A lot of things startled me when I started teaching in the UK. The accents. The ubiquity of tea. (As I like to say: “ubiquitea.”) The adorable and inexplicable pluralization of “math.” But what stunned me most was that the Brits don’t follow a sequence of math courses anything like ours.

You know the traditional American chain—Algebra, Geometry, Trigonometry, and so on?

In Britain, they make no such distinctions. It’s all “maths.”

Now, we teachers know that we inhabit imperfect systems. (Some days, we feel like we know it all too well.) I don’t think you’d say we’re unduly attached to them. If you ask most British or American math teachers, “Does your country have a well-functioning educational system?” you’ll get anything from a cynical scowl to a bout of weeping.

But ask us, “Isn’t the other country’s structure better?” and you’ll witness a sudden and righteous swell of patriotism.

This is more than a mere protective instinct, the educator’s version of “nobody beats up my little brother but me.” Whether we enjoyed our own schooling or not, those experiences shaped our vision of what an ideal math education *ought* to look like.

Should math be like science, with themed years emphasizing different branches of knowledge?

Or should it be like English, each year a cross-section of the whole subject?

Each side feels like it sees obvious weaknesses in the opponent’s armor. But having taught in both systems, I find most of the attacks are easily repelled.

Take this one:

What makes math education feel monotonous and boring, as it too often does? Not the specific choice of topic, I think.

Rather, it’s a question of lesson design.

If my only move as a teacher is to make you rehearse an algorithm all day, then it doesn’t matter if I’m leaping from statistics to calculus to combinatorics in the span of a single week. My poor students are going to be pulling their hair out and giving themselves pen-tattoos just to keep awake.

Conversely, if I’ve got a varied diet of activities—modeling tasks, challenge puzzles, card sorts, longer-term projects, open-ended questions—then kids can endure even a long unit without their will to live being snuffed out.

Or consider this attack:

Again, confusion is a lesson-by-lesson danger. Leap into an abstract problem without laying the groundwork, and you can bewilder kids in a matter of seconds. It won’t matter if you’ve been tackling similar problems all week. Kids are never so far from the cliff of consternation that a good shove can’t send them toppling over.

Meanwhile, if you ease into the day’s work with the right combination of warm-ups, built-in reminders, and intriguing questions, then kids are surprisingly quick to pick up a thread they left off months ago. (Even if it takes a while, the benefits of spreading out your experiences with a topic—see *How We Learn*, by Benedict Carey—help to compensate.)

Or check out these dueling attacks:

Empty rhetoric vs. empty rhetoric.

For the most part, neither country’s students are achieving a crisp and unified vision of the subject. We’re comparing our own country’s lofty ideals to the other’s disappointing realities—not exactly a fair fight.

If you’ve guessed that I’m building towards a wishy-washy “They’re both equally good!” conclusion, then you’re almost right. I genuinely prefer the American way, but I suspect that a twin version of me raised in Liverpool would disagree. Even so, I know we’d agree on this:

The problems intrinsic to each system are utterly dwarfed by the problems in their execution.

After bickering with my colleagues about these issues (which I do from time to time), I realize how silly it is. It’s like we’re tasting two burnt cakes and arguing which one has the better recipe.

Who cares? How would we even tell which plan is better, when all we’ve got are these monstrous piles of char?

The real question isn’t which recipe to follow. It’s how to make *either *recipe into a reality. It’s how to build a world where teachers and students can thrive and learn, without being burnt to a crisp by the ten thousand competing pressures that a society places on its schools.

Before we decide which cake is closer to perfect, let’s figure out how to bake one that’s edible.

YES! Beautiful. We need more teachers who are able to take an objective look at the education system. Thank you. When you find that edible recipe, please share.

Mmmmmmmmm – burnt maths cake! Perfect! Let’s have tea with that 🙂

EXCELLENT POINT “It’s like we’re tasting two burnt cakes and arguing which one has the better recipe” I was really hoping you would say something along those lines. Educators on both sides of the pond are getting a little ahead of themselves.

Having studied comparative K-12 mathematics curricula as well as the history of US reform effort in mathematics in the 20th century during my first year doing graduate work in math education at University of Michigan, I feel it’s worth pointing that the US notion of chopping up secondary math into “subjects” is REALLY the minority approach worldwide. Hardly anyone else does it that way. It’s not just Britain vs. its former colonies, folks. So if the Brits are way off base, so are most countries and we’re just amazingly fortunate, once again, to have been born and educated in the best possible country ever. Pardon me if I’m a bit sceptical of that likelihood. In this case, the majority may not rule, but it’s a lot closer to presenting mathematics in a way that reflects how I think professional mathematicians see things when they survey the field.

We can benefit from the reality that people specialize as they go further up the tree of mathematics and out among its branches and twigs, but to get out there, it’s generally necessary for a variety of reasons to get the sense of the unity and interrelatedness of the whole. As you move up, you see cross-connections such that the main branches of, say, combinatorics, algebra, number theory, analysis, and geometry start to interweave so that you have analytic number theory, algebraic topology, and differential geometry, for example, and there are things like category theory that then get into deep structural connections between and among various areas so that objects and morphisms and arrows and functors become ways of seeing a host of interrelationships, deep similarities, and so forth.

I can’t pretend to know enough to speak further about the above or guess how these things play out even further upward and outward, but I have some suspicion that there’s a dance that flows between merging and separating and merging more deeply before separating and then merging even more deeply. . . as one travels. Maybe that’s really wrong, but right now it feels right. And I think the US is REALLY stuck at the K-12 level (and not just in mathematics) at isolating and separating and rarely if ever seeing things at all holistically. And that’s to the great disadvantage of millions. Refusing to see holistically is, I suspect, a major flaw of western civilization. But that’s an argument for another time.

Ah, since you know more of the history, maybe you can answer my question! Do you know *why* we do it differently?

My conjecture is that, because (A) our system is very decentralized, without much national unity/control and (B) our population is fairly mobile, we have a far greater reliance than other countries on transcripts. We need transcripts to communicate what a student has learnt. And so teaching themed courses was a natural fit: that way, you can glance at a transcript, see one word (“Trigonometry”), and know more or less what the student knows.

Anyway, I suppose it’s jingoistic of me to defend our practices against the world’s consensus. But I think sometimes jingoism is justified – for example, our liberal arts approach seems like a genuinely better model for undergraduate education than the European consensus, which entails much greater disciplinary specialization. (The European approach is perhaps better for undergrads aspiring to become academic researchers, but that’s not most of ’em.)

If I thought the themed American courses represented really strict barriers between the types of mathematical knowledge, I suppose I’d agree that we’ve got the wrong approach in the US. But the walls aren’t impermeable, and I don’t think they stand in the way of the cool connections you’re talking about. Geometry classes always have lots of algebra (like those “here’s a bunch of angles; solve for x” problems, or coordinate geometry). Algebra II classes, despite the name, are really a UK-style grab-bag of topics (they often include a little combinatorics and probability, for example). And it seems to me that learning calculus as a coherent body of knowledge is the right way to do it, and doesn’t preclude appreciating its connections with other fields (econ, physics, bio) or linking its techniques to other branches of math.

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Totally agree with the final conclusion: does it really matter if we’re not executing either well?

On a side note, in Australia, we don’t differentiate Science either for the majority of schooling. Up until the final two years of school, it’s just Science. You don’t do a class called “Biology” or “Chemistry” or “Physics” until Year 11/12.

In Kenya, we follow the American though a british colony.

m=l!/n!

Algebra have ever great.

Hmm. I really like the critical look at both sides. May I reblog this?

Sure!

I’ve spent 35 years as an electronic engineer, mostly in the famed Silicon Valley. About half way through that stint, it finally dawned on me that math is NOT a subject. It isn’t a THING. It’s just shorthand. Shorthand is a set of squiggly lines on a page that represent sounds people speak. But they are a standard set that other people can read (in theory). Math is a shorthand notation with added rules for moving things around, but it all is just a way of expressing complex ideas in people’s heads. This is where math education goes wrong. The goal is to USE the complex ideas in your head to solve problems. Now here I realize that mathematicians think math is a subject in an of itself, but even people who think that serve a greater purpose or utility. The point I’m making is that in the early years a lot of math is taught as if it is an end in itself. This is wrong for most people on the planet. We should focus more on the ideas, the problems and use the math as a means of utility. Sure advanced math is complex, but only because the problems it is used to address are complex. Once I began to appreciate that math is not an end in itself, I began to appreciate the enormous beauty of the complex ideas that math is just a shorthand for! In general education is a mess because we are teaching people to memorize the RESULTS of what others have invented without teaching the problems that inspired those inventors so we have societies largely filled with “educated” morons. I’ll give you a practical case in point. Many people in marketing learned about charts and extrapolation in college. They then make business plans affecting millions of lives and dollars on the inane assumption that their business growth can extrapolated as a straight line based on two points. The same holds true for the myth of the bell curve. It took me a few years of engineering to realize that nature doesn’t do things in bell curves.

I attended an American school that, quite proudly, did not offer “pre-calculus.” They thought that pre-calculus was undifferentiated sludge, and chose to offer a class on analytic geometry instead. Which, I suppose, saved the physics teacher from having to teach it.

It seems that Algebra II has a fair amount of sludge that creeps into it that isn’t really algebra — a little bit on statics, some financial math, financial math introduces “e” as the consequence of a compound interest calculation… which is a crappy introduction….

The US system (I cannot speak for the British system) seems overly focused on a goal of getting kids ready for and through calculus, with little appreciation for anything else along the way. If you happen to take calculus at the high-school level, that course is taught with extreme focus on techniques to pass AP exam. Students fortunate enough to pass the AP exam find themselves in second year calculus at college, and for the first time taught by a professor who doesn’t care about processing students to the next link in the chain, unprepared.

“Or should it be like English, each year a cross-section of the whole subject?”

I seem to remember 1 year of American Literature, and 1 year of English Literature.

You, Ben, Said, “The problems intrinsic to each system are utterly dwarfed by the problems in their execution”. So true, so true!! Execution is the key.

As a World Languages teacher, I see so many similarities between our subjects. I started my career as a GTA at UNL and lately, I taught K-8 (yes, I saw each grade each week…phew!). Execution is the key. Are we reaching them? Are we making sense? Are we appealing to their interests? (Maybe easier in a WL class than math(s), but still… I won’t go into anecdotes…)

I follow your blog and you give me many great ways to reflect and provide many ideas on how to teach *español*. (Plus, I love math(s), too – especially «mates» or «matemáticas». How about those reversed commas/points?? … anyway… ) ¡Gracias!

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British MAth. Is it like Math Level 1, MAth Level 2, etc 😀

I believe it is called MathS in Britian (and in New Zealand) as it is a shortened version of the word MathematicS.

Historically, Mathematics became Math’s, became Maths. Bud did Economics become Econs?

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You’ve produced another very clever and insightful story. You really are a good person and I’m grateful for that.

Now, I just need to have my kids share the same enthusiasm about your stories.

One side note, Mr Goldenberg stated the US model was far superior to the rest of the world. It may pay him to check the US rankings compared to other countries. If anything speaks volumes about education standards and effectiveness, it’s world ranking.

I’m neither for or against US vs British models myself. I like the burnt cake analogy.

He actually stated that he was quite sceptical of that claim; there was sarcasm there.

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