Surprising Math Study

This article about using real world examples to teach math appeared this week in the NY Times. About the study,

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”

Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment. Their results appear in Friday’s issue of the journal Science.

Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods."

How did it work?

"In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems."

They are trying the study on elementary students next.


Charlie Mas said…
It is easier to figure out what to do in a specific case when you know the general rule than to deduce the general rule from a specific case.

When I teach my kids math I usually mess with the units. 10 times 4 is 40, whether it's 10 times 4 doughnuts (40 doughnuts), 10 times 4 trips to the movies (40 trips to the movies), 10 times 4 hundreds (40 hundred), or 10 times 4 quirktaps (40 quirktaps). It doesn't matter what a quirktap is, if you've got 10 times 4 of them, you've got 40.

If they saw only one specific case, they might not know what elements were specific to that case or what elements were part of the general rule.
zb said…
"It is easier to figure out what to do in a specific case when you know the general rule than to deduce the general rule from a specific case."

Might be true, but that's the reason to do the study. There's no reason to assume that because you can generalize from a general rule to a specific case and not the other way around, that it will work that way for everyone, and especially that it will work that way for people in different developmental stages. In early development, children can't understand general rules (in earliest version, they have no language for them). Even later, all data points towards learning the generalities from the specifics.

That being said, I thought the study reported in the NYTimes (and published in Science) was fascinating. In the journal, they've published the actual format of their rules, so eventually (whenever the stuff becomes pubic access) everyone can look at it. It made me think much harder about "experiential" learning for college students.

I eagerly await the results for elementary school kids. If they come out the same, I think we'll have some real data on which to base teaching methodologies.
reader said…
Oh big yawn. Educators and cognitive "scientists" never tire of meaningless studies pointing out the obvious. No doubt funded by our government. Yes, the train problems are hard, and are still hard. We used to call them "word problems". Kids dreaded them. What's new? Applying and generalizing a concept is always harder (or more work) than simply learning it. Learning from an application might not result in learning the underlying priciples. Jeez, this is new?
MathTeacher42 said…
I would suggest

"National Mathematics Advisory Panel. Foundations for Success: The
Final Report of the National Mathematics Advisory Panel, U.S. Department of
Education: Washington, DC, 2008."

From page 82 of the Final Report

"The Panel as a whole reviewed more than 16,000 research studies and
related documents. Yet, only a small percentage of available research met the
standards of evidence and could support conclusions.
Excerpt from the Report of the Subcommittee on
Standards of Evidence
Background: Categories of Internal and External Validity"

I've started to skim some of the documents on the main page.

As a 3rd year high school math teacher, when I consider the 30000+ who've failed the 8th grade math test called the 10 th grade WASL in 2006 and 2007, when I consider that many of the people in charge of math pedagogy in this state have been in charge for 5 or 10 or 15 years, I don't know how those currently in charge can skim these documents and then turn around and cash paychecks from the citizens who've trusted them.

They are successful at blaming teachers.

hschinske said…
I think the confusion came about because people saw children having sudden epiphanies about what fractions meant and so forth when they saw them being applied in a specific context (e.g., halving or doubling a recipe) and assumed that if they only used such practical examples in the classroom, students would learn better. But they missed the whole point, which was that it had to be a PERSONAL connection. My son's never cared a hoot about word problems and would rather do a page of plain arithmetic problems any day. Yet the problem that really clarified division with remainders for him was how to divide the pancakes fairly on Sunday morning.

I don't know how one would best get personal buy-in on such practical problems in a school environment, but just using word problems isn't enough, that's certain.

Helen Schinske
reader said…
Yes well, let's see him divide a half a pancake by a fourth of a person and come up with 2 pancakes. Sure, practical applications, as well word problems are ALWAYS going to be harder than pages of arithmetic. I don't think we need more ground-breaking studies. It also seems obvious that we need to make personal and reinforcing connections, even if it simply sparks interest.
dan dempsey said…
(1/2) pancake / [(1/4) person] =

2 pancake-person
Let us try:

(1/2) pancake /
[(1/4) pancake per person] =

2 persons
Since pancake appears in the numerator of this fraction problem
(functioning as the dividend)

You will only get the units of pancakes on the quotient,

if the divisor is without units.
dan dempsey said…
April 29th, 2008

SBE will require three math credits to graduate for students enrolling as entering 9th graders after July 2009.

For most students this will be Algebra II. Watch the uproar this causes.

It seems these SBE folks like passing rules. I wonder when they will see fit to start improving the education for all?
teacher99 said…
A math group I belong to has had about 25 emails on this article since it came out. It is seriously flawed, like most math studies as rmurphy pointed out.

Apparently the study was of 80 undergraduate students, so stretching from there to elementary/middle school is going way beyond the data.

Others commented on how a very narrow conception of "concrete" was used, effectively being a very poor use of concrete and thus no wonder there was confusion to the students. There was no concrete to abstract continuum in the study… just a narrowly conceived notion of “concrete”.

Indeed, the literature speaks about a continuum from concrete to visual representations to abstract as ages and abilities increase. Although learning style has not yet come up in the discussions I've read on this article, in my experience I've noticed that some learners do strongly prefer more concrete representations and struggle with abstract thought, whereas those who are often very good at making the abstractions quickly bore with concrete hands-on projects.

I'm also curious about why these undergraduate math students seem to be doing so much with fractions. It's unclear if they are students undergoing remediation. It’s odd that fractions is the example being reviewed at college level, although it does legitimately happen.

I have not yet been able to obtain a copy of the report but there were a lot who read it who cautioned about taking the study too far either way (pro or con concrete/abstraction) as it was a small and seemingly overstretched study.

We don't need just a lot more studies, we need fewer but better studies. Right now there are so many studies out there that anybody (teacher or district administrator) can just about proof-text anything they want by pointing to some study. Plus publishers are paying for many studies nowadays which tend to be self-serving, although at least that’s not the case here.

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