Sunday, August 28, 2011

Please visit an actual classroom before you make recommendations, VII

If our most outspoken math and science professors (Jordan Ellenberg, Brian Greene), self-styled education experts (Alfie Kohn, Susan Engel), education reporters, and even literature professors (Cathy Davidson) are to be believed, our k12 schools are still mired in the 19th century, led by drill masters who force meaningless facts and meaninglessly abstract math and science concepts down students' throats, totally oblivious to the 21st century world all around them.

Our eager diagnosticians propose the same basic cures for this supposed illness: make classrooms more student centered (preferably student group-centered), and make the curriculum more concrete and relevant to students' lives. And our major newspapers--especially The New York Times--are equally eager to give them a forum.

In the latest collusion between armchair academia and the Fourth Estate, we hear from Sol Garfunkel, executive director of the Consortium for Mathematics and Its Applications, and David Mumford, an emeritus professor of mathematics at Brown. Writing in past Thursday's New York Time OP-Ed pages on How to Fix Our Math Education, they begin by diagnosing the illness. Our current math curriculum, they write, is "highly abstract," "'pure' math with no context," with its "mysterious variable x" and its emphasis on solving quadratic equations, understanding transformations and complex numbers. This, they write, is "simply not the best way to prepare a vast majority of high school students for life":

Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.
In its place:  "a math curriculum that focused on real-life problems" and that "will lead students to appreciate how a mathematical formula models and clarifies real-world situations--what they call a "contextual approach, in the style of all working scientists,"  where instead of x and y, you have, for example, E, m and c, as in Einstein's E=mc2.

Presumably Garfunkel and Mumford have never actually looked at the latest high school math texts, which contain such problems as this and this. While arguably not good examples of real world math, this stuff is hardly abstract and "pure." For truly real-world algebra you have to go back in time to the 1960's or before.

But Garfunkel and Mumford want to do more than just tinker with the curriculum; they want to stop teaching algebra, geometry and calculus, at least to most students: 
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 short:
What we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).
We've been here before: back in 1920, when the Committee on the Problem of Mathematics, headed by William Heard Kilpatrick, argued that algebra and geometry should be eliminated from most courses of study. As Diane Ravitch describes it in Left Back:
The Kilpatrick committee recommended that mathematics be tailored for four different groups: first, the "general readers," who needed only ordinary arithmetic in their everyday lives; second, students preparing for certain trades (e.g., plumbers or machinists), who needed a modest amount of mathematics, but certainly not algebra and geometry; third, the few students who wanted to become engineers who needed certain mathematical skills and knowledge for their jobs; and last, the "group of specializers," including students "who 'like' mathematics," for whom the existing program seemed about right, although the committee proposed "even for this group a far-reaching reorganization of practically all of secondary mathematics."
This sounds an awful like the European system of vocational tracking which so many Americans rightly bemoan. Is this really what Garfunkel and Mumford want? Do they really want it determined in high school who is going to pursue physics and mathematics, or major in one of the many subjects for which college-level math is the prerequisite? 

Equally questionable are Garfunkel and Mumford's closing assertions. First:
We believe that the best way for the United States to compete globally is to strive for universal quantitative literacy: teaching topics that make sense to all students and can be used by them throughout their lives.
Perhaps Garfunkel and Mumford no longer see mathematics and science as worthwhile areas for American competition. Second:
It is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now.
Here Garfunkel and Mumford aren't repeating history, but rewriting it.  While mathematical guess and check procedures often emerge out of real-world application, in general, real-world mathematical application has lagged several hundred years behind pure mathematical theory.  As irrelevant to today's non-mathematicians as today's mathematical advancements may appear to be, they may turn out in several centuries to be crucial to solving real-world problems in anything from cryptography to quantum computing to the generation of sustainable energy.  Assuming, of course, that future scientists and engineers are still getting adequate training in basic high school math.


Deirdre Mundy said...

They also don't read textbooks. Finance, engineering, surveying, physics, even industrial planning and marketing are ALREADY part of our high school math books.

---But not until students hit Alg2/Trig, Precalc and Calc. Why, because 'real world math' is actually HARDER than "contrived problems." The only people who think applied math is "easy" are the pure mathematicians who only deal in proofs.

Our HS students are NOT doing many proofs. Heck, even most Geometry classes are trying to get rid of proofs. Kids who don't go to magnet schools don't do Calc proofs..... Our math is ALREADY applied.

It's just HARD and applied, and the reformers want to make it 'fun and easy' instead of brain-achingly hard. (Heck, even those of us who LIKE math have one kind of problem we used to hate. For instance, Related Rates took FOREVER to click for me. Eventually, they did, but it took time, effort and practice.)

They never consider the possibility that it might be GOOD for kids (especially college-bound kids!) to encounter difficult and frustrating work that needs wrestling to understand.

Honestly, I think this is a cultural issue-- we don't want ANYTHING to be hard for our kids, even the stuff that SHOULD be hard.

Except some parents DO push their kids to stretch themselves, and it's creating a class divide based on effort.

Amy P said...

"Finance, engineering, surveying, physics, even industrial planning and marketing are ALREADY part of our high school math books."


"Why, because 'real world math' is actually HARDER than "contrived problems.""

Right again.

"It's just HARD and applied, and the reformers want to make it 'fun and easy' instead of brain-achingly hard."

I think there are two different threads in reform math. On the one hand, you do have the pull toward "fun and easy," on the other hand, there's also a pull toward having kids do stupid brute force solutions of problems.

FedUpMom said...

For my somewhat different take on this article, see my post:

A Pox on the Real World

Niels Henrik Abel said...

I really don't get the inordinate fascination with "quantitative literacy." If you want to get a handle on finance, exponential growth/decay, and all the other "real world" crap that they stick in these types of textbooks, you still need to have a solid foundation of algebra, and the mathematical reasoning that goes along with it.

Can't get away from all those nasty, "irrelevant" Xs and Ys after all. And what Deirdre said is true: the "real world," "relevant" math is harder and more obtuse than the straightforward, "contrived" problems.

ChemProf said...

I teach freshman chemistry, and can tell you that students find algebra with lots of different symbols much more difficult than algebra with x. In kinetics, one of the most dreaded topics is finding a rate law from a mechanism, since it is pure symbolic manipulation.

It is also odd that they include "how computers are programs" when it is almost impossible to find a real programming class in high schools. I'm quite curious about what they think this would look like -- I'm betting it would be a lot of messing with the GUI and not much programming.

Deirdre Mundy said...

Chem-- we did real programming in High School, but we were at an STEM magnet----

More typical schools seem to equate "computer class" with word processing....

Anonymous said...

Katherine, I think you're a bit too eager to condemn European systems of vocational tracking. In the CIA World Factbook country-by-country unemployment rankings, the United States ranks behind almost every western European country and developed Latin American country. Most of those countries' educational systems separate students into college-bound and vocational tracks no later than early high school. It seems those systems do a better job than ours of training students to do jobs that are needed, which is not surprising once one considers the absurdity of foisting algebra, geometry, and trigonometry on students who would be much better served by a curriculum designed to teach them marketable skills. In addition, the current system wastes tremendous monetary resources on the very expensive myth that everyone should (be trained to) pursue higher education; at present, students who should be learning specific, job-applicable skills instead are relegated to "special education" programs which teach them nothing useful and frustrate everyone involved.

My interpretation of the NYT op-ed is that the authors want the United States to adopt a more European-style vocational tracking system, one which has evidently produced jobs with relative success. I can't say I blame them.

Hainish said...

I'd settle for a rigorous, traditional math curriculum until the end of grade 10 or so, at which point students could choose to pursue different tracks.

The end of grade 10 is also not a bad point at which to separate students into vocational vs. college-bound classes. It's not that I think the vocational students should learn less history and literature than they do now, it's just that I think they can learn about as much in 10 years as they currently do in 12.

Catherine said...

GREAT post!

I just directed the Parents Forum folks to you ---

Lsquared said...

As they say "It is through real-life applications that mathematics emerged in the past" can make a good argument for that if by "real-life applications" you mean theoretical physics, advanced cryptography, modeling recursive systems and the like. Of course, you can't learn any of that stuff without about 5 years of college math past calculus, so...

Katharine Beals said...

"It seems those systems do a better job than ours of training students to do jobs that are needed."

Agreed. In terms of actual instruction, the European system seems to me to be so much better than ours that students in all tracks there are learning more than many of our students do--even, perhaps, in math. And it's certainly far superior to ghettoizing certain students in special ed classes instead of teaching them marketable skills.

But I would not conclude that because of this it's absurd to expose all students to algebra, geometry, and trigonometry.

Or that it's because of this that we have lower unemployment rankings in the CIA World Factbook.

I agree that too many students are going college, but am troubled by any system that decides, *before students have a chance at algebra and geometry*, who should pursue which vocational path.