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) =	GIVEN
25 + 5x + 5y + xy =	FOIL (application of the Distributive Property)
25 + 10x – 5x + 10y – 5y + xy =	Substitution (of equals for equals)
10x + 10y + 25 – 5x – 5y + xy =	Rearranging terms (Associative and Commutative)
10 (x + y) + 5 (5 – x) – y(5 – x) =	Distributive Property (applied three times)
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.

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.

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."

Math Blogs Rated

It appears that folks at Blogged.com 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.

Problem Solving and Mathematics Education

Maria Miller, seen above, has an interesting and useful blog on mathematics for home schoolers, as well as other web resources, not the least of which appears to be a recent web page called "Math word problems — the do's and dont's of teaching problem solving in math" that I've just started looking at today.

The most recent entries on her blog address this vital issue of teaching problem solving, via a couple of problems for 5th graders taken from a Canadian on-line resource.

The first post Maria made on these problems reads as follows:

I'd like for you to try solve these two 5th grade problems. They are from a very nice collection of word problems for kids by Canada's SchoolNet.

I'd like to use them to illustrate problem solving. But before I do that and give you the solutions, please try to solve them yourself. And don't post solutions here for now; I will have to reject such comments.

Solutions and discussion will follow in a few days.

THE FIRST PROBLEM IS:

* How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

THE SECOND IS:

* Divide the face of the clock into three parts with two lines so that the sum of the numbers in the three parts are equal.

I set about attacking these problems as soon as I saw them, both because they appealed to me and because I wanted to be ready for the ensuing discussion. You may want to take some time to think about the first one, for which solutions and comments appeared on her blog today.

Maria's comments can be read by clicking on that last link ("today") or going to her blog and finding the entry for 16 January 2008. You will also find a response from "mathmom" and one from me. By the time you read this, there will likely be others.

SPOILER WARNING: if you read past here, assuming you've not yet clicked on the last three links, you're going to see solutions to the plus sign problem.

Here is what "mathmom" had to say:

I approached this one a little more methodically. I first summed the 9 digits to get 45. I found that the required sum was 54 greater than this number.

I then thought about what would happen if I made a two-digit number. The digit in the ones place would be added into the sum just the same as before. But the new tens digit would count for 10 times as much as it did before. So if I "promoted" the digit D to a tens place, I would add 10D and subtract D (since it would no longer be counted as a ones digit anymore) for a net increase of 9D. So if I need to increase the sum by 54, I need 9D = 54 and D=6. Thus if I "promote" the 6 by making 6+5 into 65, I arrive at the required sum.

I didn't think about the fact that I might have promoted more than one digit and used more than one two-digit number, but my son came up with the second solution, where the two-digit numbers used are 42 and 21. Here he promoted two digits, a 4 and a 2, with the same net effect as promoting just the 6. He had computed that he needed to increase the sum by 54, and then found this solution by a guess-and-check method.

To show that these are the only possible solutions, consider the possible combinations of digits that sum to 6:

6
5 1
4 2
3 3
3 2 1

We can't promote the 1, since there's nothing after it in the row to serve as its ones digit. (If you add a zero to the end of the line, this admits the 5 1 solution, but not the 3 2 1 solution, because you can't promote both 3 and 2 at once.) And we can't promote 3 twice, so the only combinations that work are 6 alone, or 2 and 4.

Here is my comment:

One correction to mathmom's comment: in the second solution, the first two-digit number is 43, not 42.

I had done something similar to arrive at her first solution (and later formalized it algebraically, though in this case I'm not convinced I gained a lot by doing so), and I also missed the second solution by not thinking further about the problem, to my embarrassment. But in reading the third paragraph of her post, I noticed before reading further that since D = 6, using her notation, and the other solution involves using D1 = 4 and D2 = 2, with D1 + D2 = 6, I then wondered about using D1 = 5 and D2 = 1, where again D1 + D2 = 6. Before I could think through why that didn't work (duh!), I saw what mathmom had added and realized the issue, which she neatly pointed out could be solved by including 0 as a tenth digit.

So a great continuation/extension question would be (for students who have been exposed to negative integers or who wouldn't be troubled by thinking about them), can we find more solutions if we append more consecutive integers to the list on EITHER end (or both ends)? Why or why not?

Since I've just thought of this, I don't have an answer. But I raise the issue because I'm always looking for extensions, not just as challenges, but because extending and generalizing problems is one of the major activities of professional mathematicians. Doing so is one way to help develop new theorems, methods, and, sometimes, entirely new areas of mathematics. I look back on my own K-12 mathematics education with a great deal of regret, not the least of which grounded in the fact that I was never led to realize any of that aspect of mathematics. Like most Americans, I was taught in a way that made mathematics appear to be a dead, completely closed subject in which everything was already known (by the professionals), and where my job was to accept received knowledge and be able to show that I "got it" by solving textbook problems that were, for the most part, strictly computational in nature. By 9th grade, the approach simply didn't grab me any longer, and not because I didn't generally do extremely well in mathematics up to that point: in fact, I was an A student through the end of 8th grade. While there were other factors having little or nothing to do with the instruction or content that led to my decreasing interest and effort in math starting in 9th grade, I wish I'd been shown something along the lines of what I mention above: the idea of doing what might be something new and original, or finding creative solutions to problems that already had known solutions, may well have kept me "in the game" at a time when I was growing restless, impatient and bored when it came to math and science.

Now, since posting the above comment this afternoon, I've had a chance to play a bit with my extension questions, as you may already have done or are about to do. I will not post my additional findings right now, but will wait to let folks have fun with the extension question on their own.

It would also be great if anyone has other extension ideas for this. Don't feel limited to the alleged "5th grade" level of the original problem, but it would bear thinking about the target for any such extensions. The one I'd suggest right away is: What happens if we change 99 to other values?

"If math were a color. . ." and other crimes against humanity.

If you're a fan of the Math Wars, you've heard about the question in the first edition of the EVERDAY MATH K-5 curriculum, "If math were a color, it would probably be ______ because ________." This particular question has probably engendered more ridicule and garnered more notoriety than anything associated with the NCTM Standards of 1989 on, and the progressive reform curricula that emerged from NSF grants in the early 1990s. After Googling on the question, I've concluded that no one on the planet has the slightest idea what could possibly be the point of asking this question. It has resulted in so much negative feedback that it has been dropped from revisions and subsequent editions of EVERYDAY MATHMATICS. Indeed, as recently as last week, Andy Isaacs, an author of the program who works at the University of Chicago Mathematics Project (UCSMP) posted on math-teach at the Math Forum:

As for the "If math were a color, ..." question, it appeared in our
first edition in a Time to Reflect section at the end of the first
unit in fifth grade, a unit that focused on prime and composite
numbers, factoring, figurate numbers, and so on. The first edition of
fifth grade was published in 1996. The "If math were a color, ..."
question was deleted in the second edition, which was published in
2001, and the entire Time to Reflect feature was deleted from the
current edition, the third, so this question has not appeared for
some years in EM.

I suppose that since Andy, whom I've met and consider a friend, is washing his hands of the question and it's been dropped entirely from the series, I should let sleeping inquiries lie, even if the folks at Mathematically Correct, NYC-HOLD, and countless "parents-with-pitchforks" groups and web-sites (as well as right-wing pundits like Michelle Malkin continue to make "big" political points about how programs like EM are sending us and our children down the fast track to enslavement by those ever-inscrutable Asians (that Ms. Malkin is herself of Filipino extraction should in no way be taken as a mitigating circumstance in her willingness to help scare the James Jesus Angleton

out of hard-working Americans with the reliable threat of hordes of Asians (Chinese, Japanese, Singaporeans, and Indians, in particular) overtaking us in the constant battle to be #1 in anything and everything. Never mind that it's actually those sneaky Finns who top the latest list of countries that outperform the US in mathematics on an international test. No one is yet recommending that we adopt the Finnish national math curriculum and load up on glug and besides, the Finns themselves are not quite so sanguine about the situation. However, racist that I am, I just can't get sufficiently frightened by images of invading Finns. Asians, of course, are notorious for "teeming," and there's no question that they're going to overwhelm American any second, no doubt due to that famous "math gene" they're all born with.*

But let's get back to the point (sometimes my own digressions frighten even me). What about that "If math were a color" question? Am I seriously planning to defend it as not only "harmless" but actually sensible? Those of you who know me are no doubt sure that's exactly what I claim.

My argument is brief, but I believe it's quite reasonable:

It is increasingly common practice for some high school teachers and some professors of mathematics and mathematics education, especially those working with future or would-be future mathematics teachers, to ask their students for a brief "mathematical autobiography" at the beginning of the term. It's a way to get to know more about students and to find out their attitudes towards and experiences with mathematics (does that seem like a trivial thing to anyone? While I wouldn't be the least bit surprised if it did given some of the folks who teach mathematics, it's nonetheless the case that some educators think it's quite valuable to know how education students think AND feel about the subject(s) they propose to teach, and how students think and feel about a subject they are expected to learn).

Now, if you're teaching K-5, especially if you're teaching K-2, how would you propose to get students to relate their beliefs about and attitudes towards mathematics? Direct questioning, like direct instruction, is not always effective, which is why many child psychologists work with their clients through playing games and other indirect methods for getting insight into experiences and feelings about them. The question"If math were a color. . .?" is a perfectly imaginative and reasonable way to get children to explore and reveal some of their feelings and beliefs about mathematics. The important thing is not, of course, to elicit a particular response or "right answer" (and isn't that a radical concept?), but to help students to find a non-threatening, non-intellectualized way to talk about math. Heaven forfend that child psychology should enter into any elementary school teacher's classroom or pedagogical repertoire, of course. Sensible, rational, intellectual questions only, most particularly in math.


*For the irony-impaired, that's satire.

Book: OUT OF THE LABYRINTH: Setting Mathematics Free

Ellen & Robert Kaplan and The Math Circle


If you only read one book about mathematics teaching and learning this coming season, let me suggest that it should be OUT OF THE LABYRINTH: Setting Mathematics Free by Robert and Ellen Kaplan, founders of The Math Circle in Cambridge, MA.

I had the distinct privilege several years ago of attending a session Bob and Ellen led at Northwestern University's Math Club. We were taken through a short series of problems that led to the main question of the day: is it possible to cover a particular rectangle with non-congruent squares? The way things were led was truly masterful, perhaps the best teaching I've ever seen or experienced. The preliminary questions helped scaffold the main one, but once we entered into trying to solve the main problem, for which most in attendance seemed to believe the answer was "No," very few comments or questions were offered to help us. And yet those "hints" that were forthcoming seemed to be just the right ones needed to steer us out of ruts or stimulate our best thinking to move forward, and within about an hour, the students (who included undergraduate math majors, graduate students in mathematics, and some faculty members from Northwestern's mathematics department, none of whom seemed to be familiar with this problem or area of math) had successfully found a correct solution. The entire process was beautiful to see.

Afterwards, I introduced myself to the Kaplans and commented on how impressed I was with the way in which the lesson had been taught with such minimal but incisive questions and comments from them. Bob stated that they did the same problem recently with a group of 5th graders in Cambridge. Before I could express any skepticism or incredulity, he added, "Of course, it took about twelve weeks!" (The problem was presented as part of an on-going Math Circles course, and a great deal of time was spent building up the necessary tools so that the students could tackle the bigger question. I had no doubt, however, that the basic approach to helping these students was the same as what I experienced in Chicago.

When I asked Bob about who else taught in the Math Circle classes, he said that it was no problem finding people who knew the requisite mathematics. The difficulty was in finding such people who could also keep their mouths shut. And indeed, that is not only a problem in Cambridge, but nationally, including in my own teaching. I struggle mightily to resist the temptation to bail students out prematurely, to succumb to the pressure, both internal and external, to show what I know and to relieve the students of responsibility for thinking and learning mathematics the only way that really makes a difference, in my view: by doing it themselves. The cliche "Mathematics is not a spectator sport" is clever, but it is also very true. It's difficult to really "own" a piece of mathematics without struggling to wrap one's brain around it. This is not to say that we can gain nothing at all from a lecture, of course. But when the money is on the line and there's no one there to help us solve a challenging problem (where "challenging" is, of course, a very personal and relative term), will we do best by recalling a step we saw someone else use, seemingly pulled effortlessly from thin air, or by calling on methods we've used ourselves and built up sufficient experience with to sense that one might be of particular value in the current case?

The story the Kaplans have to tell in their new book is far more powerful than any argument I could offer here as to why it is so crucial for students to be led to believe correctly that they CAN do it themselves and, perhaps more importantly, that they SHOULD do it themselves, collaboratively, as a community of learners, without the usual competition to be the first or the best, but out of a collective passion to know.

I will say in all honesty that what I saw in Chicago had me nearly weeping with pleasure and frustration. Pleasure at the wonderful way in which the Kaplans taught, but frustration at the rarity of the experience and my knowledge that far too few Americans ever get to see and learn mathematics (or much of anything) in this powerful way. Reading their new book, I continue to experience these strong emotions. If you care about mathematics and really empowering people to DO it, go to the Math Circles web site: check into their books, look at some of the notes you can download from recent courses, look at the photos, and generally get a taste of what they do. You'll be very glad you did.