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Revolutionizing Math at the School of the Future

(Cross-posted at Out in Left Field)

A front page article in Monday's Local News section of the Philadelphia Inquirer profiles a math class at Philadelphia's Microsoft-funded High School of the Future, whose teacher, Thomas Gaffey, placed second in Microsoft's U.S. Innovative Education Forum and was a semi-finalist in its Worldwide Innovative Education Forum. In Gaffey's ninth-grade algebra class there are:

No textbooks, no paper, no chalk, no desks, and no assigned seats.

Instead, students use laptops while sitting in rolling chairs at trapezoidal tables spaced out in hexagonal classrooms.

Just how newsworthy this sounds to you depends on whether you think chair mobility and table shape have a big influence on learning, on whether you've been following current trends in education over the last 50 years, and on how unusual you think it is for a teacher to "encourage his students to find answers to their own questions" and engage with them in exchanges like these:

"Is this an obtuse triangle?" one student asks.

"Well, what can you tell me about an obtuse triangle?" Gaffey replies.

"One of the angles has to be more than 90 degrees," the student answers.

"Are any of the angles here like that?"

"Yeah. Oh, I get it now!"

As the Inquirer explains:

This snippet of student-driven discussion is a glimpse of the style and approach that have earned Gaffey national and international recognition.

Student-driven? Who's asking most of the questions? But I'm splitting hairs here. What I should be asking is: Why does this kind of exchange warrant international recognition?

To fair, it wasn't this, specifically, that earned Gaffey his honors. Rather:

Les Foltos, one of the judges who reviewed Gaffey's work, was impressed by his emphasis on "actively engaging students in solving real-world problems." As Gaffey puts it, "If we want to teach math to learners, we should teach math how it is actually used. It doesn't matter how much you know. It matters what you can do."

Ah yes, "real world problems." Again, only if you've been out of touch with the last half century of educational reform, and with today's Reform Math in particular, will this strike you as revolutionary. Here is Gaffy's version of real world math:

In his classroom on a recent Tuesday, Gaffey's challenge to his "learners" - as students in the Parkside public school are called - was to estimate Earth's land area.

To solve the problem, the class first covered basic concepts about area and polygons - shapes with three or more straight sides.

Gaffey then asked, "If a shape has four sides, is it always a polygon?"

Learners who answered yes (the wrong answer) were asked to redefine what a polygon is, while those who answered no were asked to draw a four-sided shape that was not a polygon on the class "smart board."

Gaffey drew a shape with three straight sides and one curved side.

"Is this a polygon?" he asked.

"No," the class responded.
...

The class drew lines through each of the continents, chopping them up into complex polygons, then simple polygons.

The final phase was to derive formulas for the areas of the simple polygons, and add up the areas.

This sort of problem is not particularly new, as a quick survey through now-standard textbooks like Everyday Math and the Interactive Math Program makes clear. And it's been around long enough to have garnered some serious criticism--specifically in what Barry Garelick calls its "just in time" approach to teaching.

Among other things, "just in time" often means serious delay. For example, one would hope that students would already know the formulas for the areas of simple polygons, and how to derive them, well before they hit 9th grade.

But because so many students are so far behind where they should be, there is one thing in which I and Gaffey are in whole-hearted agreement. In Gaffey's words, as cited by the Inquirer:

"Math education, more than any other subject, is in need of drastic reform."

Your Child Left Behind

I think Mark Roulo left a link to the new article on Hanushek's research in the Atlantic.

I've posted a list of pulls on the Irvington Parents Forum.

the real world

Speaking of ditching the daily lesson plan, Allison wrote:

For those so enamored with preparing kids for the real world, why do they want school at all? Wouldn't child labor better? That would integrate everything.

Let's not and say we did, part 2

Ditch the daily lesson plan

If you think about the “real world” that we’re preparing kids for, how often is the “real world” day broken up into science moments, math moments, writing moments, etc?

I don't know about you, but for me the "real world" day gets broken up into science moments, math moments, writing moments, etc. any time I happen to be doing science, math, writing, and 'etc' all on the same day.

Say I'm studying math for the SAT.

I just study math. Without any science or writing at all. (Also, if I can find a decent worksheet, I do a worksheet.)

Or say I'm writing a book proposal or an article for the local paper.

I just write!

I don't do any math or science to speak of, unless I happen to be writing about math or science. And even then, I don't do math or science. Writing about math or science isn't the same thing as doing math or science.

...science moments, math moments, writing moments, etc? We engage all of these things at all times.

No we don't.

Also, it’s not like integrated units are anything innovative…

True.

Kids don’t need a six-week unit on mastering quotation marks; they need to learn to master the quotation marks piece in the screenplay they write collaboratively about the people of Iceland solving problems around a catastrophic tectonic event that includes the gathering and analysis of quantitative data.

oh, man

Speaking as a person who is finishing up a semester teaching English composition to college freshmen, I would have a very hard time convincing my students that what they really need isn't to learn when and where to use a comma but to write collaboratively about the people of Iceland solving problems around a catastrophic tectonic event that includes the gathering and analysis of quantitative data.

I'd get some stony looks on that one.

Real stony.

let's not and say we did
let's not and say we did, part 2
let's not and say we did, part 3

WordSmart Software

Does anyone have any comments on a product called "WordSmart High School Excellence"? I got a call out of the blue based on something my son filled out at school, although he doesn't know what that was. The implication is that this product is recommended by our school and that we can get a big discount because of that. To get the discount, they have to arrange a time to call me back. I'm trying to find out whether his high school really recommends the product or not. When I read the guidance dept. info on the school site, they talk of putting together an IEP for each child and that the parent plays an integral role. Right now, I don't feel integral.

Race to the Average

Our state will get Race To The Top money, so our town is putting together a plan that is based on the state test. The goal is:

"90% of students entering the 4th and 8th grades will be proficient in reading and math on our state assessment"

This is really a Race To The Average. Do most parents think that state proficiency levels are good enough for their own kids? I don't think so, but do they think the money will help their kids? I haven't seen our (58 page!) proposal yet, but I can't imagine that there is anything more than a guess and check approach to increasing the numbers.

wrong again

a letter to the Times:

Middle-class American children attending well-financed schools outscore nearly all other countries. But our overall scores are unspectacular because we have such a high percentage of children living in poverty.

Rich schools are good schools: the very assumption that led me to overspend on a house in an overspending town.

Maybe I should write a letter to the editor.

Here are Hanushek, Peterson, and Woessmann:

White students. The overall news is sobering. Some might try to comfort themselves by saying the [achievement] problem is limited to large numbers of students from immigrant families, or to African American students and others who have suffered from discrimination....

...[L]et us consider the performance of white students for whom the case of discrimination cannot easily be made. Twenty-four countries have a larger percentage of highly accomplished students than the 8 percent achieving at that level among the U.S. white student population in the Class of 2009. Looking at just white students places the U.S. at a level equivalent to what all students are achieving in the United Kingdom, Hungary, and Poland. Seven percent of California’s white students are advanced, roughly the percentage for all Lithuanian students.

Children of parents with college degrees. Another possibility is that schools help students reach levels of high accomplishment if parents are providing the necessary support. To explore this possibility, we assumed that students who reported that at least one parent had graduated from college were likely to be given the kind of support that is needed for many to reach high levels of achievement. Approximately 45 percent of all U.S. students reported that at least one parent had a college degree.

The portion of students in the Class of 2009 with a college-graduate parent who are performing at the advanced level is 10.3 percent. When compared to all students in the other PISA countries, this advantaged segment of the U.S. population was outranked by students in 16 other countries. Nine percent of Illinois students with a college-educated parent scored at the advanced level, a percentage comparable to all students in France and the United Kingdom. The percentage of highly accomplished students from college-educated families in Rhode Island is just short of 6 percent, the same percentage for all students in Spain, Italy, and Latvia.

The Previous Rosy Gloss

Many casual observers may be surprised by our findings, as two previous, highly publicized studies have suggested that—even though improvement was possible—the U.S. was doing all right. This was the picture from two reports issued by Gary Phillips of the American Institutes for Research, who compared the average performance in math of 8th-grade students in each of the 50 states with the average scores of 8th-grade students in other countries. These comparisons used methods that are similar to ours to relate 2007 NAEP performance for U.S. students to both TIMSS 2003 and TIMSS 2007. His findings are more favorable to the United States than those shown by our analyses. While our study using the PISA data shows U.S. student performance in math to be below 30 other countries, Phillips found the average U.S. student to be performing better than all but 14 other countries in his 2007 report and all but 8 countries in his 2009 report. (Oddly, the 2007 report takes a much more buoyant perspective than the 2009 report, though the data suggest otherwise.) Phillips also finds that individual states do much better vis-à-vis other countries than we report.

Why do two studies that seem to be employing generally similar methodologies produce such strikingly different results?

The answer to that puzzle is actually quite simple and has little to do with the fact that Phillips compares average student performance while our study focuses on advanced students: many OECD countries, including those that had a high percentage of high-achieving students, participated in PISA 2006 (upon which our analysis is based) but did not participate in either TIMSS 2003 or TIMSS 2007, the two surveys included in the Phillips studies. In fact, 19 countries that outscored the U.S. on the PISA 2006 test did not participate in TIMSS 2003, and 22 higher-scoring countries did not participate in TIMSS 2007. As a report by the U.S. National Center for Education Statistics has explained, “Differences in the set of countries that participate in an assessment can affect how well the United States appears to do internationally when results are released.”

Put starkly, if one drops from a survey countries such as Canada, Denmark, Finland, France, Germany, and New Zealand, and includes instead such countries as Botswana, Ghana, Iran, and Lebanon, the average international performance will drop, and the United States will look better relative to the countries with which it is being compared.

Teaching Math to the Talented
Eric A. Hanushek, Paul E. Peterson and Ludger Woessmann
Winter 2011 / Vol. 11, No. 1

what does Google want?

A friend of mine (ok, it was Debbie S.) talked to a person at Google about what they look for in job candidates.

He gave her a typical problem a candidate might be asked to solve during the interview. 

As I recall, it was a permutation problem.

A hard one.

So here's Tom Friedman on the three basic skills:

There are three basic skills that students need if they want to thrive in a knowledge economy: the ability to do critical thinking and problem-solving; the ability to communicate effectively; and the ability to collaborate.

Teaching for America
By THOMAS L. FRIEDMAN
Published: November 20, 2010

I guess Google-level math knowledge falls under 'problem-solving.'

equation

Economists' Grail: A Post-Crash Model
By Mark Whitehouse
Wall Street Journal November 10, 2010

I love this.

24%

Recently released data from ACT shows that only 24 percent of high school seniors knew enough in four subjects — math, reading, science and English — to do college-level work.


No More A’s for Good Behavior
by Peg Tyre
Published: November 27, 2010
New York Times

Christmas!

Aaaaw to Zzzzzd: The Words of Birds: North America, Britain, and Northern Europe

The First English Dictionary of Slang, 1699

The Classical Tradition

Six Books of Euclid

GPA Calculations

I was looking at the way my son's school calculates GPA and weighted GPA. Then I looked at how other schools calculate them. Since I was looking at high school web sites, I was only partially successful. In any case, many of them seem to be math-challenged. Even when I searched the web about how colleges calculate GPA (or not), I met with little success. We are told that colleges generally like to recalculate their own GPA. Why? If they have the numeric percentage grades, why do they have to sort them into GPA buckets (e.g. 83-87 is a 'B' = 3.0) and then come up with a new number that contains less information than the percentage grade. Perhaps some high schools send out only GPA bucket scores for each class, but do colleges know what the buckets are?

What I find more interesting are discussions that talk about how colleges like to use unweighted GPA scores. There is some sort of advantage to an 'A' in a regular course over a 'B' in an honors course. Maybe. I think it depends on whether you make it past an initial cut. Someone told me once that colleges use unweighted GPAs for the initial cut. That seems backwards to me. You want to use a weighted GPA (and rank and SAT and etc.) to make the initial cut, but then compare applicants based on unweighted GPA. Of course, they must first strip off all of the non-academic courses.

There is the Ivy League "Academic Index", which uses something called the "Converted Rank Score" (CRS) along with your SAT scores to come up with a number, that presumably, is used to determine whether you make it past a cutoff. The CRS number is based on your class rank and the size of the graduating class. Since most high schools rank based on a weighted GPA (including all fluff courses), this would indicate that class rank matters more than the unweighted GPA of core academic courses. That seems to be a poor game to play. If you are on the bubble, perhaps you might want to look at another college. I told my son that it might be nice to look at his GPA, but he should just try to get the best grades on his core academic courses. This came up because some kids at his high school are getting weird about GPA scores.

I do, however, tell him not to give away free points. For example, in science, they had a grade for whether they covered their books or not. I showed my son that if he didn't do it, his overall grade for the class would drop 3 points. I told him that it was like finding free money. Then there are the things you have no control over, like group grades. There was also the essay where he got an 80, but that was the second highest grade. Both of those things dropped his quarter grade by 5 points. There is not much you can do if a school has two different grading rules for the same course and you stuck with the wrong teacher. Things might average out in the end, but it's difficult while the student is going through it.

on beyond zebra

A friend sent me a link to this story last night:

"Infinite" is not a word you expect to find in a report on municipal spending. It's more of a science fiction–type term — Tremble, Earthling, before the infinite might of Galaxor! But there it was, in a recent report on San Francisco's finances: Spending on the city's employee retirement system in the past decade had grown at an "infinite" rate.

Naturally, that's an exaggeration. If you do the math, the city's retirement costs for employees in the past 10 years actually grew only 66,733 percent.

Still, you might call that a Galaxor-sized number.

In fiscal year 1999-2000, the city spent about $300,000 on its retirement system. In fiscal year 2009-10, it was $200.5 million. Benefits alone — not salaries, just benefits — for current and retired employees this year are budgeted at $993 million. Spending on retirees' health care and pensions is conservatively projected to triple within five years.

And after that? Infinite.

Let It Bleed
By Joe Eskenazi and Benjamin Wachs
Wednesday, Oct 20 2010
San Francisco Weekly News

I actually didn't even know there was such a thing as an infinite rate.

Warren Buffet on insuring local and state bonds...

Some quotes from this letter sent out in 2008 are rather intriguing. All of this is addressed to the stockholder's view of course, who has a different interest from the local communities that decide to insure their bonds. But it seems that Warren Buffet has some insights into the finances of many public budgets...

"Early in 2008, we activated Berkshire Hathaway Assurance Company (“BHAC”) as an insurer of the tax-exempt bonds issued by states, cities and other local entities....BHAC has become not only the insurer of preference, but in many cases the sole insurer acceptable to bondholders. Nevertheless, we remain very cautious about the business we write and regard it as far from a sure thing that this insurance will ultimately be profitable for us...


The rationale behind very low premium rates for insuring tax-exempts has been that defaults have historically been few. But that record largely reflects the experience of entities that issued uninsured bonds. Insurance of tax-exempt bonds didn’t exist before 1971, and even after that most bonds remained uninsured.

A universe of tax-exempts fully covered by insurance would be certain to have a somewhat different loss experience from a group of uninsured, but otherwise similar bonds, the only question being how different. To understand why, let’s go back to 1975 when New York City was on the edge of bankruptcy. At the time its bonds – virtually all uninsured – were heavily held by the city’s wealthier residents as well as by New York banks and other institutions. These local bondholders deeply desired to solve the city’s fiscal problems. So before long, concessions and cooperation from a host of involved constituencies produced a solution. Without one, it was apparent to all that New York’s citizens and businesses would have experienced widespread and severe financial losses from their bond holdings.

Now, imagine that all of the city’s bonds had instead been insured by Berkshire. Would similar belt-tightening, tax increases, labor concessions, etc. have been forthcoming? Of course not. At a minimum, Berkshire would have been asked to “share” in the required sacrifices. And, considering our deep pockets, the required contribution would most certainly have been substantial.

Local governments are going to face far tougher fiscal problems in the future than they have to date. The pension liabilities I talked about in last year’s report will be a huge contributor to these woes. Many cities and states were surely horrified when they inspected the status of their funding at year-end 2008. The gap between assets and a realistic actuarial valuation of present liabilities is simply staggering.

When faced with large revenue shortfalls, communities that have all of their bonds insured will be more prone to develop “solutions” less favorable to bondholders than those communities that have uninsured bonds held by local banks and residents. "

Page 13 of Berkshire Hathaway's Letter to Stockholders

(p.s. don't ignore the first page.)

money tree

Irvington School District's Financial Challenges

and: a new administrator at no extra cost to the district

from the SMO junior exam

(The competition is meant for 14 to 16 year olds.)

This was from the 2004 exam:
Find the number of digits in N where N is the product of all positive divisors of 100,000,000. (Note: For any positive integer A, we regard A itself as one of the divisors.)

Another one:
The number A=200420052006...2040 is formed when one puts the consecutive integers from 2004 to 2040 together. What is the remainder when A is divided by 9?


---

solution discussion for the first one:

Divisors come in pairs -- every divisor x, there must be a divisor n/x, and their product is n. (But if the divisor is a square root of n, then it is the same as the other divisor and it is only counted once.) How many pairs are there?

For n=10 there are two pairs (10-1 5-2), and we note the product is 10^2 = 100.
For n=100 there are five pairs (100-1, 25-4, 20-5, 10-10, 50-2) but one pair is degenerate and the product is 100^4 * 10 (=10^9).
For n=1000, there are eight pairs. Divisor product = 1000^8.
For n=10^8, the script I wrote in Mathematica tells me there are 81 divisors and the divisor product is 325.

The number of divisors for 10^n has an interesting trend, too. It seems to be (n+1)^2.

comments

I don't know what the heck has happened to the Comments sections or who ShirleyFenette2010 may or may not be.

What is a comment from whiteboydancefloor doing at kitchen table math?

déjà vu

Two parents discover Everyday Math:

As it turns out, our school district is using a controversial math curriculum called Everyday Mathematics, also known as "Reform Math." EM, as Everyday Mathematics is referred to by teachers, was developed by the University of Chicago, and according to their website, it is in use by about three million students nationwide. Here is one example of how simple addition "can" be performed using EM:

An example of what EM calls the "lattice method" for performing multiplication:

What becomes immediately clear is that several extra steps are now necessary to accomplish simple beeline computations. More steps will result in more errors -- only an idiot would claim otherwise. Eventually, EM students are taught four ways to add, five ways to subtract, four ways to multiply, and two ways to divide (traditional long division has been eschewed completely). Rote memorization is de-emphasized, and calculators (as well as estimating) are introduced in grade two.

Here is the basic rationale behind EM, directly from the University of Chicago website:

Research has shown that teaching the standard U.S. algorithms fails with large numbers of children, and that alternative algorithms are often easier for children to understand and learn. For this reason, Everyday Mathematics introduces children to a variety of alternative procedures in addition to the customary algorithms.

Links to or excerpts of said research are not provided -- we are to simply take these statements as fact. EM further claims to "make mathematics accessible to all students" by:

Incorporating individual, partner, and small group activities that make it possible for teachers to provide individualized feedback and assistance.

Encouraging risk-taking by establishing a learning environment that respects multiple problem solving strategies.

This couple is politically conservative, and as a result, the one thing they've got wrong is the idea that  liberal parents like this stuff when they don't. Not for the most part.

This passage took me aback:

What's worse, the methods purportedly being used to convince school boards to adopt EM reek suspiciously of Rules for Radicals: ^*

State that the traditional approach hasn't worked

Disparage testimony from those against the adoption as ideological and politically-motivated arguments

State that the success of any program depends on the teacher

Bring in teachers from affluent school districts as witnesses

Bring in a witness from a university

Never thought of these tactics in terms of Saul Alinsky.

Sheesh.


^* Barry's article!

On top

The top earner in Westchester County and the region, Scarsdale's Michael V. McGill, earns $372,006 in total compensation.
Source: Demand for quality school superintendents fuels high salaries

Top-earner Scarsdale superintendent Michael V. McGill on our top students:

If you listen to people like Richard Elmore, who’s a teacher at Harvard, he says the very top top American kids are scoring about the 75th percentile on international studies. So we know our top performing kids are doing very well.
Source: A bully pulpit for the Superintendent of the Year | lohud blogs

Our very top students should be performing on par with Europe and Asia's very top students.

If they're not, then our very top superintendents are overpaid.

Superintendent salaries