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...

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.

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.)

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.

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!

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.

(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... .)

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

Now imagine dealing with people who never completed high-school algebra but who are trying to calculate dosage of a drug. According to a pharmacist I knew, a child died on his watch because of incorrect dosage.

TL;DR A huge amount of college-level material, from economics to physics to chemistry depends on working knowledge of algebra.

I'd argue that algebra is fundamental - the excerpts taken involve the quadratic equation and other supposedly tedious tasks, while ignoring that these are foundations for higher level problem solving. In almost any field, somebody will ask at some point "how many X do I need for Y" - and it's usually not that clear.

I would almost posit that the author is trolling the NY Times, but today that's probably not the case. While math is hard for some students, it's an essential part of education for just about anybody that hopes to be a mildly functional member of society.

I memorized dates for history exams in high school, and that knowledge has had no practical impact on my life whatsoever.

When introducing myself to (non-math/science) academics and telling them I was a (now former) physics professor, I can't tell you how often the first thing out of their mouth was something along the lines of "I was never very good at math" or "I never liked science." I frequently have had doctors tell me how much they hated physics as a pre-med student. That kind of pride in ignorance is quite rare in the opposite direction. The most broadly educated people I know are scientists.

[edit: typo fix]

http://scienceblogs.com/principles/2008/07/26/the-innumeracy...

I often wanted to call bullshit on those people too, and reply with something like "I always hated reading" or "What use is Shakespeare, anyway?" Of course, I never did, out of both social politeness and an actual respect for the humanities and what they bring to civilization.

You can easily swap solving math problems for writing essays in that article and get the same conclusion. Also the whole article completely lacks any statistic basis. 42% of students didn't pass their bachelor exams and 57% students of one university (not saying which faculties were included in statistics) didn't pass algebra course (one math course only is mentioned). So what? How is that connected? What are the long term trends? What about other courses and other universities? Same applies for all the article - throw together random numbers that SEEM to be related (they may be, of course).

Very shallow and probably incorrect article.

The article lists arguments which can be applied to about everything else you learn in secondary education. How comes nobody ever complains about learning literature, arts, music or whatever, but people seem to insist that they'll never need math in their jobs, when in fact, math has made their lives possible as they know it.

</rant>

Nothing else leads to that either. Political opinions and social analysis have zilch value in understanding the Universe we live in. Math does.

That's the problem though. There are many steps to be learned. Things like "order of operations" and "FOIL" are necessary to get the correct answer. Its not that learning Algebra is hard, its the discipline that comes from the. Trouble is, most young people don't see the rewards in learning this discipline.

For example here's a few other disciplines that takes time but seems more rewarding to a young student (or anyone)

Swimming - can have more fun at the beach Martial Arts - self defense and confidence, and proof that you're a bad ass Learning a FPS - bragging rights, more fun online against friends and other gamers Guitar Lessons - sing your favorite songs, perform live, get lots of friends/admirers

Algebra - learn about long term goals and ... processes?

Kids are not into the "long term". They are into the short term. Thus, they are kids. Adults needs to show them why it is important to learn how to find "X". Unfortunately, most adults would rather stay far away from that stuff because they've had a hard time with it as a child as well. Its painful to them. Its easier to tell the child "because it is important" than explain why.

I personally think algebra is very cool. I like how things can workout just because you know a proven formula for solving that problem. But then again, that's why I (and other CS) make the big bucks. We do the things that no one else wants to deal with.

Using your brain is hard. Solving a tough equation is probably equivalent to running a mile. It can be done. It can be fun. Yet, you don't see many people with long slender bodies running around everywhere do we?

Everyone has their niche. Its really up to them to find it.

Having many good friends in disciplines which will never require the use of any sort of mathematics, I can safely say that none of them have had trouble with high-school level algebra.

Portions of the article strike me as a depressing appeal to entitlement amongst the lowest common denominator of students to a university education.

Well, the young woman who comes looks at the measurements I wrote down and looks at me and says, "The machine doesn't do fractions."

I was floored. She was lying to cover her innumeracy, because she didn't do fractions. If this is indicative of a trend, it's bad news for the future of the United States.

Algebra lets us describe relationships between unknown quantities. Nearly every physical law is expressed as an algebraic equation (or Calculus, which requires it). It's the lingua franca of describing the world. That's why we need it :).

If grasping Algebra is actually about attaining this developmental stage, we need to be approaching the problem on a more fundamental level. Kids will move through these stages at a different pace and if you're on the tail end of developmental pace you're going to fall through the cracks.

[1] http://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_...

Searching a little I found this interesting document addressing some of the problems which I have put on my reading list: "The Science of Thinking, and Science for Thinking: A Description of Cognitive Acceleration through Science Education (CASE)" (Philip Adey, 1999).

Assuming the idea of a "formal operational" stage of development applies, the situation looks abysmal (at least in the US and Germany, can't say much about other countries):

From the little I gathered so far (on the internet, so it has to be taken with a grain of salt) it seems that (1) a vast majority (over 60%) of people never reach formal operational maturity. (2) Ideas of how teaching can actually help with it are in it's infancy. (3) Application of said ideas is not very far along. (4) Educational systems keep leaving many (or most) students behind early, especially in math and science, while other students get bored and waste their time in class, being taught a mind-choking curriculum.

Ok, I guess I'm ranting now :-) Saddens me greatly, though.

[1] http://www.ibe.unesco.org/fileadmin/user_upload/archive/publ...

edit: spelling, grammar

It turns out that a significant number of people don't understand speed as a rate. They think of it as an intensity, as in volume of sound or brightness. (Those are flux, which are related to rate, though that's not how we perceive them.)

People like that are alien to me. I think it also explains why freeway drivers in Houston often have about 0.3 seconds of separation between cars.

Case in point - an innumerate relative of mine has been duped into selling "Nu-skin" products for virtually no money. Nu-skin gave her documents explicitly stating her odds - 99.5% of active nu-skin sellers make < $15k/year. But they also showed her videos of people who won won carribean cruises and made thousands/millions.

Guess which one she bought into?

A math literate person would recognize that her odds are better working at chipotle and reinvesting some of the proceeds in vegas.

tl;dr What a person recognizes and what a person acts on are not always the same.

A numerate person looks at things like this as a math problem. An innumerate person doesn't.

People make terrible choices all the time despite knowing and understanding all sorts of things (math, health risks, harm they may cause themselves or others) because they have emotions and because weakness of will is a real phenomenon. You seem to be completely discounting all other considerations except "understanding the numbers".

I edited this about twenty times. I suspect we have radically different views of human psychology and how knowledge affects choice. Might be interesting to discuss over a beer, but here only a distraction from the larger issue in the article. Apologies.

The lie with these kinds of things is not as much in those percentages as it is that it is not a lottery. There are people who can sell anything; they are the winners. Also, typically, there is some kind of pyramid scheme involved. You get stats on the early birds, but you cannot become an early bird yourself.

A small chance of winning, but with otherwise serious cash problems? Or high probability of a nice stable income?

I know what I would choose.

And that likely is true. If you choose any 20 hours to work each day, 7 days a week and have some talent for sales, you likely will make that 15K.

This is very true, but lowering expectations is hardly a solution. Ignorance (in the form of oh-well-that's-the-class-smart-people-take) is no better than fear. Before we set the bar too low, let's keep pushing to make mathematics less intimidating, less foreign and easier to learn for everyone.

Have a look at the TED talk linked in my top-level comment for an argument along the lines of what you're making here.

But they're important in the context of real life.

So, yes, we should keep these classes, keep having kids go through algebra and calculus and geometry...

...But you shouldn't be bound by arbitrary rules. Algebra is important, but if you find "x" by doing something other than some arbitrary thing where you subtract both sides, etc., you shouldn't get an F in the class.

Same with calculus, geometry, everything. The importance is the thinking and the logic, and what real-life application you can take from your knowledge. Making hard rules for these math courses, for example, definitely hurts this and does some of the things this article claims.

But true, honest exploration in math and thinking about it is very important, and if it's free and done in an honest way there is no question about whether it's necessary or not.

Richard Feynman covered this better than I ever could: http://www.youtube.com/watch?v=5ZED4gITL28

In middle school at least this kind of thing was allowed as long as we could show why what we did worked. The idea being that we would have to understand what we did in order to know either when it would work or when it would fail so that we could apply it correctly. If we could do that we weren't given full credit for anything because the homework/tests were meant to check that we understood how to get the correct answer (other than copying from the nerds like me).

I could be wrong though. What about famous thinkers like Jefferson or Lincoln -- did they understand algebra and statistics at all?

http://www.math.virginia.edu/Jefferson/jefferson.htm

Geometry was prominent in those days, but as a student he also learned physics with Newton's Principia and Optiks as textbooks.

Lincoln was mostly self-educated, but I have no idea what his background in math was.

If someone doesn't know anything about history, you can pass them along, claiming that they wrote a few essays or something. It's fraud, and any expert can see it, but it's easy enough to ignore it if you try.

But with algebra you can't ignore failure. It's obvious. Even the most basic tests will reveal ignorance quickly.

I conjecture that those people who drop out "because of algebra" are not proficient in any subject.

I observe that the alternatives suggested are less objective and more "hands-on". What does it mean to learn about the consumer price index without learning basic algebra? The only explanation I can think of is that it offers more opportunity to ignore educational fraud.

Algebra, as it is taught today in the U.S., is not necessary, and is probably detrimental in many of the ways stated in the article. I can't speak for non-traditional schools, but public schools, with their focus on standardized testing, have effectively destroyed the original spirit behind teaching mathematics.

Mathematics, much like other academic topics (e.g. literary analysis), was taught not as a practical skill, but as a way to improve your abstract thinking and problem solving skills. As another commenter stated, Mathematics (along with most later academics) should be a gymnasium for your brain. I like this metaphor, because if you think about solo athletics, all students are not expected to achieve at a pre-defined level. Rather students are evaluated based on improvement in performance over time.

Sadly, mathematics in recent years has become less about abstract thinking and problem solving and more about rote memorization, computation, and application. I taught basic algebra to college students for a semester. One of my most interesting experiences was with the dreaded story problem. Most of the students were simply unable to apply math to solve a problem. They were all very good all computation, but when faced with a problem in a form they didn't recognize they instantly began flailing.

Standardized testing has turned math into the process of recognizing a form, plugging in the numbers, and computing. We're now starting to see the first generation of math teachers that are a product of standardized testing, and the results are frankly, frightening. I've spoken with younger math teachers who couldn't explain the practical importance of their subject. While they loved math, they couldn't tell me why they were teaching it, other than: "It's on the [standardized] test."

This is the kind of Algebra we don't need.

* * *

Incidentally, the only way I avoided the shocking deficiencies of standard high school "math" was by fighting my way into an advanced program at the local University. It was there I was introduced to Euclidean geometry and really learned what math was all about. I think everyone should learn Geometry using the Euclidean method. I learned from a simplified book (Geometry, by Moise/Downs) which starts out with a few more postulates than Euclid did. I'd go as far to say that geometry (properly taught) will make you a better programmer/problem solver/thinker.

Equally, not everyone's opinion is equally valid, but they too may have something to add. In particular, find people with full and meaningful lives who have not done algebra, and that will show that it's not essential.

Showing them that their lives would be better with a working knowledge of algebra would be a challenge worth considering.

Mostly, Yes.

This is why I love online learning and have good hopes for the Khan academy. Projects like that can set us/our children free from the plague of bad teaching.