Friday, February 29, 2008

More On Finger Multiplication

Last night, I received the following response to my previous blog entry, Finger Multiplication, the Lattice Method, and At-Risk Students:

Dear Jerry,

I cannot find en e-mail address to Michael Paul Goldenberg, but I would like to share the following with him in response to his mail via you.

In the 1980s 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 attach here 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 ethnomathematics here!

Best wishes,

Barbro Grevholm

I wrote to Barbro and received permission to reproduce both his message to me above and what he sent from his book:

Finger Multiplication

…to obtain the products for all multiplication facts from 5 X 6 through 10 X 10

How to “read” your fingers and obtain the products can be modeled with children using base 10 manipulatives…each “up finger” is equivalent to a 10-rod, and obtaining the product of the “folded finger(s)” times the “folded finger(s)” can be modeled with unit-cubes. The final product is then obtained by summing these two (2) partial products.

To model with your fingers, begin with a clenched fist, which is worth five (5), and then each raised digit on that same hand is worth one (1) more. For example, the clenched fist is equivalent to five (5), whereas if two fingers are raised, that same hand is now equivalent to seven (7) (…b/c the five (5) for the hand, plus two (2) more for the two (2) raised fingers, 5 + 2 = 7). To more easily “read” your “folded fingers,” it is always recommended to rotate both of your wrists so that your palms are toward your face.

Assigned values:

* Each clinched fist is equivalent to five (5).
* Each additionally raised finger on that particular hand is worth “one (1) more,” therefore three (3) raised fingers on the right hand would make the right hand equivalent to eight (8).
* When “reading” your hands, each hand represents a factor from five (5) to ten (10).
* Once a factor is displayed on each hand, remember that each “raised finger” represents a ten (10), therefore skip count by ten (10) for each raised finger and place this final product in your head.
* Now it’s time to “read” the “folded fingers.” Count the number of folded fingers on the left hand, and then count the folded fingers on the right hand…multiply these two (2) smaller factors together.
* To obtain your final product for the two (2) factors, one (1) factor displayed on each hand, sum the two (2) obtained partial products (…one from the 4th step “raised fingers,” and one from the 5th step “folded” times “folded”).

Now here’s the algebra to prove this phenomenon:

Let x = the number of “raised fingers” on the left hand, and…

let y = the number of “raised fingers” on the right hand…remembering that the hand itself is equivalent to five (5), making the left hand now worth 5 + x and the right hand now worth 5 + y.

Therefore, we are finding the product of (5 + x) (5 + y).

(5 + x) (5 + y) =


  1. 25 + 5x + 5y + xy =

(application of the
Distributive Property)

  1. 25 + 10x – 5x + 10y – 5y + xy =

Substitution (of
equals for equals)

  1. 10x + 10y + 25 – 5x – 5y + xy =

Rearranging terms
(Associative and

  1. 10 (x + y) + 5 (5 – x) – y(5 – x) =

Distributive Property
(applied three times)

  1. 10 (x+ y) + (5 – x) (5 – y)

Distributive Property
(…factored out (5 – x))
and QED.

10 (x + y) = 10 times the sum of the “raised fingers” from both hands and (5 – x) (5 – y) is the product of the “folded fingers” from both hands, and therefore, the final product is the sum of these two (2) partial products.

Tuesday, February 26, 2008

Finger Multiplication, the Lattice Method, and At-Risk Students

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.

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.

Monday, February 11, 2008

Mathematics Teacher Education

"Stay away from that unproven experimental stuff. Much better to stick
with the Moving The Furniture Until He Gets Better approach." Gregory
House, MD.

The on-going debate about content, pedagogy, and pedagogical content
knowledge as they pertain to mathematics teaching and the education/
training of future mathematics teachers continues to produce more heat
than light in venues where one side continues to insist that we need
to focus on mathematics content only, that questions of pedagogy are
closed ("One way to rule them all, One way to mind them, One way to bring them all and in the darkness bind them: direct instruction!), that pedagogical content
knowledge isn't worth discussing, and that in fact all questions of
mathematics teaching and learning are closed (or at least that there
are no open ones on the table).

Contrast that attitude with what is likely to be gleaned from the
following presentation, which I plan to attend tomorrow:

Systemic school improvement through teacher learning:
The case of Japan

International comparisons repeatedly show that Asian countries such as
Japan excel in the teaching and learning of mathematics. This
presentation will consider how teacher education and professional
development in Japan contribute to these outcomes. Specific
mathematics tasks designed for “research lessons” in lesson study
groups will be examined to reveal aspects of Japanese professional
development critical for teacher and pupil learning of mathematics.
Beyond features internal to lesson study, the presentation identifies
the role lesson study plays in systemic school improvement. The
presentation compares American conceptualizations of school “reform”
to Japanese models of “continuous improvement.”

Jennifer Lewis completed her doctorate in teacher education at the
University of Michigan in 2007. She currently works on a number of
mathematics education projects in the School of Education. Jenny is
especially interested in ways teachers learn in their practice
settings, and how might they best be prepared to do so.

Educational conservatives and anti-progressives pay enormous lip
service to Asian mathematics education - in China, Taiwan, Singapore,
Hong Kong, and Japan - as long as no one looks to closely to what
actually goes on in these countries. These Americans prefer to pick
out anything that looks like it supports their viewpoints (whether in
fact it does) and ignore or spin everything else. Few of them, if any,
spend time in these countries, of course. Why bother to observe real
classrooms, kids, teachers, or teacher education? Why look at such
innovative professional development models such as Japanese-style
lesson study? After all, such practices cost: serious investments of
money, time, and personnel are needed, either through extending school
days while reducing individual teaching hours, or by paying for
substitutes to cover classes while teachers are participating in
lesson study. Wouldn't buying a set of books from, say, Singapore or
Oklahoma and throwing them at teachers suffice? Or making pre-service
teachers take more and higher-level mathematics courses taught by
professors of mathematics with little or no experience or interest in
pedagogy or pedagogical content knowledge? After all, if we're going
to spend money to pay professors to work with pre-service teachers, who
is better qualified: a mathematics Ph.D whose primary experience and
interest is in pure mathematics research, or some pretender from a
School of Education whose main claim is a minimum of three years'
actual elementary or secondary teaching experience? Clearly, wasting
time in real classrooms with kids and teaching colleagues is not as
valuable (or as lucrative for the math department) as Partial
Differential Equations for Kiddies.

Of course, this is not necessarily a meaningful dichotomy, as we see
throughout the country: there ARE mathematicians who are deeply
interested in and knowledgeable about mathematics teaching and
learning. And there are mathematics educators who know the requisite
mathematics for the relevant band(s) in K-12 curricula deeply and
well, and who can communicate it to teachers and would-be teachers
effectively. Ideally, future teachers get the best of all worlds:
enriching mathematics content courses designed specifically for future
teachers, not future Ph.Ds in pure mathematics or future engineers or
future physicists, as well as mathematics methods courses, field
experiences, practicums, and supervision from experienced and
reflective instructors with knowledge of teaching and pedagogical
content that is grounded in work with real kids out there in the
world. Both sorts of courses should be taught by people whose
knowledge crosses between the disciplines. Ideally these instructors
will have some practical knowledge of applications of the mathematics,
will have knowledge of and competence with a spectrum of the
mathematics and applications that the actual grade-level course
content points towards And also ideally, these instructors would view
one another as colleagues, rather than with fear, suspicion, or
contempt. Naturally, for such collegial relationships to work, neither
the math department-based teachers or the ed school-based teachers can
view their perspective as either best or complete: they must instead
work intimately with those from the other "world," and value the very
real contributions that each sort of teacher can make.

As long as a small, vocal, entrenched minority of mathematicians and
like-minded individuals continue to denigrate the import of teaching
knowledge grounded in subject matter knowledge, and instead pretend
that content knowledge alone, along with a slavish devotion to direct
instruction as the sole approach to teaching school mathematics, and
as long as such people have the ear of influential politicians, policy
makers, parents, and colleagues who are prone to believe what they are
told by fellow mathematicians about the "evils of Schools of Education
and those who are educated there," we're in for more years of foot-
dragging and wasted energy. Or as Gregory House so wisely observes:
"Stay away from that unproven experimental stuff. Much better to stick
with the Moving The Furniture Until He Gets Better approach."

Sunday, February 3, 2008

Math Blogs Rated

It appears that folks at have rated a bunch of math blogs, (83, to be precise) and placed mine in the top 15 (tied for 12th place), with rating 8.3 (out of 10). They evaluate blogs based on the following criteria: Frequency of Updates, Relevance of Content, Site Design, and Writing Style. Clearly, I need to update more frequently, but otherwise I guess I'm doing okay in my first year of blogging. :) Nice to get the feedback from what appears to be an apolitical source.

So here is a list of mathematics blogs, rated in a scale 0-10. If you're a math blogger, go find yours. If you're a reader of such blogs, here's a chance to find some of the best you might not know of yet.