I am honored to be here today and to be able to share my observations on the state of mathematics education in this country with the distinguished members of the Committee on Education and the Workforce.

The K - 12 teachers in this country are dedicated professionals, deeply committed to teaching our children. They persevere in the face of difficult conditions and low pay. I have the utmost respect for them. But all too often, their knowledge of mathematics is extremely superficial, and when this happens they are easily swayed by trendy and unproven programs which typically offer a superficial treatment of the subject, leading to weak backgrounds in their students.

Perhaps a local parent described this situation best when she wrote me recently that the curriculum was getting fuzzier and fuzzier, and she "concluded that by and large most teachers support it because it makes them feel OK about math - they understand language, not symbols." She continues, "I cannot tell you how many times I have heard from administrators and teachers, how, if they had had "this" math when they were in school, perhaps they, too, would have been perceived as a `math person'."

I am a research mathematician, and research in esoteric areas of mathematics is essentially all I did besides teaching graduate and undergraduate classes in mathematics at Stanford until four years ago.

Two things obligated me to spend much of my time for the last three years studying issues related to K - 12 mathematics.

The first was some courses I gave in New Mexico, where I had too many bright, very highly motivated students in my mathematics classes whose third rate K - 12 educations in mathematics could not be overcome no matter how hard these students were willing to work.

The second came from the Presidential Commission designing Clinton's proposed national eighth grade mathematics exam. The commission - including many of the foremost math education specialists in the country - distributed a list of 14 proposed problems. I and my colleagues at Stanford were amazed to find that 3 of the problems had serious errors.

One was so ill posed that it could not be solved. One had an incorrect solution included with it.

We later testified to the Clinton commission about these difficulties, and it became clear that the level of mathematical understanding on the part of the mathematics educators on this panel was unimpressive.

There is a distinction between math educators who are primarily interested in questions involving education, and mathematicians who know about mathematics. While educational issues are unquestionably important there has been a tendency recently to focus on educational questions at the expense of mathematics content. I was disturbed when I realized that it is these people who are determining the mathematics that our children learn in school. I was especially disturbed in view of the dramatic drop in content knowledge that we have been seeing in the students coming to the universities in recent years.

Since 1989 the percentage of entering students in the California State University System - the largest state system in the country - that were required to take remedial courses in mathematics have increased almost 2 1/2 times from 23% in 1989 to 55% today. And CSU admission is restricted to the top 30% of California high school graduates! This failure has important consequences for the nation. Although large numbers of US students entering the universities say they are interested in majoring in technical areas, very few actually get such degrees today.

The total number of technical degrees awarded to US citizens recently is approximately 28,000 yearly, while there are currently about 100,000 new jobs in these areas each year. Last year congress had to mandate an additional 142,000 new work visas for technically trained people, and these visas were used up by June 11, 1999, so great was the demand.

A large part of the blame rests with mathematics programs of the type recommended by the Department of Education recently as exemplary or promising.

All but possibly one of the programs in the list recommended by the Department of Education, represent a single point of view towards teaching mathematics, the constructivist philosophy that the teacher is simply a facilitator. Standard algorithms for operations like multiplication and division are not taught, but students are advised to construct their own algorithms. At all stages hand held calculators are used for arithmetic calculations. There are no means for students to develop mastery of basic arithmetic operations. Algebra is short-changed as well.

These programs all are designed to closely align with the 1989 NCTM Mathematics Standards: standards which explicitly embody all the principles above, and specifically require that skills in algebra be downplayed. Indeed, the co-chairman of the Department of Education Expert Panel on Mathematics, Steven Leinwand, recently stated that the curricula endorsed by the Department of Education "create a common core of math that all students can master." Not material that all students NEED to know or SHOULD master, imply material that HE believes all students can learn. (Incidently, Department of Education statistical analysis - C. Adelman, 1999 - show that success in algebra in high school is the single most important predictor of degree attainment in college.)

The high school programs, Core-Plus and IMP, both place heavy emphasis on topics such as discrete mathematics at the expense of basic algebra, and do not come near the level indicated in e.g., the new California Standards for most of the topics there.

However, programs such as these are completely consistent with the previous California Mathematics Standards. Consequently, at least three of them, CPM, Mathland and IMP, have been in wide use in California for up to 10 or more years. (MathLand^? and IMP were developed in the late 1980s at the same time that the 1989 NCTM Standards were being developed, and were introduced into California Schools by 1989.)

Recent studies of the SAT mathematics scores of high schools which use IMP showed a consistent and significant decline over the last ten years.

Moreover, high schools that use IMP in California scored below the state means, and those that expressed satisfaction with the program scored, on average, 10 points lower than those which were dropping the program or otherwise were dissatisfied with it.

It was the introduction of CMP and TERC (another NSF funded curriculum published by Dale Seymour -- designed for grades K - 5) in the Palo Alto school system that sparked the initial parental revolt which became the California Mathematics Wars.

It was the introduction of Everyday Mathematics in the Princeton Township School District, which led to the parental revolt in Princeton. This led to the involvement of a number of faculty members in both mathematics and physics at Princeton University and the Institute for Advanced Study in Princeton in trying to reform mathematics teaching in the district.

It was the use of TERC in one school system in Massachusetts, which prompted numerous members of the Harvard Mathematics Department to sign the open letter to Secretary Riley.

The support for these programs in the Department of Education is ultimately the responsibility of the Education and Human Resources Department, EHR, at the National Science Foundation. EHR funded the development of at least six of the "exemplary and promising" programs.

It is also probably worth noting that at the present time there is no valid research which shows that any of the programs of this type are effective.

At least equally important are the Systemic Initiatives funded by EHR, which have the objective of pushing the districts where these initiatives are awarded to adopt curricula in mathematics which align with the 1989 NCTM Mathematics Standards.

In California, there is one systemic initiative from EHR still functioning, a grant to Los Angeles Unified School District, LAUSD, the nations second largest district with 711,000 students. The people involved in this initiative resisted attempts to change the system in place there, while similar districts such as Sacramento Unified began to make major changes.

Two years ago, the two districts had equally bad scores - around the thirtieth percentile - on the California Statewide mathematics exams. This last year LAUSD had essentially the same score as previously while the Sacramento Unified scores jumped dramatically, particularly in the lower grades, due to their shift away from whole language and constructivist math.

Incidentally, I had been told two years ago that getting a grant from EHR in a mathematics related area required that one buy into the list of ideas discussed above. As a test of this I obtained all the (over 4000) abstracts for the last 9 years from EHR for awarded grants that involved mathematics.

I tested a random sample of about 200 for a few key phrases such as NCTM Standards, group learning, and discovery learning. All but four of them contained at least one of these phrases.

In conclusion, I believe that the sad state of mathematics education among high school graduates in this country is primarily the responsibility of two agencies: the Department of Education and Human Resources at the NSF, and the Department of Education. The programs they develop and push simply set too low a standard.


Written Testimony of R. James Milgram February 2, 2000









Written Testimony of R. James Milgram February 2, 2000, summary points

• "I cannot tell you how many times I have heard from administrators and teachers, how, if they had had "this" math when they were in school, perhaps they, too, would have been perceived as a `math person'."

• Two things obligated me to spend much of my time for the last three years studying issues related to K - 12 mathematics.

• The first was some courses I gave in New Mexico, where I had too many bright, very highly motivated students in my mathematics classes whose third rate K - 12 educations in mathematics could not be overcome no matter how hard these students were willing to work.

• The second came from the Presidential Commission designing Clinton's proposed national eighth grade mathematics exam. The commission - including many of the foremost math education specialists in the country - distributed a list of 14 proposed problems. I and my colleagues at Stanford were amazed to find that 3 of the problems had serious errors.

• There is a distinction between math educators who are primarily interested in questions involving education, and mathematicians who know about mathematics

• it is [math educators, not mathematicians] who are determining the mathematics that our children learn in school.

• I was especially disturbed in view of the dramatic drop in content knowledge that we have been seeing in the students coming to the universities in recent years.

• Since 1989 the percentage of entering students in the California State University System - the largest state system in the country - that were required to take remedial courses in mathematics have increased almost 2 1/2 times from 23% in 1989 to 55% today

• Although large numbers of US students entering the universities say they are interested in majoring in technical areas, very few actually get such degrees today.

• The total number of technical degrees awarded to US citizens recently is approximately 28,000 yearly, while there are currently about 100,000 new jobs in these areas each year. Last year congress had to mandate an additional 142,000 new work visas for technically trained people, and these visas were used up by June 11, 1999, so great was the demand.

• All but possibly one of the programs in the list recommended by the Department of Education, represent a single point of view towards teaching mathematics, the constructivist philosophy that the teacher is simply a facilitator

• There are no means for students to develop mastery of basic arithmetic operations. Algebra is short-changed as well.

• These programs all are designed to closely align with the 1989 NCTM Mathematics Standards: standards which explicitly embody all the principles above, and specifically require that skills in algebra be downplayed.

• the co-chairman of the Department of Education Expert Panel on Mathematics, Steven Leinwand, recently stated that the curricula endorsed by the Department of Education "create a common core of math that all students can master." Not material that all students NEED to know or SHOULD master, imply material that HE believes all students can learn

• The support for these programs in the Department of Education is ultimately the responsibility of the Education and Human Resources Department, EHR, at the National Science Foundation. EHR funded the development of at least six of the "exemplary and promising" programs

• Recent studies of the SAT mathematics scores of high schools which use IMP showed a consistent and significant decline over the last ten years.

• at the present time there is no valid research which shows that any of the programs of this type are effective [ed.: National Research Council confirmed this in On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004), pages 3 & 188

• I had been told two years ago that getting a grant from EHR in a mathematics related area required that one buy into the list of ideas discussed above. As a test of this I obtained all the (over 4000) abstracts for the last 9 years from EHR for awarded grants that involved mathematics.

• I tested a random sample of about 200 for a few key phrases such as NCTM Standards, group learning, and discovery learning. All but four of them contained at least one of these phrases.

On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004)
National Research Council

Executive Summary, page 3

Under the auspices of the National Research Council, this committee’s charge was to evaluate the quality of the evaluations of the 13 mathematics curriculum materials supported by the National Science Foundation (NSF) (an estimated $93 million) and 6 of the commercially generated mathematics curriculum materials (listing in Chapter 2).

The committee was charged to determine whether the currently available data are sufficient for evaluating the effectiveness of these materials and, if these data are not sufficiently robust, the committee was asked to develop recommendations about the design of a subsequent project that could result in the generation of more reliable and valid data for evaluating these materials.

[ellipsis]

The Quality of the Evaluations

These 19 curricular projects essentially have been experiments. We owe them a careful reading on their effectiveness. Demands for evaluation may be cast as a sign of failure, but we would rather stress that this examination is a sign of the success of these programs to engage a country in a scholarly debate on the question of curricular effectiveness and the essential underlying question, What is most important for our youth to learn in their studies in mathematics? To summarily blame national decline on a set of curricula whose use has a limited market share lacks credibility. At the same time, to find out if a major investment in an approach is successful and worthwhile is a prime example of responsible policy. In experimentation, success and worthiness are two different measures of experimental value. An experiment can fail and yet be worthy. The experiments that probably should not be run are those in which it is either impossible to determine if the experiment has failed or it is ensured at the start, by design, that the experiment will succeed. The contribution of the committee is intended to help us ascertain these distinctive outcomes.

[ellipsis]

The charge to the committee was “to assess the quality of studies about the effectiveness of 13 sets of mathematics curriculum materials developed through NSF support and six sets of commercially generated curriculum materials.”

[ellipsis]

In response to our charge, the committee finds that:

The corpus of evaluation studies as a whole across the 19 programs studied does not permit one to determine the effectiveness of individual programs with high degree of certainty, due to the restricted number of studies for any particular curriculum, limitations in the array of methods used, and the uneven quality of the studies.

source: On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004)
National Academies Press
Mathematical Sciences Education Board (MSEB)
Center for Education (CFE)
available online or purchase, pages 3 & 188

learning a year or more of math in 2 months
James Milgram on long division & lag time in math learning



-- CatherineJohnson - 02 Feb 2006

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