Welcome to Prof. David Klein’s Home Page

Welcome to Prof. David Klein’s Home Page

California State University Northridge

Department of Mathematics

 

 
 
Office: Faculty Office Building Room 127Telephone: (818) 677-7792

Email: david.klein@csun.edu

Fax: (818) 677-3634

Office Hours  Mon, Wed, 11:30-12:30 & by Appointment

Classes (Spring 2005)
Math 250 Third Semester Calculus
TTh 4 to 5:15 p.m., JR 212
Math 651B Math of Relativity & Gravitation
TTh 5:30 to 6:45 p.m., JR 215

 
B.S. Physics 1975, UCSB
B.A. Mathematics 1975, UCSB
Ph.d. Applied Mathematics,
Cornell University, 1981		 Professional InterestsMathematical Physics, Statistical Mechanics, General Relativity, Mathematics Education

Research Articles:  Mathematics, Mathematical Physics, and Physics research articles, preprints and reprints


K-12 Mathematics Education

The State of State Math Standards 2005, by David Klein
with Bastiaan J. Braams, Thomas Parker, William Quirk, Wilfried Schmid, and W. Stephen Wilson
Technical assistance from Ralph A. Raimi and Lawrence Braden
Analysis by Justin Torres
Foreword by Chester E. Finn, Jr.

  • A Case Study: An Email Debate on the quality of education research in mainstream education journals between Dr. Pendred Noyce, head of the Noyce Foundation, and David Klein focusing on the article, “The impact of two standards-based mathematics curricula on student achievement in Massachusetts” by Riordan, J. E., & Noyce, P. E. Journal for Research in Mathematics Education, 32(4), 368-398 (2001).
  • C-SPAN Panel Discussion/Debate on mathematics education. This link includes a complete transcript of a panel discussion and debate hosted by the American Enterprise Institute on March 4, 2002 and nationally broadcast by C-SPAN:
    Does Two Plus Two Still Equal Four?What Should Our Children Know About Math?

    Panelists:

    Mike McKeown, Brown University
    Gail Burrill, Michigan State University and former NCTM President
    David Klein, California State University, Northridge  (For Klein’s Presentation click here)
    Lee V. Stiff, President of the National Council of Teachers of Mathematics
    Tom Loveless, Brookings Institution

    Moderator: Lynne V. Cheney, AEI

    (“Real Player”; click here to get free download)

  • California State Adopted Middle School Math programs for 2001, official Content Review Panel Reports
  • The Freedom to Agree, by David Klein, California Political Review, Vol. 9, No. 4,  pp 15-16, 31, July/August 1998
  • Open Letter to CSU Chancellor Charles Reed in support of the California Mathematics Standards for K-12.  Endorsed by more than 100 college and university mathematicians, and by Jaime Escalante.  January 1998

Other Education Related Links

Mathematically Correct, a nationally influential parents’ organization to improve mathematics instruction in K-12

New York City Honest and Open Logical Debate on mathematics education reform, or NYC HOLD, was “formed in order to address mathematics education in the New York City schools and to provide an organization for parents, educators, mathematicians and other concerned citizens to work to improve the quality of mathematics education”

Core Knowledge Foundation, founded in 1986 by E.D.Hirsch Jr.
What Elementary Teachers Need to Know, College Course Outlines for Teacher Preparation, from the Core Knowledge Foundation.  Syllabai are available for Math, Science, History, Geography, Music, Art.

National Council on Teacher Quality
American Board for Certification of Teacher Excellence
These two organizations provide high quality teacher certification programs.
Save Our Children from Mediocre Math, a parents’ group in Thousand Oaks, California
TeachMath.net, created by math teacher E. Toby Earl

Parents Concerned with Penfield’s Math Programs, a parents’ group in Penfield, New York
Bas Braams’ education page
Bill Quirk’s home page
Ralph Raimi’s home page
Hung Hsi-Wu’s home page

CSUN and Political Issues

CSUN United for Peace and Justice, a coalition of CSUN faculty, students, and university community members

Mechanisms of Western Domination: A Short History of Iraq and Kuwait

Correcting math’s blackboard bunglers

Nov 5, 1998

Correcting math’s blackboard bunglers

(Accompanying Ramirez cartton shows a man at a desk, nameplate on desk says “public education” and the man is reading “Why Johnny can’t read” – upside down.)

They say you can’t fight city hall – but a group of California parents calling itself Mathematically Correct has taken on the statehouse itself and won the right to restore a rigorous math curriculum to public education.

They also say that as California goes, so goes the nation.

The erstwhile Golden State has indeed been the locus of a nationwide contagion of educational “reforms;’ from Whole Language to anti-American propaganda masquerading as history to science-as-radical-ecologism. Since California is the largest public school market, half the country uses textbooks originally pitched to that market.

The Whole Math fad, better known to its traditionalist enemies as fuzzy math, has since 1989 become notorious for its verbalizing and visualizing approach to mathematics, its use of concrete “manipulatives” instead of abstract number concepts, its demotion of the teacher from classroom leader to “co-discoverer” and its reliance on calculators even in the lower grades.

Hallmarks of “fuzziness” also include a fixation on “everyday, real- world” applications, on “method” or “process” rather than domain-specific content, on “higher-order thinking” rather than memorization of facts, and on team-work rather than individual achievement –

The idea of math without numbers sounded dazzlingly innovative to many educators; it was in conveying any sort of mathematical proficiency to students that the miraculous new method broke down.

Parents watched in horror as their children whipped out calculators to determine 10 percent of 470. Fuzzy algebra texts droned on trendily for a hundred pages before getting down to a single equation. State test scores nosedived. One half to two thirds of freshmen entering the Cal State University system needed at least one year of remedial math, despite being among the top third of graduating seniors.

Among those horrified parents were the founders of Mathematically Correct (MC), “dedicated to the proposition that 2+2=4.”

Cofounder Martha Schwartz recalls the night she and her husband “discovered we were not alone!” She was a college geology professor “reacting to the damage done to good kids and the suffering of the best teachers,” her husband Rick, a high school chemistry teacher whose opposition to “fuzzy science” had earned him “un-veiled threats” from district superiors.

At an American Chemical Society meeting in San Diego that the Schwartzes attended in October 1995, Michael McKeown, a molecular biologist at the Salk Institute, closed his remarks with a critique of whole math.

‘After a minute or so of this, Rick and I were almost jumping up and down in our chairs. Within a week or so we had formed Mathematically Correct!’

The fledgling force began by contacting anyone and everyone they had heard of who dissented from the new order in math education. Email and the Internet were crucial to this molecular process. All the individuals contacted had hitherto believed their school districts’ solemn protestations that they were “the only ones who had a problem with the math reform?’

Paul Clopton of UC. San Diego, instrumental in setting up the group’s website, points out that “when parents get together, the bureaucrats’ first defense fades away. As each new parent told their story, we were constantly re- energized”

At first, efforts focused on convincing local districts – Petaluma, Novato, Escondi do, San Diego, Torrance and others – to get rid of existing whole math programs. Larry Gipson, a design engineer consultant and cofounder of MC, led a (successful) fight in Escondido because “I didn’t want my kids experimented on. . . . They were telling the kids to invent their own math out of thin air.”

Mr Gipson formed Parents for Math Choice and lobbied his school board. for just tat choice between traditional and whole approaches. Today 70 percent of district parents opt for traditional, and parental per-mission is required before any experimental program is implemented.

Larry Gipson jokingly refers to himself as “the token conservative” of Mathematically Correct, and indeed, contrary to alarums sounded by the fad- ridden National Council of ‘Teachers of Mathematics (NCTM), most MC members are politically liberal even if academically traditional. Martha Schwartz, during her own local fight, found herself annoyed most by having to point out “that as a secular Jewish geology teacher and registered Democrat, I was not, as charged, a Christian fundamentalist conservative – but that those were all legal things to be!’

Mrs. Schwartz continues, “I’m always outraged when people claim females and minorities can’t learn math or science like ‘regular people! “Lest that sound like a distortion of her opponents’ real views, listen to the former head of the NCTM, Jack Price, in an April 1996 debate with Michael McKeown on San Diego radio;

“What we have now is nostalgia math. It’s the mathematics we’ve always had, good for the most part for high-socioeconomic-status white males!’

Despite some successes at the local level, MC decided to aim for an overhaul of the whole state math framework. Revised every seven years, it is this which dictate’ content and methodology prospective textbooks. The group pushed for tough new K- 12 content standards and for MC to be represented on the appointed Academic Standards Commission. They I hammered legislators with data non grata about whole math’s dismal showing on all manner of tests, wrote open letters and critiques, and repeatedly gave testimony.

The standards adopted by California at the end of 1997 are a realization of Mathematically Correct’s belief that, in Mr. McKeown’s words, “Mastery of the basic the key prerequisite for effective problem solving and one of most effective ways to build understanding!’

The new standards have been denounced by the superintendent of public instruction, who has criss-crossed the state urging teachers to simply ignore them. But an independent review by the Fordham Foundation of America’s state math standards recently rated California’s No.1 and even compared them favorably to Japan’s.

Can the group now declare victory and go home? MC member Leslie Schwarze reflects philosophically that “the battle’s been going on since the 1700s -whether children are innately good or need to be civilized through parenting.

“This baffle will never be over.”

Marian Kester Coombs is a contributing writer for The Washington Times, specializing in education topics.

Reprinted by permission

Content Review of CPM Mathematics

Content Review of CPM Mathematics

Wayne Bishop
Department of Mathematics and Computer Sciences
California State University, Los Angeles


NOTE: CPM withdrew its application to California so this report is not based on its formal submission but, instead, on the document that CPM supplies as part of the Teacher’s Version entitled, “Correlation of CPM Mathematics 1, 2nd ed. (Algebra 1, v. 6.0) and the California Mathematics Standards”, hereafter, “Correlation”. Although Professor Bishop was a member of both the 1999 and 2001 state adoption cycle Content Review Panels, any official role as a CRP member ended with the conclusion of the 2001 cycle so this report is that of an experienced private citizen, not an official CRP review. Nonetheless, the criteria used herein were developed from the state criteria that Professor Bishop used for the official reviews of the 2001 adoption cycle. He is, however, more than happy to testify informally, by legal deposition, or in person, as to the quality and consistency of this report in comparison with those which he formally helped to prepare.


Overall Summary

With regard to mathematics content, this program does not sufficiently address the content standards and applicable evaluation criteria to be recommended for adoption.

In summary, most of the program is below the specified standards level and there is too much of an assumption that work will be done in teams. Although the publisher claims that all standards are met, several are clearly not met and several more identified herein as met are, in fact, not adequately met. Finally, there is a systemic misconception as to what is meant by logical argument in mathematics. If a statement looks to be true, students are told to put it into their “Tool Kit”, then to be available in all settings of study and assessment thereafter.

Evaluation of Content Criteria

Criterion
Met Criterion
1.  The content supports teaching the mathematics standards at each grade level (as detailed, discussed, and prioritized in Chapters 2 and 3 of the framework).
NO
2.  Mathematical terms are defined and used appropriately, precisely, and accurately.
NO
3.  Concepts and procedures are explained and are accompanied by examples to reinforce the lessons.
NO
4.  Opportunities for both mental and written calculations are provided.
NO
5.  Many types of problems are provided: those that help develop a concept, those that provide practice in learning a skill, those that apply previously learned concepts and skills to new situations, those that are mathematically interesting and challenging, and those that require proofs.
NO
6.  Ample practice is provided with both routine calculations and more involved multi-step procedures in order to foster the automatic use of these procedures and to foster the development of mathematical understanding, which is described in Chapters 1 and 4.
NO
7.  Applications of mathematics are given when appropriate, both within mathematics and to problems arising from daily life. Applications must not dictate the scope and sequence of the mathematics program and the use of brand names and logos should be avoided. When the mathematics is understood, one can teach students how to apply it.
NO
8.  Selected solved examples and strategies for solving various classes of problems are provided.
NO
9.  Materials must be written for individual study as well as for classroom instruction and for practice outside the classroom.
NO
10.  Mathematical discussions are brought to closure. Discussion of a mathematical concept, once initiated, should be completed.
NO
11.  All formulas and theorems appropriate for the grade level should be proved, and reasons should be given when an important proof is not proved.
NO
12.  Topics cover broad levels of difficulty. Materials must address mathematical content from the standards well beyond a minimal level of competence.
NO
13.  Attention and emphasis differ across the standards in accordance with (1) the emphasis given to standards in Chapter 3; and (2) the inherent complexity and difficulty of a given standard.
NO
14.  Optional activities, advanced problems, discretionary activities, enrichment activities, and supplemental activities or examples are clearly identified and are easily accessible to teachers and students alike.
NO
15.  A substantial majority of the material relates directly to the mathematics standards for each grade level, although standards from earlier grades may be reinforced. The foundation for the mastery of later standards should be built at each grade level.
NO
16.  An overwhelming majority of the submission is devoted directly to mathematics. Extraneous topics that are not tied to meeting or exceeding the standards, or to the goals of the framework, are kept to a minimum; and extraneous material is not in conflict with the standards. Any non-mathematical content must be clearly relevant to mathematics. Mathematical content can include applications, worked problems, problem sets, and line drawings that represent and clarify the process of abstraction.
NO
17.  Factually accurate material is provided.
YES
18.  Materials drawn from other subject-matter areas are scholarly and accurate in relation to that other subject-matter area. For example, if a mathematics program includes an example related to science, the scientific references must be scholarly and accurate.
YES
19.  Regular opportunities are provided for students to demonstrate mathematical reasoning. Such demonstrations may take a variety of forms, but they should always focus on logical reasoning, such as showing steps in calculations or giving oral and written explanations of how to solve a particular problem.
NO
20.  Homework assignments are provided beyond grade three (they are optional prior to grade three).
NO


Notes on Individual Criteria

1.  The idea of the shortfall here is explained in more detail in the Additional Comments at the end of the review. From the Correlation, a sufficient percentage of the CA Algebra Standards are addressed in some form as to ostensibly meet Criterion 1 but there is much less present than CPM indicates. In regard to some standards, more than three-fourths of the indicated citations are stretched beyond the limit of what the standards writers clearly had in mind.

2.  CPM-1 is deliberately constructivist in regard to such things. It is unfortunate as well because, if a student has not accurately built his Tool Kit, there will be severe difficulties since there is no glossary or even clearly stated terms.

4.  Some might disagree with this assessment because too much written work is often required. The problem is that it is often a misdirected effort that does not provide sufficient mental and written calculation of a genuinely algebraic nature – for example, a Guess and Check table of values in word problems. More opportunity for standard “by-type” approaches is needed and more use and confirmation of standard skills such as arithmetic or rational function..

7.  See #4 above. The program delights in major “daily life” problems except that the forest is lost for its trees. The entire Unit 7 “Big Race” is an example, and the introductory section of Unit 12 entitled “Problems Solving with Distance, Rate, and Time.” It is almost beyond belief that students could then never have seen d = rt but it is true. The authors hold such an anti-“by type” bias that it happened; e.g., no such items are mentioned in the Assessment Handbook for either Team or Individual tests.

8.  See #7 above. Another example is the absence of I = Prt; there is not so much as a mention of the terms. Similarly with the ideas of direct and inverse proportionality that the Framework deliberately discusses in Chapter 3. The self-proclaimed goal of the program is simply not met and, ironically, somewhat by design.

9.  The program has such a pedagogical bias toward group work that it is not clear what, if anything, is expected of students outside of the classroom environment and includes such little direct instruction that it would be extremely difficult for a student who had to miss class to fill in the gaps. There is an accompanying Parent Guide but it is not clear that all parents would have a copy and, beyond that, it really is not much help.

11. Properties of exponents are shortchanged. The Pythagorean Theorem is just given (9: 12) when algebraic proofs are easily available, the quadratic formula is used without proof for a couple of chapters before a proof is given that only a leap of insight would call a proof.

12.  Nothing close.

13.  There is almost no emphasis given to standards of any kind, let alone the ideas of Chapter 3 of the Framework. For example, the second subheading is “Basic Skills for Algebra 1” and includes Standards 4.0-7.0, 9.0, and 15.0. A glance at the list in Criterion 1 shows that these basic skills are inadequately developed in CPM-1.

14.  There is a great deal of irrelevancy, especially in Volume 1, but these are not supplemental.

15.  The entire Volume 1, so the first half the course, would be better left in the closet. Almost anything mathematical is Grade 7, if not below, yet the time requirements are huge. For example, Unit 3:1 is a silly “Algebra Walk”, literally, a human graphing exercise that is at the Grade 5 standard, AF 1.4 and 1.5, yet would take an entire class period to get organized, go outside, an conduct the exercise.

19.  Regular opportunity is present but, reiterating Criterion 1, Standard 24.0 is not met. The entire course confuses heuristics and inductive reasoning, one form of mathematical reasoning, with logical argumentation.

20.  It is not clear, even from the Teacher’s Version that says “Homework begins here,” what is to be homework and what is to be done as work in class as a team.

Standard by Standard Evaluation

Standard
Evaluation
Standard 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:1.1 Students use properties of numbers to demonstrate whether assertions are true or false. Met.
Standard 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. Not met, no fractional exponents and properties of exponential expressions with the same base are not confirmed until Unit 10: 40 and 43.
Standard 3.0 Students solve equations and inequalities involving absolute values. Very weakly met, only the simplest of absolute value equations.
Standard 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12. Met, but “cups and tiles” all the way through Volume 1!
Standard 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Met.
Standard 6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). Weakly met in 12: 99 ff but weakly assessed and not in the Two-Year Final at all.
Standard 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. Met.
Standard 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. Weakly met. 12: 110, 120 meet the perpendicular  specification but they are not assessed or used regularly enough to be confirmed.
Standard 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Met.
Standard 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Met, but inadequate. No division of polynomials except simplification.
Standard 11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Met, but inadequate. Essentially no perfect square trinomials but 13: 79 hints at it.
Standard 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. Met, but too many are already in factored form and the skills are barely assessed.
Standard 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Met, but too many are already in factored form and the skills are barely assessed.
Standard 14.0 Students solve a quadratic equation by factoring or completing the square. Inadequate. There are quite a few by factoring so “or” is satisfied. (See #19).
Standard 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Not met. There is too much of a program bias against “by type.”
Standard 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Inadequate.
Standard 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. Inadequate.
Standard 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. Inadequate.
Standard 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. Not met. The standard is to “know”, while it is always available in the student’s Tool Kit and the proof is weak since completing the square is weak and not assessed.
Standard 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Weakly met and not assessed, the last topic of the course.
Standard 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Met, but weakly, more systematic methods are needed.
Standard 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. Weakly met
Standard 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Very weakly met. 13: 77 purports to address it but most won’t see the relation.
Standard 24.0 Students use and know simple aspects of a logical argument:24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

24.2 Students identify the hypothesis and conclusion in logical deduction.

24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

 Not met. The entire course confuses heuristics with logical argumentation.
Standard 25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.

25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

 Weakly met.


Comments Regarding Assessment Philosophy

Clear indication of the fact that the CPM program is not serious about students meeting the CA Mathematics Content Standards for Algebra follows from looking critically at their own words about, and examples of, assessment. For example, from the CPM Assessment Handbook, Math 1, “A good first step is trying group tests. [P. 3]” “We recommend for this course that you don’t give the usual type of in-process quiz that evaluates students’ mastery of skills while they’re still in the early stages of learning them. [P. 9]”

Why the de-emphasis on early skill mastery? It is philosophical, “When we test the students mastery of skills too early, the students focus is diverted from understanding where and how the skill is to be used.”

As shown below, CPM has carried this philosophy, that does have some element of validity, beyond logic to a point that lack of mastery of skills – ever – still allows students to get good grades in the course without having developed skill mastery. For example, the model final included in the CPM Outline for a Two-Year Program has no mention of parallel or perpendicular lines, no items that require an equation given two points or given one point and the slope, one simple item that implies factoring (solve x2 + 6x + 8 = 0) but no simplification of rational expressions, no mention of completing the square or quadratic formula, etc.

A strong student working at the level of the CA Grade 7 standards, i.e., a good pre-algebra background, would do very well on this final with no formal course, let alone two years of “algebra”! The proposed Team Final does address some of these in a minimal way but the two addition of rational expressions items are very easy and all but one of the expressions in the two quotient items are already in factored form, etc. Again, there is no mention, or use of, slopes in regard to parallel or perpendicular lines, completing the square, the quadratic formula, etc. So to argue, as CPM does, that the CA standards are met, just a bit delayed until mastery has had time to sink in, is nonsense. They are not assessed because they have not been mastered by sufficiently many students.

To put this in perspective, the reviewer’s daughter is in a school that uses one of the California approved texts at the appropriate grade level. Three-quarters of the way through her sixth grade program she was presented the first-year final of the CPM two-year program and had no trouble setting up the percentage word problems (“43 is what percent of 125”, etc.) in mathematical equation form out-loud as she read them. When she was shown the title, “Algebra – Part 1, Final Exam”, she laughed and then excused herself, “I’m sorry, but it’s kind of funny.”  Excluding the CPM specialized “Diamond Problems” that supposedly lead to eventual factoring (of which there are none, even on the Part I Team Final), she would have been able to do nearly every item correctly, and will be able to do all of them by the middle of seventh grade per the California Standards for that grade.

Indicative is the outrageous presentation of the only real word problem, “Solve the problem and write an equation. You may do this in either order. If you do not need a guess and check chart to solve the problem use it to define your variables.” That is, use algebra to “solve” it or don’t. A table of trials is perfectly OK half-way through algebra, in fact its inclusion is mandated even for those students perfectly capable of solving the problem entirely algebraically.

Looking more deeply at the assessment philosophy, “the emphasis should be on the mathematical thinking evident in the work and on what the student knows, not on what the student does not know.[P. 9]” Several pages of the Assessment Handbook are devoted to scoring holistically. “Holistic scoring means just writing the score by the problem 0, 1, 2, 3, or 4 and not making corrections on the students’ papers. [P. 9, Bold is original.]”  Still, the language, including that of the portfolios and the journals is sufficiently imprecise as to allow the possibility of clear, objective, individual student evaluation, “Assessment includes testing basic knowledge and skills, but it encompasses much more.”

Comments Regarding Research Support

The reality of this program is that the standards are not met and genuine assessment would quickly confirm that fact. The evidence and testing, both globally for all CPM students and locally as a teacher tries to assess a student’s knowledge, are entirely inadequate and the conclusions of studies as described in the Teacher’s Version is not nearly as conclusive as the writers imply.

For example, the first paragraph of the page entitled “Research Summary, Comparison of CPM and Traditional Students” is in regard to use of the CSU/UC Mathematics Diagnostic Testing Program (MDTP), data from eight schools that purport to verify that students in seven of the eight learned more in CPM-1 than in their traditional counterparts. Ignoring the fact that “traditional” is not defined and the schools are not named so it is impossible to see exactly what CPM-1 was being compared against, this study did not use the MDTP. It used only 20 of the available 50 MDTP Elementary Algebra items. Two word problems specifically designed for CPM evaluation were also included in the test. Key components of a traditional Algebra I course which are largely or completely absent from CPM -1 were omitted. The following tables indicate the breakdown of the original MDTP items into its subscales as well the distribution of MDTP questions used in this study:

MDTP Subscale
# Items
in MDTP
# Items
in CPM Test
Linear & Quadratic Equations
13
5
Arithmetic
6
5
Geometry
7
5
Graphs
6
5
Rational Expressions
5
0
Exponents and Square Roots
6
0
Polynomials
7
0

Further down the Research Summary page is “CPM End-of-Year Assessment”. The very nature of the description is indicative of the CPM approach, “we gave two questions to …” Two questions, even with “presentations of complete solutions”, is not the kind of algebra assessment most people envision when they read the subtitle.

Another measure of CPM “success” is the state SAT-9 scores, in which a page of data purports to prove that CPM schools are more successful than “their peers who use other curriculum materials,” but there is so much missing as to make the data almost meaningless. As a start, the SAT-9 is not algebra! Of course, there are some exercises, ratio and proportion problems for example, that lend themselves to nice algebraic representation, but it is not algebra at the level of then President Clinton’s assessment, “Algebra is algebra!”

A comparison of the California Algebra Standards Test would be useful data, but even that (which CPM chose not to publish) would be comparing CPM-1 students against a far less homogeneous group, some using an even more aggressively “reform” curriculum. Beyond that, some schools use CPM for regular classes and a more traditional program for more advanced ones. That could be taken as evidence, probably supportable, that CPM is preferable to “general math” but hardly an argument for using it in place of a traditional college preparatory curriculum as its name would imply.

Finally, CPM cannot be trusted to give us an honest picture. The “MDTP” study that they continue to use is a clear indicator of that fact with its 20 of the 50 MDTP items. Since the public lacks the names of the CPM schools in the CPM summary sheet, it is impossible to do a quick comparison of SES factors, for example, to see if most of the CPM schools in these counties might have had a head start even before any choice of mathematics curriculum.

Comments Regarding Assessments in CPM

A much clearer vision of how far short CPM falls on more traditional end-of-course assessments is contained within their own Teacher’s Version and, most explicitly of all, within their “Outline for a Two-Year Program”, the guide to teachers for setting up the same program but over a two-year, less-demanding schedule referred to above. This document supplements the Teacher’s Version guides for constructing unit tests and tells what the designers really have in mind for verified competence. It is far off the algebra standards of California or, beyond that, of any other set of standards for algebra.

Here is one way that the assessment materials are designed to appear to be sufficiently demanding of standards-level competence when they are not. After Unit 3, the Assessment Handbook itself does not have model exams (they are in the Two-Year Program guide), but it does have item banks, by unit, along with instructions for constructing the tests themselves. Indicative in these instructions is “If you give an individual test …” That is, even the act of having an individual, bottom-line assessment can not be taken for granted in a CPM environment.  Going onward with the quote, “it would be best to make this a very short test.”

Most indicative of all, however, is the test bank itself. The instructions recommend, “no more than one question of any type,” which would be reasonable advice if the items in each set were, in fact, of the same type. That is obviously not the case, the sets are constructed so that it is possible to avoid confirmation of the ideas involved at the level that a cursory look could imply.

Standards Representation in CPM Test Items

The test-bank items are not numbered (so as to make it easy to omit the item entirely) but some representative examples are the following from the indicated unit with item number counting from the first item in that unit looking at every reference given for the particular standard in the Correlation for the CA Standards for Algebra I .

Standard 8  Students understand the concepts of parallel and perpendicular lines and how these slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

Unit 7: 49, and 73 These words are not mentioned in these references, nor in the assessment-bank, Team or Individual. Ref. 7:83 looks at parallel for slope of 2/3 but not in general.

Unit 11: 101 This item does have students note that the slope of one specific line is -5/3 when the original line was 3/5 and that the lines are perpendicular, but without verification other than they look like it. Students are to “Record your observation in Your tool kit.” These words are not mentioned in the Study Team Questions (Team) but in #2 of the Individual Test (Individual), parallel is to be recognized by slope, without graphing. No student constructed equations are expected except when given two points.

Unit 12: 110, 122 Both of these meet the standard for perpendicular slope. Neither parallel nor perpendicular is mentioned in either the Team or Individual test banks.

Unit 13: 28, 61, 102 Items 28 and 102 meet the standard and 61 is borderline.

Standard 11.  Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

Units 0-8: diamond problems 0: 3 ff, 2: 53, 78, 4:52, 6:70, 82, 96 All of these references from Volume 1 are so far off the standard as to be open to a charge of lying. For example, the last, 6: 96 consists of 11 exercises of multiplying a monomial or binomial times a binomial, not the reverse. Granted, converting expressions in factored form into un-factored form is helpful but it is does not begin to address, let alone meet, the standard.

Unit 8: 2-4, 10, 12 These are only preliminaries to factoring; e.g., 4 is only recognizing if an expression is written in any kind of factored form, 10 and 12 are tile-pushing with the factored forms already given. 8: 18 finally is an actual factoring but only with algebra tiles, 19 and 20 tie factoring into the earlier “generic rectangle” extension thereof, 31 is factoring out monomial common factors. 8: 50-51 does treat the difference of two squares and 8: 52 is ten mixed factoring problems, including one of them. 8: 57-63 meet the standards along with 70-71 and 77-80. The Unit 8 Team #1 has six expressions to be factored, including one difference of two squares and #3 uses factoring to solve quadratic equations in one variable, none of which is a difference of two squares, exactly as in the Individual, four quadratic equations to be solved, three that still need to be factored, two that are trinomial, and none that are the difference of two squares. In the Two-Year Program, that includes models of actual tests, not “select from the following,” the Unit 8 Individual Test includes six factoring items, five with assistance and one stand alone. There is one difference of two squares item, but not “by type” but only by coincidence. The test includes no quadratic equations to be solved.

Unit 9: 90 does FOIL factoring (not in the CA Standards) and this is the only reference in the Correlation but, in fact, they missed some, 9: 21, 42, etc. However, the Team Questions only include two factoring exercises and one quadratic equation to be solved. The Individual test includes a choice of three or two factoring items, one of which includes a difference of two squares, and block of five equations to solve that includes one quadratic. By contrast, the Two-Year Unit 9 Mid-Unit Individual Test (there is no unit test because of the up-coming First Semester Test, contains no factoring items nor quadratic equations to be solved. The First Semester Individual Final has two, x2 – 7x + 12 and x2 + 5x. There are no difference of squares items and no perfect square trinomials.

Unit 10: 1-3 ff, 17 Recurring exercises do confirm the ideas. The quadratic formula is simply given in 10: 86 so it is not clear whether or not factoring will continue in solving quadratic equations. In regard to assessment, several Team and Individual items have radical expressions or decimal approximations indicating that they are not to be factored since completing the square is not introduced until three units later. Somewhat surprisingly, given the end-of-course exams, the Two-Year Unit 10 Individual Test does have several factoring problems including the advice to look for difference of two squares and a simplification of a rational expression that requires factoring both numerator and denominator (See CA Standard 11).

Unit 11-13 practice in homework 13:79 hints at perfect square trinomials but this standard is not met under almost any level of generosity. The Two-Year Part 2 end-of-course Individual Final Exam is the most indicative, here. Instead of demonstrating that students finally have mastered these ideas, there is exactly one factoring exercise, #6b) Solve: x2 + 6x + 8 = 0. There are no simplification of rational expressions, let alone multiplication or division of them.


Standard 12.  Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to lowest terms.

There are some, but see the last line of the Standard 11 remarks above. Since the assessment specifications allow for picking and choosing – they’re deliberately not numbered – it is impossible to say to what extent the program expects individual student competence.


Standard 13.   Students add, subtract multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.

There are some, but see the last line of the Standard 11 remarks above. Since the assessment specifications allow for picking and choosing – they’re deliberately not numbered – it is impossible to say to what extent the program expects individual student competence. Indicative of how far short the program is of the intended standard, there is one example given in the 1999 Mathematics Framework to demonstrate the intent of this standard: Solve for x and give a reason for each step: 2 / (3x + 1) + 2 = 2/3. There is no equation of this level of difficulty to be solved in the two volumes.


Standard 14.  Students solve a quadratic equation by factoring or completing the square.

The same remarks apply; completing the square is an afterthought at the very end of the book, Unit 13: 67, and only with the CPM insistence on an overuse of so-called “algebra tiles” belying the problem with an odd or fractional middle term, and students are simply not expected to use it. In fact, the disclaimer at the beginning of the Unit 13 Individual Test admits as much, “We really do not expect many students to begin to master the topics in Unit 13. So, we provide a little extra assistance so we can still test them on these topics,” followed by inclusion of the quadratic formula with no items that require completing the square, not even with an even middle term and a pile of algebra tile. The Part 2 Individual Final is the most indicative, of course, neither is ever needed. The one quadratic equation is already in standard form and factors easily.


Standard 15.  Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

From the Correlation comment one would expect this standard to be well met, “Word problems and investigations are at the core of the CPM program. Students regularly solve word problems without categorizing them “by type.” It is the second sentence that belies the first. Nearly all of the Unit 4-9 exercises are below the level of the algebra standards, many at the 6th grade standards, just far wordier and often with a great deal of irrelevant fluff. A good example is Unit 7, the entire “Big Race” premise. It is a complicated d = rt exercise without saying so but therefore not on standard at all, but a year or two off. Even among the last exercises that the Correlation indicates, Unit 13: 1, 12-14, 52, 62, 72, 78, 101, only 62, 78, and 101 meet the intended standard and nothing of the kind appears on the Two-Year Final. The only genuine word problem leads immediately to a pair of simultaneous linear equations. That qualifies but only as a small part of the intent of the standard. More ordinary problems are described in words and that is to be commended, but that is not an acceptable excuse for avoiding the others.

Other standards are far wide of the mark as well, and by design. CA Standard 24 is claimed to be met in the Correlation by lots of “Explain your answer…” and there is much of that in CPM-1. Most of them are not close to what the standard says and means. The books can be opened almost anywhere to see examples but using one that they chose:

Unit 10: 33 The item consist of three parts, solving a quadratic equation in standard form by factoring, graphing the corresponding function with these points as zeros and finally the supposed logical argumentation, “How are (a) and (b) related?” This not an unreasonable exercise, even good. The problem is that the informal argumentation of the exercise has nothing to do with the listed standards:

Standard 24.  Students use and know simple aspects of logical argument:

24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
24.2 Students identify the hypothesis and conclusion in logical deduction.
24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

Or another example on the same theme, 10: 96. Again, the exercise is reasonable; it is asking students to expound on the connection between the discriminant term of the quadratic formula and factorability. In fact, it would meet 24.1 if were it a bit more direct, something like, “Argue that if b2 – 4ac is a perfect square, then …” instead of “Explain what the values of sqrt(b2 – 4ac) tells you about factorability of the polynomial?” Few students would recognize that this new tool constitutes proof that the polynomial can or cannot be factored; they will not distinguish it from an earlier “logical argument” 10: 74 prior to the introduction of the quadratic formula, “Is the expression x2 + 6x + 2 factorable? Explain your answer.” Even assuming knowledge of uniqueness of factorization (which should get acknowledged in a formal setting), lacking the rational roots theorem, the ability to complete the square that (unconscionably!) is not introduced until three chapters later, or the quadratic formula, it is not at all clear what a “good” explanation would be! “I tried everything I could think of!” perhaps?

 

 

Final Observations Regarding The Curriculum Materials

In spite of being too far off the algebra standards to warrant state approval but on the positive side, the books themselves have, or rather, Volume 2 has some distinct advantages over more traditional texts. The regular, mixed review of exercises is excellent. So is requiring students to present sensible explanations in support of their conclusions. If the authors of the Teacher Version introduction materials were to abandon their strong allegiance to what’s often known as “Authentic Assessment,” i.e., team tests, journals, portfolios, and individual observation and get back to lower case authentic assessment that algebra teachers everywhere have traditionally done, the program would be greatly enhanced. Yes, there should be more “by type” word problems and they should be part of the regular mixed review exercises, more formal logical argument, actual use of completion of the square in a way that students are expected to be able to use it, etc., but that is not the worst problem. The worst problem is pedagogical.

Volume 2, in the hands of a competent traditional algebra teacher, could be reasonably effective; short of the standards, but effective. If all of it, through Unit 13, is covered and if individual students do all of the exercises, they will master enough of the skills to be considered competent at the level of algebra 1, capable of going forward successfully in mathematics that depends on these concepts and skills. The problem is, and it is convicted as fatal by the included Individual Tests in the Two-Year Program guide, there is no assurance – nor any genuine effort to confirm – that this is what will happen. A little group work from time-to-time is fine, having stronger students help weaker ones master the material likewise, but that is not the described philosophy; in fact, it is completely the reverse. Similar is the work with algebra tiles. A little at the beginning? Sure, why not. Plastic toys instead of mental manipulation far into the course is very different. But, although serious, the worst parts are not the leisure introduction to factoring, that does have supportable logic, or too much graphing calculators, or being too far off the standards. They are two; one is pretending that everyone is mastering algebra while letting strong students carry weak or lazy ones, and the other is the heavy-handed, non-algebra, time-wasting in Volume 1.

The latter problem is associated with the first pedagogically and philosophically although the supporters would phrase it differently, of course. From the Introduction and Overview, “Telling has little to do with promoting learning – students must construct their own understanding.” In a sense, this is true since we do not learn by “direct download” but neither do we learn much mathematics by activity-based insight. Being told wrongly by a convincing fellow student, a situation common in un-led team settings, is far worse than being told what is correct by a competent teacher. Yet, “The daily activities in this course will require much more work in study teams and much less introduction and explanation of ideas by the teacher.” Much of Volume 1 actually detracts from developing algebraic competence. Almost all of the mathematical content is at the level of the Grade 7 standards or below, e.g., the equations to be solved all are, but the activities are still very time consuming and sometimes frustrating. The worst of all, however, is not teaching the power of algebra itself. Unit 4: 123 is TOOL KIT CHECK UP and it is mandated that it contain Guess and Check tables, and “cups and tiles to model solving equations.” This is not algebra and it is not college preparatory math, no matter what it calls itself. Eventually, Volume 2 starts teaching some algebra but it is too little and too late.

Mathematically Correct presents The Pythagorean Theorem

Mathematically Correct presents

The Pythagorean Theorem

G. D. Chakerian and Kurt Kreith

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.

Cognitive Child Abuse in Our Math Classrooms

Cognitive Child Abuse in Our Math Classrooms
By C. Bradley Thompson

The test results are in: America’s children are flunking math. In 1996 American high school seniors finished close to the bottom on an international mathematics test. At the end of last year, American eighth-graders ranked below those of Malaysia, Bulgaria, and Latvia.

     As educators scramble to explain America’s math meltdown—as the Bush administration urges more “accountability” and a National Research Council study recommends better “training”—few are willing to look at the fundamental cause: the new, “whole-math” method for teaching.

     Inspired by a strain of progressive-education theory called “constructivism,” whole-math proponents claim that all knowledge—including mathematical knowledge—is arbitrarily constructed. They reject the idea that there are objectively demonstrable right and wrong answers, and that, consequently, there are basic skills that students must be taught. Instead, the advocates of whole math believe that each student should invent his or her own math “strategies” by using a “guess-and-check” approach. They create an inability to think beyond immediate concretes.   

  In a typical whole-math classroom, children do multiplication not by learning the abstract multiplication table, but by using piles of marshmallows. They count a million birdseeds in order to understand the concept “million.” They measure angles by stretching rubber bands across pegged boards. One whole-math program preposterously claims to foster a “conceptual understanding” of math by asking fifth-graders the following stumper: “If math were a color, it would be ______ , because ______.” Surely such exercises foster in children only conceptual stultification—along with a bewildered sense of frustration and disgust.

    Another whole-math program asks sixth-graders to address the following problem: “I’ve just checked out a library book that is 1,344 pages long! The book is due in three weeks. How many pages will I need to read a day to finish the book in time?” The proper way to solve the problem would be to use the method for long division: 1,344 divided by 21. By contrast, the whole-math approach assigns students to a group, requires them to design their own problem-solving rules, and urges them to guess if all else fails. In other words, children are told that their random “strategies” are just as good as the logically proven principles of long division. They are taught that the vote of the group, rather than the reasoning of the individual mind, is the means of arriving at the truth.

     Now imagine flying on a plane designed by aeronautical engineers who have been trained to concoct their own math schemes and to use a “guess-and-check” method.

     Whole math must lead to a miasma of confusion, boredom, and despair. Rather than encouraging independent, conceptual-level thinking, it is thoroughly anti-conceptual. It dooms children to function on a primitive, perceptual level—i.e., to flounder in a chaotic sea of concretes with no objective principles to guide them. This is cognitive child abuse. Whole-math defenders are shrinking the cognitive capacities of their students to those of infants or even animals.

     Is it any wonder that most college freshman take remedial math courses, that American universities award more than half of their mathematics Ph.D.s to foreign nationals, that for-profit math remediation companies are booming, and that 200 of the nation’s leading mathematicians and scientists signed a public letter denouncing whole math?

     Mathematics is like any other field. To master it, one must acquire basic knowledge before proceeding to more advanced stages. Proficiency in math requires that grade-school children learn the standard algorithms (i.e., the methods for addition, subtraction, multiplication, and division) and the four forms of numbers (i.e., integers, fractions, decimals, and percents). This forms the foundation upon which higher and higher levels of knowledge can then be built.

     The controversy surrounding whole math is not simply about how children are taught to deal with numbers. If we undermine the capacity of our children to learn mathematics, we undercut their ability to think. More and more, our schools are turning out students whose capacity to reason has atrophied. Students who have not learned how to add and divide are also unable to perform the more demanding cognitive tasks of understanding concepts like “justice” or “truth” or “logic.” America’s children are being turned into mindless drones, who will soon be unable to distinguish freedom from tyranny.

     Today’s “math wars,” like the controversy over how to teach reading, are at root philosophic battles that will have enormous implications for the future of America. If the advocates of whole math are allowed to win, they will be taking us a huge step away from the values of reason and science that once made America great.

 

C. Bradley Thompson is Chairman of the Department of History and Political Science at Ashland University in Ohio and a senior writer for the Ayn Rand Institute in Marina del Rey, California. The Institute promotes the philosophy of Ayn Rand, author of Atlas Shrugged and The Fountainhead. Send comments to reaction@aynrand.org.

Math Gadfly Calls Math Faddists’ Bluff

Math Gadfly Calls Math Faddists’ Bluff
– DEBRA J. SAUNDERS
Friday, August 7, 1998

WAYNE BISHOP, a professor of mathematics at Cal State L.A. is a math gadfly in his spare time. Bishop likes to call trendy new-new math boosters’ bluff by checking out their schools’ scores and seeing if their fads hurt or help students.

Bishop’s findings are not pretty, friend.

Start with Roscoe Elementary School in Los Angeles. MathLand, the ultra- trendy new-new math program, touts Roscoe as a MathLand success story on its Web page. In 1996, then-Roscoe principal Ruth Bunyan wrote a letter for MathLand boasting, “We believe that our use of MathLand materials and instructional strategies have made a significant difference during the past year.”

(For the unitiated, new-new math is math that eschews exercises that emphasize “predetermined numerical results.”)

So how did MathLand’s star school — where MathLand folks coached the teachers to increase performance — fare in California’s new Star test? In the bottom quartile. The average Roscoe student score was 21 — at the bottom 21st percentile — for second graders, 22 for third graders, 20 for fourth graders and 18 for fifth graders. Only 10 percent of fifth graders scored above the national average.

Students at the pet middle school of the former head of the National Council of Teachers of Mathematics (NCTM), Jack Price — a faddist of the first water — have suffered the same fate.

In 1995, Price told his peers, “We need to let everyone know that successes can be found in every part of the country.” As an example, he cited a program at which he worked one day a week with a dedicated math department faculty that had completely reworked its curriculum “to be in line with the (NCTM) standards.”

“He was depressed about the fact that some of us curmudgeons weren’t catching on, ” Bishop explained.

Guess what? When Bishop checked out test data for Santa Ana Unified’s Spurgeon Intermediate School, he found failure and mediocrity, then and now.

A build-it-and-they-should come success? Sprugeon’s Star scores don’t show it. Price’s pet school ranked in the bottom quartile. The average percentile for sixth graders was 23, 24 for seventh graders and 22 for eighth graders. Only 12 percent of the schools’ eighth graders scored above the national average.

Apologists might argue that the student body make-up — the school is overwhelmingly minority, a majority of students have limited English skills — mitigates this dismal showing. I can’t agree. I can find nothing understandable about minority kids failing. I’ll add that if Price wants to tell America how to teach math, he ought to be able to demonstrate that students enrolled in his model program at least can pass a math test.

(Besides, Price has boasted about the “great deal of research” of which he is aware as to how females and minorities “do not learn the same way” as white males. So a large minority pool should be a piece of cake for him.)

Price, now at Cal Poly Pomona, had the misfortune to pick up his phone yesterday when I rang to ask why most of his poor charges flunked the Star test.

“Is that surprising?” Price asked.

If I didn’t know what I know about new-new math, I would be surprised.

“It wouldn’t be surprising if you knew anything about how instruction and assessment are tied together,” He answered. “You don’t teach people apples and then give them a test for oranges.”

But in your 1995 address, you said the NCTM standards included basic skills.

“What do you want me to say?” Price asked, before he said he didn’t want to chat anymore.

No doubt he prefers an audience that is more accepting about failing innocent kids.

©2005 San Francisco Chronicle

California’s Algebra Crisis

California’s Algebra Crisis
by Paul Clopton and Bill Evers
October 6, 2003

California has had its share of educational crises—such as whole language and fuzzy math. Despite recent improvements, the state is still in the grips of an algebra crisis.

The problem became apparent twenty years ago when the report A Nation at Risk warned of a “rising tide of mediocrity” in the public schools. The report claimed that too few students were taking the more rigorous courses in high school. Twenty years later, enrollment in college-prep courses is way up. Unfortunately, evidence indicates that student learning is about the same as it was back then.

Recent reports have stressed the importance of algebra in middle school; students who succeed in algebra usually do better in the rest of school and in their careers than those who do not. Well-intentioned school administrators often hope that early enrollment in algebra will reduce the achievement gap attributed to race or family income. Hence enrollments in middle-school courses called “Algebra” have increased. But judging from results on objective statewide tests, many middle-school students are not learning the subject, even those with passing grades.

The strongest predictor of failure to learn algebra is not race or income; it is a lack of adequate academic preparation. The problem begins before students get to their first algebra class. Many school districts have watered down the content of pre-algebra courses, removing important but difficult material. The districts want more students to pass math classes, and they want to guarantee high pass rates by making the classes easy. But classes without content set students up for later failure in algebra.

The depth of the problem varies. In some schools, the percentage of eighth-grade algebra students is moderately correlated to scores on the seventh-grade California Standards Test. In those schools, algebra readiness is still being used as part of the placement decision.

In other schools, placement decisions appear unrelated to academic preparation. In the worst cases, all or nearly all students are placed in algebra by eighth grade, regardless of readiness.

No district in California is more guilty of misguided placement strategies than the San Diego City Schools. The results are disastrous. Failing to learn algebra in eighth grade results in large numbers of students repeating algebra in ninth grade, even though success is not ensured the second time around.

Admirably, California embraces learning algebra by the end of eighth grade as a long-term goal. But strengthening academics from kindergarten on is necessary before this goal can fully be met. Algebra placement rates ought to depend on student readiness. Seventh-grade student scores on the California Standards Test should guide placement in eighth-grade courses.

Another state policy adds to the problem. As of now middle schools receive more credit on California’s accountability index for eighth graders who take the algebra test than for those who take the general math test, encouraging schools to place too many students in eighth-grade algebra. The state should discourage overplacement by taking away some credit on the accountability index for algebra exam failures.

California’s algebra crisis is serious but not terminal. Schools need to concentrate on improving students’ readiness for algebra courses. Algebra for all is good, but without changes we could end up with algebra for none.

Big Business, Race, and Gender in Mathematics Reform

Big Business, Race, and Gender in Mathematics Reform

by David Klein

The mathematics reform movement may have positive attributes, but that is not what this appendix is about.  This essay is divided into three sections, each taking a critical view of what has come to be called “mathematics reform.” Rather than attempting an abstract definition of this term, I cite the  principal documents and leaders of the reform movement on particular issues.  The fault line separating the mathematics reform movement from its critics is nowhere more volatile and portentous than in California.  The third and final section of this appendix is devoted to a short history of the conflict over mathematics reform in that state, with a focus on the controversial California mathematics standards.  This set of standards has received widespread praise from prominent mathematicians and strong opposition from the mathematics reform community.  As explained in the last section, this conflict helps to define, in practical terms, the mathematics reform movement.

The second section challenges assumptions about ethnicity and gender in the reform movement. Multiculturalism and mathematics for “all students” are recurring themes among reformers. Prominent reformers claim that learning styles are correlated with ethnicity and gender.  But reform curricula, while purporting to reach out to students with different “learning styles,” actually limit opportunities. Fundamental topics, including algebra and arithmetic are abridged or missing in reform curricula without apology.

Big Business and the mathematics reform movement have at least one thing in common.  They both militate for more technology in the classroom.  Calculators and computers are regular features in reform math curricula, and technology corporations routinely sponsor conferences for mathematics educators.  The confluence of interests and the resulting  momentum in favor of more technology is the subject of the first section.

Technology, Reform, and the Corporate Influence

The 1989 report “Everybody Counts” warned:

In spite of the intimate intellectual link between mathematics and computing, school mathematics has responded hardly at all to curricular changes implied by the computer revolution.  Curricula, texts, tests, and teaching habits–but not the students–are all products of the pre-computer age.  Little could be worse for mathematics education than an environment in which schools hold students back from learning what they find natural. [NRC]


The imperative to integrate technology into the classroom goes far beyond mathematics courses. President Clinton calls for “a bridge to the twenty first century…where computers are as much a part of the classroom as blackboards.” [O]  Presently, four-fifths of U.S. schools are wired to the Internet and the rest are not far behind.  Remonstrations by well-placed technology experts and educators, based on educational considerations, seem to warrant no delays [G], [O]. A 1996 report by the California Education Technology Task Force, a group dominated by executives in high-tech industries, called on California to spend $10.9 billion on technology for schools before the end of the century.  The task force claimed that “more than any other single measure, computers and network technologies, properly implemented, will bolster California’s continuing efforts to right what’s wrong with our public schools.” [LAT1]

A corporate perspective is also sweeping American universities with  computer technology  paving the way.  The number of virtual universities and virtual courses  is increasing exponentially. In 1997 there were 762 “cyberschools,” up from 93 in 1993 and more than half of the nation’s four year colleges and universities have courses available “off site” [F]. The  second largest private university in the U.S., the University of Phoenix, offers on-line courses to 40,000 students from a faculty with no tenure.  Other examples* and a recent history of technology and the corporatization of universities may be found in David Noble’s interesting essays [N].

Will the computerization of schools improve education? The Los Angeles Times reports that “many critics worry that education policy is increasingly being driven by what companies have to sell rather than what schools need…Computer companies want more technologically savvy consumers, for example, to increase the penetration of computers beyond the 40% of homes in which they are now found.  And they argue that increased use of technology in schools will help fill a growing shortage of computer literate workers.” [LAT1]

Corporate foundations regularly fund mathematics reform projects, as for example, the “Exxon Symposium on Algebraic Thinking” for the Association of Mathematics Teacher Educators Conference, held in January 1998, with Texas Instruments hosting one of the dinners.  Conversely mathematics reformers embrace a corporate vision of education, which includes the de-emphasis of basic skills and a greater reliance on technology.  Consider, for example, the following promotional material for the K-6 curriculum MathLand from Creative Publications:

Business leaders have expressed interest in changes in education as well–changes that go beyond what a traditional standardized test can measure. Recently, the US Departments of Labor and Education formed the Secretary’s Commission on Achieving Necessary Skills (SCANS) to study the kinds of competencies and skills that workers must have to succeed in today’s workplace. According to the SCANS report What Work Requires of Schools: A SCANS Report for America 2000, business leaders see computation as an important skill, but it is only one of 13 skills desired by Fortune 500 companies. These skills are (in order of importance): teamwork, problem solving, interpersonal skills, oral communication, listening, personal development, creative thinking, leadership, motivation, writing, organization skills, computation, and reading. [ML]


The California Mathematics Council (CMC), an affiliate of the National Council of Teachers of Mathematics, boasts 12,000  members.  In an open letter to the California Board of Education dated April 17, 1996, the CMC included the same ordered list of basic skills with reading and computation given last.  Citing unspecified “educational research” and “neuro-biological brain research,” the CMC letter endorsed the direction of the 1992 reform oriented California Mathematics Framework and added:

Equally impressive is that these changes in the way we teach mathematics are supported by the business community. What Work Requires of Schools: A SCANS Report for America 2000 concludes that students must develop a new set of competencies and new foundation skills. It stresses that skills must be learned in context, that there is no need to learn basic skills before problem solving, and that we must reorient learning away from mere mastery of information toward encouraging students to solve problems.

Learning in order to know must never be separated from learning to do. Knowledge and its uses belong together (A SCANS Report) [CMC]

The NCTM Curriculum and Evaluation Standards also recommends that “appropriate calculators should be available to all students at all times” and “every student should have access to a computer for individual and group work.”  Reform texts place little restriction on technology. The second edition of the Harvard Calculus text instructs that students “are expected to use their own judgment to determine where technology is useful.” [HAL] The 1992 California Mathematics Framework recommends that calculators be available at all times to all students, including Kindergarten students, and asks, “How many adults, whether store clerks or bookkeepers, still do long division (or even long multiplication) with paper and pencil?”

None of the above is intended to suggest that technology should not be used in mathematics classes. Nor do I suggest any kind of conspiracy theory. I have incorporated the (limited) use of computers in some of my own classes at California State University, Northridge, and I agree with almost all of Professor Krantz’ balanced discussion in  Section 1.10.  My only reservation is Professor Krantz’ suggestion that  an entire “lower division mathematics curriculum [should] depend on Maple (or Mathematica, or another substitute) and that [students] need to master it right away.”  This seems to me to be premature.  It might be appropriate at some point, but a compelling curriculum with this feature should be presented, vigorously reviewed, and thoroughly tested on real students first.

The use of technology in mathematics education should be considered against the backdrop of extremely powerful business interests which seek to create new consumers of technology.  Incorporating ever more technology into the classroom may or may not be consistent with good educational practices.  Large-scale implementations of technology in the classroom receive tremendous momentum from funding agencies–at times, far beyond what the results merit.  With the huge sums of money involved in computerizing education, the educational merits of technology are rarely discussed. For politicians and entrepreneurs, no  justification is necessary, but educators should demand clear evidence of the beneficial effects of technology before it is incorporated in classrooms.

The calculator is one of the staples of the reform mathematics movement from Kindergarten through calculus and beyond.  Mathematics instructors, including calculus teachers,  regularly allow students to use calculators on examinations and contort their tests to avoid giving points for mere button pressing skills.  I agree with Professor Krantz’ assertion in Section 1.10 that “if a student spends an hour with a pencil–graphing functions just as you and I learned–then there are certain tangible and verifiable skills that will be gained in the process.”  I don’t think it is unreasonable to require students to demonstrate these and other skills on examinations without calculator assistance.

At the elementary school level, arithmetic is a victim of technology in the reform curricula. Long division in particular is frequently a target for elimination.  For example, long division with more than single digit divisors was consciously eliminated from the proposed California math standards by the Academic Standards Commission, and the California Mathematics Framework makes it clear that “clerks or bookkeepers … do [not do] long division …with paper and pencil.”  In addition to sharpening estimation skills, mastery of the division algorithm is important for understanding  the decimal characterization of rational numbers, a middle school topic, as well as quotients of polynomials and power series in later courses.

In a society that worships technology, it is all too easy to surrender the integrity of sound traditional curricula to machines, their corporate venders, and reform-evangelists.

*During the mid-1990’s the California State University administration initiated an unprecedented partnership with technology giants Microsoft, GTE, Fujitsu and Hughes Electronics.  The joint venture, called the California Educational Technology Initiative, or CETI, will, if implemented, wire up the 23 campuses of the CSU with state-of-the-art telephone and computer networks, as well as invest billions of dollars in education related electronics.   By the Spring of 1998, a dozen CSU campus faculty senates passed resolutions asking for delays and criticizing the merger.  The California State Student Association  passed its own resolution denouncing CETI and opposing any “privatization of the California State University as a whole.”  Microsoft and Hughes subsequently pulled out, but CSU Chancellor Reed continues to seek new corporate partners. The implications of such a partnership are not fully worked out, but incentives for the faculty to market computer products to students, and the creation and marketing of courseware have been seriously considered.

Gender, Race, and Ethnicity in the Reform Movement

One of the themes of the mathematics reform movement is that women and members of ethnic minority groups learn mathematics differently than white males.  The thesis that learning styles are correlated with ethnicity and gender is widely accepted in education circles and its validity is not assumed to be restricted to mathematics. One example of this ideology occurred when the Oakland School Board resolved that Ebonics is genetically based [SFC].  Main-stream views from the academy are similar. In a well-referenced study on how African Americans learn mathematics, published in the Journal for Research in Mathematics Education, one finds [MJ]:

Studies of learning preferences suggest that the African American students’ approaches to learning may be characterized by factors of social and affective emphasis, harmony with their communities, holistic perspectives, field dependence, expressive creativity, and nonverbal communication…Research indicates that African American students are flexible and open-minded rather than structured in their perceptions of ideas…The underlying assumption is that the influence of African heritage and culture results in preferences for student interaction with the environment and that this influence affects cognition and attitude…


The Journal of American Indian Education devoted an entire special issue to the subject of brain hemispheric dominance and other topics involving Native American learning styles.  Included is a reprint of Dr. A. C. Ross’ paper, “Brain hemispheric functions and the Native American,” that asserts Native Americans are “right brained.” Ross explains that the “functions of the left brain are characterized by sequence and order while the functions of the right brain are holistic and diffused.”  Elaborating, he maintains that “left brain thinking is the essence of academic success as it is presently measured.  Right brain thinking is the essence of creativity.”  Citing earlier research, Ross concludes that “traditional Indian education was done by precept and example (learning by discovery)…creativity occurs in the learning process when a person is allowed to learn by discovery.  Evidently, traditional Indian education is a right hemispheric process.” [JAIE]  The final article in the same journal takes issue with this point of view and laments that “a veritable right-brain industry has developed”  and warns of the dangers to Indian education by characterizing this entire ethnic group as right brained.

The view that women and minority group members learn differently from white males is far from marginal within the mathematics reform movement.  A radio interview of NCTM President Jack Price, independent textbook publisher John Saxon, and Co-Founder of Mathematically Correct, Mike McKeown, occurred on April 24, 1996.  The KSDO radio show on Mathematics Education, hosted by Roger Hedgecock, was held in conjunction with the annual meeting of the National Council of Teachers of Mathematics, in San Diego that year.  During the interview, President Price asserted:

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 socio-economic 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 young people, 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. [MC]


The reform movement presupposes that broad classes of non-white males learn “holistically,” that mathematics should be integrated with examples and connected as widely as possible to other human endeavors. Algebra and arithmetic are particularly short changed as “mindless symbol manipulation” and “drill and kill.” To cite one typical example of this, a mathematics educator wrote on the Association of Mathematics Teacher Educators listserve, “I know this may come as a shock to some mathematics professors out there, but few students find manipulating ‘x’ and ‘y’ engaging.”

It is clear that proponents of reform are acting out of a sincere desire to improve mathematics education for all students.  But the mathematics community should be suspicious  of trends which draw legitimacy from racial or gender theories of learning.

No one disputes that culture plays a role in academic achievement. The Los Angeles Times  published a special report entitled, “Language, Culture: How Students Cope” as part of a three day series of special reports on education [LAT2].  The Times report explicitly discounts any link between race and ability, but acknowledges that “ethnic differences [in academic accomplishments] remain, even after accounting for income, parent education or language a student speaks at home.”   High  achievement by Asian American students is a result of hard-work and a strong emphasis on the importance of education, and this contrasts sharply with the “complacency that hampers so many of California’s white students, who have shown a sharper drop on reading scores this decade than either blacks or Latinos.” “The burden of acting white” is a theory that African American students “resist schooling to protect their self-image and distinguish themselves from a majority culture that too often devalues their abilities.”  Many Latinos, for cultural and economic reasons may see pursuing an education as selfish, since getting a job instead would contribute directly and immediately to family members [LAT2].

Investigating cultural reasons for differences in academic achievement is quite different from proposing that members of different ethnicities and genders actually learn mathematics in different ways.  The latter point of view is especially serious when it leads to new, watered-down mathematics curricula.

There should be no doubt that minority students can thrive in traditional programs.  Take the case of Bennett-Kew Elementary School in Inglewood, California.  According to Principal Nancy Ichinaga, 51% of the students are African-American and 48% are Hispanic (mostly immigrant with Limited English Proficiency).  Approximately 70% qualify for subsidized lunches. Below are its 1997 California Achievement Test results, with Normal Curve Equivalent  scores (similar to percentiles):

         Grade            1         2       3        4        5
Math            62      79      81      75      68

Bennett-Kew believes in high, explicit standards for all students.  The mathematics standards are not just year-by-year, but month-by-month.  There is regular diagnostic testing of student progress and immediate remediation. The school is committed to direct instruction and does not use newer books.  While discussing the mathematics reform movement with me, Principal Ichinaga remarked, “Reform is for the birds.”

The traditional approach to teaching calculus used by the legendary teacher Jaime Escalante is another example of minority students thriving in a traditional mathematics program.  In 1974, Escalante took a job teaching basic mathematics at Garfield High School which was in danger of losing its accreditation because discipline and test scores were so bad.  Five years later, insisting that disadvantaged and minority students could tackle the most difficult subjects, he started a small calculus class. The effect was to raise the curriculum for the entire school. In 1982, 18 of his students passed the Advanced Placement calculus exam. This was the subject of the movie, “Stand and Deliver.”  Working with his fellow calculus teacher Ben Jimenez, and Garfield Principal Henry Gradillas, Escalante sent ever increasing numbers of students to leading universities with AP calculus credit.

By 1987, Garfield High School had more test takers than all but four high schools in the United States.  The number of test takers reached its peak of 143 students in 1991, the year Escalante left Garfield.  The passage rate was 61%.  The numbers have declined ever since.  By 1996 there were only 37 test takers with a passage rate of 19%.  It is interesting that former Principal Gradillas’ career declined after the spectacular successes of his high school.  After finishing his doctorate in 1987, he “expected to be given an important administrative job that would help spread the school’s philosophy to other parts of Los Angeles. Instead he was told to supervise asbestos inspections of school buildings.  District officials denied they were punishing him, but one said privately that Gradillas was refused better assignments because he was considered ‘too confrontational.”’ [WP]. Rather than studying his effective methods, Escalante is shunned by the mathematics reform community.  The disapproval is mutual.  According to Escalante, “whoever wrote [the NCTM math standards] must be a physical education teacher.” [CS]

The calculus reform movement is inextricably linked to the K-12 mathematics reform movement.  Consider, for example, the following statement from the preface of the first edition to the Harvard Calculus text [HAL]:

We have found this curriculum to be thought-provoking for well-prepared students while still accessible to students with weak algebra backgrounds. Providing numerical and graphical approaches as well as the algebraic gives the students several ways of mastering the material.  This approach encourages students to persist, thereby lowering failure rates.


Lower failure rates at the cost of eviscerating the algebra component of calculus is harmful to students of all ethnicities and both genders. Algebra and arithmetic are consistently de-emphasized in reform curricula in exchange for the more “holistic” calculator assisted “guess and check” routine. The entire reform program mortgages future opportunities to attend to the immediacy of high failure rates.  The de-emphasis of algebra in reform calculus justifies and caters to the K-12 reform mathematics program.

Calculus proofs and even definitions require students to be competent in algebra.  Calculus reform texts tend to relegate both of these to appendices, sparing students the necessity even to turn a few pages in order to avoid them.  Instructors who wish to include definitions, such as the definition of a limit and/or a few proofs, must overcome additional psychological resistance because of the location of these topics in the textbooks.  When a proof or definition is placed in an appendix, it sends the message to the student that the topic is not important and may be safely skipped.

The emergence of these trends at a time when greater numbers of previously under-represented students are attending universities should cause some reflection within the mathematics community. Are we expecting less of these students?  If so, is it because they learn mathematics differently from students of an earlier era, or is it because their mathematical preparations are deficient?  I think it is the latter, and I believe that the mathematics community would do well to purge itself of any hidden assumptions that non-Asian minority students learn mathematics differently from anybody else.  The focus should be on raising the level of mathematics education in K-12, not on how best to lower it in the universities.

The Politics of Mathematics Reform in California

Nowhere has the conflict over mathematics education reform been more contentious than in California.  California led the United States in institutionalizing K-12 mathematics reform.  The 1992 California Mathematics Framework is based on the 1989 NCTM Standards and has served as a guide for politically powerful reformers, like the California Superintendent of Instruction, Delaine Eastin (elected in 1994), as well as countless specialists in the state’s Colleges of Education who have used it as course material for K-12 student teachers.   But California’s commitment to the principles of mathematics reform  predates the NCTM Standards. For example, one finds in the 1985 “California Model Curriculum Standards, Grades Nine Through Twelve”:

The mathematics program must present to students problems that utilize acquired skills and require the use of problem-solving strategies.  Examples of strategies that students should employ are: estimate, look for a pattern, write an equation, guess and test, work backward, draw a picture or diagram, make a list or table, use models, act out the problem, and solve a related but simpler problem.  The use of calculators and computers should also be encouraged as an essential part of the problem-solving process.  Students should be encouraged to devise their own plans and explore alternate approaches to problems.


The educational philosophies behind the mathematics reform movement are canonical in America’s colleges of education and have been for most of this century [H].  The broad principles of reform have been institutionalized in California state documents for well over a decade and have taken root in the schools. Reform curricula based on these principles are ubiquitous in California’s elementary schools.  The controversial curriculum, “MathLand,” for example, has been adopted by 60% of the state’s public elementary schools, according to its publishers [T],  and there are many other similar curricula widely in use. Secondary mathematics curricula such as Interactive Mathematics Program and College Preparatory Mathematics originated in California and are widely used throughout the state at the time of this writing. The alignment of these and other self-described reform curricula with the NCTM Standards seems to be uncontested.  Indeed, much of the development and implementation of these curricula has been funded by the National Science Foundation and other powerful, reform dominated institutions.  In particular, MathLand, perhaps the worst of all reform curricula, has been promoted through the NSF funding.

California is experiencing a backlash at the grass-roots level against the general education reform movement (including Whole Language Learning and “Integrated Science”), and mathematics reform in particular. Reacting to the de-emphasis of arithmetic and algebra in the reform curricula, and the over-reliance on calculators, parents’ education organizations have emerged all over the state, several with their own web sites containing material starkly critical of “reform math”  or “fuzzy math.”  I am associated with the largest and best known of these groups, “Mathematically Correct.”

Of particular concern to parents and teachers critical of the reform movement is the lack of accountability and measurable standards of achievement  in the schools.  “Authentic assessment” in place of examinations with consequences, and little if any importance placed on student discipline and responsibility  in the reform literature, help to make reform math an object of ridicule among vocal parents’ groups. It is noteworthy that all parties acknowledge the importance of better teacher training.

The conflict between the mathematics reform movement, on the one hand, and parents’ organizations combined with a significant portion of the mathematics community, on the other hand, reached a turning point in December 1997. At that time the California Board of Education rejected the reform-oriented draft standards from one of its advisory committees– the Academic Standards Commission–and, with the help of  Stanford mathematics professors Gunnar Carlsson, Ralph Cohen, Steve Kerckhoff, and Jim Milgram, developed and adopted the California Mathematics Academic Content Standards  [CA].

Unlike the Academic Standards Commission proposal, these new standards made no pronouncements about teaching methods, only grade-level benchmarks. The reaction from the California mathematics reform community against this lack of coercion was swift and harsh.  Their response was to claim that the official math standards, written by the Stanford mathematicians, lowered the bar. Turning reality on its head, State Superintendent Delaine Eastin charged, “[The State Board of Education Standards] is ‘dumbed-down’ and is unlikely to elicit higher order thinking…” Judy Codding, a member of the Academic Standards Commission and the powerful National Center on Education and the Economy put it bluntly when she said, “I will fight to see that [the] California Math Standards are not implemented in the classrooms” [Wu].

Other Reformers with national stature echoed the outrage.  Luther Williams, the National Science Foundation’s Assistant Director for Education and Human Resources,  wrote a retaliatory letter to the California Board of Education widely interpreted as threatening to cut off funding of NSF projects in California.  The lead story in the February 1998 News Bulletin of the NCTM, “New California Standards Disappoint Many,” began with the sentence, “Mathematics education in California suffered a serious blow in December.”  The article quotes a letter from NCTM President Gail Burrill to the president of the California Board of Education that included the statements: “Today’s children cannot be prepared for tomorrow’s increasingly technological world with yesterday’s content…The vision of important school mathematics should not be one that bears no relation to reality, ignores technology, focuses on a limited set of procedures, ….California’s children deserve more.”  Presumably the accusation that technology is ignored refers in part to a policy decision of the California Board of Education that statewide exams based on the new math standards will not include the use of calculators–a serious blow to the corporate/reform ideology.

Joining the reform math community, the statewide chairs of the Academic Senates of the UC, CSU, and California Community College systems, none of whom were  mathematicians,   issued a joint statement condemning the adoption of California’s math standards and even suggested that “the consensus position of the mathematical community” was in opposition to the new standards, and generally in support of the rejected, reform-inspired draft standards written by the Academic Standards Commission.

In opposition to the reform community and in support of California’s new math standards, more than 100 California college and university mathematicians endorsed an open letter addressed to the Chancellor of the 23-campus California State University system.  The open letter disputed the existence of such a consensus and urged the Chancellor to  “recognize the important and positive role California’s recently adopted mathematics standards can play in the education of future teachers of mathematics in the state of California.”  Among the endorsing mathematicians were  several department chairs and many leading mathematicians [L].

Further contradicting the reformers’ claims against California’s math standards, Ralph Raimi and Lawrence Braden, on behalf of the  Fordham Foundation,  conducted an independent review of the mathematics standards from 46 states and the District of Columbia,  as well as Japan. California’s new board-approved mathematics standards received the highest score, outranking even those of Japan [FR].

The sharp conflict over the California math standards defined, in practical terms, the mathematics reform movement.  Reformers denounced the state’s standards in public forums and the press, while traditionalists and critics of reform defended the standards.  Based on purely mathematical considerations, the board-approved California standards are easily seen to be superior to the rejected, reform-oriented version offered by the Academic Standards Commission.   A careful and well-written comparison these two sets of standards by Hung-Hsi Wu is available on the Mathematically Correct web site [Wu].

The extent to which the California math standards will be taken seriously by school districts is difficult to predict.  The superintendent of the Los Angeles Unified School District, the second largest school district in the U.S., admonished LAUSD personnel to take no action to implement the new standards, arguing instead that the already existing LAUSD standards were superior.  A refutation and insightful comparison of the LAUSD math standards with the California standards was developed by Jim Milgram, Mathematically Correct Co-Founder Paul Clopton, and others.  It is also available on the “Mathematically Correct” web site [MC].  The LAUSD K-12 math standards are vague and repetitive, trigonometry is completely missing, and third graders are encouraged to use calculators.

Opposition to California’s mathematics standards from reform leaders continues as of this writing. Former NCTM president Jack Price wrote in a letter published by the Los Angeles Times on May 10, 1998:

...if the state board had adopted world-class mathematics standards for the 21st century instead of the 19th century, there would have been a great deal of support from the ‘education’ community.


This sententious observation encapsulates the topics discussed in this essay.  For the reformers, “world-class mathematics standards for the 21st century” eluded the Stanford mathematicians who wrote California’s 1998 math standards.  Missing are the greater emphasis on technology–an end in itself–and pedagogical directives harmonious with the reified “cognitive styles” of the racially diverse populations of the 21st century.  The “19th century” arithmetic, algebra, geometry,  and trigonometry  highlighted in California’s 1998  standards will have diminished value in the postmodern epoch of technological wonderments envisioned by math reformers.

Perhaps the academic community should consider whether the discipline of mathematics education–much more so than mathematics–needs fundamental alterations for the 21st century.

References

[CA] Available at: http://www.cde.ca.gov/board/k12math_standards.html

[CMC] California Mathematics Council Open Letter,
http://wworks.com/~pieinc/scan-cmc.htm

[CS] Charles Sykes, Dumbing Down Our Kids: Why American Children Feel Good about Themselves but Can’t Read, Write, or Add, St. Martin’s Press, 1995 p. 122

[F] Forbes, June 16, 1997, I got my degree through E-mail

[FR] Fordham Report: Volume 2, Number 3 March 1998 State Mathematics Standards by Ralph A. Raimi and Lawrence S. Braden, http://www.edexcellence.net/standards/math.html

[G] David Gelernter, Should Schools Be Wired To The Internet? , No–Learn First, Surf Later, Time  May 25, 1998

[HAL]  Hughes-Hallet et al, Calculus, John Wiley and Sons, New York, 1992, 1998

[H] E.D. Hirsch Jr., The Schools We Need; Why We Don’t Have Them, DoubleDay, New York (1996)

[JAIE] Journal of American Indian Education, Special Issue, August 1989

[LAT1] Los Angeles Times, June 9, 1997, High Tech Sales Goals Fuel Reach into Schools

[LAT2] Los Angeles Times, May 18, 1998, all of Section R

[L] Open Letter to CSU Chancellor,
http://www.mathematicallycorrect.com/reed.htm

[MC]Mathematically Correct,
http://www.mathematicallycorrect.com/

[MJ] Carol Malloy and Gail M. Jones, An Investigation of African American Students’ Mathematical Problem Solving, Journal for Research in Mathematics Education, 29, no. 2, March 1998, pages 143-163

[ML] MathLand,
http://www.mathland.com/assessInMath.html#assess_LAData

[N] David Noble, Digital Diploma Mills: The Automation of Higher Education, Monthly Review, Feb. 1998, Selling Academe to the Technology Industry, Thought and Action: The NEA Higher Education Journal , XIV, no. 1, Spring 1998

[NRC] National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education, National Academy Press, Washington, D.C., 1989

[O] Todd Oppenheimer, The Computer Delusion, The Atlantic Monthly, July 1997, http://www.theatlantic.com/issues/97jul/computer.htm

[SFC] San Francisco Chronicle, December 26, 1996, Ebonics Tests Linguistic Definition

[T] Time, August 25, 1997, Suddenly, Math Becomes Fun And Games. But Are The Kids Really Learning Anything?

[WP] The Washington Post, May 21, 1997, A Math Teacher’s Lessons in Division

[Wu]  Hung-Hsi Wu, Some observations on the 1997 battle of the two Standards in the California Math War
http://www.mathematicallycorrect.com/hwu.htm

MATH LESSONS: BEYOND RHETORIC, STUDIES IN HIGH ACHIEVEMENT

Los Angeles Times
Sunday, February 11, 2001

MATH LESSONS: BEYOND RHETORIC, STUDIES IN HIGH ACHIEVEMENT

By David Klein

Schools with low-income students tend to have low test scores. Low academic achievement, especially in mathematics, is often one of the consequences of poverty. Nevertheless, some schools beat the odds.

Bennett-Kew Elementary School in Inglewood is an example. At Bennett-Kew 51% of the students are African American, 48% are Latino, 29% are not fluent in English and 77% of all students qualify for free or reduced-price lunch, a standard measure of poverty in schools. Yet test scores at Bennett-Kew require no excuses. The average third-grader at Bennett-Kew scored at the 83rd percentile in mathematics on the most recent Stanford Achievement Test, double the score for Los Angeles Unified School District.

In the summer of 2000, the Brookings Institution, a Washington, D.C., think tank, commissioned me to find three high-achieving, low-income schools in the Los Angeles area, and to write a report about how they teach math. That report is available from www.mathematicallycorrect.com. In addition to Bennett-Kew, the report describes William H. Kelso Elementary School, also in Inglewood, and Robert Hill Lane Elementary School in Monterey Park, part of LAUSD. Students at these outstanding schools also exhibit unusually high achievement in mathematics despite modest resources.

What accounts for the high academic achievement of these schools? Can their successes be replicated?

For starters, consider how they are alike. All three closely follow the California mathematics content standards. Direct instruction, as opposed to “student discovery,” is the primary mode of instruction. All three schools focus on basic skills as prerequisites to problem solving and understanding of concepts. Calculator use is rare or nonexistent. Faculty at all three schools are well-coordinated and work together. Principals at these schools are strong leaders, and they are careful to hire dedicated teachers. The principals have found that noncredentialed teachers are sometimes better than credentialed teachers. All three schools have programs that provide remediation, and the principals closely monitor student achievement. But the most important characteristic of all three schools is that students are held to high expectations. The principals were adamant about high expectations and dismissive of excuses.

These days almost everyone uses buzzwords like “high expectations.” But Nancy Ichinaga, the former principal of Bennett-Kew and now a member of the California State Board of Education, took her students beyond the rhetoric of these words to their actual substance. The same may be said for retired principal Marjorie Thompson of Kelso and principal Sue Wong of Lane Elementary.

What prevents hundreds of L.A. schools from following suit? Part of the answer is that ideology trumps common sense in LAUSD. School administrators have long believed that “learning styles” are strongly correlated with race and gender, and that “dead white male math” is just not appropriate for minority students. As a consequence, the LAUSD board decided last year to prevent its elementary schools from buying the successful but traditional math program used at Bennett-Kew, called Saxon Math. This California state-approved curriculum is also a component of the math program at Melvin Elementary School in Reseda. Melvin, an LAUSD campus, was highlighted in Gov. Gray Davis’ State of the State speech for its dramatic improvement in test scores during the last two years.

So, what has LAUSD deemed appropriate for minority students? Following recommendations of the Los Angeles-based Achievement Council, LAUSD last year left hundreds of schools saddled with vacuous calculator-based, anti-arithmetic programs like MathLand, which is not even remotely aligned to the state standards upon which students are tested.

Perhaps the worst blunder is yet to come. Instead of focusing on California’s standards, written by world-renowned mathematicians at Stanford University, LAUSD Supt. Roy Romer is now promoting standards from the National Center on Education and the Economy, or NCEE. These standards are inconsistent with the California standards. They are faddish, low level and incoherent. Judy Codding, a vice president of the NCEE, made no secret of her organization’s hostility to California’s rigorous standards when she announced at an NCEE conference, “I will fight to see that California math standards are not implemented in the classroom.”

She might succeed. If teachers are forced to serve two contradictory masters, the high-caliber California standards and the dubious NCEE standards, the result will be more confusion and misdirection. Although LAUSD deserves some praise for recent steps to purchase state-approved textbooks, school board members should put an end to the continual bombardment of students and teachers with the latest education fads. It is far more constructive to maintain clarity of purpose, and to join successful schools that follow the state standards.

David Klein (david.klein@csun.edu) is a Professor of Mathematics at Cal State Northridge

Copyright 2001 Los Angeles Times

AN OPEN LETTER TO UNITED STATES SECRETARY OF EDUCATION, RICHARD RILEY

 

AN OPEN LETTER TO UNITED STATES SECRETARY OF EDUCATION, RICHARD RILEY

Dear Secretary Riley:

In early October of 1999, the United States Department of Education endorsed ten K-12 mathematics programs by describing them as “exemplary” or “promising.” There are five programs in each category. The “exemplary” programs announced by the Department of Education are:

 

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)

The “promising” programs are:

 

Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

These mathematics programs are listed and described on the government web site: http://www.enc.org/ed/exemplary/

The Expert Panel that made the final decisions did not include active research mathematicians. Expert Panel members originally included former NSF Assistant Director, Luther Williams, and former President of the National Council of Teachers of Mathematics, Jack Price. A list of current Expert Panel members is given at: http://www.ed.gov/offices/OERI/ORAD/KAD/expert_panel/mathmemb.html

It is not likely that the mainstream views of practicing mathematicians and scientists were shared by those who designed the criteria for selection of “exemplary” and “promising” mathematics curricula. For example, the strong views about arithmetic algorithms expressed by one of the Expert Panel members, Steven Leinwand, are not widely held within the mathematics and scientific communities. In an article entitled, “It’s Time To Abandon Computational Algorithms,” published February 9, 1994, in Education Week on the Web, 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.” (http://www.edweek.org/ew/1994/20lein.h13)

In sharp contrast, a committee of the American Mathematical Society (AMS), formed for the purpose of representing the views of the AMS to the National Council of Teachers of Mathematics, published a report which stressed the mathematical significance of the arithmetic algorithms, as well as addressing other mathematical issues. This report, published in the February 1998 issue of the Notices of the American Mathematical Society, includes the statement:

 

“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. 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.“

Even before the endorsements by the Department of Education were announced, mathematicians and scientists from leading universities had already expressed opposition to several of the programs listed above and had pointed out serious mathematical shortcomings in them. The following criticisms, while not exhaustive, illustrate the level of opposition to the Department of Education’s recommended mathematics programs by respected scholars:

 

Richard Askey, John Bascom Professor of Mathematics at the University of Wisconsin at Madison and a member of the National Academy of Sciences, pointed out in his paper, “Good Intentions are not Enough” that the grade 6-8 mathematics curriculum Connected Mathematics Program entirely omits the important topic of division of fractions. Professor Askey’s paper was presented at the “Conference on Curriculum Wars: Alternative Approaches to Reading and Mathematics” held at Harvard University October 21 and 22, 1999. His paper also identifies other serious mathematical deficiencies of CMP.

R. James Milgram, professor of mathematics at Stanford University, is the author of “An Evaluation of CMP,” “A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California,” and “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later.” This latter paper is based on a statistical survey undertaken by Gregory Bachelis, professor of mathematics at Wayne State University. Each of these papers identifies serious shortcomings in the mathematics programs: CMP, Core-Plus, and IMP. Professor Milgram’s papers are posted at: ftp://math.stanford.edu/pub/papers/milgram/

Martin Scharlemann, while chairman of the Department of Mathematics at the University of California at Santa Barbara, wrote an open letter deeply critical of the K-6 curriculum MathLand, identified as “promising” by the U. S. Department of Education. In his letter, Professor Scharlemann explains that the standard multiplication algorithm for numbers is not explained in MathLand. Specifically he states, “Astonishing but true — MathLand does not even mention to its students the standard method of doing multiplication.” The letter is posted at: http://mathematicallycorrect.com/ml1.htm

Betty Tsang, research physicist at Michigan State University, has posted detailed criticisms of the Connected Mathematics Project on her web site at: http://www.nscl.msu.edu/~tsang/CMP/cmp.html

Hung-Hsi Wu, professor of mathematics at the University of California at Berkeley, has written a general critique of these recent curricula (“The mathematics education reform: Why you should be concerned and what you can do“, American Mathematical Monthly104(1997), 946-954) and a detailed review of one of the “exemplary” curricula, IMP (“Review of Interactive Mathematics Program (IMP) at Berkeley High School“, http://www.math.berkeley.edu/~wu). He is concerned about the general lack of careful attention to mathematical substance in the newer offerings.

 

While we do not necessarily agree with each of the criticisms of the programs described above, given the serious nature of these criticisms by credible scholars, we believe that it is premature for the United States Government to recommend these ten mathematics programs to schools throughout the nation. We respectfully urge you to withdraw the entire list of “exemplary” and “promising” mathematics curricula, for further consideration, and to announce that withdrawal to the public. We further urge you to include well-respected mathematicians in any future evaluation of mathematics curricula conducted by the U.S. Department of Education. Until such a review has been made, we recommend that school districts not take the words “exemplary” and “promising” in their dictionary meanings, and exercise caution in choosing mathematics programs.

Sincerely,

David Klein
Professor of Mathematics
California State University, Northridge

Richard Askey
John Bascom Professor of Mathematics
University of Wisconsin at Madison

R. James Milgram
Professor of Mathematics
Stanford University

Hung-Hsi Wu
Professor of Mathematics
University of California, Berkeley

Martin Scharlemann
Professor of Mathematics
University of California, Santa Barbara

Professor Betty Tsang
National Superconducting Cyclotron Laboratory
Michigan State University

The following endorsements are listed in alphabetical order.

William W. Adams
Professor of Mathematics
University of Maryland, College Park

Alejandro Adem
Professor & Chair
Department of Mathematics
University of Wisconsin-Madison

Max K. Agoston
Associate Professor
Department of Mathematics and Computer Science
San Jose State University

Henry L. Alder
Professor of Mathematics
University of California, Davis
Former member of the California Board of Education
Former President of the Mathematical Association of America

Kenneth Alexander
Professor of Mathematics
University of Southern California

Frank B. Allen
Professor of Mathematics Emeritus, Elmhurst College
Former President, National Council of Teachers of Mathematics

George E. Andrews
Evan Pugh Professor of Mathematics
Pennsylvania State University

Gregory F. Bachelis
Professor of Mathematics
Wayne State University

Michael Beeson
Professor of Mathematics and Computer Science
San Jose State University

George Biriuk
Professor of Mathmatics
California State University, Northridge

Wayne Bishop
Professor of Mathematics
California State University, Los Angeles

Gary J. Blanchard
Professor of Chemistry
Michigan State University

Charles C. Blatchley, Chair
Department of Physics
Pittsburg State University

Michael N. Bleicher
Professor Emeritus,
University of Wisconsin – Madison
Chair, Department of Mathematical Sciences
Clark Atlanta University

John C. Bowman
Vice-President
National Association of Professional Educators

Khristo N. Boyadzhiev
Professor of Mathematics
Ohio Northern University

Bart Braden
Professor of Mathematics
Northern Kentucky University

Stephen Breen
Associate Professor
Department of Mathematics
California State University, Northridge

David A. Buchsbaum
Professor of Mathematics, Emeritus
Brandeis University

Frank Burk
Professor of Mathematics
California State University, Chico

Ana Cristina Cadavid
Professor of Physics
California State University, Northridge

Gunnar Carlsson
Professor of Mathematics
Stanford University

Douglas Carnine
Professor of Education
University of Oregon
Director of the National Center to Improve the Tools of Educators

Mei-Chu Chang
Professor of Mathematics
University of California, Riverside

Sun-Yung Alice Chang
Professor of Mathematics
Princeton University and UCLA

Jeff Cheeger
Professor of Mathematics
Courant Institute, NYU

Orin Chein
Professor of Mathematics
Temple University

Steven Chu
Theodore and Francis Geballe Professor of Physics and Applied Physics
Chair of Physics
Stanford University
1997 Nobel Prize for Physics

Fredrick Cohen
Professor of Mathematics
University of Rochester

Marshall M. Cohen
Professor, Mathematics
Cornell University

Paul Cohen
Professor of Mathematics
Stanford University

Ralph Cohen
Professor of Mathematics
Stanford University

Peter Collas
Professor of Physics
California State University, Northridge

Bruce Conrad
Associate Dean College of Science and Technology
Temple University

Daryl Cooper
Professor of Mathematics
University of California, Santa Barbara

Robert M. Costrell
Director of Research and Development
Executive Office for Administration and Finance
Commonwealth of Massachusetts
Professor of Economics
University of Massachusetts at Amherst

George K. Cunningham, Professor
Department of Educational and Counseling Psychology
University of Louisville

Jerome Dancis
Associate Professor of Mathematics
University of Maryland

Pawel Danielewicz
Professor, Department of Physics and Astronomy
Michigan State University

Ernest Davis
Associate Professor of Computer Science
New York University

Martin Davis
Professor Emeritus of Mathematics and Computer Science
Courant Institute
New York University

Jane M. Day
Professor of Mathematics and Computer Science
San Jose State University

Carl de Boor
Professor of Mathematics and Computer Sciences
University of Wisconsin-Madison

Percy Deift
Professor of Mathematics
Courant Institute
New York University

John de Pillis
Professor of Mathematics
University of California, Riverside

Robert Dewar
Professor of Computer Science
Courant Institute of Mathematical Sciences
Former Chair of Computer Science
Former Associate Director of the Courant Institute
New York University

Jim Dole
Professor and Chair of Biology
California State University, Northridge

Josef Dorfmeister
Professor of Mathematics
University of Kansas

Bruce T. Draine
Professor of Astrophysical Sciences
Princeton University

Bruce K. Driver
Professor of Mathematics
University of California, San Diego

Vladimir Drobot
Professor
Department of Mathematics and Computer Science
San Jose State University

William Duke
Professor of Mathematics
Rutgers University

John R. Durbin
Professor of Mathematics
Secretary of the General Faculty
The University of Texas at Austin

Peter Duren
Professor of Mathematics
University of Michigan

Mark Dykman
Professor of Physics
Michigan State University

Allan L. Edelson
Professor of Mathematics and
Vice Chair for Graduate Affairs
Department of Mathematics
University of California, Davis

Yakov Eliashberg
Professor of Mathematics
Stanford University

Richard H. Escobales, Jr.
Professor of Mathematics
Canisius College, Buffalo, NY

Lawrence C. Evans
Professor of Mathematics
University of California, Berkeley

Bill Evers
Research Fellow
Hoover Institution
Stanford University
California State Academic Standards Commission

Barry Fagin
Professor of Computer Science
US Air Force Academy

George Farkas
Professor of Psychology
Director, Center for Education and Social Policy
University of Texas at Dallas
Editor, Rose Monograph Series of the American Sociological Association

Robert Fefferman
Louis Block Professor of Mathematics
Chairman, Mathematics Department
University of Chicago

Chester E. Finn, Jr.
John M. Olin Fellow
Manhattan Institute
Former U.S. Assistant Secretary of Education

Ronald Fintushel
University Distinguished Professor of Mathematics
Michigan State University

Michael E. Fisher
Distinguished Univeristy Professor & USM Regents Professor
Insitute of Physical Sciences and Technology
University of Maryland
Wolf Prize in Physics, 1980

Patrick M. Fitzpatrick
Professor and Chair
Department of Mathematics
University of Maryland

Yuval Flicker
Professor of Mathematics
The Ohio State University

Gerald Folland
Professor of Mathematics
University of Washington, Seattle

Daniel S. Freed
Professor of Mathematics
University of Texas at Austin

Dmitry Fuchs
Professor
Department of Mathematics
University of California, Davis

David C. Geary
Professor of Psychology
University of Missouri

Samuel Gitler
Professor of Mathematics
University of Rochester

Sheldon Lee Glashow
Higgins Professor of Physics
Harvard University
1979 Nobel Prize in Physics

Simon M. Goberstein
Professor of Mathematics
California State University, Chico

Steve Gonek
Professor of Mathematics
University of Rochester

Jeremy Goodman
Department of Astrophysical Sciences
Princeton University
Co-founder, Princeton Charter School

Jonathan Goodman
Professor of Mathematics
Courant Institute of Mathematical Sciences
New York University

David Goss
Professor of Mathematics
The Ohio State University

Steven R. Goss
Chairman of the Board
Arizona Scholarship Fund
Mechanical Engineer – Raytheon Systems

Christopher M. Gould
Professor of Physics
Department of Physics and Astronomy
University of Southern California

Mark L. Green
Professor of Mathematics
University of California at Los Angeles

Benedict H. Gross
Leverett Professor of Mathematics
Harvard University

Leonard Gross
Professor of Mathematics
Cornell University

Paul R. Gross
University Professor of Life Sciences (emeritus)
University of Virginia

Dina Gutkowicz-Krusin
Principal Scientist
Electro-Optical Sciences, Inc.
Irvington, New York

Kamel Haddad
Associate Professor of Mathematics
California State University, Bakersfield

Deborah Tepper Haimo
Visiting Scholar
University of California, San Diego
Trustee of Association of Members of the Institute for Advanced Study at Princeton
Former President of the Mathematical Association of America

Joel Hass
Professor of Mathematics
University of California, Davis

David F. Hayes
Professor of Mathematics and Computer Science
San Jose State University

Dr. Adrian D. Herzog
Chairman, Deprtment of Physics and Astronomy
California State University, Northridge
Member Content Review Panel for California Science Materials

Richard O. Hill
Professor of Mathematics
Michigan State University

E. D. Hirsch, Jr.
University Professor of Education and Humanities
University of Virginia

Dr. Hanna J. Hoffman
Senior Laser Scientist
IRVision, Inc.
San Jose, California

Douglas L. Inman
Research Professor of Oceanography
Scripps Institution of Oceanography
University of California, San Diego

George Jennings
Professor of Mathematics
California State University, Dominguez Hills

Svetlana Jitomirskaya
Associate Professor of Mathematics
University of California, Irvine

Peter W. Jones
Professor and Chair of Mathematics
Yale University

Vaughan Jones
Professor of Mathematics
Mathematics Department
UC Berkeley

Peter J. Kahn
Professor of Mathematics and
Senior Associate Dean
College of Arts and Sciences
Cornell University

Sheldon Kamienny
Professor of Mathematics
University of Southern California

Ilya Kapovich
Assistant Professor of Mathematics
Rutgers, The State University of New Jersey

Hidefumi Katsuura
Professor of Mathematics
San Jose State University

Jerry Kazdan
Professor of Mathematics
Univerity of Pennsylvania

David Kazhdan
Professor of Mathematics
Harvard University

Lisa Graham Keegan
Superintendent of Public Education
State of Arizona

Sharad Keny
Professor of Mathematics
Department of Mathematics
Whittier College

Steve Kerckhoff
Professor of Mathematics
Stanford University

Robion C. Kirby
Professor of Mathematics
University of California at Berkeley

Steven G. Krantz
Chairman and Professor
Department of Mathematics
Washington University in St. Louis
St. Louis, Missouri

Sergiu Klainerman
Professor of Mathematics
Princeton University

Abel Klein
Professor of Mathematics
University of California, Irvine

Kurt Kreith
Professor Emeritus of Mathematics
University of California at Davis

Boris A. Kushner
Professor of Mathematics
University of Pittsburgh at Johnstown

Tsit-Yuen Lam
Professor of Mathematics
University of California at Berkeley

Serge Lang
Professor of Mathematics
Yale University

Benedict Leimkuhler
Associate Professor of Mathematics
University of Kansas
and Fellow, Kansas Center for Advanced Scientific Computing

Norman Levitt
Professor of Mathematics
Rutgers University, New Brunswick

Jun Li
Associate Professor of Mathematics
Stanford University

Peter Li
Professor and Chair of Mathematics
University of California, Irvine

Alexander Lichtman
Professor of Mathematics
University of Wisconsin-Parkside

Seymour Lipschutz
Professor of Mathematics
Temple University

Mei-Ling Liu
Professor of Computer Science
California Polytechnic State University

Darren Long
Professor of Mathematics
University of California, Santa Barbara

John Lott
Professor of Mathematics
University of Michigan – Ann Arbor

Tom Loveless
Director, Brown Center on Education Policy
The Brookings Institution
Washington, DC

Steve P. Lund
Professor of Geophysics
Department of Earth Sciences
University of Southern California

William G. Lynch
Professor, Department of Physics
Michigan State University

Michael G. Lyons
Consulting Assoc. Prof
Management Science and Engineering
Stanford University

Saunders Mac Lane
Max Mason Distinguished Service Professor, Emeritus
University of Chicago
National Medal of Science, 1989
Former Vice President, National Academy of Sciences, 1973-1981
Former Member, National Science Board, 1973-1979

Michael Maller
Associate Professor of Mathematics
Queens College of CUNY

Igor Malyshev
Professor of Mathematics
San Jose State University

Edward Matzdorff
Professor of Mathematics
California State University, Chico

Michael May
Co-Director, Center for International Security and Arms Control
(Research) Professor
Department of Engineering-Economic Systems and Operations Research
Stanford University

Rafe Mazzeo
Professor of Mathematics
Stanford University

John McCarthy
Professor of Computer Science
Stanford University

John D. McCarthy
Professor of Mathematics
Michigan State University

John E. McCarthy
Professor of Mathematics
Washington University

Henry P. McKean
Professor of Mathematics
Courant Institute
New York University

Michael McKeown
Professor of Medical Science
Program in Molecular Biology, Cell Biology and Biochemistry
Brown University
Former Member – San Diego Unified Math Standards Committee
Former Member – Superintendent’s Math Advisory Committee, San Diego
Co-Founder Mathematically Correct

Marc Mehlman
Associate Professor of Mathematics
University of Pittsburgh, Johnstown

Adrian L. Melott
Professor of Physics and Astronomy
University of Kansas

Aida Metzenberg
Assistant Professor of Biology
California State University, Northridge

Stan Metzenberg
Assistant Professor of Biology
California State University, Northridge

M. Eugene Meyer
Professor of Mathematics
California State University, Chico

James E. Midgley
Professor of Physics, Emeritus
University of Texas at Dallas

Dragan Milicic
Professor of Mathematics
University of Utah

Henri Moscovici
Professor of Mathematics
The Ohio State University
Clay Mathematics Institute Scholar

Govind S. Mudholkar
Professor of Statistics and Biostatistics
University of Rochester

Gregory Naber
Professor of Mathematics
California State University, Chico

Bruno Nachtergaele
Associate Professor of Mathematics
University of California, Davis

Chiara R. Nappi
Visiting Professor of Physics
University of Southern California
On leave from the
Institute for Advanced Study at Princeton

Anil Nerode
Goldwin Smith Professor of Mathematics
Cornell University

Charles M. Newman
Professor and Chair of Mathematics
Courant Institute of Mathematical Sciences
New York University

Louis Nirenberg
Professor of Mathematics
Courant Institute, New York University

Maria Helena Noronha
Professor of Mathematics
California State University, Northridge

Robert H. O’Bannon, Ph.D.
Professor, Department of Natural Sciences and Mathematics
Lee University
Cleveland, TN

Richard Palais
Professor of Mathematics, Emeritus
Brandeis University

Dimitri A. Papanastassiou
Faculty Associate in Geochemistry
Caltech

Thomas H. Parker
Professor of Mathematics
Michigan State University

Donald S. Passman
Professor of Mathematics
University of Wisconsin at Madison

Peter Petersen
Undergraduate Vice Chair and Professor of Mathematics
Department of Mathematics
UCLA

Steven Pinker
Professor of Psychology
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Author of How the Mind Works

Jacek Polewczak
Professor of Mathematics
California State University, Northridge

Dr. Ned Price
Mathematics Department
Framingham State College
Framingham,Ma.

David Protas
Professor of Mathematics
California State University, Northridge

Ralph A. Raimi
Professor Emeritus of Mathematics
University of Rochester, Rochester, New York

Douglas C. Ravenel
Professor and Chair of Mathematics
University of Rochester

Marc A. Rieffel
Professor of Mathematics
University of California, Berkeley

Tom Roby
Assistant Professor of Mathematics
California State University, Hayward

Cris T. Roosenraad
Professor of Mathematics
Carleton College

Jerry Rosen
Professor of Mathematics
California State University, Northridge

Mary Rosen
Professor of Mathematics
California State University, Northridge

Yoram Sagher
Prof. of Mathematics
University of Illinois at Chicago

Charles G. Sammis
Professor of Geophysics
University of Southern California

Mark Sapir
Professor of Mathematics
Vanderbilt University

Peter Sarnak
Professor of Mathematics
Princeton University

Stephen Scheinberg, Ph.D., M.D.
Professor of Mathematics
Clinical Assistant Professor of Dermatology
University of California, Irvine

Wilfried Schmid
Dwight Parker Robinson Professor of Mathematics
Harvard University

Dr. Martha Schwartz
Geophysicist
California Mathematics Framework Committee
Co-founder of Mathematically Correct

Albert Schwarz
Professor of Mathematics
University of California, Davis

Roger Shouse
Asst. Professor of Education Policy Studies
The Pennsylvania State University

Barry Simon
I.B.M. Professor of Mathematics and Theoretical Physics
Chair, Department of Mathematics
Caltech

Leon Simon
Professor of Mathematics and Chairman
Department of Mathematics
Stanford University

David Singer
Professor of Mathematics
Case Western Reserve University

William T. Sledd
Professor of Mathematics
Michigan State University

Alan Sokal
Professor of Physics
New York University

M.C. Stanley
Professor of Mathematics
San Jose State University

Dennis Stanton
Professor of Mathematics
University of Minnesota

Professor James D. Stein Jr.
Department of Mathematics
California State University, Long Beach

Sherman Stein
Professor Emeritus of Mathematics
University of California at Davis

Harold Stevenson
Professor of Psychology
University of Michigan, Ann Arbor

J. E. Stone
Professor of Human Development & Learning
College of Education
East Tennessee State University

Sandra Stotsky
Deputy Commissioner for Academic Affairs and Planning
Massachusetts Department of Education
Research Associate
Harvard Graduate School of Education

Robert S. Strichartz
Professor of Mathematics
Cornell University

Daniel W. Stroock
Professor of Mathematics
MIT

Justine Su
Professor of Education
Director, The China Institute
California State University, Northridge

P. K. Subramanian
Professor of Mathematics & Computer Sciences
California State University, Los Angeles

Howard Swann
Professor of Mathematics and Computer Science
San Jose State University

Daniel B. Szyld
Professor of Mathematics
Temple University, Philadelphia

Professor Sara G. Tarver, Ph.D.
Department of Rehabilitation Psychology and Special Education
University of Wisconsin-Madison

Clifford H. Taubes
Department of Mathematics
Harvard University

Abigail Thompson
Professor of Mathematics
University of California, Davis

John B. Wagoner
Professor of Mathematics
University of California at Berkeley

Bertram Walsh
Professor of Mathematics
Rutgers University–New Brunswick

Steven Weinberg
Josey Regental Professor of Science
University of Texas at Austin
1979 Nobel Prize in Physics

Steven H. Weintraub
Professor of Mathematics
Louisiana State University

James E. West
Professor of Mathematics
Cornell University

Brian White
Professor of Mathematics
Stanford University

Professor Olof B. Widlund
Courant Institute of Mathematical Sciences
New York University

Herbert S. Wilf
Thomas A. Scott Professor of Mathematics
University of Pennsylvania

Robert F. Williams
Professor of Mathematics, Emeritus
University of Texas at Austin

W. Stephen Wilson
Professor of Mathematics
Johns Hopkins University

Jet Wimp
Professor of Mathematics
Drexel University

Charles N. Winton, Professor
Department of Computer and Information Sciences
University of North Florida

Edward Witten
Professor of Physics
Institute for Advanced Study at Princeton

Jon Wolfson
Professor of Mathematics
Michigan State University

Wei-Shih Yang
Professor of Mathematics
Temple University

Shing-Tung Yau
Professor of Mathematics
Harvard University