time to throw concrete math instruction out the window?

This... could hurt.

In the first experiment, involving 80 students, some participants were given one concrete example before testing on the children’s game, while some were given two or three examples. One group only learned the generic symbols.

When tested on the children’s game, the group that learned the generic symbols got nearly 80 percent of the questions right. Those who learned one, two or even three concrete examples did no better than chance in selecting the right answers.

“They were just guessing,” Kaminski said.

In a second experiment, the researchers gave 20 participants two concrete examples and explained how they were alike. Surprisingly, this still did not help students apply the concept any better and they still did no better than chance when tested later about the game.

In a third experiment, the researchers presented 20 students with two concrete examples and then asked them to compare the two examples and write down any similarities they saw. After this experiment, about 44 percent of the students performed well on the test concerning the children’s game, while the remainder still did not perform better than chance.

This suggests that only some students, not all, benefit from direct comparison of learned concrete examples.

Finally, in a fourth experiment involving 40 students, some learned the concrete example first followed by the generic symbols, while others learned only the generic symbols. The thought here was that the concrete example would engage the students in the learning process while the generic symbols would promote transfer of that knowledge.

But even in this experiment, students who learned only the generic symbols performed better on subsequent testing than those who learned the concept using the concrete example and then the generic symbols.

The authors said that students seem to learn concepts quickly when they are presented with familiar real objects such as marbles or containers of liquid, and so it is easy to see why many advocate this approach. “But it turns out there is no true insight there. They can’t move beyond these real objects to apply that knowledge,” said Sloutsky.

Compare that with, say, the recent math-athon in the North Thurston district.

Pre-calculus students at North Thurston High School will show display boards of photographs of everyday objects that are examples of conics — circles, parabolas, ellipses and hyperboles.

“I just wanted them to see real math in the real world,” said their teacher, Julie Cassidy.

Boards showed photographs of nail polish bottles, mustard containers, bicycle racks, basketballs, garden hoses, shoe soles and more.

“I really liked it because it kind of gives you a more hands-on idea of math. I’m the type of person who likes to get out and experience in the world. I can’t sit at a desk and remember it,” said junior Raewyn Heim, 17.

Obviously, further studies are needed to see if Heim has it right. It could be that the hands-on aspect provides greater motivation for learners who aren't already college undergrads, the subjects of the OSU experiment. Also, if Piagetian principles are at play, the older students might just be more facile in the realm of abstraction. Still, as an English teacher, I have to go with this quote by one of the experimenters: "Story problems could be an incredible instrument for testing what was learned. But they are bad instruments for teaching."

Now you have a scientific reason to hate story problems.

The Math Wars Rage On

Surfing around the other day I came to the Where’s the Math? website, and was pleased to find that they’ve released their version of what our new state math standards should be. Particularly interesting is this document where they compare their version of the standards to the version the Dana Center put together for OSPI recently; math teachers especially will be interested to see the difference between what the state proposes and what an ideal could be.

I was set to go to the public meeting in Spokane two weeks back on the proposed standards, but the epic snowstorm kept me at home. If you were able to attend the roll-out in your area, please share!

summer WASL math remediation stinks

Mathwise, the astounding failure of WASL retakes has much to do with the quick-fix nature of the summer remediation offered.

The PAS program enrolled only 14 percent of eligible sophomores in summer 2006, and of those, only 28 percent passed a retake of the math WASL in August 2006. The results were only slightly better than those of students who didn't participate in the state-funded program.

The failure of the emergency remediation efforts was one factor in the Legislature's decision to suspend math as a graduation requirement until 2013. Students still must pass reading and writing to graduate.

The PAS summer math program, with a curriculum developed by the Office of Superintendent of Public Instruction (OSPI), was only four weeks. Administrators from several Snohomish County school districts said that was too little time to help most students.

"For the vast majority, what we could do in four weeks was not sufficient," said Ken Limón, assistant superintendent for Edmonds schools. Edmonds didn't offer the math summer PAS program this year because the earlier results were "so ineffective," Limón said.

Four weeks of math to close the gap? The words "optimistic" and "foolhardy" come to mind, followed closely by "ill-conceived" and "ridiculous."

The 4-week crash course might have helped more students who scored L2+ (close to passing, L3), but apparently, those students were more likely to take the WASL again without losing a precious month of summer. In the end, it may have made little difference; we have to account for the possibility that the L2's who signed up for the summer course just might have been the go-getters of the bunch.

As for the "modest" increase relative to non-participants, the WSIPP report notes [pdf]:

The gains made by PAS participants reported in Exhibit 6 are relatively modest when one considers that reading scores range from 225 to 525 points and math scores vary between 200 and 575 points. For example, PAS participants improved by only 1.3 points relative to non-participants on a 300-point reading scale.

Cost to taxpayers: $28.5 million. Cost to OSPI's pedagogical credibility: incalculable.

changes aplenty in math and grad requirements

Via KOMO, word that the state is about ready to lift the lid off new math GLE-EALR-things, including the death of the dreaded spiral:

Seeley wouldn't share many specifics about the new learning requirements before the draft is released on Tuesday, but she did offer some examples about the way the teaching of math is evolving in Washington and around the nation.

The current math learning standards offer a spiral of learning - a number of concepts are taught over a number of years with more depth added over time. The new standards will shorten the length of time students are given to master a concept like fractions, but during the years in which fractions are a major emphasis, teachers will spend more time and make more of an effort to ensure that every child understands the concept thoroughly, Seeley said.

"We're really trying to get past the spiral, so students don't get stuck spinning around," she said.

For the anti-WASL hopeful, dash yours: the WASL's still part of the program. It'll be "revised" by 2013, when it again comes perilously close to a bona fide graduation requirement before inevitable and inexorable delays. Whoops, cynicism creeping through.

Meanwhile, on the high school front:

Tuesday, State Board of Education members will be in Seattle to discuss raising the minimum number and types of classes students need to graduate from high school, and perhaps changing other state graduation requirements, such as the in-depth "culminating" project.

The board members aren't looking at removing the Washington Assessment of Student Learning (WASL) as one of the graduation requirements — that's something they support, said Edie Harding, the board's executive director.

They are considering adding to the list of courses students must take — perhaps matching what most four-year public colleges and universities expect of their applicants. And they're discussing whether to add a number of "lifelong learning skills" that students would have to demonstrate, such as leadership, civic responsibility and teamwork.

They forgot to name test-taking, but it's in there, promise.

what's right with Washington's math education

We can't get enough students to pass the WASL, but at least our ACT results are 3rd-highest in the nation. Of course, a disclaimer is required:

This fact is not incongruous with the failure of many Washington students to meet the standard in the math section of the statewide graduation test, because many students taking the ACT test are high achievers aiming toward college who probably also passed the Washington Assessment of Student Learning.

And yet...

In 2006, Washington public and private school students averaged a combined SAT score of 1059, an average score of 527 in reading and 532 in math. That put Washington at the top of a list of 23 states in which more than 50 percent of high school students take the SAT.

Even without a strict WASL math requirement, we're number one. But then there's overall "college readiness":

The percentage who met the benchmark for all four subjects — including reading and science — was 34 percent, compared to 32 percent last year. Only 23 percent met this benchmark nationally.

Bad as things might be, at least they're better in Washington.

how important are high school math and science?

If we want more scientifically competent students, they're pretty important. Biologist PZ Myers, regarding a recent study, notes:

High school physics was as effective at prepping students for college physics as high school biology was at prepping students for college biology.

Success in one area of science doesn't transfer across areas, unfortunately; a physics whiz is not a guaranteed biology nerd. What really helps, though:

Math is the #1 most effective preparation for doing well in all sciences, across the board; the more math you can get in high school, the better you're going to do in any science class you might want to take. Look at those giant gray bars — it makes almost a 2-grade point difference to be all caught up in math before you start college. Parents, if you want your kids to be doctors or rocket scientists, the best thing you can do is make sure they take calculus in high school. Please. Failing to do so doesn't mean your kid is doomed, but I can see it in the classroom, that students who don't have the math background have to work twice as hard to keep up as the students who sail in with calculus already under their belt.

streamlining Washington's math curriculum

Not enough students are passing the math WASL, but that doesn't mean our standards are too high, according to an outside consultant.

"The bottom line is that Washington's math standards need to be strengthened," wrote Linda Plattner of the Maryland-based educational research firm Strategic Teaching, which was hired by the state to assess its math expectations.

The article notes some of the proposed changes. And who benefits--besides students and teachers, of course?

After the new learning requirements are written by the Office of the Superintendent of Public Instruction, guided by this report, a Board of Education committee will recommend three to five commercial math programs that fit well with the state's standards.

This will be a change in approach for Washington state, where at least 100 different math programs are used around the state, said Corrine McGuigan, assistant superintendent for research and education development at the state Education Department and a member of the state Board of Education committee working on the math action plan.

"That doesn't mean that a district can't go off and do what it wants," she added.

One can only guess what those 3-5 privileged curricula will be, as the standardization of Washington's state education continues apace.

Math Wars, Battlefront Seattle

The debate about how best to teach mathematics is getting rehashed in Seattle as they prepare to adopt a new math curriculum. The Seattle Times and the Seattle PI have both weighed in with editorials, and we should know soon which way they decide to go: constructivist, as embodied in the much-maligned Everyday Math, or traditional, like you find in the Singapore Math curriculum.

The comments section of the PI editorial gives you a good microcosm of what the debate looks like. The traditionalist viewpoint thrives at Mathematically Correct and at Washington State's very own Where's the Math?, a great blog with frequent updates; those who believe in constructivism can find ample ammunition to support the cause at this subsite from Mathforum.org.

At the end of the day I think any of us would agree that the ultimate goal is to make sure the students are getting where they need to be. The rather nebulous part of that equation is that we're not always sure where they need to be; the recent delay of the math WASL, for example, or the NCTM's recent Curriculum Focal Points document are both pretty good reflections on the fact that the only constant is change.

What's the state of math in your district?