Karim, I didn’t want it to be a pure vs. applied argument either. What I think does matter is experience before formalization, and that experience should be in context.

But I also believe mathematics can be the context.

One of my favorite open-ended activities in Algebra 1 amounts to “Here are some number tricks. Why do they work? Now you make some.” It’s a real grabber, but has absolutely nothing to do with any application or “real-world” context. Still it serves the same purpose: opening the door for further study and conversation.

Thanks to you both!

You do both.

I have students who love solving math in the traditional “text book” style and I have students who would avoid touching one at all costs. So I differentiate and pull in real-life based instruction fairly often, but it is not my only source of math instruction. Some students respond amazingly to it, some students could care less.

Reach all types of learners and vary your styles.

Rather, it’s HOW to get kids comfortable with the abstraction. Given the way students react to math every single year, it’s really hard to make the case that the traditional (which is to say, decontextualized) approach is the best one. We may love pure equations ourselves, but we also read math blogs.

Every year students ask, “What does this mean?” and “When will I use this?”. These questions are related. If we want students to apply math to something else then we have to give them the something first. Back to the Wheel of Fortune example from earlier, students walk out knowing a lot about game shows, and that’s great. But they also walk out better at fractions, better at percents and better at probabilities than they would have been otherwise.

It’s not a choice between pure vs. applied. It’s a question of order. Do you learn grammar rules before having a conversation, or do you have a conversation, learn and refine the rules, and then apply them to a better conversation later?

I think we’re making this way too complicated.

I identify heavily with the second quote, too. I think one of the things that drew me to math as a student in the first place was the fact that no matter how crazy my over-dramatic high school life was, math was always the same, which was comforting.

Another thing I think about is how it seems that over time topics that at one point seem to be “pure math” ideas end up having unforseen applications decades or centuries down the road.

BMI = (weight in kg) / (height in meters)^2

So you have to account for converting kilograms to pounds, and meters to inches… twice.

About 2.2 pounds in a kilogram, and about 39 inches in a meter, so the conversion factor is (2.2…) / 39…^2, which ends up being about 1/703. Multiplying by 703 brings it back to the original.

Never mind that the actual numbers in BMI are just benchmarks anyway… wouldn’t it be more awesome to have a BMI of 0.041?

If we did change to a curriculum with courses in finance, data, and engineering, we’d still have to prepare students for other careers. Giving students ways to think about and solve problems is the real goal, not cos(arcsin 1/2), which is an assessed skill — I’m confident we could build an equally ugly-looking test with nothing but finance formulas or z-scores.

Also, an entire YEAR on finance? What would that even look like?? There are a lot of really deep, good applications of mathematics to finance, and at many layers of difficulty. I’d rather see finance, data, and engineering appear in all courses appropriately, and let students who want to specialize in those careers choose to take an elective course in that direction.

So, why do we tell every student in the country “you must take algebra” instead of telling them “you must take (your favorite application here)”? That’s a long conversation, but I believe the thinking skills that can be developed in mathematics have a greater pull than any specific application we could teach students. Common Core’s mathematical practices do a better job than I can on this, but it’s still a long road to change attitudes about mathematics, pushing more toward those goals than the content and skill-based goals often cited as the purpose of taking mathematics.

As a ‘pure math’ enthusiast, I am with you in heart.

I have started to question why, though. It is intellectual vanity? I imagine myself as a parent, regardless of SES level or where I live, and I’m presented with this choice:

algebra- geo – algebra 2

finance- data – basic engineering

Hmm, I think. The cool stuff about the former will certainly be incorporated into the latter (it has to be). So, uh, yeah, I’ll take the one that will prepare my kid to work, give them access to current events and ideas, and not focus on units with problems like this:

8. What is the exact value of cos(arc sin(1/2))?

(source: regents high school exam, mathematics b, 6/15/2010)

But I’m not sure dismissiveness of the Times piece is appropriate.

We take issue with the iniquities of kid’s experiences by not exposing them to said experiences? Garfunkel and Mumford don’t propose anything radical. There is nothing culturally elitist about this:

‘Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers.’

In Cuoco’s piece, he responds to this: ‘I just did the exercise and concluded that we’d end up with a program that equipped students with a set of special-purpose techniques that would likely be out of date by the time they graduated college.’

Studying the budgets of people, companies and governments is going to be out of date by the time they graduate college?

And further, what does the present sequence equip students with? Later in his post, Cuoco relates number theory to a house-building project. It was an astonishing insight. It was also one that I would expect from a very specific sub-sect of human: math and science teachers.

Garfunkel and Mumford’s proposal isn’t perfect, isn’t fully fleshed out, etc. But their sequence is not biased towards anybody except those who want a head-start on a cool career. Forget about abstraction or application or whatever. Designing curricula around these themes (finance, government, engineering) feels appropriate for this specific time.