Comment by Steuard | Hacker News Reader

Comment by Steuard | Hacker News Reader

It's easy to recognize that the author's arguments could apply just as well to any academic subject: literature, history, you name it. ("We should just teach 'citizen reading', where students learn to read recipes and furniture assembly instructions.")

But the real surprise to me is his ignorance of actual college math curricula. He says, "Why not mathematics in art and music — even poetry — along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art". I don't know about his college, but the math requirement where I teach can be met with courses such as "Math in Art and Nature" or "Liberal Arts Mathematics". It's not as if his suggestions there are novel! But here's the kicker: for both of those classes, proficiency in algebra is a prerequisite. It turns out that you can't really describe those topics that he likes without actually using some math.

On the other hand, I have come around to agree that statistics is more broadly useful than calculus. Here's a TED talk by one of my old math professors making that case: http://www.ted.com/talks/arthur_benjamin_s_formula_for_chang...

> It's easy to recognize that the author's arguments could apply just as well to any academic subject: literature, history, you name it.

The impact algebra has on graduation rates makes it unique. Something is clearly wrong: either we're teaching algebra ineffectively, or we're expecting too much and preventing students who are otherwise capable from graduating high school. It's important we figure out what's wrong and fix it. Without knowing the core problem, his proposal is no less valid than any other and, as a novel and controversial idea, might inspire research toward a real solution. I'm glad he wrote the piece, whether or not algebra should actually remain a mandatory subject.

I don't get that TED talk. For example, how are you meant to properly understand the ubiquitous Gaussian distribution without learning Calculus first?

I may not entirely understand your question. My wife has taught statistics college statistics many times (including Gaussian distributions), and her classes have never had a calculus prerequisite.

Are some topics easier to understand if you already know calculus? Sure. (I assume she has to do the same sort of brief "area under a curve" explanations there that I have to do when I teach algebra-based physics.) Can you understand the topic in greater depth using calculus? Of course. But for a first exposure to basic statistics I think you can mostly dodge the issue.

(And really, apart from already knowing the concept of an integral, does knowing calculus really buy you much when studying Gaussian distributions? You can't even do those integrals! That frustration might be even more annoying to a calculus student than to others.)

I've taken an introduction to statistics course with social science students and based on that experience, I can relate to what you are saying, 100%.

However, I think there is a broad group of students[1] for which a significantly earlier exposure to calculus would be beneficial and make learning statistics (and physics) a lot easier or at least faster.

When I took introduction to statistics as a math major, I found the subject extremely confusing because the discrete and continuous case where taught completely disconnected and useful anchors for understanding such as basic measure theory and Lebesgue integration where left out. That's certainly a good way to teach for many but for some it doesn't work.

A similar case was physics for me (classical mechanics in particular). From grade 5 to 10 (after which I avoided the subject) there was little insight gained (e.g. heavy things fall down, there may be some friction, memorize all those seemingly random formulas and if you use a long lever, make sure you pick a strong material). Then I was exposed to an introduction to physics course at university (for non-majors) and the revelation that all those random formulas have a strong grounding in just 3 general principles and can then be developed with some help from calculus was liberating. Just too late in my case. Maybe I would have loved physics and actually study it, had they told me in 7th grade that there is something tying all of it together, and the ultimate goal of the class was to reach that summit. Just trying to show the other side of the coin which should be integrated into the way math and science is taught in schools in my opinion :-)

[1] Say, the top 5-10% of middle school students.

Oh, don't get me wrong: I was served very well by today's standard math sequence. I learned fascinating stuff in precalc, and calculus was a revelation and a profound joy. Prof. Benjamin's argument favoring statistics instead was a tough sell for me.

But I'm a theoretical physicist. As much as I hate to say it, structuring the entire standard math curriculum so it works best for kids like me (or even for the top 10% of students) just isn't reasonable. (Ideally, a solid gifted program could fill that gap.) I think that we agree on that.

I'd like to think that there are ways of introducing concepts from physics or statistics that do highlight the underlying structure of the field, even if the students don't yet know all of the math they'd need to work through the details themselves. If I find a perfect way to do it, I'll let you know!

One approach I've seen and thought was interesting is the course schedule offered at the Illinois Math and Science Academy.

Their pre-calculus courses have been somewhat radically reorganized into a curriculum called "mathematical investigations" which orders the topics according to more of a practical progression. So, for example, bits of linear algebra are pulled all the way up into precalc because they're useful in geometry, and will also work better with the science curriculum. Perhaps physics teachers inheriting students who already understand vectors, for example.

Then the calculus curriculum is split into two tracks, one more basic, and a more intensive one for students who anticipate going into fields that require more calculus.

Experimental research papers in medicine, and decision making in business, are in a shambles, due to people learning and then misapplying the trappings of statistics, slamming Gaussians everywhere in plug-and-chug SAS sessions, without understanding the mathematical justifications for model selection.

Absolutely. So wouldn't it be neat if premedical students and business majors had already seen a thorough stats class in high school, so that their college-level stats classes could spend less time on the basics and more time on those subtleties?

(Speaking as someone who's taught calculus-based physics to premedical students, I think it's important to recognize that most folks who've taken a year of calculus really have not internalized those concepts enough to be fluent in applying them. I think it would take a particularly strong math background for someone to really understand the mathematical justifications for statistics, so I suspect class time would be better spent warning students about pitfalls than on hoping that they will draw meaningful conclusions from formal derivations. Heck, medical students are required to have taken calculus, but lots of them (evidently including a journal editor, peer reviewers, and 163 followup papers) apparently don't even know what an integral is: http://fliptomato.wordpress.com/2007/03/19/medical-researche... .)

Even if people don't learn to "properly" understand the Gaussian distribution, basic understanding of the nature of variation and the difference between random events and something that indicates an actual change in an underlying system would be immensely useful to people.

You don't have to know all of the fundamental underlying ideas of any concept before it has practical utility.

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