Mathematically Correct presents

The Pythagorean Theorem

G. D. Chakerian and Kurt Kreith

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At a recent school meeting, a group of Davis parents and teachers used the Pythagorean theorem to illustrate the difference between a constructivist vs. traditional approach to teaching. Their goal was to provide other parents with a basis for responding to a recent decision by the Davis Board of Education. For in Fall, 1996 Davis junior high schools will offer a choice between two different courses in Algebra 1, one emphasizing constructivist pedagogy and the other relying on a more traditional deductive approach.

As set forth in the currently used text Themes, Tools and Concepts, one constructivist approach to the Pythagorean theorem is based on the use of geoboards. A traditional approach appropriate to Algebra 1, one found in many algebra texts, is based on the dissection of a square.

 

   The former calls on students to use rubber bands to build a right
   triangle on a geoboard, use rubber bands to enclose the squares
   defined by the triangle's legs and hypotenuse, and then look for
   patterns in the areas of the squares so generated.  

   The latter asks the student to visualize two different dissections
   of a square of size (a+b) x (a+b).  Using the usual notation of a,
   b, and c for the legs and hypotenuse of a right triangle, one such
   dissection corresponds to a^2 + b^2 + 2ab and the other corresponds
   to c^2 + 2ab .  Equating these two expressions yields the usual
   symbolic representation of the Pythagorean theorem: a^2 + b^2 = c^2.
   
   [^2 notation indicates squared terms]

Laudable as the use of experimentation as a prelude to mathematics may be, there are serious dangers hidden in this constructivist approach to the Pythagorean theorem. For instance, while it is easy to construct right triangles on a geoboard by orienting the legs of the triangle along the horizontal and vertical axes, the example given in Themes, Tools and Concepts suggests that the student should use more general orientations. The question that then arises is, “how is the student to know whether a triangle with such general orientation is, or is not, a right triangle?” (It is very easy to construct geoboard triangles with one angle imperceptibly close, but not quite equal, to a right angle.)

The only mathematical solution is to use the Pythagorean theorem itself (or more precisely, its converse) to confirm that such a triangle is in fact a right triangle. However, this is precisely the knowledge that the student is being urged to construct! Thus, implicit in this particular constructivist approach to the Pythagorean theorem is the notion that the student should build his or her own knowledge by “eyeballing” right angles.

Training a generation of carpenters to rely on “eyeballing” right angles would be a national disaster (none of us would let such a carpenter touch our house). Yet, in the name of constructivism, we seem to be encouraging a generation of children to erect this pillar of mathematical knowledge on just such a basis.

Another problem arising in this constructivist approach to the Pythagorean theorem is that of calculating the areas of the squares built on the sides and hypotenuse of a geoboard triangle. The usual formula “Area = Side x Side” requires that we first determine the lengths of the sides of these squares. However, unless these squares are aligned with the geoboard’s vertical and horizontal axes, finding the lengths of their sides also requires the Pythagorean theorem!

An alternative way of finding the areas of “tilted squares” is to use an advanced mathematical result called Pick’s theorem. While children can be taught to use Pick’s theorem at an early age, any semblance of a mathematical understanding of this tool is well beyond the traditional high school curriculum. It may also be that students will be taught to calculate areas by counting unit squares, and pieces thereof.

Another possibility is to enclose such a “tilted square” within a larger square whose sides are parallel to the axes of the geoboard. While this provides an appropriate way of calculating areas, note that it corresponds to the dissection “(a+b)^2 = c^2 + 2ab” arising in the traditional proof of the Pythagorean theorem cited above. That is, this particular approach to implementing the “discovery process” takes the student half way to actually understanding the Pythagorean theorem. However, on ideological grounds, it stops short of conveying the gift of understanding.

Some may argue that it doesn’t really matter which method is used to teach the Pythagorean theorem – i.e., that both methods lead to the same result. However, this is not true. Experiments with the geoboard correspond to a cumbersome verification of the Pythagorean theorem in rather special circumstances (the geoboard’s discrete structure is well suited to experimentation, but it fails to represent the more general structure of the Euclidean plane). The traditional dissection approach corresponds to a proof of the theorem, providing an answer to the question “why.” One approach sets the stage for discoveries that lead to conjectures; the other emphasizes properties of area that lead to the understanding of an important truth.

Many parents believe that the most important end of education in any field is to raise the question “why,” to know when an answer might or might not exist, and to demand an answer when it can be given. Having children “discover” a hodge-podge of mathematical properties, without providing answers to which properties are true and why, is to deny them a real mathematical education.

While manipulatives can be powerful tools for leading students through a discovery process that reinforces mathematics, the haphazardly planned use of manipulatives can be destructive. An essential adjunct to “hands-on” mathematics is an effort to organize ideas and develop the capacity for mathematical thought and reason. Experiments performed under the tutelage of unskilled guides can lead students into a chaotic jungle, one in which their minds become entangled in an underbrush of mismatched concepts to which they, their parents, and their future teachers will be hard pressed to bring order.

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NYC MATH WARS
NYC Honest Open Logical Debate (NYC HOLD)On Math Reform
Elizabeth Carson

November 27, 2000

New York, NY. NYU mathematicians speak out against controversial new math programs being taught in NYC’s premiere School District 2 in Manhattan. Community School Board 2 will hear the professors’ preliminary report at the calendar meeting on Tuesday, November 28, 6:30 pm at 333 7^{th Ave, Seventh Floor.}

^{A groundswell of parent concern over how math is being taught in District 2 schools led to a front page New York Times article last spring, “The New, Flexible Math Meets Parent Rebellion.” CBS Weekend News brought national attention to the local struggle last May in the segment, “New, New Math = Controversy.”}

^{District 2 parents have now gained the support of mathematicians at NYU and the University of Rochester in grieving their concerns to District officials. The professors are scheduled to take part in a Math Forum sponsored by Community School Board 2 scheduled for March 1, 2001.}

^{The controversy in District 2 has gained the attention of Schools Chancellor Harold O. Levy and members of his recently established Commission on Mathematics Education charged with investigating how math is being taught in schools across the city.}

^{District 2 pilots the latest wave of experimental math programs, called the “new, new math,” echoing an earlier failed reform in the 60’s called “new math.” Memorization of math facts is no longer emphasized. Children are encouraged to use language to describe solutions and the way they feel about math.}

^{Community response in NYC and across the country has erupted in what have become known as the “math wars.” Critical parents, joined by mathematicians and scientists advocate clarity and balance in math reform: urging the inclusion of grade by grade goals, explicit teaching of standard procedures, basic skill building and rigor along with the inclusion of some of the creative exercises in the new programs. The pendulum has swung too far and must be corrected.}

^{One parent, Mark Schwartz, in testimony last year before the House Education and Workforce Committee stated: “If medical doctors experimented with our kids in the same fashion school districts do they would be in jail.” The hearings were held to review the US Department of Education’s endorsement of 10 of the experimental programs.}

^{Over 200 of the nation’s top mathematicians, including seven Nobel Laureates,} winners of the Fields Medal ^{the department heads at more than a dozen universities including Caltech, Stanford and Yale, responded to the federal endorsements with an open letter of protest to Secretary of Education Richard Riley, published in the Washington Post in November, 1999.}

^{“These programs are among the worst in existence,” said David Klein, a Cal State Northridge professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be joke if it weren’t so damaging.” Several of the cited programs, Interactive Mathematics Project (IMP),Connected Mathematics Project (CMP) and Everyday Mathematics are now being used in NYC schools.}

^{The new wave of math reform is based on a “constructivist” teaching philosophy; emphasizing creative exercises, hands-on projects and group work, with far less attention given to basic skills. Students are asked to ‘construct’ their own solutions. The use of calculators is encouraged. Teachers are instructed to serve as “facilitators” and are discouraged from explaining to students the standard solutions of basic arithmetic. Practice and drill have been eliminated. In higher grades algebra is de-emphasized. Many of the programs have no textbooks.}

^{Members of Chancellor Levy’s Math Commission will consider the new programs, which are being used in over 60% of NYC schools, including roughly 50 of the city’s weakest, which comprise the Chancellor’s District . Plans are set to expand implementation into more schools. One of the most controversial programs, the Interactive Mathematics Project (IMP), will be mandated in Bronx High Schools beginning next year.}

^{Professor Richard Askey, who holds an endowed chair in math at the University of Wisconsin, explained his motives in co-authoring the protest letter to Secretary Riley, “I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs.”}

^{High math scores remained stable in some of the privileged District 2 schools, last year; though some schools’ scores dropped significantly. Across the rest of the city, 75% of eighth grade students failed the state math test ; 66% failed the city math tests. Answers are critical at a time when graduation requires students pass a new math Regents exam.}

^{It was the introduction of Connected Mathematics Project (CMP) and Investigations in Number Data and Space (TERC), two of the District 2 programs, that sparked the initial parent revolt that led to the California Math Wars.} Six parents in Plano,Texas have filed suit in federal court against their local school district after parent requests for an alternative to CMP were denied. ^{A nuclear physicist in Okemos Michigan led the local campaign against CMP. The use of TERC in one school system in Massachusetts prompted members of the Harvard Mathematics Department to issue a public protest. Parents in Reading Massachusetts fought the adoption of Everyday Math.}

^{The “new, new math” programs are based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards. Frank Allen, past president of the NCTM and Emeritus professor of Mathematics at Elmhurst College comments: “NCTM leaders must admit that they have urged the application, on a national scale of highly controversial methods of teaching before they have been adequately debated or understood and before researchers have verified them by well-controlled and replicated studies.”}

^{=============}

November 27, 2000

To: Harold O. Levy, Chancellor

Dr Judith Rizzo, Deputy Chancellor for Instruction

Burton Sacks, Chief Executive for Community School District Affairs

Dr Irving Haimer, Member, Board of Education, Manhattan Representative

Trudy Irwin, Director of Education, Office of the Manhattan Borough President

From: NYC HOLD, Steering Committee

Re: Community School Board 2 Calendar Meeting November 28, 2000

We respectfully invite you to attend the calendar meeting of Community School Board 2 in Manhattan scheduled for Tuesday November 28, beginning at 6:30 pm. During the public session the Board will hear comment on District 2’s K-12 math programs. Distinguished faculty of the Courant Institute of Mathematical Sciences at New York University and concerned parents plan to speak.

The District’s mathematics reform is one component of the systemic instructional reform initiated by Anthony Alvarado in the previous decade. District 2 prides itself in being a leader in education reform in New York City. Our District piloted the New Standards Performance Standards in Mathematics which were adopted city-wide last year.

The implications of the relative success of our District’s math reform for the entire NYC school system should be evident.

In our District, parent concern with their children’s progress in mathematics is escalating; worries are being strongly voiced about aspects of the new math programs just as full implementation takes hold. School year 1999-2000 marked the first year the new K- 8 programs, Investigations in Number Data and Space (TERC) and Connected Mathematics Project (CMP) were mandated.

Parents are tutoring in record numbers. For many of their children, the new constructivist programs fail to provide a sound foundation in basic arithmetic. Parents recognize the value of the new programs’ creative, hands-on classroom and homework activities. However, many question the extensive time devoted to such activities, ostensibly at the expense of explicit teaching of the standard procedures and practice of skills. The absence of textbooks has exacerbated the situation.

Concerned parents have reached out to the NYU mathematics community in District 2 for analysis and opinion of our programs in hopes of finding ways to amend or extend the implemented programs to reach a better balance in the math instruction.

There are clearly implications in the course the District 2 community takes for the work of the newly appointed Commission on Mathematics Education to review math programs city-wide. We hope our experiences in District 2 can contribute to the assessment of the Committee.

Parents, teachers and administrators in District 2 share a well earned pride in our exemplary schools and strong educational community. Parents see on a daily basis, through their children’s work and enthusiasm for learning, the results of the skill and commitment of the teaching staff and administrators in our schools. Parents appreciate the formidable challenges district administrators face in guiding systemic reform in a large and diverse community school district.

We look forward to an ongoing and meaningful partnership in District 2.

We sincerely hope you will take an interest in our concerns and efforts to advance the course of math education for our children.

Sincerely,

Steering Committee
NYC HOLD

Elizabeth Carson
Christine Larson
Garry Dobbins
Margaret Hunnewell
Mary Somoza, Member, CSB #2
Granville Leo Stevens, Esq
Maureen McAndrew, DDS
Michael Weinberg

Rocky Mountain News

Seebach: Race-gap study launches 3-stage rant about readers

April 30, 2005

Everybody knows that American blacks and Hispanics are at a disadvantage to whites and Asians both in education and income. Three economists have written a paper demonstrating that the patterns of disadvantages for blacks and for Hispanics are very different, raising questions about the explanations often given for those disadvantages.

In the authors’ own words, here are some take- away points:

• “For black males, controlling for an early measure of ability cuts the black-white wage gap in 1990 by 76 percent. For Hispanic males, controlling for ability essentially eliminates the wage gap with whites. For women the results are even more striking. Wage gaps are actually reversed, and controlling for ability produces higher wages for minority females.”

• “When we control for the effects of home and family environments on test scores, the Hispanic-white test score gap either decreases or is constant over time while the black-white test score tends to widen with age.”

• “Hispanic children start with cognitive and noncognitive deficits similar to those of black children. They also grow up in similar disadvantaged environments, and are likely to attend schools of similar quality. Hispanics have substantially less schooling than blacks. Nevertheless, the ability growth by years of schooling is much higher for Hispanics than blacks. By the time they reach adulthood, Hispanics have significantly higher test scores than blacks.”

• “Our analysis of the Hispanic data illuminates the traditional study of black-white differences and casts doubt on many conventional explanations of these differences since they do not apply to Hispanics who also suffer from many of the same disadvantages.”

I know this is contrary to just about everything you’ve heard or read, so you’re asking, “Who are these people?” They’re Pedro Carneiro, University College London; James J. Heckman, University of Chicago, American Bar Foundation and University College London (and winner of the 2000 Nobel Prize in economics for developing the kind of technical statistical analysis that undergirds this paper) and Dimitriy V. Masterov. The paper was written for the Institute for Labor Market Policy Evaluation, a part of the Swedish Ministry of Industry, Employment and Communications, in Uppsala, Sweden.

The paper is “Labor market discrimination and racial differences in premarket factors” and it’s at www.ifau.se/swe/ pdf2005/wp05-03.pdf on the Web.

They don’t argue against current policies on affirmative action – though they certainly could, based on their evidence – merely that policies addressing very early skill gaps are likely to do more good than additional affirmative action policies aimed at the workplace.

One possible explanation of persistent wage gaps is that there is “pervasive labor market discrimination against minorities.” Another, which they observe is equally plausible, is that “Minorities may bring less skill and ability to the market.” And of course both could be true in varying degrees, but I think this is the most important thing they say: “The two polar interpretations of market wage gaps have profoundly different policy implications.”

And how. So if you’re a policy-maker, Go Read The Whole Thing.

Now, since I have room for only a tiny bit of what’s significant in this paper anyway, I’m going to address a different issue that invariably comes up when I write about something so contrary to received opinion.

OK, (/turn rant on/) don’t waste your time writing me that I “haven’t considered” whatever particular bee is buzzing around your bonnet. You have no information about what I have considered; you know only what I have mentioned. And let me tell you, when I’m writing an 800-word summary of a highly technical 50-page paper bristling with statistical analysis, there’s a lot I’ve considered that I don’t mention. There’s even more data that the researchers have considered that they don’t mention – although the existence of Web appendices to scholarly papers has ameliorated that problem.

Next, don’t think you’re being erudite by citing some cliche about “lies, damned lies or statistics,” which I understand is properly credited to Benjamin Disraeli, but is often attributed to Mark Twain. Yes, it is possible to lie with statistics – there’s a charming and useful little book with that in the title – but it’s a lot harder to lie with statistics than without them.

Case in point, the current flap over the number of deaths statistically attributable to obesity. If you’re one of those people who fatuously asserts that “you can prove anything with statistics” I challenge you to find me a peer-reviewed journal article proving that smoking enhances longevity, or that women are taller than men.

Last, don’t talk about motives. You have no evidence about my motives aside from what I tell you – and I could be wrong about that; lots of people are. Even if we were both right about my motives, it would have no bearing on the cogency of my arguments, which do not adduce them; that’s the ad hominem fallacy. “Fallacy,” please note, which means that even if your premises are correct, your conclusion may be wrong. (/Turn rant off/)

Oh, I feel much better now. Excellent paper.

Linda Seebach is an editorial writer for the News. She can be reached by telephone at (303) 892-2519 or by e-mail at seebach@RockyMountainNews.com.

MORE SEEBACH COLUMNS »

Copyright 2005, Rocky Mountain News. All Rights Reserved.

NYC MATH WARS
NYC Honest Open Logical Debate (NYC HOLD)On Math Reform
Elizabeth Carson

November 27, 2000

New York, NY. NYU mathematicians speak out against controversial new math programs being taught in NYC’s premiere School District 2 in Manhattan. Community School Board 2 will hear the professors’ preliminary report at the calendar meeting on Tuesday, November 28, 6:30 pm at 333 7^{th Ave, Seventh Floor.}

^{A groundswell of parent concern over how math is being taught in District 2 schools led to a front page New York Times article last spring, “The New, Flexible Math Meets Parent Rebellion.” CBS Weekend News brought national attention to the local struggle last May in the segment, “New, New Math = Controversy.”}

^{District 2 parents have now gained the support of mathematicians at NYU and the University of Rochester in grieving their concerns to District officials. The professors are scheduled to take part in a Math Forum sponsored by Community School Board 2 scheduled for March 1, 2001.}

^{The controversy in District 2 has gained the attention of Schools Chancellor Harold O. Levy and members of his recently established Commission on Mathematics Education charged with investigating how math is being taught in schools across the city.}

^{District 2 pilots the latest wave of experimental math programs, called the “new, new math,” echoing an earlier failed reform in the 60’s called “new math.” Memorization of math facts is no longer emphasized. Children are encouraged to use language to describe solutions and the way they feel about math.}

^{Community response in NYC and across the country has erupted in what have become known as the “math wars.” Critical parents, joined by mathematicians and scientists advocate clarity and balance in math reform: urging the inclusion of grade by grade goals, explicit teaching of standard procedures, basic skill building and rigor along with the inclusion of some of the creative exercises in the new programs. The pendulum has swung too far and must be corrected.}

^{One parent, Mark Schwartz, in testimony last year before the House Education and Workforce Committee stated: “If medical doctors experimented with our kids in the same fashion school districts do they would be in jail.” The hearings were held to review the US Department of Education’s endorsement of 10 of the experimental programs.}

^{Over 200 of the nation’s top mathematicians, including seven Nobel Laureates,} winners of the Fields Medal ^{the department heads at more than a dozen universities including Caltech, Stanford and Yale, responded to the federal endorsements with an open letter of protest to Secretary of Education Richard Riley, published in the Washington Post in November, 1999.}

^{“These programs are among the worst in existence,” said David Klein, a Cal State Northridge professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be joke if it weren’t so damaging.” Several of the cited programs, Interactive Mathematics Project (IMP),Connected Mathematics Project (CMP) and Everyday Mathematics are now being used in NYC schools.}

^{The new wave of math reform is based on a “constructivist” teaching philosophy; emphasizing creative exercises, hands-on projects and group work, with far less attention given to basic skills. Students are asked to ‘construct’ their own solutions. The use of calculators is encouraged. Teachers are instructed to serve as “facilitators” and are discouraged from explaining to students the standard solutions of basic arithmetic. Practice and drill have been eliminated. In higher grades algebra is de-emphasized. Many of the programs have no textbooks.}

^{Members of Chancellor Levy’s Math Commission will consider the new programs, which are being used in over 60% of NYC schools, including roughly 50 of the city’s weakest, which comprise the Chancellor’s District . Plans are set to expand implementation into more schools. One of the most controversial programs, the Interactive Mathematics Project (IMP), will be mandated in Bronx High Schools beginning next year.}

^{Professor Richard Askey, who holds an endowed chair in math at the University of Wisconsin, explained his motives in co-authoring the protest letter to Secretary Riley, “I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs.”}

^{High math scores remained stable in some of the privileged District 2 schools, last year; though some schools’ scores dropped significantly. Across the rest of the city, 75% of eighth grade students failed the state math test ; 66% failed the city math tests. Answers are critical at a time when graduation requires students pass a new math Regents exam.}

^{It was the introduction of Connected Mathematics Project (CMP) and Investigations in Number Data and Space (TERC), two of the District 2 programs, that sparked the initial parent revolt that led to the California Math Wars.} Six parents in Plano,Texas have filed suit in federal court against their local school district after parent requests for an alternative to CMP were denied. ^{A nuclear physicist in Okemos Michigan led the local campaign against CMP. The use of TERC in one school system in Massachusetts prompted members of the Harvard Mathematics Department to issue a public protest. Parents in Reading Massachusetts fought the adoption of Everyday Math.}

^{The “new, new math” programs are based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards. Frank Allen, past president of the NCTM and Emeritus professor of Mathematics at Elmhurst College comments: “NCTM leaders must admit that they have urged the application, on a national scale of highly controversial methods of teaching before they have been adequately debated or understood and before researchers have verified them by well-controlled and replicated studies.”}

^{=============}

November 27, 2000

To: Harold O. Levy, Chancellor

Dr Judith Rizzo, Deputy Chancellor for Instruction

Burton Sacks, Chief Executive for Community School District Affairs

Dr Irving Haimer, Member, Board of Education, Manhattan Representative

Trudy Irwin, Director of Education, Office of the Manhattan Borough President

From: NYC HOLD, Steering Committee

Re: Community School Board 2 Calendar Meeting November 28, 2000

We respectfully invite you to attend the calendar meeting of Community School Board 2 in Manhattan scheduled for Tuesday November 28, beginning at 6:30 pm. During the public session the Board will hear comment on District 2’s K-12 math programs. Distinguished faculty of the Courant Institute of Mathematical Sciences at New York University and concerned parents plan to speak.

The District’s mathematics reform is one component of the systemic instructional reform initiated by Anthony Alvarado in the previous decade. District 2 prides itself in being a leader in education reform in New York City. Our District piloted the New Standards Performance Standards in Mathematics which were adopted city-wide last year.

The implications of the relative success of our District’s math reform for the entire NYC school system should be evident.

In our District, parent concern with their children’s progress in mathematics is escalating; worries are being strongly voiced about aspects of the new math programs just as full implementation takes hold. School year 1999-2000 marked the first year the new K- 8 programs, Investigations in Number Data and Space (TERC) and Connected Mathematics Project (CMP) were mandated.

Parents are tutoring in record numbers. For many of their children, the new constructivist programs fail to provide a sound foundation in basic arithmetic. Parents recognize the value of the new programs’ creative, hands-on classroom and homework activities. However, many question the extensive time devoted to such activities, ostensibly at the expense of explicit teaching of the standard procedures and practice of skills. The absence of textbooks has exacerbated the situation.

Concerned parents have reached out to the NYU mathematics community in District 2 for analysis and opinion of our programs in hopes of finding ways to amend or extend the implemented programs to reach a better balance in the math instruction.

There are clearly implications in the course the District 2 community takes for the work of the newly appointed Commission on Mathematics Education to review math programs city-wide. We hope our experiences in District 2 can contribute to the assessment of the Committee.

Parents, teachers and administrators in District 2 share a well earned pride in our exemplary schools and strong educational community. Parents see on a daily basis, through their children’s work and enthusiasm for learning, the results of the skill and commitment of the teaching staff and administrators in our schools. Parents appreciate the formidable challenges district administrators face in guiding systemic reform in a large and diverse community school district.

We look forward to an ongoing and meaningful partnership in District 2.

We sincerely hope you will take an interest in our concerns and efforts to advance the course of math education for our children.

Sincerely,

Steering Committee
NYC HOLD

Elizabeth Carson
Christine Larson
Garry Dobbins
Margaret Hunnewell
Mary Somoza, Member, CSB #2
Granville Leo Stevens, Esq
Maureen McAndrew, DDS
Michael Weinberg

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MATH PROBLEMS
Why the U.S. Department of Education’s
recommended math programs don’t add up

By David Klein

---

What constitutes a good K-12 mathematics program? Opinions differ. In October 1999, the U.S. Department of Education released a report designating 10 math programs as “exemplary” or “promising.” The following month, I sent an open letter to Education Secretary Richard W. Riley urging him to withdraw the department’s recommendations. The letter was coauthored by Richard Askey of the University of Wisconsin at Madison, R. James Milgram of Stanford University, and Hung-Hsi Wu of the University of California at Berkeley, along with more than 200 other cosigners. With financial backing from the Packard Humanities Institute, we published the letter as a full-page ad in the Washington Post on Nov. 18, 1999, with as many of the endorsers’ names and affiliations as would fit on the page. Among them are many of the nation’s most accomplished scientists and mathematicians. Department heads at more than a dozen universities–including Caltech, Stanford, and Yale–along with two former presidents of the Mathematical Association of America also added their names in support. With new endorsements since publication, there are now seven Nobel laureates and winners of the Fields Medal, the highest award in mathematics. The open letter was covered by several newspapers and journals, including American School Board Journal (February, page 16).

Although a clear majority of cosigners are mathematicians and scientists, it is sometimes overlooked that experienced education administrators at the state and national level, as well as educational psychologists and education researchers, also endorsed the letter. (A complete list is posted at http://www.mathematicallycorrect.com.)

University professors and public education leaders are not the only ones who have reservations about these programs. Thousands of parents and teachers across the nation seek alternatives to them, often in opposition to local school boards and superintendents. Mathematically Correct, an influential Internet-based parents’ organization, came into existence several years ago because the children of the organization’s founders had no alternative to the now “exemplary” program, College Preparatory Mathematics, or CPM. In Plano, Texas, 600 parents are suing the school district because of its exclusive use of the Connected Mathematics Project, or CMP, another “exemplary” program. I have received hundreds of requests for help by parents and teachers because of these and other programs now promoted by the Education Department (ED). In fact, it was such pleas for help that motivated me and my three coauthors to write the open letter.

Common problems

The mathematics programs criticized by the open letter have common features. For example, they tend to overemphasize data analysis and statistics, which typically appear year after year, with redundant presentations. The far more important areas of arithmetic and algebra are radically de-emphasized. Many of the so-called higher-order thinking projects are just aimless activities, and genuine illumination of important mathematical ideas is rare. There is a near obsession with calculators, and basic skills are given short shrift and sometimes even disparaged. Overall, these curricula are watered-down math programs. The same educational philosophy that gave rise to the whole-language approach to reading is part of ED’s agenda for mathematics. Systematic development of skills and concepts is replaced by an unstructured “holism.” In fact, during the mid-’90s, supporters of programs like these referred to their approach as “whole math.”

Disagreements over math curricula are often portrayed as “basic skills versus conceptual understanding.” Scientists and mathematicians, including many who signed the open letter to Secretary Riley, are described as advocates of basic skills, while professional educators are counted as proponents of conceptual understanding. Ironically, such a portrayal ignores the deep conceptual understanding of mathematics held by so many mathematicians. But more important, the notion that conceptual understanding in mathematics can be separated from precision and fluency in the execution of basic skills is just plain wrong.

In other domains of human activity, such as athletics or music, the dependence of high levels of performance on requisite skills goes unchallenged. A novice cannot hope to achieve mastery in the martial arts without first learning basic katas or exercises in movement. A violinist who has not mastered elementary bowing techniques and vibrato has no hope of evoking the emotions of an audience through sonorous tones and elegant phrasing. Arguably the most hierarchical of human endeavors, mathematics also depends on sequential mastery of basic skills.

The standard algorithms

The standard algorithms for arithmetic (that is, the standard procedures for addition, subtraction, multiplication, and division of numbers) are missing or abridged in ED’s recommended elementary school curricula. These omissions are inconsistent with the mainstream views of mathematicians.

In our open letter to Secretary Riley, we included an excerpt from a committee report published in the February 1998 Notices of the American Mathematical Society. The committee was appointed by the American Mathematical Society to advise the National Council of Teachers of Mathematics (NCTM). Part of its report discusses the standard algorithms of arithmetic. “We would like to emphasize that the standard algorithms of arithmetic are more than just ‘ways to get the answer’–that is, they have theoretical as well as practical significance,” the report states. “For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials.”

This statement deserves elaboration. How could the standard algorithms of arithmetic be related to algebra? For concreteness, consider the meaning in terms of place value of 572:

572 = 5 (10²) + 7(10) + 2

Now compare the right side of this equation to the polynomial,

5x² + 7x + 2.

The two are identical when x = 10. This connection between whole numbers and polynomials is general and extends to arithmetic operations. Addition, subtraction, multiplication, and division of polynomials is fundamentally the same as for whole numbers. In arithmetic, extra steps such as “regrouping” are needed since x = 10 allows for simplifications. The standard algorithms incorporate both the polynomial operations and the extra steps to account for the specific value, x = 10. Facility with the standard operations of arithmetic, together with an understanding of why these algorithms work, is important preparation for algebra.

The standard long division algorithm is particularly shortchanged by the “promising” curricula. It is preparatory for division of polynomials and, at the college level, division of “power series,” a useful technique in calculus and differential equations. The standard long division algorithm is also needed for a middle school topic. It is fundamental to an understanding of the difference between rational and irrational numbers, an indisputable example of conceptual understanding. It is essential to understand that rational numbers (that is, ratios of whole numbers like 3/4) and their negatives have decimal representations that exhibit recurring patterns. For example: 1/3 = .333…, where the ellipses indicate that the numeral 3 repeats forever. Likewise, 1/2 = .500… and 611/4950 = .12343434….

In the last equation, the digits 34 are repeated without end, and the repeating block in the decimal for 1/2 consists only of the digit for zero. It is a general fact that all rational numbers have repeating blocks of numerals in their decimal representations, and this can be understood and deduced by students who have mastered the standard long division algorithm. However, this important result does not follow easily from other “nonstandard” division algorithms featured by some of ED’s model curricula.

A different but still elementary argument is required to show the converse–that any decimal with a repeating block is equal to a fraction. Once this is understood, students are prepared to understand the meaning of the term “irrational number.” Irrational numbers are the numbers represented by infinite decimals without repeating blocks. In California, seventh-grade students are expected to understand this.

It is worth emphasizing that calculators are utterly useless in this context, not only in establishing the general principles, but even in logically verifying the equations. This is partly because calculator screens cannot display infinite decimals, but more important, calculators cannot reason. The “exemplary” middle school curriculum CMP nevertheless ignores the conceptual issues, bypassing the long division algorithm and substituting calculators and faulty inductive reasoning instead.

Steven Leinwand of the Connecticut Department of Education was a member of the expert panel that made final decisions on ED’s “exemplary” and “promising” math curricula. He was also a member of the advisory boards for two programs found to be “exemplary” by the panel: CMP and the Interactive Mathematics Program. In a Feb. 9, 1994, article in Education Week, he wrote: “It’s time to recognize that, for many students, real mathematical power, on the one hand, and facility with multidigit, pencil-and-paper computational algorithms, on the other, are mutually exclusive. In fact, it’s time to acknowledge that continuing to teach these skills to our students is not only unnecessary, but counterproductive and downright dangerous.”

Mr. Leinwand’s influential opinions are diametrically opposed to the mainstream views of practicing scientists and mathematicians, as well as the general public, but they have found fertile soil in the government’s “promising” and “exemplary” curricula.

Calculators

According to the Third International Mathematics and Science Study, or TIMSS, the use of calculators in U.S. fourth-grade mathematics classes is about twice the international average. Teachers of 39 percent of U.S. students report that students use calculators at least once or twice a week. In six of the seven top-scoring nations, on the other hand, teachers of 85 percent or more of the students report that students never use calculators in class.

Even at the eighth-grade level, the majority of students from three of the top five scoring nations in the TIMSS study (Belgium, Korea, and Japan) never or rarely use calculators in math classes. In Singapore, which is also among the top five scoring countries, students do not use calculators until the seventh grade. Among the lower achieving nations, however, the majority of students from 10 of the 11 nations with scores below the international average–including the United States–use calculators almost every day or several times a week.

Of course, this negative correlation of calculator usage with achievement in mathematics does not imply a causal relationship. There are many variables that contribute to achievement in mathematics. On the other hand, it is foolhardy to ignore the problems caused by calculators in schools. In a Sept. 17, 1999, Los Angeles Times editorial titled “L.A.’s Math Program Just Doesn’t Add Up,” Milgram and I recommended that calculators not be used at all in grades K-5 and only sparingly in higher grades. Certainly there are isolated, beneficial uses for calculators, such as calculating compound interest, a seventh-grade topic in California. Science classes benefit from the use of calculators because it is necessary to deal with whatever numbers nature gives us, but conceptual understanding in mathematics is often best facilitated through the use of simple numbers. Moreover, fraction arithmetic, an important prerequisite for algebra, is easily undermined by the use of calculators.

Specific shortcomings

A number of the programs on ED’s list have specific shortcomings–many involving use of calculators. For example, a “promising” curriculum called Everyday Mathematics says calculators are “an integral part of Kindergarten Everyday Mathematics” and urges the use of calculators to teach kindergarten students how to count. There are no textbooks in this K-6 curriculum, and even if the program were otherwise sound, this is a serious shortcoming. The standard algorithm for multiplying two numbers has no more status or prominence than an Ancient Egyptian algorithm presented in one of the teacher’s manuals. Students are never required to use the standard long division algorithm in this curriculum, or even the standard algorithm for multiplication.

Calculator use is also ubiquitous in the “exemplary” middle school program CMP. A unit devoted to discovering algorithms to add, subtract, and multiply fractions (“Bits and Pieces II”) gives the inappropriate instruction, “Use your calculator whenever you need it.” These topics are poorly developed, and division of fractions is not covered at all. A quiz for seventh-grade CMP students asks them to find the “slope” and “y-intercept” of the equation 10 = x – 2.5, and the teacher’s manual explains that this equation is a special case of the linear equation y = x – 2.5, when y = 10, and concludes that the slope is therefore 1 and the y-intercept is -2.5. This is not only false, but is so mathematically unsound as to undermine the authority of classroom teachers who know better.

College Preparatory Math (CPM), a high school program, also requires students to use calculators almost daily. The principal technique in this series is the so-called guess-and-check method, which encourages repeated guessing of answers over the systematic development of standard mathematical techniques. Because of the availability of calculators that can solve equations, the introduction to the series explains that CPM puts low emphasis on symbol manipulation and that CPM differs from traditional mathematics courses both in the mathematics that is taught and how it is taught. In one section, students watch a candle burn down for an hour while measuring its length versus the time and then plotting the results. In a related activity, students spend a whole class period on the athletic field making human coordinate graphs. These activities are typical of the time sacrificed to simple ideas that can be understood more efficiently through direct explanation. But in CPM, direct instruction is systematically discouraged in favor of group work. Teachers are told that as “rules of thumb,” they should “never carry or grab a writing implement” and they should “usually respond with a question.” Algebra tiles are used frequently, and the important distributive property is poorly presented and underemphasized.

Another program, Number Power–a “promising” curriculum for grades K-6–was submitted to the California State Board of Education for adoption in California. Two Stanford University mathematics professors serving on the state’s Content Review Panel wrote a report on the program that is now a public document. Number Power, they wrote, “is meant as a partial program to supplement a regular basic program. There is a strong emphasis on group projects–almost the entire program. Heavy use of calculators. Even as a supplementary program, it provides such insufficient coverage of the [California] Standards that it is unacceptable. This holds for all grade levels and all strands, including Number Sense, which is the only strand that is even partially covered.”

The report goes on to note, “It is explicitly stated that the standard algorithms for addition, subtraction, and multiplication are not taught.” Like CMP and Everyday Math, Number Power was rejected for adoption by the state of California.

Interactive Mathematics Program, or IMP, an “exemplary” high school curriculum, has such a weak treatment of algebra that the quadratic formula, normally an eighth- or ninth-grade topic, is postponed until the 12th grade. Even though probability and statistics receive greater emphasis in this program, the development of these topics is poor. “Expected value,” a concept of fundamental importance in probability and statistics, is never even correctly defined. The Teacher’s Guide for “The Game of Pig,” where expected value is treated, informs teachers that “expected value is one of the unit’s primary concepts,” yet teachers are instructed to tell their students that “the concept of expected value is nothing new … [but] the use of such complex terminology makes it easier to state complex ideas.” (For a correlation of lowered SAT scores with the use of IMP, see Milgram’s paper at ftp://math.stanford.edu/pub/papers/milgram.)

Core-Plus Mathematics Project is another “exemplary” high school program that radically de-emphasizes algebra, with unfortunate results. Even Hyman Bass–a well-known supporter of NCTM-aligned programs and a harsh critic of the open letter to Secretary Riley–has conceded the program has problems. “I have some reservations about Core Plus, for what I consider too shallow a coverage of traditional algebra, and a focus on highly contextualized work that goes beyond my personal inclinations,” he wrote in a nationally circulated e-mail message. “These are only my personal views, and I do not know about its success with students.”

Milgram analyzed the program’s effect on students in a top-performing high school in “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later,” based on a statistical study by G. Bachelis of Wayne State University. According to Milgram, “…there was no measure represented in the survey, such as ACT scores, SAT Math scores, grades in college math courses, level of college math courses attempted, where the Andover Core Plus students even met, let alone surpassed the comparison group [which used a more traditional program].”

And then there is MathLand, a K-6 curriculum that ED calls “promising” but that is perhaps the most heavily criticized elementary school program in the nation. Like Everyday Math, it has no textbooks for students in any of the grades. The teacher’s manual urges teachers not to teach the standard algorithms of arithmetic for addition, subtraction, multiplication, and division. Rather, students are expected to invent their own algorithms. Numerous and detailed criticisms, including data on lowered test scores, appear at http://www.mathematicallycorrect.com.

How could they be so wrong?

Perhaps Galileo wondered similarly how the church of Pope Urban VIII could be so wrong. The U.S. Department of Education is not alone in endorsing watered-down, and even defective, math programs. The NCTM has also formally endorsed each of the U.S. Department of Education’s model programs (http://www.nctm.org/rileystatement.htm), and the National Science Foundation (Education and Human Resources Division) funded several of them. How could such powerful organizations be wrong?

These organizations represent surprisingly narrow interests, and there is a revolving door between them. Expert panel member Steven Leinwand, whose personal connections with “exemplary” curricula have already been noted, is also a member of the NCTM board of directors. Luther Williams, who as assistant director of the NSF approved the funding of several of the recommended curricula, also served on the expert panel that evaluated these same curricula. Jack Price, a member of the expert panel is a former president of NCTM, and Glenda Lappan, the association’s current president, is a coauthor of the “exemplary” program CMP.

Aside from institutional interconnections, there is a unifying ideology behind “whole math.” It is advertised as math for all students, as opposed to only white males. But the word all is a code for minority students and women (though presumably not Asians). In 1996, while he was president of NCTM, Jack Price articulated this view in direct terms on a radio show in San Diego: “What we have now is nostalgia math. It is the mathematics that we have always had, that is good for the most part for the relatively high socioeconomic anglo male, and that we have a great deal of research that has been done showing that women, for example, and minority groups do not learn the same way. They have the capability, certainly, of learning, but they don’t. The teaching strategies that you use with them are different from those that we have been able to use in the past when … we weren’t expected to graduate a lot of people, and most of those who did graduate and go on to college were the anglo males.”

Price went on to say: “All of the research that has been done with gender differences or ethnic differences has been–males for example learn better deductively in a competitive environment, when–the kind of thing that we have done in the past. Where we have found with gender differences, for example, that women have a tendency to learn better in a collaborative effort when they are doing inductive reasoning.” (A transcript of the show is online at (http://mathematicallycorrect.com/roger.htm.)

I reject the notion that skin color or gender determines whether students learn inductively as opposed to deductively and whether they should be taught the standard operations of arithmetic and essential components of algebra. Arithmetic is not only essential for everyday life, it is the foundation for study of higher level mathematics. Secretary Riley–and educators who select mathematics curricula–would do well to heed the advice of the open letter.

David Klein is a professor of mathematics at California State University at Northridge.

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Marks of a good mathematics program

It is impossible to specify all of the characteristics of a sound mathematics program in only a few paragraphs, but a few highlights may be identified. The most important criterion is strong mathematical content that conforms to a set of explicit, high, grade-by-grade standards such as the California or Japanese mathematics standards. A strong mathematics program recognizes the hierarchical nature of mathematics and builds coherently from one grade to the next. It is not merely a sequence of interesting but unrelated student projects.

In the earlier grades, arithmetic should be the primary focus. The standard algorithms of arithmetic for integers, decimals, fractions, and percents are of central importance. The curriculum should promote facility in calculation, an understanding of what makes the algorithms work in terms of the base 10 structure of our number system, and an understanding of the associative, commutative, and distributive properties of numbers. These properties can be illustrated by area and volume models. Students need to develop an intuitive understanding for fractions. Manipulatives or pictures can help in the beginning stages, but it is essential that students eventually be able to compute easily using mathematical notation. Word problems should be abundant. A sound program should move students toward abstraction and the eventual use of symbols to represent unknown quantities.

In the upper grades, algebra courses should emphasize powerful symbolic techniques and not exploratory guessing and calculator-based graphical solutions.

There should be a minimum of diversions in textbooks. Children have enough trouble concentrating without distracting pictures and irrelevant stories and projects. A mathematics program should explicitly teach skills and concepts with appropriately designed practice sets. Such programs have the best chance of success with the largest number of students. The high-performing Japanese students spend 80 percent of class time in teacher-directed whole-class instruction. Japanese math books contain clear explanations, examples with practice problems, and summaries of key points. Singapore’s elementary school math books also provide good models. Among U.S. books for elementary school, Sadlier-Oxford’s Progress in Mathematics and the Saxon series through Math 87 (adopted for grade six in California), though not without defects, have many positive features.–D.K.

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For more information

Askey, Richard. “Knowing and Teaching Elementary Mathematics.” American Educator, Fall 1999, pp. 6-13; 49.

Ma, Liping. Knowing and Teaching Elementary Mathematics. Mahwah, N.J.: Lawrence Erlbaum, 1999.

Milgram, R. James. “A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California.” ftp://math.stanford.edu/pub/papers/milgram

Milgram, R. James. “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later.” ftp://math.stanford.edu/pub/papers/milgram/andover-report.htm

Wu, Hung-Hsi. “Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education.” American Educator, Fall 1999, pp. 14-19; 50-52.

Seebach: Race-gap study launches 3-stage rant about readers

April 30, 2005

Everybody knows that American blacks and Hispanics are at a disadvantage to whites and Asians both in education and income. Three economists have written a paper demonstrating that the patterns of disadvantages for blacks and for Hispanics are very different, raising questions about the explanations often given for those disadvantages.

In the authors’ own words, here are some take- away points:

 “For black males, controlling for an early measure of ability cuts the black-white wage gap in 1990 by 76 percent. For Hispanic males, controlling for ability essentially eliminates the wage gap with whites. For women the results are even more striking. Wage gaps are actually reversed, and controlling for ability produces higher wages for minority females.”

• “When we control for the effects of home and family environments on test scores, the Hispanic-white test score gap either decreases or is constant over time while the black-white test score tends to widen with age.”

• “Hispanic children start with cognitive and noncognitive deficits similar to those of black children. They also grow up in similar disadvantaged environments, and are likely to attend schools of similar quality. Hispanics have substantially less schooling than blacks. Nevertheless, the ability growth by years of schooling is much higher for Hispanics than blacks. By the time they reach adulthood, Hispanics have significantly higher test scores than blacks.”

• “Our analysis of the Hispanic data illuminates the traditional study of black-white differences and casts doubt on many conventional explanations of these differences since they do not apply to Hispanics who also suffer from many of the same disadvantages.”

I know this is contrary to just about everything you’ve heard or read, so you’re asking, “Who are these people?” They’re Pedro Carneiro, University College London; James J. Heckman, University of Chicago, American Bar Foundation and University College London (and winner of the 2000 Nobel Prize in economics for developing the kind of technical statistical analysis that undergirds this paper) and Dimitriy V. Masterov. The paper was written for the Institute for Labor Market Policy Evaluation, a part of the Swedish Ministry of Industry, Employment and Communications, in Uppsala, Sweden.

The paper is “Labor market discrimination and racial differences in premarket factors” and it’s at www.ifau.se/swe/pdf2005/wp05-03.pdf on the Web.

They don’t argue against current policies on affirmative action – though they certainly could, based on their evidence – merely that policies addressing very early skill gaps are likely to do more good than additional affirmative action policies aimed at the workplace.

One possible explanation of persistent wage gaps is that there is “pervasive labor market discrimination against minorities.” Another, which they observe is equally plausible, is that “Minorities may bring less skill and ability to the market.” And of course both could be true in varying degrees, but I think this is the most important thing they say: “The two polar interpretations of market wage gaps have profoundly different policy implications.”

And how. So if you’re a policy-maker, Go Read The Whole Thing.

Now, since I have room for only a tiny bit of what’s significant in this paper anyway, I’m going to address a different issue that invariably comes up when I write about something so contrary to received opinion.

OK, (/turn rant on/) don’t waste your time writing me that I “haven’t considered” whatever particular bee is buzzing around your bonnet. You have no information about what I have considered; you know only what I have mentioned. And let me tell you, when I’m writing an 800-word summary of a highly technical 50-page paper bristling with statistical analysis, there’s a lot I’ve considered that I don’t mention. There’s even more data that the researchers have considered that they don’t mention – although the existence of Web appendices to scholarly papers has ameliorated that problem.

Next, don’t think you’re being erudite by citing some cliche about “lies, damned lies or statistics,” which I understand is properly credited to Benjamin Disraeli, but is often attributed to Mark Twain. Yes, it is possible to lie with statistics – there’s a charming and useful little book with that in the title – but it’s a lot harder to lie with statistics than without them.

Case in point, the current flap over the number of deaths statistically attributable to obesity. If you’re one of those people who fatuously asserts that “you can prove anything with statistics” I challenge you to find me a peer-reviewed journal article proving that smoking enhances longevity, or that women are taller than men.

Last, don’t talk about motives. You have no evidence about my motives aside from what I tell you – and I could be wrong about that; lots of people are. Even if we were both right about my motives, it would have no bearing on the cogency of my arguments, which do not adduce them; that’s the ad hominem fallacy. “Fallacy,” please note, which means that even if your premises are correct, your conclusion may be wrong. (/Turn rant off/)

Oh, I feel much better now. Excellent paper.

 

 

Linda Seebach is an editorial writer for the News. She can be reached by telephone at (303) 892-2519 or by e-mail at seebach@RockyMountainNews.com.

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Where’s the Math?
– DEBRA J. SAUNDERS
Sunday, October 17, 1999

 

THIS MONTH, the U.S. Department of Education came out with a list of 10 “exemplary” or “promising” math education programs. Kings County fourth-grade teacher Doug Swords was shocked at the department’s bad choices.

Some three years ago, his school district adopted MathLand, a math curriculum that prefers not to give lessons with “predetermined numerical results.” The department of Educrats, oops, I mean, Education, rated MathLand as “promising.” Today, he said 14 out of 18 teachers use MathLand only as a supplement. “I stashed away my Addison-Wesley textbooks, as did a few other teachers,” he explained.

Do you teach your students how to multiply? I asked him. (You wouldn’t think that would be something I’d have to ask, but these days, it is.) Yes, he said. Is MathLand helpful in teaching kids to multiply? “No, quite frankly,” Swords answered.

UC Berkeley math professor Hung-Hsi Wu couldn’t believe the department described MathLand as “promising.” He’d describe MathLand as “execrable.”

Or how about: “I can’t believe it’s math class.” A second-grade MathLand exercise called Fantasy Lunch instructs students to think up their fantasy lunch, draw it on paper, then cut out the “food” and place their drawings into a bag.

A frantic teacher wrote to me two years ago, furious that she had spent 75 minutes on that exercise and there was no math in it. It was “like therapy,” she said. On more than one occasion, her students asked her, “Can we do some real math now?”

Wu had problems with the other nine picks as well. While there were things he liked about the high school programs, they lacked what he called “mathematical closure. You start something, you ought to finish it.”

He said almost all of his students took more traditional math classes not cited as “exemplary” or “promising” by the Department of Education. That wouldn’t surprise Melissa Lynn, who got As in high-school math, then placed in the bottom 1 percent in the University of Michigan math placement test. She blames the Core-Plus program which the department rated as “exemplary.” “It had very good intentions, and wanted you to apply real principles to real life scenarios,” she explained this spring, “but it was missing the crucial element of algebra.“

Wayne Bishop, a math professor at Cal State L.A. who is the Ralph Nader of math curricula, sees the department’s move as a reaction against California’s return to math sanity — after a mad fling when state educrats embraced “there is no right answer” new-new math curricula.

He’s right. The selection panel appoint ed by the department had as a main criterion that the math series ascribe to trendy standards put out by the National Council for Teachers of Mathematics (NCTM).

Don’t ask me why. Last year Bishop looked at the scores of some of the students subjected to the brilliance of new- new math wizards. In 1995, NCTM Chairman Jack Price boasted about a program on which he worked. Turns out, Price’s star school ranked in the bottom quartile nationally in the STAR test last year. Only 12 percent of the school’s eighth graders scored above the national average. Price called that a successful program.

The department cited data that show schools whose test scores improved with MathLand. Bishop isn’t impressed. “They appear to have excluded data where MathLand scores dropped,” he noted.

An administrator from an urban district that stopped using MathLand had just visited a school that had seen a 27 percent increase in its math scores after buying a traditional math series that didn’t rate in the department’s Top 10. Under ideal circumstances, he said, MathLand could work, but urban districts don’t have too many ideal circumstances.

Bill Evers of Stanford’s Hoover Institution called the department’s Top-10 picks “unconditional surrender to fuzziness.”

Fuzziness? The department praised one K-6 math program because, “Features include problem solving; linking past experience to new concepts; sharing ideas; developing concept readiness through hands-on explorations; cooperative learning through small-group activities; and home-school partnerships.”

Sounds more like marriage counseling than math class.

The problem: It’s not the kids who need counseling here. It’s the adults who care so little about children’s success that they would assert that Fantasy Lunch makes for a “promising” math program.

 

You can reach Debra J. Saunders on The Gate at sfgate.com.

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©2005 San Francisco Chronicle

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DEBRA J. SAUNDERS — Man of Science Has a Problem With Real Math
– DEBRA J. SAUNDERS
Friday, December 19, 1997

THIS STORY demonstrates why you can’t trust Clinton’s education gurus to write national tests for America’s students. If there’s a sure thing in life, it’s that D.C. educrats will dumb down any subject, given half a chance and millions of dollars.

The tale begins this month as California’s state Board of Education was about to vote on math standards for public school students. A standards panel had written a document rich in trendy educratese. (“Show mathematical reasoning in solutions in a variety of ways.”) The board wanted — and ultimately approved — a meatier document with solid standards for computation and less fluff about writing about math. [an error occurred while processing this directive]

By injecting more math into math — actually expecting kids to memorize multiplication tables in the third grade and master long division in the fourth — the board invited the ire of state schools chief Delaine Eastin and the federal government. On December 11, the day before the final vote, Luther S. Williams, assistant director of the federally funded National Science Foundation, fired off a letter to board president Yvonne Larson. Basics wags call it “the blackmail letter.”

Williams, who didn’t call me back, criticized the new standards for not “elevating problem-solving and critical thinking.” His letter chided the board for preferring the “wistful or nostalgic `back-to-basics’ approach,” which he wrote, “has chronically and dismally failed.”

He then reminded Larsen that his bureaucracy gives grants totaling more than $50 million of taxpayer money to six California school districts, including Oakland. “You must surely understand,” he wrote, that his group “cannot support individual school systems that embark on a course that substitutes computational proficiencies for a commitment to deep, balanced, mathematical learning.”

On what planet does this man of science live?

First, Williams has a little jurisdictional problem. President Clinton says he doesn’t want the federal government to butt into local school business. Also, the guy works for a science — not math — agency. But he is so arrogant and power drunk that he feels free to sic his Science Foundation on California math dissidents.

Second, the state’s commitment to “deep, balanced mathematical learning” — aka new-new math — has resulted in computational deficiencies, as well as general arithmetical idiocy. For some years, trendy California educators have focused on students writing about math, repeatedly explaining how equations work and exploring their feelings about math. They’ve also taken to giving students credit for wrong answers. Thus, “critical thinking” has come to mean not being critical of students.

The result: In the last National Assessment of Educational Progress math test, California fourth-graders scored behind students from every state but Mississippi and Louisiana. Only 13 percent were rated proficient. Eastin has suggested that the state board should “get out of the dark ages.” She ought to get the schools out of the dark ages.

No wonder some parents are “nostalgic,” as Williams put it, for the days when basics were emphasized, and cash registers all had numbers on them instead of pictures of hamburgers. Back in the days of what Williams classified as failure, students scored an average of 22 points higher on math SATs.

Here’s a novel thought. Let the National Science Foundation give a grant to solve the great mystery of modern education: How is it that swells like Williams can look at the 1950s as years of math failure, but see no problem with high- school kids needing a calculator to compute 10 percent? How can you say you stand for problem solving without being able to recognize a problem?

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1999 Conference on Standards-Based K-12 Education California State University Northridge Transcript of R. James Milgram (edited by the speaker) Biography
.		Return to conference page Return to transcript of Norman Herr   Mr. Milgram: I would like to start by again thanking David Klein and Cal State Northridge for arranging and organizing this wonderful opportunity to get together and compare ideas on the incredibly challenging times ahead of us.  Professor Wu brought up a number of critical points in his discussion and one of them that he mentioned — that this is a long term challenge — is particularly important. I’d like to fill in somewhat what the problem is here. First of all, “long term” has generally been understood to be in the order of perhaps three years, and there seem to be real expectations of being able to meet the standards in that time frame.  But this is very unrealistic! A realistic long term is maybe 15 years. If we are lucky, in 15 years the average student may get near the standards if everything goes just right. In a shorter time than that, it is almost inconceivable to believe that this will happen. California today ranks just about at the bottom in the United States, in terms of the level of mathematical achievements of students in K-12. The United States ranks near the bottom among all the developed countries in the world in terms of math achievements of students. We have an incredibly long way to go because you have to remember that the new California Mathematics Standards were written to match the levels of the standards of the top achieving countries in the world. Meeting these standards is a daunting challenge and we had better take it seriously. We now look at the reasons we clearly needed new standards in mathematics.  They can be subsumed in three main areas. REASONS FOR NEW STANDARDS The increasing failure of the present system to produce enough technically skilled graduates to meet national needs  Curricular problems which leave more and more students without the prerequisites needed for their majors, particularly in technical areas  Lack of a clear understanding – on the part of teachers and math educators – of the major goals of the mathematics component of K-12 education   The next three slides explain a little bit about how we see some of this so we cannot escape from these issues.  The facts quoted in these slides come from recent newspaper articles for the most part. INDICATIONS OF FAILURES From 1990 to 1996 there has been a 5% decline in high-tech degrees — engineering, math, physics, computer science — in this country and the trend is continuing.  Of the decreasing number of high-tech degrees awarded a significant and growing proportion go to foreign nationals.  At the doctorate level 45% of high-tech degrees were granted to non-U.S. Citizens   From 1990-96, there’s been a 5% decline in high-tech degrees overall in this country. And the trend is continuing — in fact, the trend is accelerating.  Even though the number of high-tech degrees is decreasing, it is vital to note that an ever increasing portion go to foreign nationals. At the doctorate level, for example, 45% of high-tech degrees are granted to non-U.S. Citizens and at Stanford, in the mathematics department, two thirds  of our graduate students are foreign-born. Even 10 years ago, less than half were. As a result of this situation it has been impossible to fill all our technical jobs with United States citizens.  This is particularly true in Silicon Valley.  To find qualified people to fill these positions Congress was intensely lobbied by Silicon Valley, and Congress was forced, much against their will, to provide 142,500 more visas for foreign nationals to fill jobs in Silicon Valley. Currently it is estimated that the number of foreign-born residents of Silicon Valley is about 25% of the population. Among all the states as I said in the beginning, California colleges showed the greatest decline in high tech degrees. INDICATIONS OF FAILURES – II Last year Congress was forced to provide 142,500 more visas for foreign nationals with high-tech skills  Currently it is estimated that the number of foreign born residents of Silicon Valley is about 25% of the population  Among all states, California’s colleges showed the greatest decline in high-tech degrees awarded.   So the first point is that the system today is simply failing to produce enough technically qualified graduates to meet national needs. The foremost problems and most dramatic declines are here in California. Curricular problems are overwhelming here and leave more and more students without prerequisites needed for developing and learning technical skills in college. When they come to us, even at Stanford, more and more of them are just not able to become engineers and scientists, even though this is their original intent. They just don’t have the background any more. It is a dramatic change. Finally, and sadly, because I have the utmost respect, and I think we all do, for the practicing teachers, the level of understanding on the part of teachers and above all of math educators — that is members of the educational schools throughout the country — that is required for teaching mathematics in K-12 is just not there any more. Look at the effect of this lack of understanding on our students. INDICATIONS OF FAILURES – III The percentage of entering students in the California State University System who are place into remedial mathematics courses after taking the ELM placement exam is about 88%  Overall, well over 50% of entering students are placed into remedial mathematics courses.  The average level of the questions on the recent version of the ELM is about grade level 6.9 according to the new California Standards.   This 88% is a statistic that astounded me. And it is correct, differing from the failure rates commonly reported (which are bad enough). The percentage of entering students in the California State system who are placed into remedial mathematics courses after taking the ELM placement exam is 88%.  Let me emphasize this: 88% of those students taking the exam fail it. Some of you may know a statistic of about 55% for the failure rate.   Unfortunately, this is calculated by counting the 40% of the entering students who are not required to take the exam as having passed it. These 40% are counted as passing it probably so the statistic will look reasonable.  I reiterate that the actual statistic is 88% taking the ELM fail it, and it is not that hard an exam overall.  In any case, well over 50% of entering students in the California State University system are placed into remedial math courses. Those are some of the reasons for our current problems. They stare at us. We can’t avoid or deny them. Now, I would like to give you an idea of the real complexity of the problem and the consequent difficulty with trying to fix it. On our first slide the second problem with mathematics that I indicated is the lack of understanding of curricular development on the part of math educators. Curricular development is a very complicated issue.  As an illustration, I’m going to look at one topic, long division, now. Long division is something that a lot of professional math educators want to take out of curriculum. So let’s just look at why it is in the curriculum.   CURRICULAR PROBLEMS The recent fashion of not teaching material like long division and factoring polynomials is based on claims that such skills are no longer useful.  This reflects a deep lack of understanding of the role of mathematics in fields like science, engineering and economics.  In mathematics many skills must be developed for many years before they can be used effectively or before applications become available.   First of all, I claim that taking — even asking to take it out of the curriculum — shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they’ve been introduced.  Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples. SOME SKILLS DIRECTLY ASSOCIATED WITH LONG DIVISION Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division.  Long division is essential in learning to manipulate and factor polynomials.  Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial fractions and canonical forms.   So just to start, understanding that decimals represent rational numbers if and only if they are terminating or ultimately repeating — a skill that was requested be put into the standards by math educators — cannot be understood without long division.  It is only in understanding of the process of taking the remainder in long division that you see the periodicity or termination happen. I regard the repeating decimal standard as relatively minor, but some people seem to think it is important. The next topic is critical and almost everyone thinks it’s minor (Laughter). Long division is essential to learning to manipulate polynomials. Without it, you simply cannot factor polynomials. So what, you ask?  Again, this is a question that doesn’t come up until the third year in college.  At this point the skills that have come from long division through handling polynomials become essential to things like partial fraction decomposition which is important in calculus but finds its main applications in the study of systems of linear differential equations, particularly in using Laplace transforms, which is the critical construction in control theory.  It is also essential in linear algebra for understanding eigenvalues, eigenvectors, and ultimately, all of canonical form theory — the chief underpinning of optimization and design in engineering, economics, and other areas. The previous slide indicated  what I call the static applications of long division. The next slide illustrates  some of the “dynamic” applications.   DYNAMIC SKILLS ASSOCIATED TO LONG DIVISION The process of long division is one of successive approximation, with the accuracy of the answer increasing by an order of magnitude at each step.  The skills associated with this process become more and more fundamental as students advance.  They include all infinite convergence processes, hence all of calculus, as well as much of statistics and probability, to say nothing of differential equations.  Long division is the main application of the previously learned skills of approximation.   Long division is the only process in the K – 12 mathematics curriculum in which approximation is really essential. The process of long division is a process of repeatedly approximating and improving your estimates by an order of magnitude at each step. There is no other point in K – 12 mathematics where estimation comes in as clearly and precisely as this. But notice that long division is also a continuous process of approximation, the answer keeps getting more and more accurate and when the students learn how to do long division with decimals they learn to carry the process to many decimal places.  This leads naturally — in a well conceived curriculum — to students understanding continuous processes, and ultimately even continuous functions and power series. The development of these skills are all contingent on a reasonable development of long division.  I don’t know of any other or any better preparation for them. What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I’m referring to here is the experience of my students in a differential equations class in the fall of 1998.  The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations.   Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school. But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster.  Moreover, it was very difficult for them to fill in the gaps in their knowledge.  It seems to take a considerable amount of time for the requisite skills to develop.   APPLICATION OF THE SKILLS ASSOCIATED TO LONG DIVISION The combination of these skills is used critically in economics, engineering and the basic sciences via Laplace transforms and Fourier Series.  Without a thorough grounding in these topics it is impossible to do more than routine work in most areas of engineering, the most active current areas of economics and generally, any area involving optimization.   So you see the problem. The problem is that the scope of things in mathematics is so long that an ordinary second, third, fourth grade teacher is not equipped to make a judgment about whether a subject is needed or not needed.     SCOPE IN THE MATHEMATICS CURRICULUM The long division story illustrates one of the chief problems with curricular development in mathematics. The period needed before a learned skill can be fully utilized can be as long as eight to ten years.  It takes real knowledge of mathematics as well as how it is applied to make judgements regarding curricular content.   I think the long division problem illustrates the problem described on the slide above very well. And I put that dragon up there advisedly.   EDUCATORS TELL US OF THE NEED FOR CONCEPTUAL UNDERSTANDING AND MATH REASONING SKILLS IN OUR STUDENTS These skills ARE critical in todays technological society.  What many math educators tell us represent examples and exercises for developing these skills are NOT relevant and/or NOT correct.   The first slide mentioned a third aspect of the problem, which was the lack of knowledge of the subject on the part of math educators. To make it clear, I’m talking about math educators and not teachers. Teachers learn what they are told in the education schools and just hope that this background prepares them sufficiently.  They do the best they can and have the most demanding job that I know of.  As a group I believe they are the most dedicated people I know of. But if you do not provide teachers with the proper tools, they can’t do a proper job.   MATH EDUCATORS OFTEN HAVE LIMITED KNOWLEDGE OF MATHEMATICS For example, three of the 14 problems originally proposed by the presidential commission on the eighth grade national mathematics text and/or the “solutions” they gave were INCORRECT. This commission included many of the best known math education experts in the country.  The next slides discuss one of these problems.   I just want to spend a few minutes, now, looking at some of the problems that we have seen in the last few years when we — as professional mathematicians — have looked at some of the things that math educators are trying to tell the world is mathematics.  I will concentrate on problems that these people suggest for testing mathematical knowledge.   A PROBLEM FROM THE NATIONAL EIGHTH GRADE EXAM   We are given the following pattern of dots: At each step more dots are added than were added at the last step. How many dots are there at the twentieth step?   This is a problem from the original proposed 8th grade national exam, produced by a presidential commission including most of the best known math educators in the country.  The problem appears to be simple and every person I’ve asked, who I haven’t warned to think hard and carefully about it, has answered immediately, “Oh, it’s of the form n times n plus 1, so you are looking at the 20th stage, therefore the answer is 20 times 21.” But that’s not right. The words need to be read carefully. The point is, the words tell you the only thing you are actually given — namely, that there are more dots added at each stage than the previous stage. That’s all you are given, and the picture is just a picture.   ANALYSIS OF THE PROBLEM The answer given by the Presidential Commission on the National Eighth Grade Exam was 20 X 21 = 420 This is incorrect! The correct answer is that any number of dots is possible as long as there are at least 267.  As was pointed out, the Presidential Commission that proposed this problem included many of the best known math educators in the country.       ANALYSIS OF THE PROBLEM – II This can be seen by considering that you must add at least seven dots to get to the fourth stage, eight to get to the fifth, nine to get to the sixth, and so on, but, of course, you can always add more.  So the formula for the number of dots at the nth stage with n>2 becomes: any number at least as big as 6+(6+7+8+…+(n+3)) which equals any number at least as big as (n+3)(n+4)/2 – 9 = 267   Hmm?  Actually that problem was about as complicated as any problem I’ve seen at this level, and it was proposed for the 8th grade national exam! When you read it carefully, it is a problem a 12th grade senior would have trouble solving. So what is the moral here? If you want to learn mathematics, you must learn it precisely. Mathematics is precision and one of the first objectives in teaching K – 12 mathematics is for students to learn precise habits of thought. The next slide presents a problem that Wu is very fond of (Laughter).  It can be found in many sources, but in particular it was included as part of the original Mathematics Standards Commission’s proposed California Mathematics Standards.   A PROBLEM FROM THE ORIGINAL STANDARDS COMMISSION STANDARDS   You have a friend in another third grade class and want to determine which of your classrooms is bigger. How do you do it? This problem is often proposed as an example which shows that “there is no single correct answer” since you could use perimeter or volume or area to measure size. Of course, this is incorrect!   The trouble is that bigger is not precisely defined. And if every term is not precisely defined, your problem is not well posed. So technically this is not a well-posed problem. Of course, we realize that is a little technical.  We have an idea that bigger has certain connotations — but unfortunately, a lot of them: perimeter, area, volume, and maybe even combinations of the three such as 3A + 2.4P + 7V.   ANALYSIS OF THE COMMISSION PROBLEM   The difficulty here is that bigger is not precisely defined, and to do mathematics you generally have to know exactly what each term means. However, mathematics does provide for the situation where terms can have different meanings. There is still a single “correct” answer. It consists of the set of all answers. But since bigger can mean anything, the set of answers is uncountably infinite, and this problem is totally inappropriate for any but the most advanced high school students.   You see, when you put in a linear combination of the three, you get an uncountable number of possible definitions of bigger. That’s all right. Mathematics allows for this, as long as you can make some sense of the problem.  Mathematics says the correct answer to the problem is all possible answers to the problem (Laughter). If you are going to take that problem at face value, you have to give me an uncountable number of answers.   MORE DETAIL ON SOLUTIONS   Here is an example which illustrates the point that the “answer” is a collection of “all solutions”. Consider the system of two equations in three unknowns: 2x + y + z = 1 x + 2y + z = 0 A solution is x = 1, y = 0, z = -1. The answer is x = 1 + y z = -1 – 3y     So, what is the point? One of the most important things, as I indicated, that students should learn in doing mathematics is precise habits of thought. Suppose we start with a “real world problem”, given, as is typical for such problems, very imprecisely.  We want students to be able to break the problem apart into smaller problems, make sense of them, and solve them or recognize that it is not possible to solve them with the information given.  One of the first things that mathematics should prepare student for is making the best possible (rational) decisions when faced with real problems.   SUMMARY – I   One of the most important things that students should learn from studying mathematics is precise thinking. They should understand how to recognize when a problem is well-posed. They should be able to decompose a possibly ill-posed problem into pieces which can be made well-posed, and solve the individual sub-problems.   Now, I don’t for a minute want to minimize the fact that students have to learn basic number skills, certainly they have to do that too. And they have to learn things like statistics, I mean, this is critical in our world today, and it is a wonderful thing that it is commonly taught today.  It helps prepare students to defend themselves from tricky claims and fake uses of statistics.  Students also have to learn how to survive in the monetary world. So a key part of our request for changes  when the State Board of Education asked some of us at Stanford to help revise the California Math Standards was that compound interest be put back into the 7th grade standards.   SUMMARY – II   They should also learn the basic mathematical skills needed to survive in today’s society. These include basic number-sense They also include skills needed to defend themselves from sharp practices, such as being able to determine the real costs of borrowing on credit cards. Additionally, they include being able to recognize illegitimate uses of statistics.   I think everybody has the idea now. I have many more problems here, all of which are incorrect and all of which are due to some of the top math educators in the country. But I think you all get the idea of what the level is here and what we are trying to deal with, so I think we can skip most of them.  But there is one more example that is  worth noting (Laughter).   A PROBLEM FROM THE NEW NCTM STANDARDS   The following is proposed as a Kindergarten problem: How big is 100? This suffers from exactly the same difficulty. I asked one of our best graduating seniors this problem (he has a fellowship to study in Germany for next year and the year afterwards will continue his graduate work at Harvard).   This is from the current new proposed version of the NCTM standards. “How big is 100?” It suffers from every one of the flaws I mentioned before. But I loved the response from the student above. A PROBLEM FROM THE NEW NCTM STANDARDS II   Without even a moment’s hesitation he answered: Oh, about as big as 100! Indeed, any other answer would involve elements of perception and psychology, not mathematics.   Okay. I think probably I’ll finish up now and say again that it’s a long process ahead. It is a serious, serious thing we are trying to do. But I think it is something that we can do. It’s just something we cannot treat lightly and cannot treat in any way as a casual enterprise.  For example if you hear someone say something to the effect that “Oh, we’re going to give the teachers the Standards. We are going to say, now teach — and it’s over — no problem,” be very suspicious. IMPLEMENTING THE MATH STANDARDS   Problems California students rank at or near the bottom among all the states in average mathematics competency Generally teachers in grades K-4 have little competence in mathematics above their grade levels Expectations We cannot solve these problems all at once Time is needed, and skills and competencies should be introduced gradually. The new California Math Framework shows the most important skills that must be learned first.     It is a huge process — of re-education on everyone’s part, it is a process we all have to contribute to and work on with full attention. But I think there are grounds to hope that we can actually do it. And the one thing that has the potential to help with this process is the Framework.  The Framework is something that Wu and I worked on with Janet and the Curriculum Commission, and with many of the best people in many aspects of education throughout the country.  The Framework has been designed to ease our way into the teaching to the Standards. It’s something that I think we have to focus on a lot more in the next few months as we try to figure out how to reach the levels needed. I would like to just say one word about one of the ways in which the new Framework can help. IMPLEMENTING THE STANDARDS   In first grade there are only five emphasis topics in the Framework out of 30 total topics: Count, read and write whole numbers to 100 Compare and order whole numbers to 100 using symbols for less than, greater than or equal to Know the addition facts and corresponding subtraction facts (sums to 20) and commit to memory Show the meaning of addition and subtraction Explain ways to get the next element in a repeating pattern     The critical thing about this is that the Standards for first grade have about 30 basic topics. Well, those topics are, for the most part, quite difficult at the first grade level and will take a great deal of time and effort to teach properly. Fortunately, it turns out that only 5 or so of them are essential. The Framework identifies the essential standards and makes your jobs as teachers and your jobs as curriculum developers much easier because the textbooks in the next textbook adoption will be focused on the emphasized topics, rather than the entire 30 topics in the Standards. So this will allow us to focus on just a few pieces and make your job of reaching the levels needed a little simpler. I think this is where I’ll stop (Applause).
.		Contact the organizers Postal and telephone information: 1999 Conference on Standards-Based K12 Education College of Science and Mathematics California State University Northridge 18111 Nordhoff St. Northridge CA 91330-8235 Telephone: (Dr. Klein: 818-677-7792) FAX: 818-677-3634 (Attn: David Klein) email: david.klein@csun.edu clipart: http://www.clipartconnection.com/, http://web2.airmail.net/patcote1/partydan.gif

NYC MATH WARS
NYC Honest Open Logical Debate (NYC HOLD)On Math Reform
Elizabeth Carson

November 27, 2000

New York, NY. NYU mathematicians speak out against controversial new math programs being taught in NYC’s premiere School District 2 in Manhattan. Community School Board 2 will hear the professors’ preliminary report at the calendar meeting on Tuesday, November 28, 6:30 pm at 333 7^{th Ave, Seventh Floor.}

^{A groundswell of parent concern over how math is being taught in District 2 schools led to a front page New York Times article last spring, “The New, Flexible Math Meets Parent Rebellion.” CBS Weekend News brought national attention to the local struggle last May in the segment, “New, New Math = Controversy.”}

^{District 2 parents have now gained the support of mathematicians at NYU and the University of Rochester in grieving their concerns to District officials. The professors are scheduled to take part in a Math Forum sponsored by Community School Board 2 scheduled for March 1, 2001.}

^{The controversy in District 2 has gained the attention of Schools Chancellor Harold O. Levy and members of his recently established Commission on Mathematics Education charged with investigating how math is being taught in schools across the city.}

^{District 2 pilots the latest wave of experimental math programs, called the “new, new math,” echoing an earlier failed reform in the 60’s called “new math.” Memorization of math facts is no longer emphasized. Children are encouraged to use language to describe solutions and the way they feel about math.}

^{Community response in NYC and across the country has erupted in what have become known as the “math wars.” Critical parents, joined by mathematicians and scientists advocate clarity and balance in math reform: urging the inclusion of grade by grade goals, explicit teaching of standard procedures, basic skill building and rigor along with the inclusion of some of the creative exercises in the new programs. The pendulum has swung too far and must be corrected.}

^{One parent, Mark Schwartz, in testimony last year before the House Education and Workforce Committee stated: “If medical doctors experimented with our kids in the same fashion school districts do they would be in jail.” The hearings were held to review the US Department of Education’s endorsement of 10 of the experimental programs.}

^{Over 200 of the nation’s top mathematicians, including seven Nobel Laureates,} winners of the Fields Medal ^{the department heads at more than a dozen universities including Caltech, Stanford and Yale, responded to the federal endorsements with an open letter of protest to Secretary of Education Richard Riley, published in the Washington Post in November, 1999.}

^{“These programs are among the worst in existence,” said David Klein, a Cal State Northridge professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be joke if it weren’t so damaging.” Several of the cited programs, Interactive Mathematics Project (IMP),Connected Mathematics Project (CMP) and Everyday Mathematics are now being used in NYC schools.}

^{The new wave of math reform is based on a “constructivist” teaching philosophy; emphasizing creative exercises, hands-on projects and group work, with far less attention given to basic skills. Students are asked to ‘construct’ their own solutions. The use of calculators is encouraged. Teachers are instructed to serve as “facilitators” and are discouraged from explaining to students the standard solutions of basic arithmetic. Practice and drill have been eliminated. In higher grades algebra is de-emphasized. Many of the programs have no textbooks.}

^{Members of Chancellor Levy’s Math Commission will consider the new programs, which are being used in over 60% of NYC schools, including roughly 50 of the city’s weakest, which comprise the Chancellor’s District . Plans are set to expand implementation into more schools. One of the most controversial programs, the Interactive Mathematics Project (IMP), will be mandated in Bronx High Schools beginning next year.}

^{Professor Richard Askey, who holds an endowed chair in math at the University of Wisconsin, explained his motives in co-authoring the protest letter to Secretary Riley, “I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs.”}

^{High math scores remained stable in some of the privileged District 2 schools, last year; though some schools’ scores dropped significantly. Across the rest of the city, 75% of eighth grade students failed the state math test ; 66% failed the city math tests. Answers are critical at a time when graduation requires students pass a new math Regents exam.}

^{It was the introduction of Connected Mathematics Project (CMP) and Investigations in Number Data and Space (TERC), two of the District 2 programs, that sparked the initial parent revolt that led to the California Math Wars.} Six parents in Plano,Texas have filed suit in federal court against their local school district after parent requests for an alternative to CMP were denied. ^{A nuclear physicist in Okemos Michigan led the local campaign against CMP. The use of TERC in one school system in Massachusetts prompted members of the Harvard Mathematics Department to issue a public protest. Parents in Reading Massachusetts fought the adoption of Everyday Math.}

^{The “new, new math” programs are based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards. Frank Allen, past president of the NCTM and Emeritus professor of Mathematics at Elmhurst College comments: “NCTM leaders must admit that they have urged the application, on a national scale of highly controversial methods of teaching before they have been adequately debated or understood and before researchers have verified them by well-controlled and replicated studies.”}

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November 27, 2000

To: Harold O. Levy, Chancellor

Dr Judith Rizzo, Deputy Chancellor for Instruction

Burton Sacks, Chief Executive for Community School District Affairs

Dr Irving Haimer, Member, Board of Education, Manhattan Representative

Trudy Irwin, Director of Education, Office of the Manhattan Borough President

From: NYC HOLD, Steering Committee

Re: Community School Board 2 Calendar Meeting November 28, 2000

We respectfully invite you to attend the calendar meeting of Community School Board 2 in Manhattan scheduled for Tuesday November 28, beginning at 6:30 pm. During the public session the Board will hear comment on District 2’s K-12 math programs. Distinguished faculty of the Courant Institute of Mathematical Sciences at New York University and concerned parents plan to speak.

The District’s mathematics reform is one component of the systemic instructional reform initiated by Anthony Alvarado in the previous decade. District 2 prides itself in being a leader in education reform in New York City. Our District piloted the New Standards Performance Standards in Mathematics which were adopted city-wide last year.

The implications of the relative success of our District’s math reform for the entire NYC school system should be evident.

In our District, parent concern with their children’s progress in mathematics is escalating; worries are being strongly voiced about aspects of the new math programs just as full implementation takes hold. School year 1999-2000 marked the first year the new K- 8 programs, Investigations in Number Data and Space (TERC) and Connected Mathematics Project (CMP) were mandated.

Parents are tutoring in record numbers. For many of their children, the new constructivist programs fail to provide a sound foundation in basic arithmetic. Parents recognize the value of the new programs’ creative, hands-on classroom and homework activities. However, many question the extensive time devoted to such activities, ostensibly at the expense of explicit teaching of the standard procedures and practice of skills. The absence of textbooks has exacerbated the situation.

Concerned parents have reached out to the NYU mathematics community in District 2 for analysis and opinion of our programs in hopes of finding ways to amend or extend the implemented programs to reach a better balance in the math instruction.

There are clearly implications in the course the District 2 community takes for the work of the newly appointed Commission on Mathematics Education to review math programs city-wide. We hope our experiences in District 2 can contribute to the assessment of the Committee.

Parents, teachers and administrators in District 2 share a well earned pride in our exemplary schools and strong educational community. Parents see on a daily basis, through their children’s work and enthusiasm for learning, the results of the skill and commitment of the teaching staff and administrators in our schools. Parents appreciate the formidable challenges district administrators face in guiding systemic reform in a large and diverse community school district.

We look forward to an ongoing and meaningful partnership in District 2.

We sincerely hope you will take an interest in our concerns and efforts to advance the course of math education for our children.

Sincerely,

Steering Committee
NYC HOLD

Elizabeth Carson
Christine Larson
Garry Dobbins
Margaret Hunnewell
Mary Somoza, Member, CSB #2
Granville Leo Stevens, Esq
Maureen McAndrew, DDS
Michael Weinberg