Monday, February 16, 2015

Sort of a progress report on Singapore Math Figured Out for Parents, and a whole lot about why I think it's important.

The work on Singapore Math Figured Out for Parents, which is my book about exactly that, is proceeding very nicely, and I thought I'd give a bit of progress report here. For those of you who like the social commentary essays or want me to talk about my science fiction writing, all things in time—I'm trying to maintain a rotation. I'm not a brand. I'm trying not to become one. I'm probably going to be something other than a brand, and when I become that, I'll pretend I became it on purpose.
Meanwhile, in my non-brandish kind of way:
Quick catch-up on what this is about: the Singapore Math system was developed in that nation in the 1980s, and has undergone continual development and improvement since that time. In the international math achievement rankings, the top 5-6 nations for the last few years have consistently been places that adopted Singapore Math a decade or more previously (as you might expect, students who start out in Singapore Math reap more benefit than those who have to cross over when they're older). Almost entirely because of those international statistics, many high-achievement-oriented charter schools in the United States have tried to adopt Singapore Math in the last few years, with varying degrees of success. Some mostly-affluent public school districts are in the early process of adoption, and a few of the charter systems that are dedicated to raising student achievement in low income students and/or students of color are exploring it as well.
Even if your kid's school is not going to Singapore Math directly, it's going to get closer to it. The incoming Common Core math standards have borrowed some material from Singapore Math. Furthermore, the necessity of teaching to the test in the present climate of No Child Left Behind plus Common Core standards has caused the main Singapore Math publisher in the United States to bring out a line of Singapore Math books which are "aligned with the standards." *
Singapore Math is better math for elementary students, but it can't be adopted by just handing the teachers copies of the new textbooks and telling them to stay a chapter ahead. That's the main thing I'll be talking about later on in this piece—why it's better and why it's also hard to implement. For some parents trying to help kids with homework, it is somewhere between terrifying, confusing, and enraging.
Hence my project: a book that figures out this Singapore Math thing for parents. By figuring out, I mean several different things:
•figuring out why your kid's school is changing to Singapore Math, or might, or should
•figuring out how Singapore Math is going to make your kid much more math-proficient; there's a reason it works, and a reason it works better than approaches you may be more familiar with
•figuring out why they do that**
•figuring out how to help your kid with homework
•figuring out whether your kid is actually getting the benefits of Singapore Math, or just being pushed through a different set of worksheets.***
One way to my conclusion: reading and theory. Lately I've been polishing and clarifying that first part, about how Singapore Math is the better way to teach math. Since to write it, I have to understand it myself, much of my recent reading has been in understanding the deep reasoning that underlies Singapore Math, and the fundamental problems with the alternatives.
The tradition of arguing about how kids ought to learn math goes back at least to Reverend Thomas Vowler Short in 1840, accelerating into a full-blown uproar by the turn of the twentieth century. **** So lately my research reading has been devoted to getting caught up enough to say something reasonably intelligent (or at least excusably not-stupid) in the long-running discussion about how people in general and kids in particular ought to learn mathematics. I've read a great number of highly influential books that only experts have ever heard of—sometimes feel like I'm reading the secret history of elementary school.
Where it has all led me is to a much deeper conviction that:
• the Singapore Mathematics methods are flat-out the best system ever devised for teaching mathematics to students whose abilities are in the range from about one standard deviation below to about two standard deviations above average, which embraces about 80% of all math students.
•Singapore Math is better for a reason that is subtle, complex, and, to use a scary word, deep—not as in "deep philosophy" but as in "deeply embedded."
Another way to my conclusion: practical experience. In the last few months, this has also been confirmed by personal experience. My wife is a reading and math intervention specialist; she helps kids who are having reading or math problems get back on track before they fall too far behind. She has a tutoring business on the side which is usually booked up with a waiting list, and since my math background goes a good deal further than hers, and she gets occasional calls from parents who are mainly worried about math, she started working me in to cover math tutoring on more advanced subjects, as another pair of eyes while she was administering diagnostics, and in general to help out on math interventions.
Naturally enough, since I've been spending months understanding what's up with Singapore Math, I reached for that toolbox, and I've been delighted with the results. I'm a reasonably good tutor—I know the math, I know the educational theory, I get along with children, I have a large bag of tricks for getting the math and the kid onto better terms with each other. But with the Singapore Math concepts and approaches, I'm a much better tutor, and kids who once seemed hopelessly blocked are moving ahead fast enough to eventually catch up with and pass their classmates. (It's also a major kick to hear them say, "I like math," sometimes in a tone of astonishment, or, to quote one of them directly, "It's math but it's actually kind of cool.") I've said for many years that the ability to use math well is a basic human gift like poetry, dance, music, storytelling, drawing, dressing well, making small talk, coding, or cooking, something kids should be able to do competently and confidently, and appreciate intelligently, by the time they grow up.
In the short run, I'm taking on much more tutoring work. As I coach more kids through their personal math walls, I learn vastly more about how Singapore Math works at the level where ultimately any system of math instruction has to work: in the mind of the individual kid. I'll end up with a better book because of that.
But in the longer run, here's what I've really learned between the reading and the tutoring experience: Singapore Math is the one system of math instruction that really understands and uses the deep relationship between procedural proficiency and conceptual understanding.
The meeting of theory and practice: why Singapore Math is really better.
Quick definitions: procedural proficiency is the ability to execute an algorithm quickly and accurately; it's quite literally know-how. If you can do a long division of a 4-digit number into a 9-digit number, keeping the decimal point where it belongs, out to 6 digits of accuracy, in less than 2 minutes, you are more procedurally proficient at long division than about 95% of American adults (and about as procedurally proficient as a run-of-the-mill Japanese adult, or a top-of-the-first-quartile Singaporean). There are several components to procedural proficiency: knowing which algorithm goes with which problem, remembering all the steps correctly, and executing each step quickly and accurately. If you know how to hand-extract an nth root, you're more procedurally proficient than someone who only knows how to hand-extract a square root. If you know four ways to find the roots of a quadratic equation in one variable******, you're more procedurally proficient than the guy who only knows the quadratic formula. If you know the inside-out pattern for multiplying two-digit numbers, but you sometimes reverse the inside and outside products, you are less procedurally proficient than the person who never does. If you've got multi-digit lattice multiplication down cold but have to stop and count out any time there's an eight or a seven, you're less proficient than the person who knows the whole multiplication table.
For our purposes here, procedural proficiency applies to any procedure and any problem. Whether a student is doing division of whole numbers by counter-rectangles (a method which especially irritates some parents) or by short division (most parents don't even know there's such a thing nowadays), if he's doing it quickly and reliably, he's procedurally proficient at it. If a student always checks to see whether the coefficients of x1 add up to zero and uses the trivial ±√c/a when they do, that's a procedurally proficient decision even if he doesn't know the quadratic formula.
Conceptual understanding is being able to make a simple mathematical argument about why an algorithm works, why a thing is true, or in general, "why." If, off the top of your head and without having to think about it, you can quickly and clearly explain why, to divide a fraction by a fraction, you invert the divisor and multiply, you've got conceptual understanding. If you can prove the Pythagorean theorem or that there's an infinite number of primes, you have conceptual understanding of those ideas, no matter how slowly you use the Pythagorean algorithm to find the diagonal of a rectangle or how difficult you find it to do a simple factorization.
Traditional methods: going really fast till you sock into that wall. Those two concepts are the heart and soul of why teaching mathematics has been such a thorny problem, ever since Reverend Short first made a good guess on the subject. Young kids love patterns, rhythms, repetition, and so forth and learn them very easily (consider clapping games, nursery rhymes, songs like "B-I-N-G-O," games like hopscotch and jacks, just to start with, or just wait till you're on a city bus next to a little kid who has a favorite commercial jingle s/he sings over and over).
So the traditional method of instruction, learning highly patterned algorithms (write this here, put that there, cross that out and write the next highest, etc.) produces very quick, easy procedural proficiency. Nearly every adult who has math trouble (which is, truthfully, most Americans) will tell you sadly that they "loved math" or "were good at math" up till ... and that up till is almost always some point where the fading memory of a maturing brain was no longer able to keep all the patterns straight, or the patterns became too complex (think how much more complicated long division is than two-digit addition), or there just wasn't any reliable pattern any more. Usually the same people will tell you they loved math but hated word problems, which is something like loving playing scales on the clarinet but hating music, or loving counting out the box step but hating to go dancing.
Mary Boole seems to have been the first person to figure out and articulate clearly that if kids learn it algorithms exclusively as aconceptual patterns for manipulating meaningless symbols, they will inevitably hit some wall later. When that happens, if they don't have the tool of referring to the underlying principles and concepts, they're done; they can go no further. Some few kids are lucky enough that they acquired concepts all along (very often on their own, by simply enjoying playing with numbers); those are the ones we think are "naturally good at math." Other kids, driven by one kind of necessity or another (wrath of parents, lure of a career, etc.) begin belated and partial conceptual learning, and get enough of it to go on for a while, at least until they hit the limits of their conceptual learning skills. And the great majority just declare themselves "not good at math" and give up, spending the rest of their lives evading situations that math could make easier.*******
Now, a good math teacher in the early grades has always been able to point out concepts as the students progressed. A stack of blocks, by not getting any taller or shorter when blocks are moved around within it, beautifully illustrates the commutative and associative principles (whether they're called by that name or not). The number of tiles on the floor doesn't change whether you stand on the west or the north side, and that can teach the commutative principle for multiplication. But there have also always been too many teachers out there who just wanted to get done with the worksheet, and whose answer to "Why?" was "Because you don't want to stay in from recess."
So essentially, the traditional style of teaching math has been shown, under all sorts of conditions, for a good hundred years and more, to produce early procedural proficiency but expose many students to a later conceptual block. You get more kids who can make change quickly but fewer who can go to engineering school. And when they hit the blocks, it's painful and frustrating and most of them come away hating math.
Reforms: don't go there, there's a wall; or here's a key, so why do you need a lock?
Naturally enough, reformers who wanted to fix the hitting the wall problem either tried to avoid the hitting, or tried to avoid the wall, i.e. most reform math movements involved either:
1.           simplifying and dumbing down math so that kids can learn a basic set of patterns and let it go at that (never a very good option and a disaster in the 21st century when so much of the better part of the job market requires math)
2.           teaching conceptual understanding as an alternative. That's what New Math, inquiry-based math, and several other systems try to do.
That second idea makes sense on the surface; if the reason little Sammy can't grasp fractions is that s/he only knows multiplication and division as procedures, teach them to him/her as concepts in the first place.
But as is well-known, in practice this leads to kids who can define the cardinality of a set but can't figure it out without resort to their fingers. Concept-heavy math education often fails to lead to procedural proficiency; worse yet, because the concepts are ungrounded in any experience, the students seem to know them only as names, and not to be able to apply them or see what they refer to. If we were producing calculus wizards who couldn't make change, we might live with that by automating change-making or training change-specialists; but the embarrassing truth for conceptualists has been that without a procedural base, people don't seem to acquire the concepts either.
So there's the dilemma: emphasize procedural proficiency and lose large parts of every cohort to frustration and despair when they don't have the conceptual basis to go on. Emphasize conceptual understanding and lose even larger parts of every cohort to learned helplessness and a propensity to name things but not be able to work with them.
Where Singapore Math is really a revolution. The mathematicians (Singapore has some superb ones) and the educational psychologists (ditto) looked at the problem something like this:
Procedural proficiency approaches are focused on manipulating symbols quickly and accurately. Conceptual understanding approaches are focused on connecting meanings accurately. But a student can only really know for sure that a procedure is accurate if s/he understands the concept behind it, and a concept isn't really understood till the student sees it happening.
In short, procedures are what concepts mean about, and concepts are the things that govern and allow us to remember procedures. It's a dialog.
Concepts are how we remember procedures. The last few decades of memory research have shown that memories are not recordings, but a set of cues from which we structure and rebuild a narrative, visualization, or other coherent thing we need to refer to. If you're truly procedurally proficient, you know that already; if your mind slips for a moment while doing long division, the concept that explains why you multiply and then subtract (and go back a step if the partial remainder is larger than the divisor) is there in your memory too, ready to activate if you do something in the wrong order or get stuck.
Also, procedural proficiency is essential before the next level of concept can be learned effectively. A student who can only add by counting forward has to do far too much work for a 7-year-old memory (which is highly accurate and retentive but works in tiny chunks) to be able to also grasp multiplication. There are only so many processors and registers available, and they have limited capacity; to grasp higher concepts, lower ones have to be automated. Let me give you three very fast analogies:
1)        You control where you point your eyes (high level concepts about your environment) but you usually leave depth of focus "automatic" and you have no choice at all about whether to see with your rods or cones. If you had to decide how to balance the signals coming in from your eyes, you couldn't see at all.
2)       If you've ever watched a young kid learning to play baseball, you know there's not much use telling him/her "The play's at first" until s/he knows that a "play" is "a situation into which you should throw the ball," and all that's useless until s/he can throw a ball somewhere reasonably close to first base. The concept of a "play" is meaningless until the kid is actually someone who can play the game as a player who can throw into a play.
3)       A college science instructor friend tells me that he has to coach students to read every formula and equation (many of them seem to think that stuff is just decoration and the meaning is in the words around it), and strongly urges them to solve each one for all the variables. If they can easily see that F=ma implies m=F/a and a=F/m, and keep all those in mind as they read on, he figures they'll probably make it, though not as easily as the students who didn't need to be told. If they don't see much relation between those three, or think of them as three separate facts, he suggests they change majors to "something words-only."
And there's the great thing with Singapore Math. There's a structured pathway from concept to procedure to higher concept to more complex procedure to even higher concept, on and on; the concepts are not just window-dressing and decor to keep the smart students from getting bored with algorithms. Nor are the algorithms mere passing examples of the to-be-admired concepts. It's understood that the ground for a concept has to be prepared with procedures that have become largely automatic—you can't have an a-ha! where everything clicks if the pieces of it can't move fast enough to click. And it's equally understood that the focus for a procedure has to be on implementing a concept; a kid who is chanting "four times nine equals thirty-six" needs to be thinking of what "times" and "equals" mean every time he chants it.
What great teachers have always done anyway, Singapore Math is built around.  So unlike any other system, Singapore Math doesn't try to "emphasize the really important thing, which is" {procedures, concepts}. It builds in that vital alternation, that you acquire the concept to help remember the procedure that will allow you to see the next concept that will enable you to remember/construct/apply the next procedure, over and over—like a person walking forward on two feet instead of spinning in a circle around a stationary foot, or a climber going hand over hand, not limited to the height of the first handhold.
And as a corollary, Singapore Math training materials make it clear that we acquire procedures by procedural methods (i.e. practice, including both repetition and variation, or to use the dreadfully old-fashioned words, drills and exercises). We acquire concepts by conceptual methods (inquiry, pattern-finding, reasoning, extrapolation, and so forth). And we join procedures to concepts, and concepts to procedures, and move on.
Contemplate that phrase, if you will. And we join procedures to concepts, and concepts to procedures, and move on.
How's it work in practice? Let me take an example I've now seen three times: some kids cling to very simple but time-consuming procedures like counting up to add, counting down to subtract, or counting by factors to multiply (e.g. to multiply 8X5 they tick off 5,10,15,20,25,30,35,40 under their breath as they pop up eight fingers in sequence). In each case, it turned out that one reason the kid did this was because s/he thought the construction caused the answer to be right—in other words, the equal sign did not express that 8X5, 5X8, 10X4, and 40 were all the same number. Rather, the equals sign was a command to "calculate now," and 8X5 would not be 40 until you calculated it.
I found that out by slowly walking them through their mental process, until I confirmed that for them, calculation was what made the result true, rather than a procedure for finding the truth.
The cure was then to demonstrate, in up to a dozen different ways, what = actually means—that 8X5 is not "how you make 40" but "another name for 40." Once they saw that, they also saw that 40 could be right without doing the calculation—and with a bit of salesmanship on my part, that it would be much easier to just know that when you needed it (just as it's easier to get a pizza if you know it's called a pizza, rather than to say, "I want you to make me a circle of dough and put cheese and a mixture of spiced crushed tomatoes on top of it, then heat the oven ..." every time).
Then, pushing them to remember every single time what the math facts meant, so that they would know when and where to use those automatic bits of knowledge, we worked out a mutually agreed on plan and timetable for memorization. They all did it faster than expected/required. And the next step was that we used their newfound procedural proficiency to start exploring the next concept up the ladder ... from which would flow more procedures, with which they would be able to understand more concepts.
One last thing: tutoring students almost always move on, as they catch up with regular students or as they become secure in running ahead of their classmates. The biggest thing they take with them from that dose of Singapore Math, I hear, is that once they have that model of how to learn math, they can apply it whether their teacher does so consciously or not. They have, in short, acquired the ability to learn mathematics—really learn real mathematics—whether the teacher is good or not. And if knowledge is power, that's the very source of power.

* quick translation of that for those of you who don't read American educationese fluently: the order and timing of mathematical topics from Singapore Math has been partly copied into the new guidelines (Common Core) that many schools are adopting. (Feel free to insert several hours of reading up about politics of Common Core right here, if you insist. Go ahead, I can wait. I've done it myself. I'll even feel some sympathy for you after you're done). In general, the international version of Singapore Math far exceeds what Common Core asks for, so within the Common Core process, Singapore Math has been a force for higher standards and against dumbing-down. No Child Left Behind is a Bush administration holdover law that the Obama Administration has found useful as a tool against underperforming schools, and so has not given it up despite heavy pressure from their close political allies, the teacher's unions. NCLB is a system in which poor student performance on standardized tests makes schools liable to pressure and sanctions that can go as far as reallocating their funds or students to the point of closing them down. Many terrified school boards and school administrators therefore insist that any curriculum be "aligned with the standards" by which they mean an exhaustive list of "right here on this page is where we teach the third graders the material they'll need to correctly answer the type of question that a specific standard requires." Because Singapore Math is generally more advanced at comparable ages than the standard American math curriculum, most of the "alignment" consists of verifying that by the time students are tested on any topic, they'll already have been taught it; almost no material has had to be added, and little existing material has had to be moved earlier. This checklist/bookkeeping process seems to have produced little or no actual change from international Singapore math, but it's probably been very successful at selling new editions of the old textbooks.
** for many different values of that. In general, Singapore Math presents the material in an order and manner that has sound reasons in educational psychology, mathematical rigor, and preparation for more advanced work—often in all three. For obvious reasons, though, this isn't always explained in student textbooks, and sometimes even the teacher may not be clear on it, especially in the many places where Singapore Math is being thrown at them on the "stay a chapter ahead" basis. For most parents with kids in Singapore Math, there seems to be an occasional that, as in, "Well, she seems to be learning math, and I see why they have her do most of it, but what is that about?" Hence, a whole section of the book dedicated to thats.
*** subject for a future blog. I think most of us realize that a really good math teacher can use many of the not-very-good textbooks and materials on the market, and still really teach math to most or all of the kids in the room, by supplementing, emphasizing, or sometimes teaching against the materials. Unfortunately, underprepared, mathphobic, rigid, insecure, or otherwise weak math teachers can also use a great curriculum to baffle, confuse, frustrate, and turn off a whole roomful of kids. Some teachers who are being handed Singapore Math to teach are being thrown in over their heads, and it really doesn't matter whether they're in over their heads because they aren't getting enough support, didn't get enough preparation, never learned the underlying math, or just hate math. Whoever's fault that is, it's the kids who pay the price. So that section of the book will be about making sure that if your kids' school is wise enough to offer them Singapore Math, what they get is actually Singapore Math.
**** see, for example, the surprisingly modern-sounding arguments in Mary Boole's Lectures on the Logic of Arithmetic (1903) or John Dewey's The Psychology of Number (1895).
***** and you gosh-danged well should, too.
******* subject for another future blog after I do some more research: it is extremely well established that illiterate adults have an enormous number of tricks for not getting caught being unable to read (getting other people to fill out forms for them, sticking to restaurants and canned goods that have pictures of the food, listening to class discussion and repeating things other students have said, saying they forgot their glasses and asking someone to read the text to them, etc.) There are probably as many tricks, and as clever tricks, for innumerates, but they are probably used much more often. Unlike illiterates, however, I suspect that there are many, many more innumerates who just live with being cheated, pay too much, having to use trial and error, etc.