But how many high school calculus teachers spend their evenings

doing calculus?(What would that evenmean?) Can you imagine a geometry teacher spending his evenings writing faux-formal proofs with modus ponens? Do algebra teachers even use algebra? Do theydependon it?Can you trust a teacher who doesn’t use what he teaches? Who has

neverused what he teaches?Can you trust a teacher whose only connection to a subject is

teachingit?How can such a teacher know if what he’s teaching is valuable, or how well he’s teaching it? (“Curricula” and “exams”, respectively, are horrendous answers to those questions.)

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I love this Bret Victor, wherever he may roam.

Can we replace “teacher” with “curriculum,” though? A teacher may never use decimal division, but it’s not because she doesn’t

wantto but because she doesn’thaveto. We should trust the teacher…and just throw the standard away.So how about this:

Can you still call it a “standard” if it isn’t?One should distinguish between performing mathematical calculations and using mathematical concepts. You use mathematics every time you try to make sense of quantitative information. I cannot read a newspaper without encountering functions and variables, proportions, rates of change (derivatives), uncertainty (probability), and so on. These concepts have shaped my view of the world in a fundamental way.

Is he criticizing math teachers who have never done formal mathematical research? There’s a tone to this passage that I don’t like. I have never for the life of me ever solved a quadratic equation for a practical purpose (unless you consider simulating gravity in a computer game to be ‘practical’). Yet I feel the space of problems that quadratic equations occupy and the various methods of solution provide a good framework for students to practice deductive reasoning. On a daily basis, I don’t “use” 99.9% of the things I’ve ever learned. I do use the thinking and reasoning skills I developed while learning them.

OMGGGGGG yup i’ve never solved a quadratic equation for a practical purpose…..

I don’t think the central critique of the piece is at the level of individual skills. I think it’s more of an orientation. I want my writing teacher to be writing. I want my science teacher to be investigating natural phenomena. I want my math teacher to be exploring mathematical terrain new to him/her. I don’t want my writing/science/math teacher to only be connected to the subject by way of the textbook.

If I recall correctly, Hardy expressed a similar concern in

A Mathematician’s Apology.I love the question,

How can such a teacher know if what he’s teaching is valuable?On that note, I can’t say I share your perspective on quadratics,

Tao Wang. I definitely don’t feel that way about quadratics. Indeed, I feel they only occupy such an enormous part of the algebra curriculum because they comprise the only family of polynomials that permits itself to be studied in great detail. If the quadratic formula were as hard as the cubic formula, they’d have a much smaller place in the curriculum.@Tao My initial reaction is that if we want students to be good deductive thinkers and have great reasoning skills, then we should teach those things directly. I have a hard time with “let’s teach A and hope that we get B out of it.” Especially when the A we are teaching only has .1% usability.

In general, this kind of thing is what keeps me up at night. I feel like a fraud trying to convince high school kids that factoring quadratics is somehow useful. Or that simplifying big fractions full of negative exponents is something worth doing. It just isn’t. (Maybe for that 10 to 15% that might go on to pursue a mathematical career, but not for the 85% that won’t.) Ugh. I feel sick.

Wow, 15%? Where is this? :)

Sorry

Aaron B., I have that effect on people sometimes.Teaching college seniors and grad students, I generally teach only what I use myself, generally on a daily basis. (OK, there are a couple of lectures a year on topics that need to be covered but which I don’t personally use—but I come very close to teaching just the stuff I use.)

I do, however, rely on the students having already been taught a lot of the basics (like algebra and calculus and grammar and punctuation and computer programming and probability and the fundamental dogma of biology and genetics and …), so that I only have to do the parts that are useful for the problem at hand. I do reteach and review things that are immediately needed (especially since many of my grad students are re-entry students and have not seen some of the ideas for a decade or two).

So, in the ideal world, everyone would always be taught by people who cared so much about what they taught that they spent much of the rest of their time using that material (or stuff that depended on it). I don’t think we’re in such an ideal world, but I would be somewhat suspicious of a math teacher who never did recreational math.

I’m afraid that Bret Victor, like many other engineers (see also Salman Khan), doesn’t really understand what mathematics is. As he discusses in the link, he was taught many mathematical methods by engineers. Engineers are not mathematicians. Engineers do not, for the most part, “do mathematics.” They simply use some of the wonderful byproducts of mathematics.

It’s especially troubling that Mr. Victor references Paul Lockhart, who sees mathematics (rightly so) not primarily as a set of useful tools, but more as an expressive form of art, and also as a liberation from the “real world.” As far as I recall, G. H. Hardy, who has also come up in this conversation, held a similar view.

I would be more inclined to give Mr. Victor the benefit of the doubt here, and try to come up with some mostly-agreeable interpretation of his comments, if I weren’t already familiar with some of his work. Check out his video titled “Interactive Exploration Of A Dynamical System,” if you haven’t already. Beyond the strange technical mistake of identifying dynamical systems entirely with differential equations, I can’t help but get the strong impression that Mr. Victor thinks that giving students a black box simulation of a dynamical system can, in some way, teach them mathematics. More generally, I consider his expressed desire to remove abstract symbolism and exact methods from mathematics (see “Scrubbing Calculator,” for example) to be nothing short of a direct attack on the power and beauty of the subject. His belief that mathematical symbols were introduced because of the constraints of working on paper is a gross absurdity (though also, coming from someone with his technical training, perhaps an indictment of the very unnatural way those symbols are forced upon students).

Sorry if this is a bit tangential and long-winded, but I think Mr. Victor’s background and views of mathematics should be taken into account when interpreting the comments quoted here.

These comments indicate that Bret Victor would like teachers who understand the importance of the material they teach; who use the material on a regular basis, seemingly preferably as a career but alternatively as recreation and hobby. However, in the examples he provides, he seems to, as Roy Wright has stated, provide two different examples of “understanding” and “using” mathematics, namely understanding its importance to engineering and other applications and using it to solve real-world problems (or simulations of them), and the view of Lockhart, namely understanding its beauty and connections to other areas of higher mathematics and using it as a way to energize and enrich one’s brain. So which is more important? Which is “real math”? Or are they both “real math”? In that case, which is more important for students to learn? Which is more important for the teacher to engage in?

I should say, though, that I don’t disagree with the sentiment. I agree that teaching should be about communicating a way of thinking, and that it is important for teachers to be able to think in that way in order to do this job. However, I guess what I’d like to know, as someone who is planning to become a teacher, is what’s more important? Thinking about teaching mathematics or teaching mathematics? Given all of the minutia and other complications which go into teaching, how can a teacher effectively practice the right way of thinking and also have time for thinking about how to teach the actual material? And I guess I wouldn’t be surprised if the answer was just that it’s a difficult task…

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There’s something about the tone of that essay that bugs me. Overall, I agree with Roy.

One thing that’s nagging at me in the essay is that math teachers are on the hook in his mind because he cannot picture someone doing math recreationally, the way he can picture someone researching history for fun, or writing fiction for the love of it. Wow. That’s evidence.

And it bugs me that he feels that the geometry teacher must be doing “faux-formal proofs with modus ponens” to count as using the subject she teaches. In that case, the writing teacher in his analogy should be writing lame five-paragraph essays with explicit introductory and conclusion paragraphs to send to that literary journal.

I guess I have to echo John’s “I don’t disagree with the sentiment,” but if that’s what we’re looking for in a teacher– someone who devotes a great deal of energy to the field itself, as well as to teaching it– then we should be paying teachers a whole lot more, and expecting them to spend no more than 15 or so hours a week in the classroom, so they have time (and more importantly energy) for their research.

I actually typed out that last sentence, and then realized that I’m basically describing college. (Although you would need to knock the 15 down to 6 to 9.) Meh. I was an undergrad at a large research university. The quality of instruction… well, some of it was fantastic, and some of it was… not. Passion for the subject does not translate to skill in inspiring love for it in students at a lower level.

I think Bret’s expectations are too high. You can’t compare teaching high school academic subjects to teaching university level courses or extracurricular activities. The motivations and priorities of both students and teachers are different at each level. I agree that effective high-school teachers likely engage with the subject matter they teach outside of the classroom, but they are probably more apt to do it with the priority being how to improve their lessons or engage their students more, rather than how they can advance the field of study.

“Your students won’t care how much you know until they know you care about them”, as the saying goes. Some educators take that too literally and think their job is nothing but caring, but obviously balance is needed.