I’ve always dreaded being asked for my “teaching philosophy.”

For years, I gave nonsense or scattershot answers. “Logic and critical thinking are paramount.” “I care more about conceptual understanding than computational skill.” “A balanced, student-centered approach is always best.” “We buzzword to buzzword, not for the *buzz*word, but for the buzz*word*.” At best, each of my disjointed half-theories captured only a piece of the puzzle.

Worse still, none of my replies explained why I devote so much class time to plain old practice. If I was such an enlightened liberal educator, why did I assign repetitive computations for homework? On the other hand, if I was a traditionalist at heart, why did I fall head-over-heels for high-minded progressive rhetoric? Was I an old-school wolf, a new-school lamb, or some strange chimera?

Well, I’ve finally got my answer, and it only takes eleven words: **Math is big ideas, approached from as many angles as possible.**

What do I mean by “big ideas”? Well, here are a few examples:

And what do I mean by “many angles”? I mean that, in our best moments, my students and I come at these ideas like undergraduates approaching a dessert buffet: relentlessly, purposefully, and from all sides.

First: the historical angle. Even when the names-and-dates history doesn’t fit into my lesson plans, I try to contextualize each idea as part of a long lineage, to show how it answers a question, unlocks a door, fills a hole. I want my students to see each idea as one scene in a grand narrative of mathematical discovery.

Second: the verbal angle. English class isn’t the “opposite” of math class, as too many students think. Rather, good language skills empower us to discuss ideas of all types and stripes, especially mathematical ones. A precise and evocative vocabulary is beyond precious. Language allows us to debate productively, to learn as a collective, to think as a team.

Third: the scientific angle. Math’s most explosive ideas send shockwaves throughout the sciences. Physics, obviously—but also economics, biology, geology, chemistry, even psychology and sociology. Math has a symbiotic relationship with the sciences: it furnishes them with a powerful toolkit, and they provide it with concrete examples, a corporeal form for its abstracted soul.

Fourth: practice. Math without any computational practice is a mushy math, a math with no spine. To understand what makes, say, linear equations tick, you’ve got to solve ‘em, graph ‘em, play with ‘em in a hundred different ways. You can’t grasp patterns until you’ve worked through examples. Without multiplication facts at your fingertips, you’re unlikely ever to apprehend deep truths about the distributive property. If you’ve never spent a day multiplying out products of the form (ax + b)(cx + d), then you’ll never internalize the methods for factoring quadratics.

So there you have it. Big ideas from many angles.

I don’t always succeed from every angle—I might botch the history, or shortchange students on practice, or be just plain ignorant of the relevant applications. But that’s why I’m glad my students have *other* teachers—each with their own philosophy, each in their own distinctive way enriching our students’ understanding of those crucial big ideas.

This is a great read. Not so long ago I had to write my philosophy for my internship and your example of “buzzword for buzzword and not buzzword” is pretty much what it was. I’m planning on evolving it and finding a more perfect explanation of my beliefs as I grow in my career.

I also wanted to thank you for that fantastic explanation of what math is to you. That alone leads me to sharing this on my Facebook.

Thanks for reading! It’s hard to avoid buzzwords, partly because they often helpfully gesture towards big, meaningful ideas – it’s just that, like any word, they lose meaning when overused. Anyway, good luck with the evolution of your own articulation of purpose.

No kidding! While some buzzwords were created to buzz, many of them started out as a good idea.

I just want to compliment you with the interesting articles you write, with funny ‘bad drawings’. Keep writing and I will keep reading. 🙂

Thanks for reading! 🙂

I always approach teaching math as being a personal trainer at the brain gym.

It’s about overcoming obstacles and steadily increasing the weight and performing tons of repetitions in different ways.

It’s too abstract to give a fun explanation to teenagers, but math class really does affect everything, even English class. One of my LEAST favorite things teachers do is denounce other subjects — “Oh, I was so terrible at English, I don’t know how to put together a sentence!” as if it’s a point of pride. It’s probably why kids will gravitate to one subject instead of trying to see how each subject relates to the others…

Ah well, nice article, though!

That’s a nice metaphor – I find kids are often pretty responsive to mind/body analogies. “The brain is a muscle” may not be literally true, but has a nice figurative truth to it.

I also dislike the badmouthing of other subjects (except, of course, for the fun to be had bashing economics). Wrote a post not long ago (“Confessions of a Math Major”) to that effect, though without the venom for econ.

Anyway, thanks for reading!

Can you give us your sources about the history of mathematics ?

So far I’ve only read “the history of pi” from Petr Beckman, and oddly, it really touched me.

He proposed to use the number of accurate digit of pi used by a civilization to see his evolution in science.

It was really odd to see the knowledge going backward has the Roman empire crush the Greeks, loosing also square relationship, and understanding linear relationship, with all the problems it can cause when building aqueducts.

I loved reading about the Laplace who got his “job” as a mathematician by telling the king he would find the Philosopher’s stone, but knowing it was an irrational belief, he rather worked on astrophysics.

Reading again about our history from an other perspective is really interesting.

Sure! I love reading pop math books, actually. Some good ones:

1. Fermat’s Enigma, by Simon Singh

2. Joy of X, by Steven Strogatz

3. Anything by John Allen Paulos (less historical, but full of nice anecdotes and tidbits)

4. Anything by Ian Stewart (again, not necessarily historical, but great at contextualizing ideas)

5. A Drunkard’s Walk, by Leonard Mlodinow (and he’s got other books that also look great but which I haven’t gotten around to reading)

6. If you stop by a bookstore, they should have a small “math” section within the science section, probably with nice accessible books on the history of 0, e, the golden ratio, etc.

I didn’t know that about Laplace – that’s a great story. Reminds me of the mathematician (I think it was Euler?) who was asked by the royal court to debate an atheist. The atheist didn’t know anything about math, so the mathematician said, “Sir, a = b^n / c; therefore, God exists. Reply!”

Hahahahahaha! XD Euler really proved his point didn’t he.

Anyways, thanks a lot for the interesting articles and for this bool list 😀 I always enjoy when I come here.

Thanks – come visit again soon! 😉

I just saw your answer, many thanks for all the recommendations 🙂

Glad to help!

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Loved it! Really brilliant and to the point!

The problem with traditional maths teaching is that it’s mostly practice but students never learn what they practice doing, therefore feel it’s meaningless, irrelevant and boring or just technical. Some people like it. I only understood what maths was about years after I’ve finished school. For me the conceptual part was WAY more important to my understanding than boring old practice.

I’m definitely with you. Despite a century of pushback against rote learning, there are a lot of classrooms where it’s still the norm. I think practice is really valuable, and has its place, but that place isn’t “the entire curriculum.”

We never discussed history in math class. Never got beyond doing equations. When we finally got to word problems I was totally lost. Besides adding pennies and dimes in 1st grade we never did much/any real world application of what we were learning.

I’m so grateful that I stumbled through a stats class in college. I use those concepts all the time and laugh at journalists often.

Stats is so valuable. There are lots of journalists that do great statistical work (and not just at places like Five Thirty Eight, where it’s built into the mission statement), but lots who still botch basic concepts, too.

Even w/ my sophomore level understanding of stats, I see and hear misinterpretations all the time.

I keep telling my kids that stats is the most importsnt math class they will take in college. They avoid it like Organic Chem!

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When I was in teacher college I was asked to explain why I became a teacher. The first time I was asked I took the questions seriously and spouted off some buzzwords. The 5th time I was asked I cringed. The 10th time I was asked I casually informed the professor that I had a world domination plan that involved minions and children were the most vulnerable and therefore mouldable asset available. So in short, I just had to get my teaching certificate to hide my brainwashing intentions.

The laugh was hesitant and awkward. Thanks for this post. Especially the bit about teachers of all philosophies working together to help educate children – beautiful! Keep up the inspiration!

Thanks! The brainwashing answer is a pretty legit one, if you ask me. 😉

And I definitely mean that bit about teachers of all philosophies. I work hard on my teaching, and I’m sometimes even proud of it, but a world where all math teachers are like me would be an awful, impoverished world missing lots of what makes it interesting.

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I think you need to go with a philosophy of principles, with only a hint of what you think is wrong with common methods.

For example: I love the way mathematical thinking and scientific reasoning can enrich all of life, and give me a way to connect seemingly unrelated things. I teach other people these things so that they can see the picture too, and get some context of how that fits into the historical story of maths and science. I understand that the leaps of understanding that STEM brings can make it hard for teachers to imagine learning them for the first time, so I try to think carefully about how to demonstrate the core of a concept without losing people in the terminology or details of computation techniques that have been shown to fail students in the past.