But I've never seen anyone take these facts to mean that there is something intrinsically challenging about multiplication and division in a way that is not the case with addition and subtraction.
Why?
I don't know, but I have some thoughts.
Which.... will have to wait. It's getting late, and I'm still trying to edit the body of this post into shape, so I'm going to set that aside and skip to the end, and just say that I think children should probably be taught to solve problems like the one above, which appeared on the October SAT. I'm pretty sure problems this can be used to find out whether students are suffering associative interference between addends and factors, which I bet an awful lot of students are no matter how quickly and accurately they can construct factor trees.
I'm also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like. I think -- I don't know -- that fluency with the terminology might help reduce associative interference. "All math looks alike": the 5 and the 2 in 5+2 look exactly like the 5 and the 2 in 5x2. But the words addend and factor have nothing in common whatsoever.
More later.
7 comments:
"I'm also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like."
That sounds like the New Math program that I went through as a kid. Which actually was a fine program - I learned from it - but evidently others disagreed.
Learning the times tables is hard because it is a skill completely unrelated to understanding multiplication. I don't think the concept of multiplication is hard, nor division. One of the things I liked about Singapore math was that they introduce multiplication right along with addition.
One of the things I liked about Singapore math was that they introduce multiplication right along with addition.
Interesting.
I think multiplication is an 'easy' topic that turns out to be hard later on....
I suspect a lot of children who had no trouble learning multiplication have trouble with the concept of factors and factorization down the line.
I wonder if introducing multiplication along with addition helps prevent confusion between addends & factors??
It might ----
I think that all the complicated names (addends, subtrahends, divisors, dividends, …) make simple ideas complicated. I never could remember which name went with which thing. In fact, I couldn't tell you now what "dividend" refers to, but I can certainly manipulate fractions and rational functions.
I think that the naming actually interferes with the understanding, as too much working memory is taken up with the names.
It may look like a cliff at certain filter points, like the 7th grade tracking point and the high school math class that you finally flunk. However, the process starts the first day of school.
Kids climb the failure cliff year-by-year because schools don't ensure proper grade-level mastery of skills. Our lower school (K-4) talks about (finally!) trying to make sure kids know their adds and subtracts to 20 by the middle of 3rd grade. While they are doing that, proper 3rd grade material is not being mastered. At some point (7th grade) the delayed and weak attempt at mastery stops and kids are tracked with all sorts of left-over gaps and lack of skills. Those gaps lead to a final failure or low peak in math.
They talk about balance, but they don't ensure a proper level of mastery even for the basics. Talk of the complexity of math or the difficulty of learning multiplication or fractions lets them off the hook. Schools teach math for an hour each day. Of course learning can be difficult, but they have plenty of time.
I call it low expectations, ignorance, and incompetence.
Oy, such a good post. Nunes and Bryant have done work in this area. The short story is that children often exit the 3rd grade (thereabouts) with misconceptions about what multiplication is (and hence what division is) and this can affect virtually all future mathematical learning. (Of course it would.)
If I could hope for any idea to take hold in education, it would be this one: The problems that your students are having with math right now in your class might not be due to lack of motivation, the complexity of the topic, bad parenting, crummy resources, video games, or the administration. It might be that we didn't teach them the right math from the beginning. Mastery is not a unidimensional concept. There are masteries that can hobble you for life. "Mastering" multiplication the wrong way is an example that is widespread, I think.
--JD
The triggering question
"r^2 is a multiple of 24 and 10. What is the smallest value?"
is supposed to invoke the prime factorization:
24 = 2^3 * 3
10 = 2 * 5
Squares must have even exponents on all prime factors, so the smallest
r^2 is 2^4 * 3^2 * 5^2
and the smallest r is
2^2 * 3 * 5
That is, r=60.
One could get at this more slowly by doing guess and check on the exponents, but recognizing that 2, 3, and 5 must be factors of r seems essential. Note that only easy multiplication facts (2,3,5) are needed for this problem. They could have made it much harder by using 7 or 13, without testing mathematical reasoning nearly as well (more opportunity for careless error).
A note on Singapore -- they introduce multiplication in second grade, but do it intelligently. They introduce multiplication by 2, 3, 5 and 10, then when students really know what multiplication is, they introduce the "ugly" ones. So students don't wind up blindly memorizing the times table, but do wind up mastering it!
I will say, the r^2 problem didn't seem too bad to me, but I really liked factoring when we first learned it, and would do it for fun (yes, I am a weirdo).
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