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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