NctmReformsAgain < Kitchen

12 Sep 2006 - 18:20

new new new math

Arithmetic Problem
New Report Urges Return to Basics In Teaching Math
Critics of 'Fuzzy' Methods Cheer Educators' Findings;
Drills Without Calculators Taking Cues From Singapore
By JOHN HECHINGER
September 12, 2006; Page A1

The nation's math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.
In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.
The council's advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.
Those recommendations horrified many educators, especially college math professors alarmed by a rising tide of freshmen needing remediation. The council's 1989 report influenced textbooks and led to what are commonly called "reform math" programs, which are used in school systems across the country.
The new approach puzzled many parents. For example, to solve a basic division problem, 120 divided by 40, students might cross off groups of circles to "discover" that the answer was three.
Infuriated parents dubbed it "fuzzy math" and launched a countermovement. The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals.
Nevertheless, the council's new guidelines constitute "a remarkable reversal, and it's about time," says Ralph Raimi, a University of Rochester math professor.
Francis Fennell, the council's president, says the latest guidelines move closer to the curriculum of Asian countries such as Singapore, whose students tend to perform better on international tests.

So maybe it wasn't such a great idea after all for IUFSD to ban my Singapore Math course.

new timeline

According to their report, "Curriculum Focal Points," which is subtitled "A Quest for Coherence," students, by second grade, should "develop quick recall of basic addition facts and related subtraction facts." By fourth grade, the report says, students should be fluent with "multiplication and division facts" and should start working with decimals and fractions. By fifth, they should know the "standard algorithm" for division -- in other words, long division -- and should start adding and subtracting decimals and fractions. By sixth grade, students should be moving on to multiplication and division of fractions and decimals. By seventh and eighth grades, they should use algebra to solve linear equations.

Here's the Singapore sequence.

Lutherans turning into Catholics

A recent study by the Thomas B. Fordham Foundation, a Washington nonprofit group, found that only two dozen states specified that students needed to know the multiplication tables. Many allowed calculators in early grades.
Chester E. Finn Jr., the foundation's president and a former top official at the U.S. Department of Education, blamed the earlier math-council guidelines for state standards that neglect the basics. He described the new advice as a "sea change," saying that "it's a little bit like Lutherans deciding to become Catholics after the Reformation."
Understanding math, rather than parroting answers to poorly understood equations, was the goal of the council's controversial 1989 standards. Those guidelines called on teachers to promote estimation, rather than precise answers. For example, an elementary-school student tackling the problem 4,783 divided by 13 should instead divide 4,800 by 12 to arrive at "about 400," the 1989 report said. The council said this approach would enable children using calculators to "decide whether the correct keys were pressed and whether the calculator result is reasonable."
"The calculator renders obsolete much of the complex pencil-and-paper proficiency traditionally emphasized in mathematics courses," the council said then. In 2000, in another report, the council backed away somewhat from that position.
Still, in response to the earlier recommendations, many school systems required children to describe in writing the reasoning behind their answers. Some parents complained that students ended up writing about math, rather than doing it.
As the debate heated up, concern grew about U.S. students' math competence. In 2003, Trends in International Mathematics and Science Study, a test that compares student achievement in many countries, ranked U.S. students just 15th in eighth-grade math skills, behind both Australia and the Slovak Republic. Singapore ranked No. 1, followed by South Korea and Hong Kong. Fueling concern about the quality of elementary and high-school instruction: one in five U.S. college freshmen now need a remedial math course, according to the National Science Board.

low-income students

This is very exciting. The AIR report (pdf file) led me to believe that Singapore Math had been a flop in low-income schools because the student mobility is so high (and see Hirsch on this subject, too):

If school systems adopt the math council's new approach, their classes might resemble those at Garfield Elementary School in Revere, Mass., just north of Boston. Three-quarters of Garfield's students receive free and reduced lunches, and many are the children of recent immigrants from such countries as Brazil, Cambodia and El Salvador.
Three years ago, Garfield started using Singapore Math, a curriculum modeled on that country's official program and now used in about 300 school systems in the U.S. Many school systems and parents regard Singapore Math as an antidote for "reform math" programs that arose from the math council's earlier recommendations.
According to preliminary results, the percentage of Garfield students failing the math portion of the fourth-grade state achievement test last year fell to 7% from 23% in 2005. Those rated advanced or proficient rose to 43% from 40%.
Last week, a fourth-grade class at Garfield opened its lesson with Singapore's "mental math," a 10-minute warm-up requiring students to recall facts and solve computation questions without pencil and paper. "In your heads, take the denominator of the fraction three-quarters, take the next odd number that follows that number. Add to that number, the number of ounces in a cup. What is nine less than that number?" asked teacher Janis Halloran. A sea of hands shot up. (The answer: four.)
Ms. Halloran then moved on to simple pencil-and-paper algebra problems. "The sum of two numbers is 63," one problem reads. "The smaller number is half the bigger number. What is the smaller number? What is the bigger number?" (The answers: 21 and 42.)
In this class, the students didn't use the lettered variables that are so prevalent in standard algebraic equations. Instead, they arrived at answers using Cuisenaire rods, sticks of varying colors and lengths that they manipulate into patterns on the tops of their desks. The children use the rods to learn about the relationship between multiplication and geometry. The goal: a visceral and deep understanding of math concepts.
"It just makes everything easier for you," says fifth-grader Jailene Paz, 10 years old.

Cuisinaire rods for bar models!

That's so cool!

TERC time

The Singapore Math curriculum differs sharply from reform math programs, which often ask students to "discover" on their own the way to perform multiplication and division and other operations, and have come to be known as "constructivist" math.
One reform math program, "Investigations in Number, Data and Space," is used in 800 school systems and has become a lightning rod for critics. TERC, a Cambridge, Mass., nonprofit organization, developed that program, and Pearson Scott Foresman, a unit of Pearson PLC, London, distributes it to schools.

parents don't get it part 1

Ken Mayer, a spokesman for TERC, says many parents have a "misconception" that Investigations doesn't value computation. He says many school systems, such as Boston's, have seen gains in test scores using the program. "Fluency with number facts is critical," he says.

parents don't get it part 2

Polle Zellweger and her husband, Jock Mackinlay, both computer scientists, moved to Bellevue, Wash., from Palo Alto, Calif., two years ago so their two children could attend its highly regarded public schools. She and her husband grew suspicious of the school's Investigations program. This summer, they had both children take a California grade-level achievement test, and both answered only about 70% of the questions correctly. Ms. Zellweger and her husband started tutoring their children an hour a day to catch up.
"It was a really weird feeling," says their daughter, Molly Mackinlay, 15. "I do really well in school. I am getting A-pluses in math classes. Then, I take a math test from a different state, and I'm not able to finish half the questions."
Eric McDowell, who oversees Bellevue's math curriculum, says parents misunderstand Investigations.

If it weren't for the parents, teaching would be a great job.

math wars and war wars

In the Alpine School District in Utah, parent Oak Norton, an accountant, has gathered petitions from 1,000 families to protest the use of Investigations. His complaints began more than two years ago, when he discovered at a parent conference that his oldest child, then in third grade, wasn't being taught the multiplication tables.
Barry Graff, a top Alpine school administrator, says the system has added more traditional computation exercises. Over the next year, Alpine plans to give each school a choice between Investigations or a more conventional approach. Mr. Graff, who says Alpine test scores tend to be at or above state averages, expects critics to keep up the attacks and welcomes the national math council's efforts to provide grade-by-grade guidance on what children should learn.
"Other than the war in Iraq, I don't think there's anything more controversial to bring up than math," he says. "The debate will drive us eventually to be in the right place."

wow

I bet things are hopping over at math-teach & math-learn.

[pause]

hmm

No action thus far.

Once Wayne Bishop posts this baby, we'll be in a shooting war.

update: Bishop's got it!

let the fun begin

what Singapore students can do at the end of 7th grade

-- CatherineJohnson - 12 Sep 2006

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"The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals."

Indeed, Oceania has always been at war with Eastasia.

-- DougSundseth - 12 Sep 2006

"The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals."

Oh really. So why do they happily approve of so many curriculums that don't even teach some of those fundamentals?

That is some serious backtracking. About time. I was just about to go warn a friend about what her child will NOT learn with Trailblazers by the end of 5th grade. Just to be safe, I think I'll still warn her.

-- SusanS - 12 Sep 2006

I recommend reading Ken's post at D-ed Reckoning.

-- KarenA - 12 Sep 2006

Isn't it nice that NCTM is saving the world from NCTM?

-- BarryGarelick - 12 Sep 2006

Barry!

I just sent you a copy of the full text - can you get it out to folks?

-- CatherineJohnson - 12 Sep 2006

I'm waiting for math-teach to blow up.

-- CatherineJohnson - 12 Sep 2006

So, we've had, what, (grabs calculator: [2006] [+] [1989] [-] [=]) ~~16 oops!~~ 17 years of educational malpractice? Grrrr....

-- OldGrouch - 12 Sep 2006

hey, Susan - if you've got time take a look at my Comment on Ken's blog.

MATH TRAILBLAZERS is perfectly positioned to say it's already doing everything this report says should be done.

-- CatherineJohnson - 12 Sep 2006

Old Grouch

We're not hearing much in the way of an apology, are we?

Just stuff like "Parents don't understand."

-- CatherineJohnson - 12 Sep 2006

Trailblazers doesn't officially teach long division, I believe. I forgot what else it doesn't officially teach. The teachers have to stop and teach what's missing.

-- SusanS - 12 Sep 2006

You're right!

It doesn't!

Remember that whole long passage about how it's too hard to teach long division but it takes the same amount of time to teach long division AND forgiving division?

-- CatherineJohnson - 12 Sep 2006

Here it is

-- CatherineJohnson - 12 Sep 2006

A friend of mine has a daughter who made A's in 4th and 5th grade Trailblazers. She did not know her time tables, ditto division facts, and her fraction knowledge was very basic to nonexistent.

She did not know that you could divide a fraction.

This was the A student. I can't imagine what the C student looks like in this class.

Another friend's gifted kid dropped dramatically in the standardized tests knocking him out of the high math track for middle school. I told her that it was probably math facts and fractions. She found out through an online assessment that, in fact, it was.

If a teacher looks at the good stuff that curriculums like Trailblazers offer, supplementing what is missing, then it is probably alright. But when they follow it to the letter, it appears to me to be a disaster waiting to happen.

-- SusanS - 12 Sep 2006

She did not know that you could divide a fraction.

Did she have any sense visually of "friendly fraction division"?

I mean, did she have any sense that when you divide a pizza into 6 pieces that's a form of division?

Let's see....that's not right; that's a whole being divided into parts.

Did she have a sense that you can divide a half of a pizza into smaller parts - that that is division of a fraction?

-- CatherineJohnson - 13 Sep 2006

What online assessment did she use?

-- CatherineJohnson - 13 Sep 2006

I skimmed the 5th grade book. There was just about nothing in it - especially compared to a book like Saxon or Singapore.

Actually, compared to SRA there was nothing in it.

-- CatherineJohnson - 13 Sep 2006

TRAILBLAZERS is pure counting. The whole thing is "Integrated Math & Science"; hence the "TIMS."

The idea was to teach all of math as a form of statistics.

-- CatherineJohnson - 13 Sep 2006

heck

math-teach and math-learn are still in the dark

-- CatherineJohnson - 13 Sep 2006

If Barry doesn't get that article to Wayne Bishop or Jerry Becker I'm going to track them down myself!

-- CatherineJohnson - 13 Sep 2006

My mother in law gave me her copies of NTCM's Teaching Elementary Math magazines. Nearly EVERY article in these publications showcased constructivist teaching. I can't believe that NTCM is doing this much of a flip. I wonder how their faithful followers are going to take the news.

-- NicksMama - 13 Sep 2006

I'm stunned by the whole thing.

I'd love to know how this came about.

-- CatherineJohnson - 13 Sep 2006

And yes, NCTM is pure constructivism.

You've got to look at NEA's website (Barry cites it in his new article in Ed Next, which isn't online yet.)

NEA actually says that today's student doesn't do tons of problems.

Instead, today's student might be assigned one problem that takes him one or two hours to do.

-- CatherineJohnson - 13 Sep 2006

Wayne Bishop doesn't think too much of the document.

-- CatherineJohnson - 13 Sep 2006

Hey Catherine or Carolyn --

Someone needs to add "D-ed Reckoning" to the links list at the left. <-----------

-- GoogleMaster - 13 Sep 2006

I don't think too much of the document either. Think of The Who: "Meet the new boss; same as the old boss."

Wait a minute. Isn't the boss Bruce Springstein? I'm confused.

-- BarryGarelick - 13 Sep 2006

Catherine,

A teacher has created a new site based on the bar models in Singapore Math. Students can move blocks around the screen to model their math problems. Looks like whole number operations are up now. Site says ratios, fractions, etc are coming up. This looks VERY interesting!

http://www.ThinkingBlocks.com

Liz B. The publisher got right on the new NCTM guidelines,too.

-- KtmGuest - 13 Sep 2006

What online assessment did she use?

She printed out the Saxon 50 question one (ten for each level) and he fell apart at the division of fractions part, along with a couple of other spots.

I honestly am not sure what my other friend's child knows because she paused so much when I asked her basic things. I couldn't always tell. I could see her trying to hide a bit, something every math phobe learns to do well. This is where those assessments are so helpful. They just clear things up right then and there. You know what to do or you don't.

I think it's the "shifting sand" problem, also. She just doesn't have a firm enough grasp and so it just "went away" over the summer.

-- SusanS - 13 Sep 2006

Focal Points is a good step for math curriculum, but there's still a lot left unsaid. I think I see a lot of Liping Ma in the new 4th and 5th grade fraction/decimal focal point. I hope the publishers really follow through, but the key is going to be to get States to realign their curriculum frameworks. Until that happens, fuzzy state tests will continue to force fuzzy curricula into elementary schools.

-- LynnGuelzow - 13 Sep 2006

D-ed Reckoning isn't there????

aaacccckkkk!

-- CatherineJohnson - 13 Sep 2006

Google Master - thanks for the heads up. I just shot Carolyn an email so I hope she can get Ken's blog & Linda Moran's Teens & Tweens up pretty soon.

You may have noticed that Carolyn is WAY less active on the blooki of late.

That's because Carolyn has a life.

-- CatherineJohnson - 13 Sep 2006

Actually, I'm serious about that.

In case any of you missed it, Microsoft bought Carolyn's company last spring, and her job is way better, more interesting, more challenging, etc. than it was.

So most of her energies are going into her work now, which is good. Not so good for us, but very good for her.

As for me, I ought to be churning out books instead of writing a "blooki," but ktm continues to be a first priority for all kinds of reasons (i.e. structured procrastination, afterschooling, learning math in middle age, practice writing.....and there are probably others).

-- CatherineJohnson - 13 Sep 2006

One day last summer I started asking myself, Why am I doing this?

Why am I learning math in middle age?

Why am I doing math every day?

WHY?

I was feeling baffled until I realized that I may be doing this almost entirely for Christopher. I'm loving math, but I think it's unlikely I'd be doing this if I didn't believe that it's up to me to teach math to my kid.

Now my study of math has taken on a life of its own, and ktm helps me with me as much as with Christopher.

But at heart I'm a parent who has concluded that I can't count on my school district to teach math well enough that Christopher will be able easily to take math courses in college or major in a math-related field.

I'm dead serious about that.

-- CatherineJohnson - 13 Sep 2006

Joanne Jacobs has a link to KTM in her post today about the Wall Street Journal article.

-- KarenA - 13 Sep 2006

I think I see a lot of Liping Ma in the new 4th and 5th grade fraction/decimal focal point. I hope the publishers really follow through, but the key is going to be to get States to realign their curriculum frameworks. Until that happens, fuzzy state tests will continue to force fuzzy curricula into elementary schools.

I think everything will stay fuzzy until charter schools & homeschooling take over.

The ed schools are the problem.

Our new young teachers are pure constructivists. Ms. Kahl's teaching is radical constructivism, in fact.

Remember when she rolled her eyes at my friend because my friend had assumed Ms. Kahl taught the procedures for solving a problem before she assigned the problems.

Christopher's English teacher, Ms. Kozak, also young & freshly minted, told me last year that "the only kind of learning that's meaningful is learning children discover for themselves."

She didn't say this forcefully or argumentatively.

She said it casually, in passing, as if this were a Known Fact.

-- CatherineJohnson - 13 Sep 2006

hey ktm Guest

Thank you for the link!

I wonder whether something like this would be good for Andrew.

He can't use a pencil to save his life; he can't even use a mouse.

Putting blocks together won't be easy, but it's not hard for him, either.

Cool!

Thank you so much!

-- CatherineJohnson - 13 Sep 2006

ok, math-teach is heating up

Here's a post about the thinkingblocks from "Suzanne" (I think I've learned a lot from her posts in the past....assuming she's the person I'm remembering)

-- CatherineJohnson - 13 Sep 2006

hmmm...

The math warriors at math-teach are grumpy about the Cuisinaire rods.

Bishop seems to be saying that the math problems ("The sum of two numbers is 63," one problem reads. "The smaller number is half the bigger number. What is the smaller number? What is the bigger number?") isn't a Singapore-type problem.

In my experience it absolutely is; moreover, this is the type of problem that, when solved through bar models, completely transformed my understanding of elementary math.

Have to go find that post .....

-- CatherineJohnson - 13 Sep 2006

ok, let me think.....YES, that problem is CLASSIC SINGPORE MATH. Kids do zillions of them in Challenging Word Problems Book 3. I'm still trying to make sure Christopher grasps this concept absolutely.

Here's a different version of it:

-- CatherineJohnson - 13 Sep 2006

The teacher's problem is even better in its way, because you end up with one number divided into 3 equal parts.

That problem teaches the nature of division more simply; it's the "starter" problem, the one you should probably begin with before moving on to the "Dr. Johnston" problem.

-- CatherineJohnson - 13 Sep 2006

I should add that back when we first started ktm Carolyn found evidence that manipulatives don't work for grade school kids - who don't seem to be able to dissociate "toy" from "manipulative" - but do work for middle school kids working on fractions.

So I don't know that Cuisinaire rods are necessarily a good way to go about doing bar models for younger kids.

Nevertheless, I still think the idea is cool.

Don't know why.

-- CatherineJohnson - 13 Sep 2006

BUT I'll add that I've come to have some trust in my "intuitions" when they're this strong.....

Some.

-- CatherineJohnson - 13 Sep 2006

Actually, I know why I like the idea so much.

Little kids who've been given no practice-to-mastery for handwriting, and that is all kids today, can have a TERRIBLE time drawing those damn bar models.

One of my kids in Singapore Math couldn't begin to get a bar model down on paper; it was SO hard for him.

That's probably the main thing I'm liking about the Cuisinaire rods.

You're taking part of the load off of the brain, giving more over to the math.

Something about the 3-dimensionality appeals to me, too - though that's probably wrong!

-- CatherineJohnson - 13 Sep 2006

I had one kid flat-out refuse to draw any bar models at all.

I talked to his mom, and we decided to have him do what he wanted to do, which was the Brain Maths book.

I don't know whether he disliked the bar models because of the drawing element; his drawing was just fine as far as I could tell.

He may have disliked them because they were novel.

Kids hate new stuff!

-- CatherineJohnson - 13 Sep 2006

Carolyn is absolutely right. Contrary to what developers of manipulatives and the programs that use them would have us believe, children cannot distinguish manipulatives from toys. My colleagues and I have tried everything from unifix cubes to fraction burgers to teach math concepts but, in the minds of children, they are all glorified building materials.

That's why I like Thinking Blocks. Children receive the benefits of creating models onscreen without the tower and bridge building that takes place with Cuinenaire rods, base 10 blocks, etc.

It's interesting that you mentioned using Thinking Blocks with Andrew. There is interest in Thinking Blocks from folks working with students with disabilities. The LDResources site has it listed in its directory and http://www.k8accesscenter.org is reviewing the program now.

Suzanne

-- KtmGuest - 13 Sep 2006

Have you tried the Thinking Blocks?

I guess I don't know exactly what they are.

I'm not surprised to hear that kids can't separate "Lego" from "math manipulative."

That's called "environmental dependency" in neuro-speak, and the younger you are the more environmentally dependent.

That study found that middle school kids learned a lot from fraction manipulatives, which also makes sense because their frontal lobes are quite a bit more developed at that point.

-- CatherineJohnson - 13 Sep 2006

hey!

Are you the "Suzanne" from math-teach?

(Assuming I've spelled your/her name right....)

-- CatherineJohnson - 13 Sep 2006

Thanks so much for the info - I'll be interested to see what they learn.

-- CatherineJohnson - 13 Sep 2006

It's good to know about the Access Center, too - thanks!

-- CatherineJohnson - 13 Sep 2006

Here's Carolyn's post about the math manipulative study.

She was writing about the big CA review of math ed research.

This link should probably have all of the posts we've written about manipulatives. To find people's comments you have to click on the title of each post (and unfortunately we don't have a way to indicate whether there are or are not comments...)

-- CatherineJohnson - 13 Sep 2006

These are the fraction manipulatives I like (very helpful with Christopher):

I used a labeler to add equivalence labels to each tile, e.g. 1/5; 20%; .2

-- CatherineJohnson - 13 Sep 2006

One of all-time favorite books is The Berenstain Bears book, titled, The Trouble with School (or something like that). Brother Bear misses school because he is sick and therefore misses the explanation about division. As a consequence, he doesn't do his homework and doesn't ask for help. Naturally, unpleasant consequences follow.

When the truth comes out that Brother Bear didn't do his homework because he didn't understand the concept of division, Gramps takes a plate of ten cookies and gives two cookies to each of the five family members present. He then connects the dots for Brother Bear and explains that distributing ten cookies equally to five "Bears" was division.

Granted, that's an oversimplification and I would be the first to suggest that it would also be necessary to show Brother Bear HOW to do division on paper. But what I liked about the approach, especially for a young child, was that it was both a "hook" and a concrete, real-life application of a complex, abstract mathematical principle.

-- KarenA - 13 Sep 2006

Yes, I'm Suzanne from math-teach.

I have worked through a few of the practice sets in ThinkingBlocks^?. Each set contains two or three related models to prevent children from "plugging in" numbers as they do with textbook word problems. The challenge set in the multiplication and division program shows just how powerful bar models can be. I don't think many fourth or fifth grade students would have any idea how to solve the problems in that set without the models. The program deviates a bit from Singapore Math in its treatment of remainders. Also, two step addition and subtraction problems are not represented by typical bar models. However, these topics appear frequently in US texts so the publisher was probably making a compromise.

I do like the fraction tiles but in a classroom situation you have to make certain everyone is on task else it becomes a giant, colorful mosaic. When used on a one-to-one basis, manipulatives can be very worthwhile.

Suzanne (I'll register next time through)

-- KtmGuest - 13 Sep 2006

I do like the fraction tiles but in a classroom situation you have to make certain everyone is on task else it becomes a giant, colorful mosaic.

They fly all over the place just working with one kid sitting at the dining room table.

Would paper ones do better?? (only because of the higher friction...)

Doug Sundseth made several sets for us.

I'll find the link.

-- CatherineJohnson - 14 Sep 2006

fraction manipulatives to print

fraction manipulatives

fraction manipulatives in black & white

These are all pdf files easy to print out on card stock.

The beauty of these things is that you don't have to keep track of pieces. You can print out a couple of sheets per child, have extra "tiles" when some go missing, etc.

If you want to use glue or scotch tape that's fine, too, since you'll be throwing them out afterwards.

disposable manipulatives!

I'm not inside the classroom, but seeing as how we have disposable everything else, I see a market.....

-- CatherineJohnson - 14 Sep 2006

Doug also made some beautiful number line sheets.

I've used these a lot.

-- CatherineJohnson - 14 Sep 2006

Karen I got a copy of the Tomie DePaolo book you liked -

-- CatherineJohnson - 14 Sep 2006

YO CATHERINE! Check your email. We finally have a link to the article.

-- BarryGarelick - 14 Sep 2006

fantastic!!!!

I'll go look —

-- CatherineJohnson - 14 Sep 2006

WebLogForm
Title:	new new new math
TopicType:	WebLog
SubjectArea:	ConnectedMath, ConstructivistTeaching, DirectInstruction, EverydayMath, ImpCurriculum, IrvingtonMath, IrvingtonSchools, MathLand, MathWars, SaxonMath, SingaporeMath, TercInvestigations, TrailBlazers
LogDate:	200609121419

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