COMET • Vol. 4, No. 14 – 25 April 2003


(1) Professional Conferences–Calls for Speakers

Information for prospective speakers at upcoming mathematics education conferences is available at the following Web sites:

* Association of Mathematics Teacher Educators

Dates:  Jan. 22-24, 2004

Location:  San Diego, CA (Marriott Mission Valley Hotel)

Proposal deadline: May 31, 2003


* National Council of Supervisors of Mathematics

Dates:  April 19-21, 2004

Location: Philadelphia, PA

Proposal deadline: June 1, 2003


* National Council of Teachers of Mathematics

Dates:  April 21-24, 2004

Location:  Philadelphia, PA

Proposal deadline: May 1, 2003


URL (Regional Meetings):

* Research Council on Mathematics Learning

Location: Oklahoma City, OK

Dates:  February 19-21, 2004

Proposal deadline: June 30, 2003


* School Science and Mathematics Association

Dates: October 23-25, 2003

Location: Columbus, OH (Radisson Airport Hotel and Conference Center)

Proposal deadline:  April 30, 2003


(2)“Ammunition for Backers of Do-or-Die Exams” by Greg Winter

Source  The New York Times – 23 April 2003


Two new studies make the case that do-or-die exams–which decide whether students graduate, teachers are dismissed or schools are shut in more than half the states in the nation–have brought about at least a modicum of academic progress, especially for minority students who may get scant attention otherwise.

The studies entirely contradict what some other scholars have found and are bound to feed an already fiery debate over the phenomenon known as high-stakes testing, a course of educational change that teachers resent, the Bush administration embraces and states are hurriedly adopting.

Neither the authors nor their peers contend that the new research ends the dispute, given the many remaining open questions. But taken together, the studies appear to push the research pendulum away from critics who have argued that the fixation with make-or-break exams undermines teachers, stifles analytical learning and squeezes out struggling students, all without providing any clear benefits.

“If I were gambling on whether to put in a high-stakes system or not, I would put one in,” said Martin Carnoy, the Stanford University professor who co-wrote one of the studies. “There’s some probability I would be wrong. But if I were to put my money on something right now, I would try this.”

In his study, published next month in Educational Evaluation and Policy Analysis, a peer-reviewed journal, Mr. Carnoy and a colleague, Susanna Loeb, examined whether states with serious test consequences did better on a nationwide math assessment than their counterparts bearing none at all.

While there seemed to be little to no difference in the performance of white students, the study found that the more consequences a state imposed, the better its minority students typically did.

In fact, for every additional layer of sanction or reward placed on schools, teachers and children, about 3.5 percent more black students and 3 to 4 percent more Latinos grasped the basics of eighth-grade math.

The same pattern did not prove true for Latinos in math in the lower grades, but for black students it did, leading Mr. Carnoy to speculate that the threat of consequences may compel schools to demand more from students whom they may have otherwise written off.

No less importantly, the study found that do-or-die exams did not lead to more dropouts, as other researchers have argued. Still, there was no evidence that they improved graduation rates.

“If that’s not the aim, what is all this about?” asked Mr. Carnoy, whose study was financed by the federal government. “Why do we care about raising test scores if more people aren’t going to finish high school and go on to college?”

Another peer-reviewed study to be published next month found that when states imposed consequences on their own exams, their students tend to do better on nationwide math assessments as well.

Margaret E. Raymond and Eric A. Hanushek of the Hoover Institution at Stanford, two supporters of high-stakes tests, found that national math scores between 1996 and 2000 rose an average of seven-tenths of 1 percent in states with no consequences, 1.2 percent in those that simply published the results in the newspaper and 1.6 percent in states that either rewarded success or penalized failure.

Small though they may seem, the differences between the gains are meaningful, the authors argue, because national math scores have barely moved in decades.

“The systems that we had in place by 2000 are not going to revolutionize our schools, that’s clear,” Mr. Hanushek said. “But they are an element in moving our schools forward.”

The study was paid for by the Packard Humanities Institute and the Smith Richardson Foundation, neither of which claims a position on the issue.

There is a caveat to all this data, educational researchers warn. Just because a correlation may exist between make-or-break exams and achievement does not mean the exams are to thank for any progress. Indeed, there are so many factors influencing test scores, be they economic or curricular, that proving a causal link between consequences and results may never be done.

“That goes for both sides of the debate,” said Robert L. Linn, president of the American Educational Research Association. “To be able to definitively attribute results to the stakes of a test is a stretch for any of them.”

Underneath the academic dispute over the efficacy of make-or-break exams is a strong distaste for them among teachers. A survey of more than 4,000 teachers last February found that about three quarters of them, whether or not they taught under the threat of serious consequences, said that state testing programs “were not worth the time and money involved.”

Perhaps more worrisome, 76 percent of teachers facing the highest stakes and 63 percent of those encountering the lowest said that mandatory testing led to teaching in ways that contradicted their own ideas of sound educational practice.

“Teachers are saying look, we’re not opposed to high standards, but when you put so much emphasis on this one testing measure, it really becomes problematic,” said Joseph J. Pedulla, who helped coordinate the survey for National Board on Educational Testing and Public Policy at Boston College. It was financed by The Atlantic Philanthropies, which takes no position on the testing question.

The idea of teaching to the test bothers many educators, but others argue that it does not preclude learning thinking skills that can be transferred from one context to another.

A study last February by the Manhattan Institute, which was not peer-reviewed, found that the results on state exams closely paralleled those on other, independent tests.

Though it looked at less than 10 percent of the nation’s schools, the study suggested that whatever controversy surrounds make-or-break exams, they can be reliable instruments for gauging student learning.

“Which should we believe, the teachers or the tests?” said Jay P. Greene, the study’s author. “We might wonder whether the teachers have it right and the test has it wrong. But it’s also quite possible that the test has it right.”

(3) Quantity of Coursework Rises Since 1983″ by Lynn Olson

Source: Education Week – 23 April 2003


It’s the most familiar recommendation in that most familiar of reports, A Nation at Risk.

Twenty years ago this week, the National Commission on Excellence in Education recommended that all high school students complete four years of English, three years of mathematics, three years of science, three years of social studies, and a half-year of computer science. College-bound students were encouraged to add two years of a foreign language.

Warning of a “rising tide of mediocrity” in American education, the report called on the nation to raise its expectations for students, develop rigorous and measurable standards for academic performance, and better prepare and reward teachers.

Two decades later, students are taking more academic courses than before. When computer science is excluded, according to the National Center for Education Statistics, the percentage of students taking the so-called “new basics” curriculum advocated in the federal report roughly quadrupled from fewer than 14 percent of graduates in 1982 to 56 percent in 1998.

But research shows it’s the level and quality, not just the quantity, of courses that count. And while more students, particularly poor and minority students, are taking higher-level courses than ever before, significant gaps remain.

By 1998, for example, 45 percent of white and 56 percent of Asian high school graduates had completed some advanced math or science courses, but only 30 percent of black and 26 percent of Hispanic graduates had done so.

“In saying we ought to take more courses, we probably said too little about the quality of the courses,” Milton Goldberg, who served as the executive director of the commission for the U.S. Department of Education, said recently.

Since 1983, studies have shown that students who take advanced courses, particularly in math, generally achieve at higher levels, attend and finish college at higher rates, and earn more as adults than their peers who take lower-level courses.

Finishing a course beyond the level of Algebra 2 “more than doubles the odds that a student who enters postsecondary education will complete a bachelor’s degree,” concluded researcher Clifford Adelman in “Tools in the Toolbox,” a 1999 analysis of longitudinal data from the federal High School & Beyond survey. The sequence of math courses a student took in high school was even more important than socioeconomic status in predicting a student’s odds of finishing college, he concluded…

Today, minority students and those from poor families remain underrepresented in the most rigorous high school courses. And progress in closing the gap in performance between minority and nonminority students has slowed.

Ronald F. Ferguson, a lecturer in public policy at Harvard University’s John F. Kennedy School of Government, notes that black teenagers made rapid improvement on the National Assessment of Educational Progress reading and math tests during the 1980s, but that their progress had stopped by 1990.

He attributes the earlier gains, in part, to the fact that from 1982 to 1990, black and Hispanic students on average increased the number of math courses they took at the level of algebra or higher by almost a full course; white students raised theirs by half a course. Yet far more remains to be done.

(4) Articles about H. S. M. Coxeter

(a) “Harold Coxeter, 96, Who Found Profound Beauty in Geometry, is Dead” by Stuart Lavietes

Source:  The New York Times – 7 April 2003


Dr. Harold Scott MacDonald Coxeter, a mathematician who was hailed as one of the foremost geometricians of his generation and whose ideas inspired the drawings of M. C. Escher and influenced the architecture of R. Buckminster Fuller, died on March 31 in his home in Toronto. He was 96.

Dr. Coxeter, whose childhood fascination with symmetry led to his career in mathematics, was driven by the idea that beautiful explanations exist for all puzzles. Several mathematical concepts have been named for him, including Coxeter groups.

He also made major contributions to the theory of polytopes, which are complex objects of more than three dimensions that, while not existing in the real world, can be described mathematically.

Dr. Coxeter met M. C. Escher in 1954 at a mathematics conference in Amsterdam. Escher, a Dutch artist who had grown tired of repeating images of birds and fish on a flat plane, had heard about Dr. Coxeter’s work on shapes in multidimensional space and sought him out. After the conference, Dr. Coxeter sent him a copy of his paper “Crystal Symmetry and Its Generalizations,” which was illustrated with complex geometric figures, including a circle containing a pattern of objects that grew smaller and smaller as they neared the edge.

Inspired, Escher used this figure as a source for his series of “Circle Limit” etchings. Many of his most famous works, including “Ascending and Descending,” his 1960 illusion of interlocking staircases, reflect Dr. Coxeter’s ideas about polytopes.

Dr. Coxeter and Escher remained friends until the artist’s death in 1972. In 1996, Dr. Coxeter published a paper proving that Escher’s “Circle Limit III” was mathematically perfect.

Dr. Coxeter was also friendly with R. Buckminster Fuller, who cited the influence of Dr. Coxeter’s ideas on the development of the geodesic dome. Mr. Fuller dedicated his book “Synergetics” to Dr. Coxeter…

A child musical prodigy, he was an accomplished pianist who composed various pieces for the piano, a string quartet and at the age of 12 an opera…

(b) Source:


The aroma of antiseptic and crisp sheets mingles with the sooty smell of a small coal-burning fireplace at the end of the infirmary room. Two thirteen-year-old boys are in side-by-side beds recovering from the flu in their private school’s sick-room. “Coxeter, how do you imagine time-travel would work?” asks John Petrie, one of the boys.

“You mean as in H.G. Wells?” says Donald Coxeter, the other boy. H.G. Wells’s classic science fiction book The Time Machine is a popular topic of conversation. Both boys believe time travel will eventually be possible. After a few seconds Coxeter says, “I suppose one might find it necessary to pass into the fourth dimension.” That is the moment when he began forming ideas about hyperdimensional geometries.

Both boys were very bright. They started using the books and games by their beds to play around with ideas of higher dimensional space–spaces and dimensions that go beyond the ordinary three dimensions of natural space as we see it. These early musings lead Coxeter to later discoveries about regular polytopes–geometric shapes that extend into the fourth dimension and beyond.

Soon after he recovered from the flu, Coxeter wrote a school essay on the idea of projecting geometric shapes into higher dimensions. Impressed by his son’s geometrical talents and wishing to help Coxeter’s mind develop, his father took him to visit Bertrand Russell, the brilliant English philosopher and educator. Russell helped the Coxeters find an excellent math tutor who worked with Coxeter enabling him to enter the famous Cambridge University.

Coxeter has always been known as H. S. M. Coxeter. Though his first name is actually Donald, he does not use it formally. At 19, in 1926, before Coxeter had a university degree, he discovered a new regular polyhedron, a shape having six hexagonal faces at each vertex. He went on to study the mathematics of kaleidoscopes which are instruments that use mirrors and bits of glass to create an endlessly changing pattern of repeating reflections. By 1933 he had counted and specified the n-dimensional kaleidoscopes…

[Note:  This site also contains several video clips of Coxeter.]

(c) “Donald Coxeter Dies; Leader in Geometry” by Martin Weil

Source:  Washington Post – 5 April 2003


…Dr. Coxeter had been a teacher and researcher at the University of Toronto since 1936, and he continued to be active in mathematics almost until the hour of his death.

“He was doing it to the very end,” said a daughter, Susan Thomas, who had been taking care of him….

The plane geometry with which many people become familiar in high school occupies a small niche in what mathematicians consider to be geometry. It is a broad study that finds highly abstract counterparts–often in many dimensions, often impossible to visualize–for points, lines and shapes and for the relations among and between them.

Explorations of these relationships have found many applications in theoretical physics, in such practical matters as robotics, computer graphics and photographic interpretation and in areas of art as well.

Some of Dr. Coxeter’s work on symmetries of abstract multidimensional geometric objects has been credited with leading to discoveries in molecular structure. In particular, it was linked to research that led to the Nobel Prize-winning discovery of the molecule of a complex isotope of carbon with potentially valuable properties…

Known for his scores of articles and treatises, he was the author of several books prized by mathematicians. They include The Real Projective Plane, Introduction to Geometry and Geometry Revisited as well as Non-Euclidean Geometry. The last gives an account of the geometries that result when, in violation of the postulates learned in school, parallel lines do meet, or more than one parallel line may be drawn through a given point…

His daughter said that a few years ago he was interviewed for a television program on his secrets of longevity. As he related them, they included proper diet (he was a vegetarian) and exercise (he did 50 pushups regularly).

In addition, his daughter recalled, he said, “You have to be passionately interested in something.”

For Dr. Coxeter, that was mathematics. “Absolutely,” his daughter said…