kitchen table math, the sequel

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From Nicholas Kristoff's column in the Times today:

What is the sum of the three consecutive whole numbers with 2n as the middle number?

A. 6n+3

B. 6n

C. 6n-1

D. 6n-3

More than three-quarters of South Korean kids answered correctly (it is B). Only 37 percent of American kids were correct, lagging their peers from Iran, Indonesia and Ghana.

~~~~~~~~~~

A piece of wood was 40 centimeters long. It was cut into 3 pieces. The lengths in centimeters are 2x -5, x +7 and x +6. What is the length of the longest piece?

Only 7 percent of American eighth graders got that one right (the answer is 15 centimeters). In contrast, 53 percent of Singaporean eighth graders answered correctly.

~~~~~~~~~~

How many degrees does a minute hand of a clock turn through from 6:20 a.m. to 8 a.m. on the same day?

A. 680 degrees

B. 600 degrees

C. 540 degrees

D. 420 degrees

Only 22 percent of American eighth-graders correctly answered B, below Palestinians, Turks and Armenians.

~~~~~~~~~~

Correlation isn't causation, but the absence of correlation is meaningful.

Fifteen years of constructivist mathematics programs adopted in virtually every public school in the country, fifteen years of teacher-training in authentic problem solving and guide-on-the-sidery, and here we are.

At a minimum, we can say that constructivist math has not been a blinding success.

~~~~~~~~~~

Back to Kristoff, I love this brain teaser for some reason:

You’re in a dungeon with two doors. One leads to escape, the other to execution. There are only two other people in the room, one of whom always tells the truth, while the other always lies. You don’t know which is which, but they know that the other always lies or tells the truth. You can ask one of them one question, but, of course, you don’t know whether you’ll be speaking to the truth-teller or the liar. So what single question can you ask one of them that will enable you to figure out which door is which and make your escape?
Are You Smarter Than an 8th Grader?

from Science Daily:

Texas A and M University researchers ... have found that not fully understanding the "equal sign" in a math problem could be a key to why U.S. students underperform their peers from other countries in math.

"About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students," note Robert M. Capraro and Mary Capraro of the Department of Teaching, Learning, and Culture at Texas A&M.

[snip]

The problem is students memorize procedures without fully understanding the mathematics, he notes.

"Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer," he explains. "So the work would look like 4+3+2=(9)+2=11.

"This response has been called a running equal sign...

[snip]

The Texas A&M researchers examined textbooks in China and the United States and found "Chinese textbooks provided the best examples for students and that even the best U.S. textbooks, those sponsored by the National Science Foundation, were lacking relational examples about the equal sign."

Students' Understanding of the Equal Sign Not Equal, Professor Says
August 11, 2010

They had me until that last line.

How do Everyday Math and Terc teach the equal sign?

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

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.'

The top earner in Westchester County and the region, Scarsdale's Michael V. McGill, earns 372,006ドル 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

cross posted to the Irvington Parents Forum

Waiting for Superman

Tue, October 12 5:10 || 7:25 || 9:40
Wed, October 13 5:10 || 7:25 || 9:40
Thu, October 14 5:10 || 7:25 || 9:40

Jacob Burns Film Center Theater
364 Manville Rd., Pleasantville, NY 10570
Info-line: 914.747.5555
914.773.7663

============

Hi everyone -

Waiting for Superman is incredible. So moving. Entertaining, too; Geoffrey Canada in particular is riveting. The title comes from a story he tells about the day his mother told him Superman doesn’t exist.

The film follows 5 or 6 children whose families are trying to find good schools for their kids, including one mom who can no longer afford the Catholic school her daughter has been attending. The lottery scenes at the end are excruciating.

Amazingly, the film does **not** give wealthy, white suburban schools a pass. About three quarters of the way in, the film tells us that suburban schools have the same underachievemement problems urban schools do; then we see data showing that the top 5% of U.S. students rank far below the top 5% of students in other countries. Which is true.

Here’s a picture of the suburban girl waiting to see if her number will be called:

Here’s the trailer.

Catherine

What about the US's better students? When asked, Schmidt replied, "For some time now, Americans have comforted themselves when confronted with bad news about their educational system by believing that our better students can compare with similar students in any country in the world. We have preferred not to believe that we were doing a consistently bad job. Instead, many have believed that the problem was all those 'other' students who do poorly in school and who we, unlike other countries, include in international tests. That simply isn't true. TIMSS has burst another myth - our best students in mathematics and science are simply not 'world class'. Even the very small percentage of students taking Advanced Placement courses are not among the world’s best."

TIMSS - Trends in International Mathematics and Science Studies

from Erin Johnson:

The 2007 TIMSS does break out the US results by race. In 4th grade the US Asian subgroup performs only slightly less than the top Asian countries but by 8th grade the US Asian subgroup performs significantly lower.

4th grade US Asians: 582
4th grade Singapore: 593

8th grade US Asians: 549
8th grade Singapore: 599

Highlights from TIMSS (pdf file)

Note that the math scores of Singapore stay the same from 4th to 8th grade, but the US Asian subpopulation (as well as the entire student population) declines.

If math performance was purely an IQ phenom, we would expect that those 4th and 8th grade levels would be comparable, but this is not the case.

Also, there is evidence from Whitehurst that math curricula does matter. So it is not inconceivable that differences between math in Asia vs the US might account for performance differences.

Anecdotally, having used Singapore math it is easy for me to see why Asian math programs would be significantly better at enabling kids to learn math.

[I]mmigrant students actually do better if they begin their American education in high school rather than in elementary or middle school.

Foreign-Born Students New to N.Y. Outshine Native Born
by Sarah Garland
New York Sun
February 11, 2008 Edition

the nativity gap

from Bostonian:

Where is the evidence, adjusting for race, that American math curricula are worse than those used in Asia?

William Schmidt, director of the U.S. national Research Center for the Third International Mathematics and Science Study, is the person to read on this subject:

The Role of Curriculum
A Coherent Curriculum: The Case of Mathematics (pdf file)

and see:
The Sequence of Mathematics Topics in Top-Achieving Countries (pdf file)


nativity gap

This Stiefel, Schwartz, and Conger study, while not an analysis of curriculum, supports Schmidt's argument:

Do Immigrants Differ from Migrants? Disentangling the Impact of Mobility on High School Completion and Performance
Leanna Stiefel, Amy Ellen Schwartz, and Dylan Conger

from the New York Sun's story:

Foreign-born newcomers to New York City's public schools are performing better than native-born newcomers, a New York University study shows.

The working paper by NYU education researchers titled "Do Immigrants Differ From Migrants?" has also deflated any notions that immigrant students tend to do worse in American schools the older they are when they arrive. On the contrary, the findings demonstrated that immigrant students actually do better if they begin their American education in high school rather than in elementary or middle school.

[snip]

Ms. Schwartz, speaking at an NYU symposium last week, said she and her colleagues had hypothesized that immigrant students "who come late are the ones who are really disadvantaged," guessing that their lack of language skills, the stress of moving to a new country, and institutional differences between the schools they came from and the New York schools might hurt their graduation rate and performance on tests.

Instead, they found that the foreign born "just do remarkably better," Ms. Schwartz said.

Foreign-Born Students New to N.Y. Outshine Native Born
by Sarah Garland
New York Sun
February 11, 2008 Edition

Here is Andrew Wolf's take:

Immigrant children outperform some native-born children in New York schools, my colleague Sarah Garland reported the other day. Indeed, it seems the longer newly-arrived children attend our schools, the worse they do. These conclusions come from a new study, "Do Immigrants Differ From Migrants?"

"The foreign born are whizzing by the native born at every level," one of the researchers, Amy Ellen Schwartz, said.

[snip]

If one drives past the Bronx High School of Science any given school day morning, one will encounter a seemingly endless caravan of yellow school buses. Most of these buses come from Queens, the borough known for its immigrant population. Sixty percent of those recently offered a spot at Bronx Science as a result of their test scores come from Queens, as are 40% of the new freshmen at Stuyvesant. It [sic] it turns out that a good number of these commuter students are recent immigrants, why do the city's specialized high schools seem to have such a disproportionate number of newly arrived students?

Whizzing By
BY ANDREW WOLF
March 3, 2008

I miss the Sun.

America’s future math teachers, on average, earned a C on a new test comparing their skills with their counterparts in 15 other countries, significantly outscoring college students in the Philippines and Chile but placing far below those in educationally advanced nations like Singapore and Taiwan.

The researchers who led the math study in this country, to be released in Washington on Thursday, judged the results acceptable if not encouraging for America’s future elementary teachers. But they called them disturbing for American students heading to careers in middle schools, who were outscored by students in Germany, Poland, the Russian Federation, Singapore, Switzerland and Taiwan.

On average, 80 percent to 100 percent of the future middle school teachers from the highest-achieving countries took advanced courses like linear algebra and calculus, while only 50 percent to 60 percent of their counterparts in the United States took those courses, the study said.

U.S. Falls Short in Measure of Future Math Teachers
by Sam Dillon

Here's the report: The Preparation Gap: Teacher Education for Middle School Mathematics in Six Countries (pdf file)


NCTM response

“There are so many people who bash our teachers’ math knowledge that to be honest these results are better than what a lot of people might expect,” said Hank Kepner, professor of mathematics education at the University of Wisconsin, Milwaukee, who is president of the National Council of Teachers of Mathematics. “We show up pretty well here, right in the middle of the pack.”

NCTM motto: Good enough is good enough.





I could do this problem, which is a good thing, given that I've completed ALEKS geometry.

On the other hand, I had to think about it -- and I don't remember having learned that the angle bisector also bisects the opposite side in a parallelogram.

Take the guideline that most college-bound students should proceed through the K-12 math curriculum--a hierarchy of classes stepping from simple arithmetic to calculus--at a pace that lands them in algebra in ninth grade. This is too easy for most children. In a regular middle school two of my children attended, all of the students took algebra in or before eighth grade and over 80 percent mastered it, putting them one or more years ahead in math.

Entire countries of students accomplish this routinely. The Third International Mathematics and Science Study, conducted in 1996, found that the material taught in U.S. eighth-grade math classes was taught in the seventh grade in many other developed countries and even earlier in Japan and Germany. As a result, U.S. eighth graders performed significantly poorer on a standardized math test than eighth graders in twenty other countries, and far poorer than Japanese students, who scored highest. Overall, U.S. elementary and middle school math education lags a full year behind that in dozens of countries and one and a half years behind Japan and Germany.

Math Coach by Wayne Wickelgren, p. 4

comment thread here

I could write a book on this topic. My AP calculus students are for the most part exceptional math students. Many have never been taught to think. The emphasis in all their classes has been on scoring well on the Regents exam. So much emphasis is put on calculator use that some have forgotten basic arithmetic. One of my students has been in my class in 9th grade and I remember her doing all her calculations by hand with no difficulty. It took me months to wean her away from her calculator dependency.

Good students will be able to pass on their own, but they need guidance as much or even more than the weak ones. By ignoring this group, we are creating a bunch of intellectual idiots.

pissed off teacher

repeat, repeat

Good students will be able to pass on their own, but they need guidance as much or even more than the weak ones.

For Americans, this notion -- that good students need teaching as much or more than weak students -- is sooooo counterintuitive.

I don't say this as a criticism of the U.S., particularly. It's just the way things are. I first learned that other cultures don't think about talent & achievement the same way we do reading Stevenson & Stigler's The Learning Gap.

The cultural difference between American and Asian cultures on this question is so significant that the National Mathematics Advisory Panel report addressed it in its Fact Sheet:

Student Effort Is Important

Much of the public's "resignation" about mathematics education is based on the erroneous idea that success comes from inherent talent or ability in mathematics, not effort. A focus on the importance of effort in mathematics learning will improve outcomes. If children believe that their efforts to learn make them "smarter," they show greater persistence in mathematics learning.

Here's a study by David Uttal:

Abstract The poor mathematics performance of children in the United States has become a topic of national concern. Numerous studies have shown that American children consistently perform worse than their counterparts in many parts of the world. In contrast, children in China, Japan, Taiwan, and other Asian countries consistently perform at or near the top in international comparisons. This paper examines possible causes of the poor performance of American children and the excellent performance of Asian children. Contrary to the beliefs of many Americans, the East Asian advantage in mathematics is probably not due to a genetically-based advantage in mathematics. Instead, differences in beliefs about the role of genetics may be partly responsible. Asians strongly believe that effort plays a key role in determining a child's level of achievement, whereas Americans believe that innate ability is most important. In addition, despite the relatively poor performance of their children, American parents are substantially more satisfied with their children's performance than Asian parents. The American emphasis on the role of innate ability may have several consequences for children's achievement. For example, it may lead children to fear making errors and to expend less effort on mathematics than their Asian counterparts. As research on genetic influences on behavior, traits, and abilities increases scientists should be careful to ensure that the public understands that genetics does not directly determine the exact level of a child's potential achievement.

Beliefs about genetic influences on mathematics achievement: a cross-cultural comparison
Genetica November 02, 2004

rich schools & biology

This leads me to a topic I have meaning to get posted for months: Richard Elmore has found that wealthy schools are particularly committed to biological explanations of student performance:

In more affluent communities, I also found that variations in student performance were frequently taken for granted. Instead of being seen as a challenge to the teachers’ practice, these differences were used to classify students as more or less talented. Access to high-level courses was intentionally limited, reinforcing the view that talent, not instruction, was the basis of student achievement.

What (so-called) low-performing schools can teach (so-called) high-performing schools
by Richard Elmore
National Staff Development Council VOL. 27, NO. 2 SPRING 2006 schools by

Ditto that.

My all-time favorite experience of this phenomenon (I've had many) was the day Ed and I met with the Earth science teacher and the chair of the science department discuss C's erratic grades in the class, which ranged from A to F.

Their explanation: "C. can't think inferentially."

Unfortunately I wasn't quick enough on my feet to ask why it was he could think inferentially on A & B days but not on C, D, & F days.

Probably because I am a real American.


and see: Carol Dweck: The Secret to Raising Smart Kids

from Paul H:

It's instructive to put constructivism in context. I put it smack dab in the middle of; Spiraling Curricula, Curricula Bloat, ** Constructivism **, Inclusion/Immersion, and Grade Level Placement. I'll call these the five horsemen of the apocalypse.

Horse #1, Spiraling Curricula (the Trojan horse), lays the natural hierarchy of a subject on its side, preferring to teach everything in a strand as a set of increasingly complex parallel universes that never need to be mastered. We spend, on average, 6 years on concepts that other countries dispense with in 3.

Once you install horse #1 you observe that, without mastery, every stovepipe in the spiral is easier to achieve success in (because you don't really measure it anymore). This leads to the evolution of horse #2, Curricula Bloat, where you get to fill your newly invented extra time with bloated concept development (2-3 times more concepts per year than is common in the TIMMS countries that surpass us).

Horses #3 and #4, Inclusion/Immersion and Grade Level Placement, differ in motivation but produce the same noxious result, an enormous range of student capabilities in a single classroom, with an attendant reduction in the number and type of teachers that address them. And finally with horses 1,2,4, and 5 teamed up you're ready for the lead horse, horse #3,Constructivism.

With an exquisitely complex spiral, delivering an overwhelming number of concepts to a highly diverse population grouped by virtue of their hat size, there is no other choice but to have the kids teach each other. Horse #3 is inevitable.

Every argument you hear for constructivist philosophy is no more than rationalization, devised to make palatable, the misbegotten notion that kids can teach themselves the things that mankind learned over thousands of years, driven by the need to address the maelstrom created by the four horsemen that accompany it. Once you figure out where it comes from you're better able to appreciate why it's so popular. Unfortunately, it's very hard to change the direction of the lead horse when the rest of the team is not cooperating and these horses all tend to be discussed in isolation, where they can be made to sound plausible. Together though, they are undeniably toxic.

And yes I know there are only four horsemen but I'm invoking my 21st century skills to invent my own literary reality.

from Flypaper:

I would point out that Minnesota showed dramatic gains on TIMMS not because of “new, more rigorous standards,” but because of that state’s decision to implement a coherent grade-by-grade curriculum in mathematics. William Schmidt took the lead in developing that curriculum (pdf file) and deserves to bask in glory for what he has done for the children of Minnesota. That is the most important lesson of 2007 TIMSS for the United States.

I'd love to hear more about this.

TIMSS has been released.
View a quick chart here: Average mathematics scores of fourth- and eighth-grade students, by country: 2007

Singapore moves to #2 at fourth grade and #3 at eighth grade in Mathematics. The U. S. improves. (Hello reform math?)

Singapore stays #1 for Science.

When looking at the trends in average mathematics scores of fourth grade students from 1995 to 2007, the U.S. score increases by 11 and Singapore by 9 . At the eighth grade, the U.S. improves by 16 and Singapore drops by 16.

Big jump by England too!

More analysis later.

2007 TIMSS results will be released tomorrow. It will be interesting to see if Singapore students remain at the top. In 2004, Singapore implemented the "Teach Less, Learn More" initiative. From the MOE website:

"TLLM would mean less dependence on rote learning, repetitive tests and a ‘one size fits all’ type of instruction, and more on experiential discovery, engaged learning, differentiated teaching, the learning of life-long skills, and the building of character through innovative and effective teaching approaches and strategies."

Sounds a lot like some American mathematics programs.

The Singapore based materials used in the United States haven't been used in Singapore since 2001 and this will be the first TIMSS to truly gauge the effectiveness of the newer materials.

My guess is that Singapore will continue to be a mathematical powerhouse. There is so much more behind their success than bar model drawing and books.

Houghton Mifflin's Great Source division may be very eager for these results. They have creatied an Americanized version of the materials currently in use in Singapore that is expected to be available next fall. Tomorrow's results may impact the perception of their program.

Take a TIMSS test!

I was pondering the disconnect between two very recent "news items".

On the one hand, everyone is gushing about the improvements in math scores on the 2007 National Assessment of Educational Progress. I'm reading about it everywhere, it seems. On the other hand, the U.S. is sitting out the Advanced TIMSS which is designed to show how our advanced students compare to those in other countries. Strangely enough, I'm NOT reading about that everywhere.

It seems we've managed to raise the floor (every so slightly) while letting the ceiling come crashing down. I guess that puts "good enough" somewhere in the middle.

Apparently, the goal is mediocrity. Based on those parameters, I'd say we're right on track.

Cross-posted at Mindless Math Mutterings.