Tuesday, July 1, 2014

The Power of Dots

On June 29, the New York Times ran a story about the Common Core Mathematics Standards. If ever you wanted proof of the dismal mathematics education most Americans have been provided, you will find it in the story’s “human interest lede,” which described one mother’s response to seeing her daughter’s homework. By taking the daughter out of school to teach her herself “the old fashioned way” she herself had been subjected to, this well-meaning parent was ensuring that, as had clearly been the case for the mother, the daughter too would not be exposed to real mathematical thinking—the kind that in today’s world is a key to the most attractive jobs. Instead she would be subjected to the same, dreary, rote-skills-drills inflicted on previous generations—a process designed to train people for routine work in the pre-computer era, but so hopelessly inadequate for the 21st century that parents are un-equipped to figure out for themselves the simple (albeit unfamiliar) math homework their children are assigned.

Surely, if mathematics education should achieve one thing, it is develop the ability to figure things out for yourself. We’re not talking the Riemann Hypothesis here; the focus is basic school arithmetic, for heaven’s sake.

To continue with the Times article, arrays of dots seemed to figure large in this parent’s dislike of the Common Core. She felt it was pointless to spend time drawing and staring at arrays of dots.

True, it would be possible—and I am sure it happens—to generate tedious, and largely pointless, “busywork” exercises involving drawing arrays of dots. But the image of a Common Core math worksheet the Times chose to illustrate its story showed a very sensible, and deep use of dot diagrams, to understand structure in arithmetic. Much like the (extremely deep) dot array at the top of this article, which I’ll come to in a moment.

To the girl’s parent, mathematics is about numbers, but that’s just a surface feature. It’s really about structure. And throughout the ages, mathematicians have used the most simple symbols possible to bring out and understand that structure: namely, dots and lines.

The Times’ parent, so dismissive of time spent drawing and reflecting on dot diagrams, would, I am sure, think it a waste of time to devote any effort trying to make sense of the dot diagram I used to open this post. She would, I have no doubt, find it incomprehensible that an individual with a freshly-minted Ph.D. in mathematics would spend many months—at taxpayers’ expense—staring day-after-day at either that one diagram, or seemingly minor variations he would start each day by sketching out on a sheet of paper in front of him.

Well, I am that mathematician. That diagram helped me understand the framework that would be required to specify an infinite mathematical object of the third order of infinitude (aleph-2) by means of a family of infinite mathematical objects of the first order of infinitude (aleph-0). The top line of dots represents an increasing tower of objects that come together to form the desired aleph-2 object, and each of the lower lines of dots represent shorter towers of aleph-0 objects. In the 1970s, a number of us used those dot diagrams to solve mathematical problems that just a few years earlier had seemed impossible.

That particular kind of dot diagram was invented by a close senior colleague (and mentor) of mine, Professor Ronald Jensen, who called it a “morass.” He chose the name wisely, since the structure represented by those dots was extremely complex and intricate.

In contrast, the simple, rectangular array implicitly referred to in the New York Times article is used to help learners understand the much simpler (but still deep, and far more important to society) structure of numbers and the basic operations of arithmetic, as was well explained in a subsequent blog post by mathematics education specialist Christopher Danielson. The fact is, dot diagrams are powerful, for learners and world experts alike.

The problem facing parents (and many teachers) today, is that the present student generation is the one that, for the first time in history, is having to learn the mathematics the professionals use—what I and many other pros have started to call “mathematical thinking” in order to distinguish it from the procedural skills so important in past times.

The reason for that is that in the world today’s students will graduate into, computation is as plentiful as water or electricity. The smartphone we carry around with us is much faster, and more accurate, in carrying out mathematical procedures than any human.

In a single generation, society’s need for mathematical mastery has gone from procedural computation, to being able to make effective and reliable use of an effectively unlimited amount of automated computation. To put it bluntly, mastery of computational skills is no longer a marketable asset. The ability to make good use of computational power is where it’s at in math today.

For almost all the three thousand years of mathematical development, the focus in mathematics was calculation (numerical, symbolic, or geometric). Learning mathematics meant learning how to perform those calculations, which boiled down to achieving mastery of various procedures. Mastery of any one procedure could be achieved by rote learning—doing many examples, all essentially the same—leaving the only truly creative mental task that of recognition of which procedure to apply to solve which problem.

Numerical and symbolic calculation (arithmetic and algebra) are so simple and routine that we can program computers to do it for us. That is possible because calculation is essentially trivial. Perceiving and understanding structure, on the other hand, is something that (at least at the present time) requires human insight. It is not trivial and it is difficult. Dot diagrams can help us come to terms with that difficulty.

When movie director Gus Van Sant was faced with introducing the lead character, Will Hunting (played by Matt Damon) in the hit 1997 film Good Will Hunting, establishing in one shot that the hero was an uneducated (actually, self-educated) mathematical genius, the first encounter we had with Will showed him drawing a dot diagram on a blackboard in an MIT corridor.

You can be sure that when an experienced movie director like Gus Van Sant selects an establishing shot for the lead character, he does so with considerable care, on the advice of an expert. By showing Will writing a network of dots on a blackboard, Van Sant was right on the button in terms of portraying the kind of thing that professional mathematicians do all the time.

The one bit of license Van Sant took was that the diagram we saw Matt Damon writing was not the solution to a problem that had taken an MIT math professor two years to solve. (Unless MIT math professors are a lot less smart than we are led to believe!) It was a real solution to a real math problem, all right. I am pretty sure it was chosen because it fitted nicely on one blackboard and looked good on the screen. It absolutely conveyed the kind of (dotty) activity that mathematicians do all the time—the kind of (dotty) thing I did in my early post-Ph.D. years when I was working with Prof Jensen’s morasses.

But it’s actually a problem that anyone who has learned how to think mathematically should be able to solve in at most a few hours. Numberphile has an excellent video explaining the problem.

So, New York Times story parent, I hope you reconsider your decision to take your daughter out of school to teach her the way you were taught. The kind of mathematics you were taught was indeed required in times past. But not any more. The world has changed dramatically as far as mathematics is concerned. As with many other aspects of our lives, we have built machines to handle the more routine, procedural stuff, thereby putting a premium on the one thing where humans vastly outperform computers: creative thinking.

Those dot diagrams are all about creative thinking. A computer can understand numbers, and process millions of them faster than a human can write just one. But it cannot make sense of those dot diagrams. Because it does not know what any particular array of dots means! And it has no way to figure it out. (Unless a human tells it.)

Next month I’ll look further into the distinction between old-style procedural mathematics and the 21st-century need for mathematical thinking. In particular, I’ll look at an excellent recent book, Jordan Ellenberg’s How Not to be Wrong.

The book’s title is significant, since it recognizes that the vast majority of real-world mathematical problems do not have a unique right answer, and that the real power of mathematical thinking is making sure you are not wrong. (The book’s subtitle is “The power of mathematical thinking.”)

I’ll also look at a new mathematics video game that also focuses on mathematical thinking, this time, school-room Euclidean geometry. It’s called DragonBox Elements.

You might want to check out both.


Ze'ev Wurman said...

I am eagerly waiting for your defense of the use of Roman number system instead of the decimal one in elementary grades. After all, we do use the letter X a lot in higher level mathematics, so surely it must be good for young children to also use it in their arithmetic.

Jon Awbrey said...

The real problems of CCSS have more to do with the commercial exploitation of public school tax dollars than anything else. There is a wealth of information and informed commentary on this score at Diane Ravitch's blog:


Keith Devlin said...

Jon Awbrey's comment hits the nail on the head.

Ze'ev Wurman has clearly missed the entire point I was making and appears to have a very misguided (though not unique) view of doing mathematics, probably a result of poor education.

R. Craigen said...

In my work as a professional mathematician I too quite often use dot diagrams of various sorts. Some look exactly like the ones Devlin displays here; we call them Ferrers diagrams and they have a somewhat different purpose. In advanced math many imaginative notations and representations are used as excellent, precise tools to aid thinking.

Devlin's invocation of this fact in reply to the parent of a 9-year-old bored to tears with doing arithmetic in the most boring possible way reminds me of the canonical story about the lost balloonist who shouts to someone on the ground, "Sir, can you tell me where I am?"

The bystander shouts back, "You're in a balloon!" He replies, "I perceive that you are a mathematician, sir!" When queried, "How can you tell?" he says, "Because what you told me is absolutely true ... and completely useless".

It's quite true that advanced math may, at times, use complicated-looking diagrams involving dots. And this is quite useless as an explanation of why little boys and girls need to be forced to engage in boring, wasteful and insightless tedium to perform tasks for which there are enlightening, easy-to-learn and efficient procedures near at hand.

If Dr. Devlin was found to be staring at his dot diagrams for days when a well-known elementary and efficient process would obviously have arrived at the same insight in a few lines and at very little expenditure time and effort, either he or his colleagues would consider him to have been foolish for doing so, or perhaps ready for retirement.

Let us teach 9-year-olds the best and most suitable mathematics for their level of mathematical development and not lead them down garden paths to inefficient thinking and mindless tedium in service of an abstract and unsubstantiated belief that "deep understanding" will magically arise if they are limited to performing arithmetic essentially as our cave-dwelling ancestors did (substituting pages of dots for piles of rocks).

Keith Devlin said...

R Craigen confuses learning mathematics with doing mathematics. Methods that are efficient for doing mathematics are not necessarily well suited for learning mathematics.

In particular, computational efficiency is definitely not a good metric for learning mathematics. In terms of doing mathematics, the most efficient way to multiply in today’s world is use a calculator, so if efficiency were the metric, we should just train kids to use a calculator well. (Which we should do, but wisely we do more.)

To prepare students for life in the 21st Century, learning researchers have developed methods that are optimized not for efficiency but for learning. And dot diagrams play a useful role in that learning.

True, as I noted in my article, it is possible to overdo it and generate a lot of dotty busywork. But the illustration of a dot diagram the Times used showed an excellent use of a dot diagram to develop an understanding of one of the main instantiations of multiplication -- a particularly important example, since multiplication is a notoriously complex operation that many adults do not understand well, let alone school kids.

The article by Christopher Danielson that I referenced explains how dot arrays can help kids come to understand (some important aspects of) multiplication.

R Craigen may be right, and some of my colleagues may have thought me foolish for spending months staring at my morass diagrams, but I was not trying to impress those colleagues, I was trying to understand something that I found complex and difficult to grasp. Those diagram helped me do that. In fact, I was not alone. Some of those colleagues also spent a lot of time staring at similar diagrams, and likewise seemingly making no progress for a long time. (I guess those would be the ones who did not think me foolish!)

Foolish or not, many of us did eventually mange to achieve the required understanding, and we made progress, so I would argue that the method is not without merit. In fact, I find it had to imagine a better one. The brain seems to need that external stimulus, impoverished and skeletal though it may appear to someone who has never had such an experience.

The commentator’s final comment seems to indicate the degree to which he has missed my point. Neither I, nor any child in a school classroom, is working with dot diagrams to *carry out* calculations. The goal is understanding.

Once someone understands, say, multiplication, she or he has various choices of efficient ways to do it. The classical Hindu-Arabic algorithms are one way. But there is a much faster and more accurate way: use a calculator.

In my school days, calculators were not available, so it was important for us to master the classical algorithms. In contrast, in the 21st century we are preparing today’s students to go out into, unlimited computation power is as freely available as running water and electricity, so it makes more sense to ensure our students can make good use of that utility.

Audrey Tan said...

I find it ironic that someone who, some years ago, insisted that multiplication is not repeated addition, is now supportive of children drawing so many dots to "study" multiplication.

Personally, I have no problem with drawing dots to support a child who is learning the concept of multiplication (even if it does have a whiff of repeated addition about it). However, when drawing dots becomes a prescribed method, then we have a serious problem.

To describe the image chosen by The NY Times as a "sensible and deep use of dot diagrams" and "all about creative thinking" is akin to describing a baby "experimenting with space" when he/she throws a toy across the room. I think the only sense in which this example is deep is the way the author continues to dig himself!

Mary O'Keeffe said...

I certainly don't advocate going back to the old ways in which the mother was educated. There is, however, a huge difference between a research mathematician analyzing dot patterns because s/he has decided that it looks like a promising way to approach an unsolved problem and a child being *assigned* to draw massive numbers of dots to solve problems s/he may feel she already knows how to do without being given any clearly explained motivation for WHY she should be doing this exercise. There are many other approaches to do this same kind of learning in ways that might be more engaging as well as easier for young children who may have fine motor skills issues or visual alignment difficulties (eg crossed eyes) which could make it hard to draw and count dots. Arranging pennies or Lego squares or other manipulatives into arrays could serve the same purpose. The problem is not the pedagogy behind the dots. The problem is mass-produced mindless worksheets which do not motivate the kind of mindful problemsolvers we need.

Keith Devlin said...

Not sure what Audrey Tan finds as ironic – dot diagrams of different kinds provide ways to help gain understanding, by focusing attention on the possible structures and the patterns. The Times illustration showed a dot diagram that focuses on one of several important instantiations of multiplication. I think Ms Tan is too focused on the dots themselves, and missing the depth of structure that a dot diagram can lead to. We do, after all, view Newton’s (mythical) observation of the falling apple as having scientific significance – he was indeed, if not “experimenting with space”, then very definitely “observing and reflecting on space”. The depth lies not in the activity but is what is going on inside the person’s head. Of course, mindless drawing of dots has no benefit, and a worksheet that demands students use dot diagrams as the solution method would surely not be productive. But that would be a problem with the teaching, not the CCSS.

I agree with everything Mary O’Keefe writes, and never suggested anything at odds with what she says. What the CCSS set out to do, and what I was advocating, is shift the focus from learning specific procedures (now done for us by machines) to acquiring the ability to think like a mathematician. And the fact is, mathematicians spend a lot of time reflecting on concepts and problems until they achieve sufficient understanding (not infrequently using dots diagrams to aid the cognitive process). That kind of thinking is the marketable mathematical ability for the 21st Century.

Barry Garelick said...

Mary O'Keefe states: "I certainly don't advocate going back to the old ways in which the mother was educated."

Can you elaborate on the ways you believe the mother was educated and why you think they were bad? I can provide you examples of textbooks from the 30's through the 60's which show that not only was math NOT taught by rote, but provided the contextual and conceptual explanations that are continually mischaracterized as missing.