Sunday, March 16, 2008

Gaps and Redundancies

There is some irony to be found in the title of Tamar Levin's excellent article in Friday's edition of The New York Times, "Report Urges Changes in Teaching Math." To do anything other than what the report recommends would hardly qualify as teaching math. Here's the crux of the matter:

Closely tracking an influential 2006 report by the National Council of Teachers of Mathematics, the panel recommended that math curriculum should include fewer topics, spending enough time to make sure each is learned in enough depth that it need not be revisited in later grades. That is the approach used in most top-performing nations, and since the 2006 report, many states have been revising their standards to cover fewer topics in greater depth.

It was the frequent revisiting of earlier topics in later grades, with little increase in the sophistication of the approach, that drove me crazy in primary and secondary school. And it wasn't just math--it was virtually every subject. And despite this revisiting in later grades, students' achievements lag those in other countries. So much redundancy in instruction, and yet so many gaps in knowledge. That's strong evidence of the possibility of making gains in outcomes without additional resources.

There is more of interest in the article, particularly in this passage:
After hearing testimony and comments from hundreds of organizations and individuals, and sifting through a broad array of 16,000 research publications, the panelists shaped their report around recent research on how children learn.

For example, the report found it is important for students to master their basic math facts well enough that their recall becomes automatic, stored in their long-term memory, leaving room in their working memory to take in new math processes.

“For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,” the report said.

Dr. Faulkner, a former president of the University of Texas at Austin, said the panel “buys the notion from cognitive science that kids have to know the facts.”

We needed cognitive science to figure that out? There was some competing notion, masquerading as an educational philosophy, that suggested that kids did not have to know the facts? The recommended approach all sounds very familiar, if not widely utilized.

Read the whole thing.


Anonymous said...

In the US we educate everybody, and we don't track until late in the educational process. For this reason topics must be revisited over and over again, even after the top and middle of the class have achieved mastery. And this isn't just a public school problem -- the same model is used in most private and parochial K-12 schools. Lately we call this teach-to-a-dumbed-down test system "No Child Left Behind." That would be clever except it leaves our best and brightest bored and ready to hang themselves in the back of the classroom.

In contrast, other countries use a more utilitarian system (India, China, Japan, etc). This is one reason they are so good at producing engineers.

The biggest mistake we made was replacing the meritocracy with the aristocracy in this country. You think those idle rich legacy students at Dartmouth are gonna save the planet?

Tom said...

Andrew - This concoction is known as a "spiral curriculum" and has been shown in studies from the 70's-90's to be better for average and below average students to achieve standards; it makes bright children crazy. (I would assume that you qualify here!)

There is some merit in the previous commenter's first few paragraphs; tracking earlier would enable brighter students to escape the spiral and move on to other things. This generally begins to happen by middle school, or certainly by high school, so it's not too big a deal.

As to the vision of lazy, indolent legacies filling the seats at'll have to use some other "fact" to comfort yourself...even the legacy kids are much smarter than the average college student, and I doubt the professors could pick one out reliably. The competition for "legacies" is as high as for regular admissions (fewer kids, yes, but higher average intelligence, more desire to go to Alma Mater, and a limit on the # of legacies admitted) and while no one would claim that the student body is perfect, it's a standard deviation better than 90% of schools.

And, no, I don't work there. :)

Anonymous said...

the U.S. needs to consider how to use summers - vacation bible school, math boot camp, science camp, sports camps, etc, and the resources required

anecodtally, many of the problems at my employers in the US have not been math related. it is something else.

Richard Schwartz said...

As the father of two girls making their way through the school system in Nashua, I have been utterly appalled by the math curriculum, which I understand to be a rather popular program that is quite representative of the way math is now taught in a large number of districts around the country. The approach taken, which is known variously as "whole math" or "Chicago math" (because it came out of research at the University of Chicago -- but the Education department, not the Mathematics department), seeks to make students comfortable with math by showing them multiple ways to approach basic operations and letting them choose which one to use in any given situation. This touchy-feely method doesn't just fail to give students that automatic response, it actively discourages it by making the first reaction to any problem a choice between (supposedly) equally valid approaches. I strongly believe that for anyone who has not already achieved an intuitive understanding of basic math operations, and especially for those who are math phobic, having to make such choices is as likely to lead to "analysis paralysis" as to actual progress in solving the problem or learning anything new. I'm hopeful that this report will lead to dramatic reform, and the sooner the better.

Anonymous said...

Sorry about the comment on legacies . . . I was thinking about Bush Jr. at Yale . . . Dartmouth likely has higher standards than Yale . . .

Independent Accountant said...

Did you read the article in the NYT about Finland a few weeks ago? I realize I am racist, but note that in the absence of egalitarianism, many more kids would drop out of school. There was a lively debate about this from about 1955 to 1960, which really got going after Sputnik. This means: if as I believe, here we go, racism again, you need at least a 90 IQ to graduate from high school, only about 40% of Negroes could if they wanted to. But everyone must graduate from high school, no child can be left behind. Balderdash.

HoosierDaddy said...

I know in our public shool in Indiana they did not want the kids to learn the multiplication table. They had one taped to the desk for them to refer to. We finally worked with our kids at home to get them up to speed. They are just setting kids up to be helplessly dependent on Calculators.

Tom said...

We sometimes need to be open to advancements in the art/science of teaching. Just because something worked for us 40 years ago does not mean it will work today, or that most parents will accept it.

Take "creative spelling" for example. Parents are horrified when teachers do not correct spelling in first-third grade, deploring current standards, blah blah blah. But guess what? The kids turn out to be better writers AND better spellers by 8th grade. Any school that does it the old way is harming their students, and there are statistics to prove it.

So we need to be careful with our reactions to "new math" methods. 40 years ago, we did not have calculators in our phones, spreadsheets everywhere, and a standards requirement that children pass calculus in 11th grade to be considered for top schools. Maybe instantaneous mental calculation is a tradeoff we can make. Maybe not, but it's not the slam dunk people think it is.

Dr. T said...

"So we need to be careful with our reactions to "new math" methods."

No, we don't. The new methods of teaching math have proved to be failures. I see this from two sides: the father of two girls now in their teens and as a college and medical school professor. At home, I've seen my bright daughters struggle with algebra because the text and the teachers used idiotic approaches instead of straightforward teaching (which I provided). At the college end, I was astonished to learn that Old Dominion University was offering a for credit remedial algebra course. Virginia mandated two years of high school algebra. I assumed the remediation was for students who barely qualified for college and needed algebra for a business major or something. No. The remedial algebra was for science majors so they could get through Chemistry 101. When students accepted as science majors do not know enough algebra for a general chemistry course, you know that math teaching has failed.

Anonymous said...

My kids had the new math. No problems.

The biggest factor in their success was quality teaching.

There are no remedial math classes at my daughter's college. If you didn't score high on the SAT math section you don't get into that school.

Maybe Old Dominion needs to look at their entrance requirements ? I'd worry about the rigor of any class at a college that's providing remedial math or English courses.

SuperMonkeyCube said...

The things that bug me about math education is that they never seem to teach the whole thing, and they seem to cut it up into parts that satisfy curriculum requirements. Teachers want to teach more - or at least in my experience - but the risk of giving lots of supplemental material to round out the things that are being presented is a rash of students repeatedly uttering the phrase "Is this on the test?"
Sure, you need to teach one thing at a time, but I think it's also important to show how it relates to the whole of mathematics so the bigger picture is apparent. One of the things that in my experience helped students get past some of the difficulties of the material is drill, drill, drill. Not just addition and multiplication, but other reasonable concepts. I don't want to have to explain to a high school mathematics student that 1/2, 0.5 and 50 percent are all the same idea, but - for example - fractions are so compartmentalized in the way that they're taught that many students fail to see them as small quantities, but as some sort of strange math in two layers with special rules.
As I recall, I went to school with a couple of Chinese kids that weren't just drilled on computation, but rudimentary geometric proofs as well. When it was time to learn something new, there was so much foundation already laid that the new idea had a firm footing to fit it in with the other ideas.
I'm probably ranting - the only other thing I was going to say was the usual "They needed a study to figure this out?"