I have begun, quite informally, an unusual collaboration with a friend who is in her first year teaching mathematics to at-risk students in Saginaw, MI. (This is not her first year as a teacher, however, as she has prior experience teaching theater in public schools). I will be posting in more detail about some of what we're doing and how it's working out for her and her students in another entry quite soon, but I wanted to look at a specific method for finding the product of two single-digit numbers from 5 to 9 that she showed me recently. The context of this method is that I had shown her how to do lattice multiplication with multidigit numbers, but she realized that many of her students were weak with the necessary single-digit multiplications that are needed to complete the lattice and hence would not clearly benefit from the lattice approach (nor from any other, since it's rather difficult to do multiplication of larger numbers by hand if you don't know the tables past four or so). Thus, she first taught them the finger method outlined below, then introduced the lattice method, and the students appear to be doing well with this combination, as I will report in more detail in the subsequent entry.
Becca's method was not familiar to me, though I had long ago (back in the 1980s) read a book on Korean finger math ("chisenbop") that I found interesting. In looking for images for this post, I found that there are many other "finger-math" methods out there, some of which many readers have likely heard of (like the "trick" for the 9's table), others perhaps less familiar. I have tentative plans to attend some workshops in Seattle next month by Alice Ho, a Singapore-based mathematics teacher, one of which will deal with a finger-math method she teaches to kids, adults, and educators in Singapore.
In any case, Becca's approach for multiplying, say, 7 x 8 is to have students first make two fists with the backs of their hands facing up. To represent 7, extend the number of fingers more than 5 needed to represent the number (in this case, 2). On the other hand, do the same thing for 8 (resulting in this case in 3 extended fingers). Now add the extended fingers and append a 0 on the right to the sum (in this case giving 50). For the rest of the product, multiply the number of NON-extended fingers on the left hand (3) times the number of NON-extended fingers on the right hand (2) and add the results to the previous product (50 + 6) giving the originally-desired product, 56.
I tried another example and satisfied myself that I could do the method, but of course was curious as to why this worked. Becca didn't know, and I quickly set out to satisfy myself. Here is what I came up with:
Let your two numbers be x and y.
What you're doing with the fingers up business is subtracting 5 from each of the two digits (multiplier and multiplicand), so you start with (x-5) and (y-5).
You add them and multiply the result by 10 (since you're using the result as your tens digit):
10[(x-5) + (y - 5)]
The second part gives you the difference between 10 and the two digits, which is (10 - x) and (10 - y) and you multiply them together: (10 - x) (10 -y) to get your units digit.
So altogether, you have 10[(x-5) + (y - 5)] + (10 - x) (10 -y).
That's 10[ x + y - 10] + 100 - 10x - 10y + xy
which equals 10x + 10y - 100 + 100 - 10x - 10y + xy
which simplifies to simply xy, the product you wanted in the first place.
Please note that this method actually works for ANY one digit times one digit number, but it's physically harder to make use of for digits less than 5.
For example, 2 x 3; Let x = 2 and y = 3.
Plugging into the formula, you have 10[ 2-5 + 3-5) + (10-2)(10-3)
= 10*(-5) + 8*7
= -50 + 56
= 6
But of course, if you know how to find 8*7 in your head, I would assume you'd be able to find 2*3 in your head as well. And how you physically represent negative numbers of fingers is beyond me. ;)
It also works for numbers greater than 9, but becomes increasingly cumbersome and the purpose of the approach (provide a useful tool to do and/or practice the upper half of the multiplication tables from 5 to 9) is defeated.
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I sent the above analysis to Becca and she responded by wondering how I came up with it and whether some of her students would be able to do a similar analysis. The latter is an open question that I hope she pursues with them. As for the first inquiry, it seemed obvious that the issues was representing the tens digits as x - 5 and y - 5 since we were subtracting 5 from the original two digits (or adding onto 5, if you prefer) to decide how many fingers to put up on each hand. The digits used to calculate the units digit of the answer were clearly what was left if we subtracted the original digits from 10, individually. In other words, this became rather simple algebra (I use the word "simple" somewhat ironically, of course) once the way to represent the digits was settled correctly.
I mention the above in part because of the similarity in my experience with my early encounters with lattice multiplication: I hadn't heard of it, saw a teacher do it, understood the process but couldn't believe that he hadn't any clue why it worked and worse that he didn't seem in the least bit curious to know, even when I figured it out. Becca, on the other hand, was eager to find out why, and once she knew, immediately wanted to see if she could get her students interested in and successful with figuring it out as well. That's just part of the difference between our informal "coaching/mentoring" relationship and the more official one I had with a few of the teachers whom I was coaching without their having much choice in the matter. While this doesn't prove anything, it does point a bit towards Virginia Richardson's work on having professional development for teachers that originates from the teachers, rather than being imposed from above.
In any event, I look forward to getting more information on how these at-risk high school kids do with these methods, whether they are curious about how they work and, if so, whether that leads them to want to learn more algebra. I will report on that if I get a report from her. Meanwhile, I will complete the blog entry I began on what else Becca has been up to and my contribution, such as it's been, to her work in Saginaw. Much more to come, I hope.
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10 comments:
Some teachers may find the following rewritten analysis of "finger multiplication" slightly simpler than what Michael gave.
Let L and R be the number of extended fingers on the left and right hands.
Then the problem is to compute the product
Prod =(5+L)*(5+R)= 25+(L+R)*5 +L*R.
Notice that the product of the numbers of bent fingers is
P2 = (5-L)*(5-R)= 25 -(L+R)*5 +L*R.
And the difference is
Prod - P2 = (L+R)*10.
This kind of representation and calculation might seem fairly natural to a Mayan since the numbers from 5 to 9 are written as a bar followed by zero to four dots.
This is an interesting method, and I applaud Becca's interest in understanding why it works. But I do not think the method should be taught, except as interesting enrichment, for several reasons: (1) it is not easy to learn, (2) it is very hard to explain, and (3) it helps with very few of the basic number combinations.
The method is appropriate when both factors are between 6 and 9, and there are only 16 such products to learn. Allowing for commutativity, there are really only 10 products here. But 4 of these products are learned much more easily with the 9's trick, which has an associated finger trick that is easy to use, easy to learn, and not hard to explain.
Of the remaining 6 multiplication combinations, 3 are squares that are not very hard to learn anyway.
That leaves us with only 3 multiplication combinations that this method might help with: 6x7, 6x8, and 7x8.
But these are the products that are always hardest to learn. For these products, rather than spending time on a method that students are unlikely to be able to explain, I would prefer that students use skip counting, "nearby facts," and the idea of "near squares" for 6x8 as near 7x7. And after a while, perhaps the kids could be encouraged to memorize these three, as long as they also have other ways of checking their answers.
Brad Findell
Ladnor: thanks for your revision. I reproduced my initial analysis without revision because I wanted my thought process available for Becca and others. In my experience, hiding the process of figuring things out is one of the problems we have in math: we make things look cleaner, clearer, and easier than it usually is. There is no dearth of "clean" presentations, but a definite shortage of transcriptions of thought processes.
Brad: My interest is in results, and Becca is getting them, apparently. From that perspective, this method is "useful." I hardly see it as powerful or generally wonderful, but as we're looking at something that appealed to and worked for at-risk kids in the grade range 7 - 10 who had no previous engagement with mathematics, were operating at about a fourth to fifth grade level, and were not big fans of school in general, there's reason enough to take this seriously.
I definitely think there are better way to help students gain mastery of addition and multiplication facts. I've often cited the approach John van de Walle suggests in his work as a sensible model. But he's looking mainly at teaching young kids in a developmental context. Becca's kids are "educationally retarded" in the literal sense: they're slow relative to where we would expect them to be if they'd learned in developmentally appropriate ways. This isn't a comment on their capabilities, but on the reality of where they are. So I am impressed greatly by the mere fact that Becca found ANYTHING that worked for them, that they were willing to try, and that they are currently using successfully. I see this method as a means to an end, not the end in itself. Perhaps now that they are engaged in doing these calculations and getting success in doing them through the finger math and lattice method, they will be more open to revisiting the tables from the sort of perspective you and I and van de Walle and others would prefer them to do. If not, they're still miles ahead of where they were a couple of months ago.
Michael,
Thanks for your response. I agree completely that Becca should be praised for finding ANYTHING that gets her students engaged. Let me modify slightly what I said before.
I don't think that this method should be taught in, say, 3rd grade, when a primary goal of instruction is developing proficiency in multiplication.
But if the audience is, say, 9th graders, then the method might be taught with a goal of using algebra to explain how the method works. Improved multiplication proficiency is then gravy, for the students who still need it.
With a little creativity, a lot of algebra instruction can serve to solidify things the students should have learned earlier.
Brad Findell
Thanks again, Brad. She's got at-risk kids in inner-city Saginaw, MI, currently grades 7-10, planning to expand to 7-12 over the next two years. They are pretty much at the 4th-5th grade level coming in, which is typical of my experience with similar groups in Wayne-Westland, MI. However, her students are about 50/50 black and Latino, whereas I had about 60% white, 30% black, 10% Latino. It's important to note that the grade level they test at holds for literacy as well, which makes it very important to scaffold things beyond just the mathematical content. Finding materials that are suitable in print is probably a fool's errand. Adapting good problems and practice materials however, is part of what she's doing now, as I will be blogging about shortly.
Here is one further thought on the basic identity behind the finger multiplication. That is, if
P1 =(5+L)*(5+R)= 25+(L+R)*5 +L*R, and
P2 = (5-L)*(5-R)= 25 -(L+R)*5 +L*R,
then P1 - P2 = (L+R)*10.
Generalize by replacing 5 by any number M. That is, if
P1 =(M+L)*(M+R)= M^2+(L+R)*M +L*R, and
P2 =(M-L)*(M-R)= M^2 -(L+R)*M +L*R,
then P1 - P2 = 2*(L+R)*M.
We are probably more familiar with the symmetric version of this where L=R. If
P1 =(M+L)*(M+L)= M^2+ 2*L*M +L^2, and
P2 =(M-L)*(M-L)= M^2 -2*L*M +L^2,
then P1 - P2 = (2*M)*(2*L).
This is the difference of squares factorization. Just let B=M+L and A=M-L then the above says
B^2 - A^2 = (B+A)*(B-A).
For a geometric version of this we could allow M and L to be vectors and take multiplication to be dot product, then this is called the polarization identity. It shows that in the parallelogram Q with one vertex at the origin and adjacent edges the vectors M and L, the difference of the squares of the diagonal lengths is connected to the measure of the angle between M and L.
|M+L|^2 - |M-L|^2 = 4*(M*L).
Note that if you add the squares of the diagonal lengths of Q you will get instead 2*(M^2 +L^2), which is the sum of the squares of the diagonal lengths in the rectangle whose side lengths are |M| and |L|.
In the 1980ies I was approached by a mathematics teacher who taught adults early mathematics. In his class he had a group of gypsies and they had taught him how they multiply numbers between 5 and 9 with the fingers.
The teacher could not figure out why it worked and wanted me to explain it, so I did. Later I gave this task to my secondary students to solve and I see it as a good modeling task and exercise in use of algebra for them.
I have published the task in a book in 1989 called The Challenge -
Problems and Mind-Nuts in Mathematics (in Swedish) and will attach in an e-mail the one
page of that book translated into English of the task I call Handy
multiplication (fingerfärdig matematik in Swedish).
This seems to be old folk knowledge and I have seen old Swedish books
where the method is explained and also a method for how to multiply
numbers between 11 and 15. We are into ethno-mathematics here!
Best wishes,
Barbro Grevholm
Michael, when once showing this method to a math class for prospective elementary teachers, an engineering student waiting outside the classroom for the next class became very excited when he saw what I was doing. After class he told me that he was from Africa and that was the way school kids learned their multiplication facts in his country. Unfortunately I don't remember what country he was from. I wonder if it is still taught in countries in Africa or elsewhere?
By the way, I'm sure you know the easy way to remember the "nines" facts: lay your hands palm down on the table. For 1x9, fold down your little finger on your left hand, and you have 9 up. For 2x9, fold down just the ring finger on your left hand. Note that you have 1 finger up left of the "down" finger, and 8 on the right hand side. Sure enough, 2x9 = 18. For 3x9, fold down your middle finger on your left hand, and you get 2 up on the left side and 7 on the right, etc. all the way to 10x9. Many kids (and teachers) love this memory device, and it also brings up the nice little question of why the digits always add to 9.
Cheers - Marj Enneking
One interesting extension to me -- since I am fond of imagining aliens (with bilateral symmetry, of course; has Star Trek taught us nothing? :) who have who have k fingers on each hand and use a base 2k -- is that this method also works in any even base 2k. The justification above extends fine; just replace all mentions of 10 with 2k and of 5 with k.
Regarding the second example, I'd say the method basically maps the {6-10}x{6-10} times table onto the simpler {1-5}x{1-5} times table, which is very nice! Though the algebraic representation makes it clear that the mapping works both ways (we're 'allowed to' use the method with {1-5}), I think we wouldn't /want to/ do so because it moves in the not-simpler direction.
(I am curious about modifying this to handle {1-5}x{6-10} multiplication; I don't think it can be used as is, because it seems like it would just swap the complexity from one multiplicand to the other.)
I totally agree about the situation where a teacher "hadn't any clue why it worked and worse that he didn't seem in the least bit curious to know"... I can never believe it either! The curiosity is so ingrained in my psyche. But I disagree that the difference here is primarily a "difference between our informal "coaching/mentoring" relationship and the more official one". It seems like the difference is more between the particular people involved, and perhaps of the culture of the department or school, rather than just the mentoring relationship (though that is certainly one aspect).
Briefly, if you want to see extensions of this that allow for larger products (37 * 38 and, I believe, mixed tens such as 37 * 78) get a copy of the old book Cheaper by the Dozen.
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