Last month, as I read Christopher J. Phillips’ brief and engrossing The New Math: A Political History, I found myself reciting the ominous line from Battlestar Galactica:
Early in my teaching career, I spent a lot of time and life-force railing against the shortcomings of a rote math education. The mindless manipulations. The paper-thin comprehension. The lack of critical thought. I saw it as my duty to name (and blame, and shame) these patterns.
As the years went by, I realized these critiques were not as fresh as they felt. People like me had been decrying methods like those not just for years, but for centuries. Such critiques did not really disrupt the system; they were a longstanding element therein.
Far from challenging the status quo, I was playing a comfortable role within it.
These thoughts came rushing back as I read Phillips’ pithy and potent history. There is nothing new under the sun—at least, not in our philosophies of math pedagogy. The arguments just go round and round.
Witness this passage, about the rival textbooks of Pike and Colburn:
Pike emphasized the importance of memorizing arithmetic rules and then applying them to various examples…
Colburn’s [approach] was to reverse rule and example: instead of presenting rules, he presented simple examples in an effort to lead children to form rules for themselves….
Contemporaries understood the differences between the textbooks to be about differences in reasoning…
[One critic] proclaimed… that rule-based methods failed because a student would not have “been called upon, in this process, to exercise any discrimination, judgment, or reasoning…”
[Another critic] claimed that inductive methods would… ultimately undermine authority by erasing the traditional grounding of rigorous knowledge in rules.
Is this about the Common Core battles of the 2010s? Sure sounds like it.
But no, it’s about the New Math controversy of the 1960s. Right?
Wrong again. Colburn published his book in the 1820s. Pike wrote his in the 1780s.
All of this has happened before. All of this will happen again.
As Phillips elucidates, a silent assumption underlies both sides of the Colburn/Pike debate. “Even—perhaps especially—at the most elementary levels,” he writes, “evaluating mathematical methods entailed assumptions about the virtues of intellectual training.”
Let me spell that out: the shared assumption, the axiom that both sides accept, is that math education shapes the intellect. Arithmetic is not just arithmetic. However you manage matters of multiplication, that’s how you’ll also approach matters of democracy.
In the 1960s, New Math reformers worried that rote drill would breed blind deference to authority. They hoped instead to create a society of mini-professors, seeing the world in terms of flexible, abstract structures.
In the 1970s, “back to basics” counter-reformers held the opposite hope, and the opposite fear. They believed rote drill inculcated discipline and diligence, and that the New Math would breed a feckless generation that was forever confusing true with false, right with wrong.
The rival camps favored opposite kinds of minds, and opposite kinds of math. But they shared a deep principle: Math makes minds.
I’ve long operated on this same principle. Vital to a free and thriving intellect—and thus, to a free and thriving society—is great mathematical thinking, whatever that is.
At the moment, I can’t help wondering if we’ve all got it wrong. Maybe math education isn’t about broader intellectual habits. Maybe it is not, as 17th-century Jesuits believed, a model of how divine authority flows forth from unquestionable axioms. Maybe it is not, in Underwood Dudley’s lovely phrase, about “teaching the race to reason.” Maybe it’s none of those things.
Maybe, if we want to break the Battlestar Galactica cycle of endless “math wars,” we need to embrace a new axiom: math education is just about math. Maybe those stakes are high enough.
Except that it’s not really just about math education, is it? It’s just that in math education the two approaches can be so easily distinguished, and the rote learning method is so extreme. But the link with overall development of critical thinking extends to very similar arguments about how to teach other subjects.
Yes, I think that’s right! The question of “what kind of thinking do we want to cultivate?” is there in every subject. I suspect math stands out for a few additional reasons:
1. Math is (along with English/language) the subject that gets the most time during schooling.
2. Math is (along with English) less about *content* than *skills.* When there’s more content to master (as in, say, science or history), it takes some of the spotlight away from the “what kind of thinking?” question.
3. Math is (in contrast to English) a subject that students will not necessarily use every day, and that many people still find alien and unfamiliar. I imagine that moves the focus toward these big, abstract questions of thinking style.
I never was much of a Battlestar Galactica fan. About 7 years ago, a local large university was having difficulty within their honors cybersecurity program. The students who had high school GPAs over 4.0 were getting admitted to the program and were having great difficulty succeeding in the honors college program. They were fantastic at following directions, but they could not think for themselves at all. I would posit that they were not even successful in rote arithmetic, if you consider that they couldn’t determine the best approach to a given problem, without having identified it for them. If the goal is to have them endlessly check columns of numbers which someone has already checked and have someone else check those same columns of numbers again, then maybe they were successful. (I don’t recall the film, but that sticks in my head as an old movie reference.)
The university asked me to develop the curriculum for a course to teach critical thinking skills, but we used subterfuge in the course design. It was an undergraduate math survey course, disguised as a course exploring how to solve puzzles, that was designed to teach critical thinking skills. After developing the curriculum, I taught the course for a few years. I think that it was successful and if I took the time, I could see revising the course for years tom come as it is all fertile territory, as I was probably learning as much as the students.
I think that there is probably a middle ground between the rote arithmetic and extreme critical thinking. But unless there is a little cricket sitting on everyone’s shoulder, as they go through life, telling them when to apply which tool they learned in grade school, there is a need to teach enough to figure which tool in the toolbox they need to pull out. Now where is that 8/16″ wrench, I don’t see it in my toolbox?
If I had to guess, I would say that your movie reference was from Brazil (1985).
Sounds like a great course! What sort of work did you ask students to do?
I agree that perhaps the worst of all educations is (1) atomizing math into itty-bitty tools, and then (2) not giving students any practice or instruction on how to choose the right tool for an occasion. Gotta help them find their own cricket.
I tend to agree with your provisional conclusion “Math education is just about math” — but the comment I would add is that no matter how you learn material, you do so by pushing against the boundary of what you already know and building upon your accumulated skills.
What I’ve learned by volunteering in schools for 13 years, grades 3-12, is that yes, there are basically two ways to teach math: through memorization (whether that’s times tables or trig identities) or through comprehension (basically, not using a short cut until you can explain to your class why it works). The first way works fine until it doesn’t: as soon as a student is confused, there’s nothing he or she can do *except* memorize new things, and that’s a miserable educational experience. That’s how we teach kids to hate math.
So let’s teach through comprehension.
Pretty sure I agree with this!
My point in this post isn’t to forsake understanding. It’s more to discipline myself against a certain kind of wishful thinking: that if my students understand math in rich and flexible ways, this will somehow shape and transform their way of thinking about other domains (ethics, democracy, etc.)
Folks sometimes credit math with this kind of transformation in their own lives. And I don’t want to minimize those narratives. But I think they’re just that, narratives — people making sense of their singular intellectual journeys — rather than any kind of generalizable social scientific result.
Great post, Ben.
It has been thought that math proficiency is required for intellectual pursuit. Because of the unreasonable effectiveness of mathematics in explaining natural phenomenon and its ubiquity in our practical lives, the axiom (as you call it) that being better at math is somehow ‘necessary’ holds sway in our minds.
But I like Dudley’s [other] reminder (http://geofhagopian.net/M54/IsMathNecessary.pdf) better: Is math necessary (for anything)? No, but it is sufficient!
In general, several people (Freeman Dyson, Bryan Kaplan included) have argued that formal education is overrated. Since math education in schools is formal education, how could it escape the dire consequences of formally educating kids especially in a cookie-cutter style?
More practically, it is clear that you do not need to teach any math to mathematically inclined minds; they will figure it out.
The practical challenge is for young minds who are late mathematical bloomers or who are intelligent minds who just wouldn’t enjoy mathematics as adults. If your child or your student(s) represent these classes, how do you go about it? Would you profess the ‘rote’ route or the ‘understanding-first’ route?
As a father who has tried to teach his two kinds mathematics (while learning it himself), I have tried to ask a more mundane question: Is the present being spent in a lively manner in a kind, rather disorganized, but helpful environment? Is the passing moment being a drudergy, or is one having fun?
I have done it for about 6 years. I am happy to report that by just being a kind facilitator, I have able to kindle an interest in my childrens’ mind about mathematical thinking and problem-solving. The fear has gone and a more humane mental picture of mathematics has started to emerge in their minds. I am not positing that they will become math majors and that anybody can (rather, far from that).
But I believe that when they will look back at the time we spent on learning mathematics from great books (e.g. Gelfand-Shen’s Algebra), they will have some fond memories. Isn’t this what Keith Devlin’s vision for a mathematics classroom was about?
Thanks for sharing your experience. It sounds like a good one, for you and your kids!
Interesting post, but I think that the conclusion that traditional mathematics education (focused on the basics, on mastering the standard algorithms, etc.) has the goal of, as you say, “inculcat[e] discipline and diligence”, and avoid “breed[ing] a feckless generation that was forever confusing true with false, right with wrong”, is a bit of a strawman, and a bit of a failure on the ideological Turing test (that is, a traditionalist might say this is caricaturing their beliefs). In my experience, a traditionalist in terms of mathematics education (a position I am sympathetic to) would probably rather frame their position in terms of the mathematics themselves instead of a presumed impact on broader society, and say that competency in doing mathematics is what creates motivation for students (in other words, being able to do things encourages students to persist), that procedural and conceptual understanding are not in opposition but instead feed into each other, so mastering procedures can foster understanding, and recall Paul Kirschner’s distinction between the epistemology and pedagogy of a field: a novice’s way of learning is different from an expert’s, so it’s probably impossible to “create a society of mini-professors”. Learning must go through steps instead of jumping directly to doing what experts do.
But it is quite right that this debate is absolutely not new and keeps popping up from time to time using different names for centuries. It dates back at least to Rousseau, if not earlier.
In that phrasing, I actually wasn’t trying to characterize traditionalist educators en masse, but just the specifics of the grassroots 1970s “back to basics” movement. The language I used is pretty similar to what appeared in lots of op-eds and letters to the editor at the time. I agree that it’s not a particularly defensible vision — but then again, as you point out, the “society of mini-professors” vision is just as dubious.
Your approach is of course a much stronger one, and strikes me as a healthy synthesis of “traditionalist” and “progressive” ideas. Notably, you question the same assumption I’m questioning here: the idea that math education’s core purpose is to shape one’s thinking about subjects other than math. You focus instead (correctly, I think) on the question of how to build mastery *within* the discipline.
I think we need both the following rules and the critical thinking elements of math instruction. Without learning arithmetic (addition and multiplication facts) essentially by rote, most students struggle greatly with fractions and exponents. Beyond that, I would argue that a particular topic (exponents, for example) should be introduced in a critical thinking/intuitive understanding context, but by the end of the unit (or when we teach it again next year), we should introduce the rules, and perhaps offer some suggestions on which are or aren’t worth memorizing.
Students who focus too much on the rules will apply the same rule all the time, even when it’s not appropriate, because they are missing the critical thinking piece. They tend to struggle with applying their math skills outside the narrow context in which they learned them. Students who focus too much on critical thinking tend to do problems in an inefficient way because they don’t have enough background to do them quickly. They may figure out an appropriate strategy to solve novel problems and still get the wrong answers.
Another vote for both approaches instead of just one or the other!
Yes, I like this synthesis! It sounds quite similar to my own approach. One thing I like is that you avoid making the sneaky assumption I’m calling out in this post: that approaches to math education ought to be judged by some standard external to math, such as what kind of “democratic mind” they inculcate.
Tiny comment: I have not seen much of Battlestar Galactica (in either iteration), only about a dozen episodes altogether. But the line you are quoting has appeared elsewhere.
Maybe the oldest is in the book Peter Pan by J.M. Barrie:
https://www.goodreads.com/quotes/468991-all-of-this-has-happened-before-and-it-will-all
“All of this has happened before, and it will all happen again.”
That phrasing also appears in a song “Seek 200” by the band Information Society:
https://genius.com/Information-society-seek-200-lyrics
“All of this has happened before, and it will all happen again”
Yeah, thanks for pointing that out!
I actually saw the Peter Pan version when I was writing the post. I decided to stick with the BSG iteration because it felt thematically right — an ancient, grueling war between opposing factions, stretching from the mythic past to the sci-fi future, but always fought on the same terms…
“The first time as tragedy, then as farce.” Which BSG does not pick up on.
Okay, it is still a tragedy but it seems to me that that in each cycle the farcity increases.
What I really find absurd is the binaryness of both sides. Neither side seems to account for different learner goals (budding mathematician or physicist, STEM, or general familiarity with quantitative reasoning) nor for different learning processes requiring different mixes of these two and other approaches. It is all focused on the teaching and not on the learning.
Yeah, as my friend Michael Pershan likes to point out (and as I’ve written about too), the math classroom is a big muddled compromise between various competing and conflicting goals (several of which you nicely separate). Some of the farce is that we keep wanting a crisp sense of purpose from a place that has none.
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“The first time as tragedy, then as farce.” Which BSG does not pick up on.
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Eh. Looking back on that modern-day coda in the series finale, I think they nailed "farce" pretty well.
(And yes, I'm still miffed that a show that opened with the statement "And they [the Cylons] have a plan" itself in fact had no plan at the outset or, apparently, at any point, and ended up with a trite and incoherent aesop.)
Well, I’m one of the half-dozen people on the internet who actually enjoyed and appreciated the BSG finale (for reasons too long to go into), but I do agree that the “they have a plan” line was hilariously infuriating (or infuriatingly hilarious). While watching with my wife I’d usually ad-lib substitute lines, like “and they have no clue what they’re doing” or “and they are making this shit up as they go.”
The more I teach math the more I seem to agree with Paul Lockhart’s view ( math as an art – https://maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf )… Not that I can actually use this approach in class…
What I think is most interesting about Lockhart is that he recognizes that real math (after we’re done with a little bit of algebra and geometry) is essentially useless for “real life”, and that if we are to teach it, we shouldn’t try to shove it into supposedly real-world contexts, what some would call “mundanisation” (ex. per Greg Ashman, https://fillingthepail.substack.com/p/mundanisation). This tendency is very common in textbook writing, and I feel that it is transparently forced and serves to cheapen mathematics and make it seem as even less useful than it is. But where I think Lockhart errs is that he goes too far in the other direction, trying to build authentic mathematical experiences intended for novices without taking care that they have mastered the skills that they need to have in ordre to have these authentic mathematical experiences.
Don’t get me wrong, I think Lockhart is much more right than the mathematics education specialists who insist on teaching being done through real-world contexts (which they would call “authentic”, even though authentic mathematics is anything but that); but there is something missing from his views.
I found Lockhart’s Lament utterly electrifying when I first read it. It’s such a searing, beautiful vision of what it can be like to learn math.
These days, I tend to agree with Marc on both the strengths and the weaknesses. To achieve such a singular and coherent vision, Lockhart needs to pick a specific purpose for math education — to the exclusion of the various other purposes that actual math education must serve. I think your comment captures it exactly, Georgios: we all love the lament, but don’t (and can’t) actually teach in accordance with it!
I hate this idea that math education should be about EITHER learning rules OR deep thought. It should be BOTH-AND. For example, help kids understand how multiplication works AND have them memorize the multiplication tables. Help kids understand how quadratic equations work, AND teach them the formula. Help kids understand proofs and axioms AND reinforce that the angles in a quadrilateral add up to 360 degrees.
Agreed! And I think this becomes clearer when you reject the assumption I highlight in this post: that math education is about something other than math.
If you think math education is about “thinking skills” more generally, then it’s tempting to organize it around whatever “kind of thinking” we believe to be most generally valuable.
But if you think math education is about mastering mathematics, then it’s clear that it will involve many kinds of thinking, just as a dancer must master many kinds of motion, or a musician many kinds of sound.
What is unfortunate is that when it comes to an age that students can really understand why e.g. long multiplication works, they are usually not interested in that question because “I know it because it I saw it before”.
Is there any text or school of process which teaches different STYLES of mathematical proof? I mean a good deal more than inductive vs deductive. I mean there are LOTS of mathematical proofs and if one wants to establish a new theorem, there are lots of ways to begin and go about it. Are there different styles? That kind of thing would be interesting to know.
“Let me spell that out: the shared assumption, the axiom that both sides accept, is that math education shapes the intellect. Arithmetic is not just arithmetic. However you manage matters of multiplication, that’s how you’ll also approach matters of democracy.”
Could not agree more with the above passage!
I really enjoyed your post, mind if I link to it? Feel free to link to mine as well.
My site address: https://bunchiesblog777.com
I just finished teaching the first half of Plato’s Republic, and I’m sending this post to my students. You touch on so many things that run through Plato’s writings.
Fascinating post. And are my 10- and 11-year olds ready for BSG yet?
But something I am not getting is how it matters, in practice of some kind, whether one is teaching math because of some imagined transfer to other disciplines or habits of thought, or if one is doing it for the sake of teaching math itself?
How does it change anything for the teacher or students? For the system it could, certainly, diminish the extraordinary amount of required hours. But for the individuals and groups in the system?
Ah, that’s such a good question!
I guess my premise is that experiences are shaped by expectations. What we get out of an experience depends (in ways large and small) on what we imagine we’re supposed to get out of it.
More concretely: a lot of super math-anxious and math-averse students seem to be guarding themselves against the threatening idea that their mathematical failures reflect something deep about their brain and its shortcomings. There’s this ambient idea that mathematical struggle indicates a domain-general deficiency in thinking itself. And so math puts them into a defensive crouch, from which learning math is basically impossible.
But if we could banish that idea — and replace it with the idea that math draws upon many kinds of thinking, all of them lovely, and that even these various kinds do not begin to exhaust the variety of human thoughts — in short, that math is at once too internally diverse and too domain-specific to correspond to the idea of “intelligence” — maybe that would lower the stakes enough for some of those students to let down their guard?
On BSG: it’s been a few years, but I feel like they may need a few more years! Maybe at 12 and 13.
Julia, see my wall of text below. You can’t expect to teach children how to build a house (critical thinking) when they are mystified about how to pour the foundation. The approach absolutely changes things for Students, and that drives outcomes. The critical thinking skills emerge as they learn enough math and math “rules/properties/structures” to structure their thinking. Critical Thinking IMO, is an emergent result of effective math teaching – and the only way to do that with kids (and adults) is by rote. The transition from rules to critical thinking starts to emerge when applying math to real-world problems (i.e. math word problems.) Trying to jump directly to critical thinking without laying the foundation is really putting the cart before the horse so to speak.
You cannot just give people a canvas of ambiguity and say, go think critically, identify the key drivers and first principles, and figure it out without first building sufficient skills or mental scaffolding to make such and attempt possible. Otherwise, it’s an exercise in futility and ends in confusion, reduced self-esteem and a belief that they are not smart enough to do it – which is hugely limiting when authoring their future.
WRT BSG – the old/original BSG is probably ok for 10/11 year olds. The new one, not so much. The remake has a lot of very suggestive situations and some nudity and bad language, but the story is good and some of the themes they touch on are good. I was a big fan of both versions.
Personally, as a parent with two boys who’s elementary school textbooks in Ontario, Canada, featured the “Figure out the rules yourself” method. IMO, *this method is stupid and doesn’t work*. Particularly when trying to teach children the fundamentals of arithmetic and algebra. I’m not a math major; I’m a computer science grad.
Every day, my kids would come home completely baffled by their math homework and with poor Grades in math. As a Computer Scientist, and Asian Father, I was horrified. Every day they needed help with their homework b/c the teachers direction and the textbook were nonsensical. So we’d open the textbooks, and I’d look at the chapter they were in, what it was attempting to teach them, and how. It didn’t take long before I chucked the textbook aside with disgust and taught them myself (Which I did for years – every day after school reteaching what their math teacher should have taught.)
The instructions in the textbooks were, even to me, inane, puzzling, nonsensical, and done in such a way that wild guessing was the result, and of course, if they guessed wrong, they were marked accordingly. Eventually, I took them both to Kumon (pure Asian rote math learning.) They excelled to the degree that both of them went from having Math as their worst subject (and most hated) to having Math as their best subject (and being happy about it.) – THey weren’t necessarily happy about doing Kumon – but with an Asian father demanding that no grade be less than 92%, they were happy with the outcomes. Now one has a B.Sc. in Statistics and Data Science working in AI, and the other has a Bachelors in Economics and is finishing his MBA at an esteemed college in the US. I get the argument about teaching kids how to use math to think for themselves, but that should wait until High School at least and be applied more through teaching how to solve math problems. Despite decades of the “Thinking Skills” method in Canada, kids are still graduating or going to University/College without the ability to think critically. Perhaps there should be a class on the philosophy of logic (deductive/inductive,) first principles thinking, and logical fallacies, instilled in the curriculum instead of spending a month teaching about ancient Egypt. Being able to understand key drivers and principles is highly valued in the business world – which is why companies like McKinsey with the McKinsey method exist.
I mean, Egypt is interesting, but I don’t feel it is helpful for equipping kids to succeed in today’s world. If they are interested in that kind of subject, they can study it in Anthropology/Archaeology at University or as a High School elective class.
If I were the education minister in Canada (or the US), I’d buy the Kumon company and make it a mandatory part of the K-7 curriculum for all kids. Then, I’d create and combine that with robotics and computer science classes so that the application of what they were learning could be made fun and hands-on—and real. That would be the fastest way to improve math results across the country (Which, last time I checked, were pretty abysmal).
My kids got the outcomes they did because they had an Asian (Tiger-Dad) father with enough of a background in Math to be helpful and recognize it’s importance, and who did anything and everything he could to help them excel (The Asian part), including reteaching them all their 3-7th math at home in a way they could understand. I sometimes think about the rest of the kids who did very poorly in math, found it confusing and as a result hated it and felt anything Math based was not for them. As a result, some of these kids are now working in the coal mines or laborer jobs b/c they never achieved the aptitude required to get into a good University and never thought that Science or Engineering was something they could be successful at. I know this b/c some of them are still friends with my boys. In today’s Canadian Economy under Turdeau, they are getting crushed, and all of them want to leave Canada. If only they had an excellent Math education and could have been encouraged to become Engineers, Computer Scientists, etc. And no, you will not successfully retrain all of today’s coal miners to be Coding (that’s ridiculous, as computer science is heavily based on math like linear algebra, and some of these folks are coal miners because they flunked out in Math. Thanks idiotic “Common core” nonsense!)
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