kitchen table math, the sequel: 4/27/08 - 5/4/08

Saturday, May 3, 2008


While talking with a student about politics after economics class:

"Who are you going to vote for this year? Obama, McCain or Hillary?"
"Actually ... I can't vote cuz I'm not a citizen."
"If it's any comfort, I would have leant towards more libertarian candidates like Gravel or Paul."


Hope people won't mind me making a slightly off-topic, tangentially-related-to-education post here. :-)

After weathering the oh-so-relaxing process of getting a financial aid package two days before the May 1st deposit deadline (because my expired green card nearly derailed the college application process), I've come to ponder naturalisation to become a US citizen. The tricky thing is that the government of my birth country (Singapore) doesn't allow dual citizenship. ("We have yet to reach the stage of nationhood where a Singaporean with a second citizenship would still retain his identity and loyalty to Singapore as his homeland wherever he goes, with his second citizenship being only of secondary importance," a particularly hurtful government officer once wrote.)

I have been contemplating the question, "Well if I did naturalise, how would the Singapore government know anyway?" I feel strongly attached to both nations, and for a while I thought the current arrangement was satisfactory, allowing me to live in both places. This year's close shave has made me think otherwise.

If I do naturalise, I would end all risk of future deportation and in general stop being treated like a second-class citizen (well, technically *not* a citizen, but I digress. I wouldn't have to renew my green card every ten years and pay exorbitant renewal fees (which is why my mother took so long to renew our cards). Being able to vote at least once in my lifetime is a plus, for my birth country's elections are at the present moment, a joke. Being able to run for office is a bigger plus. Not being denied job opportunities in the public sector especially in sensitive areas of government might be a good thing. (The FBI once offered my mother an attractive position ... only to find out she wasn't a citizen.) If I ever get drafted I won't be prevented from becoming an officer. Finally, access to all the fun things that citizens enjoy.

On the other hand, I could risk the possibility of losing my Singaporean citizenship. Political reform in my birth country is a big thing for me, and it would be nice not being disqualified from participating in the political process there. One day the Opposition after all might gain enough strength to pass laws allowing dual-citizenship in Parliament. There is also the small issue of being able to visit my extended family, revisit my friends, eat the cuisine, reminisce about childhood, and all the fun things about coming back to your place of birth.

I am wondering if anyone would possibly have any idea if the US government would inform my birth country of my act?

If there's a significant chance to the otherwise, do you all think I could also naturalise in time to cast a ballot for oh, I don't know, Obama?

Friday, May 2, 2008

Allison on "the naturals"

The "naturals" will be fine, I think. Kids with natural math talent can come from behind. I could be too sanguine about this, of course...
You are. The kids with natural math talent who are not utter prodigies DO NOT come from behind at a school like Harvard, MIT, Caltech in the math or sciences. They are completely outclassed by the Russians, Czechs, Estonians, Koreans, Japanese, Singaporeans, etc. In physics at MIT, the Russian kids were an order of magnitude ahead of the brightest American math kid in physics. In math, it was the same.

What American kids have going for them is an escape hatch: the "naturals" in math can more easily go into Investment Banking and other places that math skills are wanted than the foreigners, such as the Russians can, it seems (because of a lack of H1 B visas, maybe?)

But the only thing keeping more of the best math and science foreign kids out of MIT and the Ivies are restrictions on percentages of foreign students.

In grad school, it's almost a lost cause. There are virtually no Americans in the top programs, and white American men are almost unheard of. They have to have been the real prodigy (skipped high school, or college at 15, etc. and never fell off the train of perfection) to get there.

meanwhile back on planet earth

I have concluded, belatedly, that pundits and policy wonks Have. No. Idea.

Last year females attained 58% of 4year college degrees and males 42% - the ratios for african american and latino females are more than 2 to 1.The causes are unknown and under researched. We have indicators but scant hard evidence. One symptom was released thursday by the National Assessment - NAEP. 41% of the eigth grade girls compared to 20 % of the boys were proficient in writing! Can this be explained by patterns of instructional interactions between teachers and boys? I doubt it.

The College Puzzle

And that's that. He doubts it. It's a big mystery!

Couldn't this expert maybe have Googled the whole issue before deciding school couldn't possibly have anything to do with the fact that boys aren't learning in school?

Speaking as a person who has actually had boys in school for lo these many years, I have no problem imagining that patterns of instructional interactions between teachers and boys might be just the tiniest bit non-optimal for boys. Especially seeing as how homeschooled boys seem to be doing just fine.

why we're not going to be getting more guys teaching school any time soon
In this paper, we compare subjective principal assessments of teachers to the traditional determinants of teacher compensation – education and experience – and another potential compensation mechanism -- value-added measures of teacher effectiveness based on student achievement gains. We find that subjective principal assessments of teachers predict future student achievement significantly better than teacher experience, education or actual compensation, though not as well as value-added teacher quality measures. In particular, principals appear quite good at identifying those teachers who produce the largest and smallest standardized achievement gains in their schools, but have far less ability to distinguish between teachers in the middle of this distribution and systematically discriminate against male and untenured faculty.

Principals as Agents: Subjective Performance Measurement in Education


sauce for the goose
Happy Father's Day
questions & answers for Niki Hayes

Wednesday, April 30, 2008

Arnold Kling on the mathefication of economics

An Important Emerging Economic Paradigm

By Arnold Kling

"… three types of activity generate a process of continuing and cumulative change. Trading creates new opportunities for innovation and institutional change. Innovation creates new opportunities for institutional change and trading. Institutional change creates new opportunities for trading and innovation. ...the process does not converge on or 'discover' a known or predictable outcome."
-- Meir Kohn

The Dartmouth University economics professor Meir Kohn has identified some important characteristics of an emerging alternative paradigm in economics. This approach, which he calls "the exchange paradigm" and which I prefer to call Learning Economics, is an important new direction in the field. The focus shifts from how an economy allocates a given set of resources to how an economy functions as a learning mechanism, sorting through innovations to find those that provide genuine improvements in living standards.

For those like Kohn and myself, trained in late-20th-century academic economics, appreciating the learning economy has brought with it a skepticism about the value of mathematics as a tool for economic analysis. If Kohn is correct, then economists of the past two or three generations may have committed one of the greatest blunders in intellectual history.

The most distinctive trend in economic research over the past hundred years has been the increased use of mathematics. In the wake of Paul Samuelson's (Nobel 1970) Ph.D dissertation, published in 1948, calculus became a requirement for anyone wishing to obtain an economics degree. By 1980, every serious graduate student was expected to be able to understand the work of Kenneth Arrow (Nobel 1972) and Gerard Debreu (Nobel 1983), which required mathematics several semesters beyond first-year calculus.

Today, the "theory sequence" at most top-tier graduate schools in economics is controlled by math bigots. As a result, it is impossible to survive as an economics graduate student with a math background that is less than that of an undergraduate math major. In fact, I have heard that at this year's American Economic Association meetings, at a seminar on graduate education one professor quite proudly said that he ignored prospective students' grades in economics courses, because their math proficiency was the key predictor of their ability to pass the coursework required to obtain an advanced degree.

The raising of the mathematical bar in graduate schools over the past several decades has driven many intelligent men and women (perhaps women especially) to pursue other fields. The graduate training process filters out students who might contribute from a perspective of anthropology, biology, psychology, history, or even intense curiosity about economic issues. Instead, the top graduate schools behave as if their goal were to produce a sort of idiot-savant, capable of appreciating and adding to the mathematical contributions of other idiot-savants, but not necessarily possessed of any interest in or ability to comprehend the world to which an economist ought to pay attention.

Where Math Works

Kohn points out that mathematical economics describes a world with a given set of goods to trade. The mathematical models answer the question: when all opportunities for profitable trade have been exploited, what will be the end result? This approach is a perfect fit for international economics, where we often are interested in tracing through the implications of opening a previously-closed country to trade. Many of Samuelson's classics are in this area, as are the works of Eli Heckscher, Bertil Ohlin (Nobel 1977), and James Meade (Nobel 1977).

The theory of finance offers another opportunity for mathematical economics to shine. By looking at how various financial claims will pay off under different circumstances, economists can make useful predictions about the relationships among various security prices. The portfolio theory of Harry Markowitz (Nobel 1990) and William Sharpe (Nobel 1990), the corporate finance model of Franco Modigliani (Nobel 1985) and Merton Miller (Nobel 1990), and the option pricing model of Fischer Black, Robert Merton (Nobel 1997), and Myron Scholes (Nobel 1997) combine mathematical elegance with empirical validity and practical application.


Schumpeter and Hayek

Friedrich Hayek (Nobel 1974) captured some aspects of the learning economy paradigm. Appreciating the importance of moving from the unknown to the known, Hayek focused on the "discovery procedure" embedded in market processes.

In some respects, the evolutionary mechanism of the learning economy was anticipated in Joseph Schumpeter's phrase creative destruction. As University of Reading, England economics professor Mark Casson put it,

"In their mathematical models of economic activity and behavior, economists began to use the simplifying assumption that all people in an economy have perfect information...That leaves no role for the entrepreneur..."

According to Schumpeter, the entrepreneur is someone who carries out 'new combinations' by such things as introducing new products or processes, identifying new export markets or sources of supply, or creating new types of organization. Schumpeter presented an heroic vision of the entrepreneur as someone motivated by the 'dream and the will to found a private kingdom'; the 'will to conquer: the impulse to fight, to prove oneself superior to others'; and the 'joy of creating.'"

The creative destruction that took place in Schumpeter's day was a relatively gradual process. As this Wired story reports,

"'Schumpeter probably was right all along,' says [then FCC Chairman] Michael Powell, 'but it's only now, at Moore's law speed, that you can actually observe it.'"

The theories of Schumpeter and Hayek do not translate well into mathematics. One can write down an equation and call it a "Schumpeterian" model or a "Hayekian" model, but such sterile exercises lose the flavor of their theories. As a result, there is not much room for their theories in the conventional economics Ph.D program.

The Role of Government

Many (but not all) disciples of mathematical economic theory are quite paternalistic. Markets result in the best allocation of resources only under a narrow and implausible set of assumptions. Most real-world markets possess features that make them unlikely to produce such an optimum, even after all opportunities for profitable trades are exhausted. By taking a sufficiently optimistic view of government's ability to correct such "market failures," one can become quite expansive about the scope for government involvement in the economy.

On the other hand, many (but not all) of the proponents of the learning economy are quite adamant that government's role ought to be reduced. I am in that camp.

My view is that innovation by trial-and-error is the most valuable economic process, and government intervention tends to be inimical to such innovation. Even when the market is producing unsatisfactory outcomes, my view is that eventually innovators will come along with better ideas. In that way, the market's errors tend to be self-correcting. Government's errors tend to perpetuate and to deepen.

Bureaucracies tend to resist innovation. [ed.: you can say that again] In fact, in a corporate setting, that is their function. External companies have lots of goods and services to sell to corporate America. Their own middle managers have lots of ideas. How can a company sift through all of these possibilities? The answer is bureaucracy.


However, simply being large and bureaucratic is not the chief problem with government. The main way that government impedes innovation is by siding with those who are threatened by innovation. The incumbents in an industry always look to government for protection. When their wishes are granted, economic progress is thwarted.

When you see a sector in the economy that lags in economic performance, either in relation to other sectors or to similar sectors in different countries, chances are that incumbent protection is at the root of the problem. In the United States, the biggest cost increases are in industries where government combines protection of incumbents against competition with subsidies in demand.


In higher education, no well-known university or college has gone out of business in my memory. Over this same period, countless corporate giants have bitten the dust. To me, this says that the incumbent protection racket in higher education works really well. There is hardly any entry, and hardly any exit. No surprise, then, that dousing this sector with subsidies leads primarily to inflation.

Mathematical Hazing

One of the best incumbent-protection rackets going today is for mathematical theorists in economics departments. The top departments will not certify someone as being qualified to have an advanced degree without first subjecting the student to the most rigorous mathematical economic theory. The rationale for this is reminiscent of fraternity hazing. "We went through it, so should they."

Mathematical hazing persists even though there are signs that the prestige of math is on the decline within the profession. The important Clark Medal, awarded to the most accomplished American economist under the age of 40, has not gone to a mathematical theorist since 1989.

These hazing rituals can have real consequences. In medicine, the controversial tradition of long work hours for medical residents has come under scrutiny over the last few years. In economics, mathematical hazing is not causing immediate harm to medical patients. But it probably is working to the long-term detriment of the profession.

I believe that I've observed -- and lived through -- the same process in the humanities, the difference being that in the humanities the move is toward theory.

Mark Roulo on the mathefication of the universe

"What I've seen in academia is that the various disciplines seem to 'complexifiy' themselves, either by becoming 'theoretical,' in the case of language-based disciplines, or by becoming 'mathified,' in the case of the social sciences."

The mathefication is not just an academic thing. The real world is becoming more quantitative. Some examples:

(1) Major League Baseball has (slowly ... with a lot of resistance) started paying attention to statistics *much* more when evaluating players. The key here isn't just that people are paying attention to things like batting average. They have *always* done that. The key is that people are starting to try to correlate different things with winning. So, the modern baseball stats-head mostly doesn't care about batting average.

The stats are easily available to someone with a *solid* high school math education. The point, however, is that even baseball is getting much more quantitative than it historically has.

(2) Drug Design. More and more companies are trying to design drugs using computer models (not just physics/chemistry simulators). This requires *lots* of math beyond what a normal PhD chemist would know.

(3) Quantitative supply chain management. Think about what Wal-Mart does. Very mathy. Wasn't done by anyone 30 years ago.

(4) Wall Street Trading. I'd say that today if you don't have a very heavy math background, you basically *can't* trade stocks/bonds/etc on Wall Street today. There may be some exceptions, but the field is much more quantitative than 30 years ago. Read up on LTCM (which went down in flames!) for more details.

(5) SQM (Statistical Quality Control), used by many manufacturing companies, is a mathematical approach to quality control.

So it isn't just academia. A lot more jobs have gotten quantitative in the last 30 years.

Which sorta makes things like TERC even more scary.

Tuesday, April 29, 2008

Multiplying Fractions with Prof. Wu

This is the 6th in a series of posts fleshing out the material written by Hung Hsi Wu in Critical Concepts for Understanding Fractions. See also Part I , Part II, Part III, Part IV , and Part V .

We are now progressing rapidly through operations on fractions! In this post, we discuss multiplication of fractions.

We recall our definition for a fraction k/l. k/l was defined as the concatenation of k pieces when the unit length is divided into l parts. That is, the fraction k/l is the same as saying the fraction k/l of the number 1.

Generalizing this, we can define a fraction of a number as follows:

k/l of a number m is the sum of k l-length pieces of m, that is, when m is divided into l equal parts, and we concatenate k of those parts.

But a fraction is just a number, as indicated by its length on the number line (or its location on the number line), so the definition of k/l of a number m can is true whether m is a whole number or a fraction. If m is a whole number, then k/l of that whole number is defined as above. If m is a fraction, say m = a/b, then k/l of that fraction means: we take the length [0,a/b], partition it into l pieces, and then concatenate k of them.

We now define the product, or the multiplication of two fractions as:
k/l * m/n = k/l of a segment of length m/n.

This fits our intuition, if we are comfortable with multiplication: 1/3 of 6 is 2 = 6/3, 2/3rds of 9.5 is 2/3 * 9.5 = (2* 9.5 )/3 = 19/3 = 6 and 1/3. But we still need a rule for how to multiply such products.

The product rule for fractions is: k/l * m/n = k*m/l*n.

Now, this should make sense: the denominator is telling us how many partitions we need to make, and the numerator tells us how many of those partitions we concatenate together. When we multiply fractions k/l * m/n, we simply partition m/n into l pieces again, and concatenate k of them. If we remember order of operations, the rule should also become clear: m/n was when we partitioned 1 into n pieces, and concatenated m of them, and (a/b)/c is the same as a/(b * c): partitioning first into n and then each of them into l is the same as partitioning into n*l = l*n same sized pieces in the first place. Likewise, d*(a/(b*c)) is the same as (d*a)/(b*c): counting up d pieces of a pieces of size (b*c) is the same as counting d*a pieces of size b*c.

More formally, though, we can prove this rule as follows. We begin by going to our definition: k/l of m/n means we divide the length m/n into l parts, and concatenate k of them. Partitioning the segment m/n into l pieces is done as following: using equivalent fractions and association, m/n = l*m/l*n = l*(m/l*n). That is, m/n is the concatenation of l pieces, each of size m/ln. In other words, dividing m/n into l parts means creating parts each of size m/ln. Concatenating k of them means k*(m/ln) = km/ln.

Such a simple formula helps clarify the multiplication algorithm of decimals. Recall that the formula is:
* Multiply two numbers as if they are whole numbers, ignoring the decimal points,
* Count the total number of nonzero digits to the right of the decimal places in the two numbers, call it P
* Put the decimal point in position so that the resultant product has P nonzero digits to the right of the decimal place.

This follow naturally as a consequence of the powers of ten in the denominator:

1.25 * 0.0067 = 125/100 * 67/10000 = (125 * 67)/ 1000000 = 8375/1000000 = 0.008375

See, we’re getting the hang of these fractions! Onto division next time.

Singapore Math 6B exit test

Here is a problem from the Primary Mathematics exit test for grade 6B (second semester, 6th grade):

An empty rectangular tank, 60 cm long by 50 cm wide, contains 3 metal cubes of edge 10 cm. The tank is being filled with water flowing from a tap at a rate of 10 liters per minute. If it takes 6 minutes to fill up the tank, find the height of the tank.

How many Scarsdale kids could do this problem?

Few. (I'll ask my friend the next time I see her.)

Here's another:

A car left Town A at 10:00 a.m. and travelled towards Town B at the average speed of 70 km/h. At the same time a truck left Town B and travelled towards Town A over the same road at an average speed of 50 km/h. If the distance between Town A and Town B is 420 km, at what time would the car and the truck pass each other?

I talked to the Admissions Director at the Masters School, which admits kids from Asian countries, about their knowledge of mathematics. She basically said: no comparison. (She said it politely. But that's what she said.) The kids they see, some of them, started algebra in 5th grade.

They're good at it, too.

Elite private schools in Manhattan, some of them, start algebra in 5th grade, too. Here in Westchester, however, not only do public schools not start algebra in the 5th grade, they are adopting constructivist curricula that progress even more slowly than the curricula they replaed. When C. went through grade school here he learned long division in 4th grade. Now the school, using Math Trailblazers, is teaching long division in 5th, and they're mostly not teaching long division at all. They're teaching forgiving division.

Fareed Zakaria needs to get a clue.

the deathless meme of the high-performing school

I can have it for you Thursday

comment from lsquared re: Thursday

How discouraging. No wonder kids hate proofs--they get tested on them without ever having a chance to learn them (OK--this isn't true of all schools/teachers). I love proofs and love teaching them, but there's nothing I know that can be effectively done by Thursday.

There's just one word for this comment: droll.

I love droll!

Speaking of what can and cannot be done by Thursday, if I hurry up and order now I could, by Thursday, have 5 or 6 or 7 books about proofs.

Any suggestions?

Also, does anyone know anything about Math Dictionary With Solutions by Chris Kornegay? It looks and sounds fantastic (way expensive, but fantastic - exactly what a person at my stage of the game desperately wants and needs).

the deathless meme of the high-performing school

The school system, the line goes, is in crisis, with its students performing particularly badly in science and math, year after year, in international rankings. But the statistics here, although not wrong, reveal something slightly different. The real problem is one not of excellence but of access. The Trends in International Mathematics and Science Study (TIMSS), the standard for comparing educational programs across nations, puts the United States squarely in the middle of the pack. The media reported the news with a predictable penchant for direness: "Economic Time Bomb: U.S. Teens Are Among Worst at Math," declared The Wall Street Journal.

But the aggregate scores hide deep regional, racial, and socioeconomic variation. Poor and minority students score well below the U.S. average, while, as one study noted, "students in affluent suburban U.S. school districts score nearly as well as students in Singapore, the runaway leader on TIMSS math scores." (pdf file) The difference between the average science scores in poor and wealthy school districts within the United States, for instance, is four to five times as high as the difference between the U.S. and the Singaporean national average. In other words, the problem with U.S. education is a problem of inequality. This will, over time, translate into a competitiveness problem, because if the United States cannot educate and train a third of the working population to compete in a knowledge economy, this will drag down the country. But it does know what works.

The U.S. system may be too lax when it comes to rigor and memorization, but it is very good at developing the critical faculties of the mind.

The Post-American World
Fareed Zakaria
excerpt posted at:
Foreign Affairs

This is the logic of NCLB: white schools good; black schools bad. Affluent white parents exercise choice by moving to affluent white suburbs where they can enroll their children in affluent white schools.


That is exactly what affluent white parents do, affluent white parents and their camp followers, a hardy band of non-affluent white, black, and Hispanic parents who move heaven and earth to get their kids into these schools, too.

But the story doesn't end there. If it did, kitchen table math wouldn't exist.

number one

Number one, let's review the meaning of the word study.

The words "students in affluent suburban U.S. school districts score nearly as well as students in Singapore, the runaway leader on TIMSS math scores" do not appear in a study.* They appear in a trade publication published by Tau Beta Pi, The Engineering Honor Society.

Tau Beta Pi, interestingly enough, sponsors a K-12 initiative to improve the public schools: (pdf file)

...significantly large percentage of our high-school graduates lags behind their peers in many developed countries, with respect to performance in mathematics and science. In many cases, these countries are our competitors in the global marketplace, and hence this situation has a direct impact on our economy in the long term.

Now that is a world many parents whose children are currently attending affluent suburban schools will recognize.

The article Mr. Zakaria quotes, which appeared in the Winter 2007 issue of The Bent of Tau Beta Pi, cites the same studies everyone cites when speaking of U.S. kids' math and science knowledge in an international context: TIMSS & PISA. Neither study found that advanced students in the U.S. are on par with advanced students elsewhere. They found the opposite. Advanced students here are not advanced elsewhere. The only U.S. students who are competitive with their peers in Europe are those taking Advanced Placement calculus, which is 5% of the total.

And, of course, we have no idea how AP calculus students would fare in a head to head competition with students in Singapore because Singapore wasn't included in the comparison of advanced students. Given the fact that Singapore students performed significantly better than all 16 of the countries that were included, I think it's safe to assume that America's AP students would take a shellacking if they were to go up against Singapore students studying calculus at the same age.

Here is what the Department of Education has to say about advanced students in the U.S. and how they compare to advanced students in Europe:

Overview and Key Findings Across Grade Levels

At the fourth grade, U.S. students were above the international average in both science and mathematics. In the eighth grade, U.S. students scored above the international average in science and below the international average in mathematics. At the end of secondary schooling (twelfth grade in the U.S.), U.S. performance was among the lowest in both science and mathematics, including among our most advanced students.


Achievement of Advanced Students (16 Countries) **

The advanced mathematics and physics assessments were administered to a sample of the top 10-20 percent of students in their final year of secondary school in each nation. In the advanced mathematics assessment, this included the 14 percent of U.S. students who had taken or were taking pre-calculus, calculus, or AP calculus compared to advanced mathematics students in other countries. In the physics assessment, this included the 14 percent of U.S. students who had taken or were taking physics or AP physics compared to advanced science students in other countries.

  • The performance of U.S. physics and advanced mathematics students was below the international average and among the lowest of the 16 countries that administered the physics and advanced mathematics assessments. The U.S. outperformed no other country on either assessment.
  • When you compare U.S. twelfth graders with Advanced Placement calculus instruction (about 5 percent of the U.S. cohort) to all advanced mathematics students in other nations, their performance was at the international average and significantly higher than 5 other countries.
  • When you compare U.S. twelfth graders with Advanced Placement physics instruction (about 1 percent of the U.S. cohort) to all advanced science students in other nations, their performance was below the international average and significantly higher than only 1 other country.
  • More countries outperformed U.S. students in physics than in advanced mathematics. This differs from results for mathematics and science general knowledge, where more countries outperformed the U.S. in mathematics than in science.

update: I'd forgotten that the U.S. has decided to "sit out" the TIMSS test of advanced students in 2008:

This article reports on reactions to the U.S. Department of Education's first time decision to sit out an international study designed to show how advanced high school students around the world measure up in math and science. Mark S. Schneider, the commissioner of the department's National Center for Education Statistics, which normally takes the lead in managing the U.S. portion of international studies of student performance in those subjects, said budget and staffing constraints prevent his agency from taking part in the upcoming study, which is known as the "Trends in Mathematics and Science Study-Advanced 2008." The study, in which nine countries have so far agreed to participate, will test students taking physics and upper-level math classes, such as calculus, at the end of their secondary school years. It comes as national leaders in the United States are promoting improved math and science education as critical to protecting the nation's economic edge. The statistics agency is still overseeing the regular administration of Trends in Mathematics and Science Study (TIMSS), which got under way in the United States this year. The larger of the two studies, the regular TIMSS assesses 4th and 8th math and science achievement in 62 nations.

Singapore Math exit test 6B

* At least, they do not appear in a study that is findable by Google.
** Australia, Austria, Canada, Cyprus, Czech Republic, Denmark, France, Latvia, Lithuania, Norway, Germany, Russian Federation, Slovenia, Sweden, Switzerland, United States NOTE: none of the Asian countries are included in this list

math? who said anything about math?

This has nothing whatsoever to do with anything going on around here, past, present, or future.

(Niki Hayes sent this thing to me, so, really, it's all her fault.)

balanced literacy in France

Liz Ditz left a link to an editorial that appeared last fall in Le Figaro (right of center, supported Sarkozy in the election, good newspaper), which was discussed by Language Log. Quite a lot of this sounds wrong to me; it sounds so wrong, in fact, that I'm not going to take the time to track down everything I have around here on left-brain, right-brain, language, mood, etc. to sort it out. I have 8 months of filing to do, after all. Not to mention another 2 or 3 hours of geometry homework.

[update: oh, good. Language Log says it's wrong, wrong, wrong. That's what I thought. Ed, translating over my shoulder, "Is this right?" "Can this possibly be right?" "This sounds off-base to me."]

Nevertheless, the section on whole language -- and what I take to be balanced literacy -- is useful to read because it could have been written by Louisa Moats.

Definitely take time to read the entire post at Language Log, as well as Liz Ditz's post.

À quand une vraie réhabilitation de l'enseignement primaire ?
Par Lucien Israël * «Des quantités de choses échapperont à tout jamais à ceux qui n'ont pas accès à la grammaire »
15/10/2007 | Mise à jour : 03:25 |
Les difficultés que rencontre l'institution scolaire et, plus généralement, l'évolution des comportements des jeunes ont fait couler beaucoup d'encre ces dernières années. Étonnamment, en dépit de l'importance des enjeux, ce sujet n'a pas vraiment été abordé jusqu'à maintenant par les candidats à l'élection présidentielle. L'ampleur de la tâche en effraie sans doute plus d'un ! Pour l'aborder, voyons sereinement les faits, les conséquences et les causes.

Je commencerai par le constat suivant : en décembre 2001, l'OCDE a mené une étude dans trente-deux pays sur la capacité de lecture et de compréhension à l'entrée en 6e. Pour la compréhension de l'écrit, la France était au 14e rang. Et ce n'était guère plus brillant dans les domaines techniques et scientifiques, domaines dans lesquels les pays anglo-saxons nous devancent largement.

Un niveau médiocre ou faible en lecture, écriture, grammaire, etc., compromet l'avenir des jeunes et de la société : il existe quelques dizaines de milliers de mots dans une langue qui servent à comprendre, à s'exprimer et à s'imprégner d'une culture. La richesse du vocabulaire et l'usage de la grammaire sont les principaux moyens d'acquérir le sentiment d'appartenance à un groupe culturel. Celui-ci est en effet basé sur son histoire mais aussi sur sa langue. Et cela concerne tous les enfants, et pas seulement les enfants d'immigrés.

Par ailleurs, nous ne cessons de nous parler à nous-mêmes. Un vocabulaire restreint, des significations imprécises, empêchent de se parler à soi-même : non seulement on ne lit pas, non seulement on ne communique pas correctement avec autrui, mais on ne communique pas non plus avec soi-même, donc on ne se connaît pas. Si on n'a pas de subjectivité soi-même, on n'a pas la notion de l'existence d'une subjectivité chez autrui. Par conséquent, en cas de désaccord avec autrui, on ne discute pas : on tape dessus !

La neurophysiologie est à cet égard très éclairante : elle permet de faire le lien entre les faits constatés plus hauts et leurs causes : le cerveau gauche est celui de l'analyse, en particulier de l'analyse des mots (cela est valable pour les droitiers et pour un certain nombre de gauchers. Pour les autres, c'est l'hémisphère droit qui remplit ce rôle). L'hémisphère gauche est celui de l'analyse des idées, de leur perception, de leur enregistrement, de leur comparaison à d'autres, de leur critique ; celui, aussi, de la mémorisation. L'hémisphère droit, au contraire, est celui de l'émotion - positive ou négative -, de la perception non analysable, du sentiment. Les enfants, par exemple, perçoivent par leur cerveau droit ce qu'ils regardent à la télévision. S'ils ont un cerveau gauche « en bon état », ils sont capables de comprendre et de critiquer ce que leur cerveau droit reçoit, car les deux hémisphères communiquent. Si au contraire le cerveau gauche a été « abandonné », ils sont entièrement livrés aux images qui leur sont montrées.

Il se trouve qu'une révolution pédagogique a eu lieu à la fin des années 1970, qui concernait l'ensemble des enseignements de l'école primaire. Si l'on prend le cas précis de la méthode d'apprentissage de la lecture et de l'écriture, on sait que la méthode utilisée aujourd'hui est celle de la méthode globale (la semi-globale revenant exactement au même). La méthode syllabique fait appel au cerveau gauche puisqu'elle consiste à décortiquer les mots en syllabes et en lettres. La méthode globale, qui consiste à reconnaître la forme des mots, s'appuie au contraire sur l'hémisphère droit puisqu'elle est basée sur l'intuition.

Les méthodes d'apprentissage actuelles laissent en friche l'hémisphère gauche. Il ne reçoit que peu d'informations et de sollicitations. Le registre lexical est pauvre et, par conséquent, la compréhension du monde, de soi-même et des autres bien moindre. Je prendrai l'exemple concret des Esquimaux : leur langue comporte une soixantaine de mots différents pour évoquer la neige : ils perçoivent, par conséquent, une foule de nuances que nous-mêmes ne voyons pas. Des conséquences sont déjà visibles et ne peuvent que s'aggraver : la place est libre pour l'impulsif, la violence et la capacité d'être dominé par autrui ou de se donner à lui sans réfléchir. De même, des quantités de choses échapperont à tout jamais à ceux qui n'ont pas accès à la grammaire. Qu'attendent les candidats à l'élection présidentielle pour annoncer une véritable réhabilitation de l'enseignement primaire ?

* Professeur émérite de cancérologie.

October 15, 2007
Many things have forever escaped those who don’t have access to grammar

When will we have a true reform of primary schooling?
by Lucien Israel *

The difficulties encountered by public schools and more generally the evolution of young people’s behavior: both of these things have caused a great deal of ink to be spilled in recent years. Surprisingly, despite what’s at stake here, this subject hasn’t really been taken up by the candidates in the presidential election. The size of what’s at issue undoubtedly terrifies more than one of the candidates. In order to take the subject on, let’s look calmly at the facts and consequences and the causes.

I’ll start by saying the following thing: December 2001, the OECD undertook a study in 32 countries about the ability to read and understand à l'entrée en 6e. [Ed say: not sure which grade this is, but it’s “still young”] In comprehension and writing, France was number 14 out of the 32. Our country was hardly more brilliant in the areas of science and technology, realms in which Anglo-Saxon countries are far ahead of us.

A mediocre or weak ability or result in reading, writing, and grammar compromises the future of the young people of our society. There are a few 10s of 1000s of words in a language which allow people to understand, express themselves and participate in a culture. The richness of the vocabulary and the usage of grammar are the main ways of acquiring a sense of belonging to a cultural group. This group is in a sense based on its history but also on its language. And this concerns all children and not only the children of immigrants.

In addition, we never stop talking about ourselves. A narrow vocabulary with imprecise meanings prevent us from talking about ourselves. Not only do we not read, not only do we not communicate correctly with others, but we don’t even communicate correctly with ourselves, which means that we don’t come to know ourselves. If we don’t have a subjective understanding of ourselves, then we can’t have an idea of the subjectivity of others. As a result, whenever there’s a disagreement with someone else, we don’t discuss, we come to blows.

The neuropsychology in this case is enlightening. It allows us to make connections between things that are said and their causes. The right brain is the part that analyzes and in particular it analyzes words (and this is true for all righthanded people and a certain number of left handed people; for others it is the right hemisphere that fulfills this role). The left hemisphere is the one that analyzes ideas, their perception, their memory or their recording, and their comparison with others and their criticism. It’s also the hemisphere where memory takes place. The right hemisphere, by contrast, is the hemisphere of emotion, positive and negative; it’s the hemisphere of non-analyzeable perceptions, of sentiment. Children, for example, perceive in their right brain what they see on television. If their left brain is in good shape, they are capable of understanding and criticizing and understanding what their right brain perceives because the two hemispheres have good communication with each other. If on the other hand the left brain has been “abandoned,” kids are completely at the mercy of images shown to them.

There was a pedagogical revolution that took place at the end of the 1970s and it involved the primary education in its entirety. If we take the precise case of the method of teaching reading and writing we can see that the method used today is méthode globale (partially whole language turns out being exactly the same thing). The méthode syllabique [phonetics] activates the left brain because it forces the child to disaggregate or untangle words into letters and symbols. Whole language, which involves only the form of the words themselves, activates by contrast only the right hemisphere because it’s based on intuition.

Current pedagogical methods leave the left hemisphere out of the picture. It receives only a small amount of information and is rarely activated. As a result, our whole lexical register is poor, our understanding of the world and of ourselves and others reduced from what it once was. I’ll take the concrete example of eskimos. Their language includes over 60 different words to evoke snow. They can see, as a result, a whole array of nuances that we ourselves don’t see. The consequences are already visible and can only get worse. There is now a much greater possibility to be governed by impulses and violence and a much greater likelihood of being dominated by others, or of allowing ourselves to be unwittingly dominated by others. At the same time, a great many things will escape forever all those who don’t have an understanding of grammar. What are the presidential candidates waiting for? We need to begin a real reform of primary education.

Professer emeritus of oncology

It's a global conspiracy.

Monday, April 28, 2008

The 'Choke Tester'

Today was just too much to bear. This school year will not be over soon enough, if you ask me. Between the endless projects, back-to-back creative writing assignments, and Everyday Math in second grade, there just seems to be no time to learn multiplication or division, what a paragraph or pronoun is, or real science for that matter. I used to be able to laugh at some of the silly assignments my children would bring home from school, but It's not even funny anymore.

Just when you thought it couldn't get any worse, it does.

Right after he filled me in on how he got an incomplete because he didn't use partial products on his math work today (despite the annoying fact that all his answers were correct), my second grade son handed me his health assignment which reads as follows:

Directions: Using the 'Choke Tester' you made in class, make a list of items in your home that can fit through it. Make a plan with your family where to safely store the listed items in your home to prevent choking hazards.

[lines for the list]

I talked about this with my ___________________________.

Of course, that explains what the weird paper towel tube thingie covered in stickers, pipe cleaner, and heaven knows what else that he brought home today was supposed to be. I thought it was some strange art project. But no. It's a choke tester, of course. What else would it be?

Why, in the name of Zeus, does our family need to have a plan to store our choking hazards when we haven't needed to child-proof our home in at least five years? And why, oh why, is this considered homework?

If I should be worried about my eight year old son putting some random choking hazard in his mouth and trying to swallow it, I probably have bigger problems than I realized. Last time I checked, we were well past this stage.

Even my son knows how ridiculous this health assignment is. He said, "I knew you wouldn't like this, Mom." To which I responded, "You're right. I don't." Then he asked, "We won't do stuff like this when I'm homeschooled, right?"

Boy, you can say that again.

kitchen table math

I have just spent 3 hours doing Chris’ new homework assignment. The problem set is drawn from two short lessons on parallelograms appearing in one short chapter of Glencoe Geometry New York. The two lessons cover NINE theorems about parallelograms, not one of which I’m able to prove although I am apparently expected to be able to prove all 9 now that I've seen them tidily numbered and listed in an attractive Glencoe Geometry Theorem Chart enhanced with a red-and-tan color scheme.* One two-column proof of Theorem 8.4** (Opp. angles of parallelogram are congruent) and one paragraph proof of Theorem 8.10 (If both pairs of opposite angles are congruent, quadrilateral is a parallelogram) and it's off to the races.

That's not all. Having read nine theorems about parallelograms & 2 proofs, I am now Glencoe-ready to solve homework problems involving PARALLELOGRAMS ON THE COORDINATE PLANE USING THE MIDPOINT FORMULA, etc.

This is a textbook written by math educators.***

Speaking of math educators, thank God I have the Teacher Wraparound Edition. Unfortunately, what I really need now is the Teacher Solution Manual (ISBN: 0078602041). Too bad I didn't think of that.

* I feel about tan the way I feel about beige, only more so.
** And, yes, I did spend time Googling the known universe to find out whether these numbers are official: is Opp. angles of parallelogram are congruent always and everywhere Theorem 8.4? Apparently not.
*** written by math educators, but sounding suspiciously like contemporary geometry textbooks authored by actual mathematicians...

Bob Dixon on the worst textbook you could possibly imagine

What if Zig Engelmann set out intentionally to write the worst textbook he possibly could? ...If you think about it, Zig should be able to pull this off better than anyone alive. ...Extremely confusing concepts all would be introduced at the same time and/or in close approximation. Stuff would be "taught" and then dropped, or more accurately, "covered" and then dropped. New material being covered would logically require mastery of prerequisite knowledge that most of the students most likely wouldn't have.


If Zig were to engage in this little heuristic exercise, the result, I believe, would be a textbook that would sell like crazy and generate a fortune in royalties, and it would take about one-twentieth of the time that it would take to write an instructionally sound textbook.... [T]his ... came to me while looking at my daughter's geometry textbook. The thought hit me that if Zig had tried to write the worst possible geometry textbook in the world, it would end up looking a lot like Emily's geometry text, published b one of the few major educational publishing companies still standing.


Within two lessons of a single chapter of a best-selling geometry text, eight major, similar concepts are introduced, with eight more or less similar names or labels. Absolutely no one associated with this textbook ever took a single minute to look at all of this stuff through the eyes of the learner.

It's same ole, same ole. A few kids get A's on the tests, which passes as proof that the book is fine, but there is something wrong with the kids who flunk the test or get C's or D's. Zig could have written this book without having seen it just playing the game of trying to make everything as difficult and confusing as humanly possible. I've looked carefully for spots where Zig could, in fact, make the book worse than it is, but they are few and far between.

Isn't that strange? Really. You'd think that any best-selling textbook would get a few things right, would do a few things that actually take a student perspective into consideration. You'd think that Zig's horrible geometry text would be far worse than a best-selling geometry text. This is both astounding and depressing. Someone is making a fortune off a textbook that could be just slightly better than the worst textbook we could imagine, the worst textbook Zig could write. Get out the thesaurus: overwhelmed, dismayed, incredulous, confounded...

What Would You Get if You Set Out to Write the Worst Textbook You Could Possibly Imagine? A Best Seller
by Bob Dixon
Direct Instruction News
Spring 2008

Of course, I'd love to know what textbook he's talking about.

I was thinking Glencoe Geometry, but it can't be because the lead author on the textbook Dixon's talking about is a mathematician.

I just checked the author page for Glencoe Geometry & all 5 authors are math educators.

kitchen table math

dog paintings & photos

I met this artist at the Rivertowns Studio Tour this weekend. Her paints & photos are incredible.

Sunday, April 27, 2008

Schema-Based Instruction for Mathematics

Kitchen Table Math hasn't previously addressed the work of Asha K. Jitendra or the "Schema-Based Instruction" approach

Adrienne Edwards wrote a precis of Jitendra's recent article. What follows is her introduction:
In the Spring 2008 Issue of Perspectives, the quarterly publication of IDA, Asha K Jitendra describes a workable way to teach math to LD (indeed all) students. This post is adapted from it.

Since problem solving is not well addressed in many mathematics textbooks, Jitendra and colleagues have developed a conceptual teaching approach that integrates mathematical problem solving and reading comprehension strategies (e.g., reading aloud, paraphrasing, questioning, clarifying and summarizing).

Called “Schema-Based Instruction” (SBI), the system was tested and perfected for a decade. The goal: to improve student learning of word problems, especially students with LD and those at risk for math failure.
I recommend that you read Edwards' whole summary and the journal article reprinted at LD online before commenting here (Edwards doesn't allow comments on her blog, but I've emailed her a link to this post -- perhaps she will come over and comment).

Elsewhere on Schema-Based Instruction for Mathematics

Schema-based instruction improves math skills
(APA Monitor Online, Monitor on Psychology Volume 38, No. 4 April 2007

Students who learn to identify three different kinds of word problems—and what strategies to use for each—do better on math tests than students who learn only one general-purpose model, finds a study in the February Journal of Educational Psychology (Vol. 99, No. 1, pages 115–127). Stopping to categorize a word problem before picking a plan to solve it may be especially effective for low-achieving students, says study author Asha Jitendra, PhD, a special education professor at Lehigh University.
LD Online: An Exploratory Study of Schema-Based Word-Problem-Solving Instruction for Middle School Students with Learning Disabilities: An Emphasis on Conceptual and Procedural Understanding (reprint of article in The Journal of Special Education (2003): Vol. 36/NO. 1/2002/pp. 23-38)
Implications for practice: The findings from this study have several implications for practice. First, the schema-based intervention, with its emphasis on conceptual understanding, helped students with learning disabilities not only acquire word-problem-solving skills but also maintain the taught skills. Therefore, results of the study highlight the effectiveness of strategy instruction for addressing mathematical difficulties evidenced by students with learning disabilities (Montague, 1995, 1997b). Second, the results of this study suggest that teaching students to identify the relationships present in each word problem promotes generalization to other, untaught skills (e.g., multistep problems). Students with learning disabilities should receive instruction that teaches them to understand the key features of problems prior to solving them. Third, the effectiveness of the strategy when implemented by the classroom teacher may indicate the importance of researchers' collaborating with practitioners to adapt instruction to meet students' individual needs. Involving the classroom teacher in the implementation of this study was important because the teacher is now more likely to invest effort in continuing to use a strategy that had beneficial effects for her students.

Journal of Learning Disabilities, v29 n4 p422-31 Jul 1996 (ERIC digest)

Paper presented at the Annual International Convention of the Council for Exceptional Children (73rd, Indianapolis, IN, April 5-9, 1995). (ERIC digest)

A comparison of single and multiple strategy instruction on third-grade students' mathematical problem solving Journal of educational psychology (2007), vol. 99, no1, pp. 115-127 [13 page(s) (article)]
The purposes of this study were to assess the differential effects of a single strategy (schema-based instruction; SBI) versus multiple strategies (general strategy instruction, GSI) in promoting mathematical problem solving and mathematics achievement as well as to examine the influence of word problem-solving instruction on the development of computational skills. Eighty-eight 3rd graders and their teachers were assigned randomly to conditions (SBI and GSI). Students were pre- and posttested on mathematical problem-solving and computation tests and were posttested on the Pennsylvania System of School Assessment Mathematics test, a criterion-referenced test that measures student attainment of academic standards. Results revealed SBI to be more effective than GSI in enhancing students' mathematical word problem-solving skills at posttest and maintenance. Further, results indicate that the SBI groups' performance exceeded that of the GSI group on the Pennsylvania System of School Assessment measure. On the computation test, both groups made gains over time.
Google Books: Teaching Mathematics to Middle School Students with Learning Disabilities ;
here's the link to the Amazon page

Solving Math Word Problems: Teaching Students with Learning Disabilities Using Schema-Based Instruction
(12452) ISBN: 9781416402459 ($53.00)
This is a detailed-scripted program using Schema-Based Instruction (SBI), designed as a framework for instructional implementation. It is primarily for school practitioners (e.g., special and general education teachers, school psychologists, etc.) teaching critical word problem solving skills to students with disabilities, grades 1-8.
I can see that I will really need to wrap my brain around this approach.

Left-Brained Epiphanies

Our blogging colleague, Out in Left Field (who posts here as Lefty) is asking a question: Are all epiphanies right-brained? In other words, the common story is "introverted, scholarly type learns to embrace extroversion, play, and lack of structure."

Lefty is looking for stories of left-brain epiphanies:
How often, for example, does an outgoing musician learn that what makes him truly happy is retreating to his study to analyze business cycles and market equilibria?
Go and contribute your stories.

Some prompts:
Left-brain: logical, systematic, analytical, one-at-a-time, abstract, verbal, introverted.

Right-brain: emotional, incidental, intuitive, holistic, relational, nonverbal, social.

Reminder: don't post your stories here, go over to Out in Left Field