DOES TWO PLUS TWO STILL EQUAL FOUR?

  WHAT SHOULD OUR CHILDREN KNOW ABOUT MATH?

March 4, 2002 

Transcript prepared from a tape recording

2:45 p.m.	Registration
3:00	Presenter:	Mike McKeown, Brown University
	Discussants:	Gail Burrill, Michigan State University
		David Klein, California State University at Northridge
		Tom Loveless, Brookings Institution
		Lee V. Stiff, National Council of Teachers of Mathematics
	Moderator:	Lynne V. Cheney, AEI
5:00	Wine and Cheese Reception

Proceedings:

MRS. CHENEY:  Good afternoon.  I'm Lynne Cheney, and I'd like to welcome you to the American Enterprise Institute.  This is an organization dedicated to scholarly research and open debate, and in that spirit, we are here today to talk about a method of teaching mathematics that supporters call "reform math" and opponents call "fuzzy math."

I myself have been a critic of this method of math teaching, but my role today is not as partisan but as moderator.  And in the name of neutrality, I will call this method of teaching "NCTM math" since both sides agree that this approach has been inspired by reports from the National Council of Teachers of Mathematics, the NCTM.

Instructional programs based on NCTM recommendations generally de-emphasize drill and memorization, encourage the use of calculators from kindergarten on, and recommend that students discover methods of addition, subtraction, multiplication, and division for themselves.

I have an example to illustrate this last point.  It is from an NCTM-inspired math program called MathLand, and in this example, students are asked to solve the following problem.  Here's the problem:

I 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.

Many adults, having learned long division, would use that algorithm, that particular long division algorithm, to divide 1,344 by 21.  But students today are often not taught long division, and in the process of trying to figure out for him- or herself how to deal with this problem, the particular student featured in MathLand and held up as an example of how things should be, this particular student added up 21s until reaching 1,344.  So this is what the solution looked like, and I'll show it to our panelists and then just for a minute or two put it up here.  I don't want to distract the rest of the presentation.  But it's an example of a student inventing a way to solve this problem, and the result looks like this, and here is the student's explanation.

Twenty-one days, there are 21 days in three weeks, the student explained, so I added 21s until I got 1,344.

Now, proponents of NCTM math argue that such an approach encourages conceptual understanding.  Many parents, on the other hand, look at this kind of problem solving with dismay and some concern.  They, of course--we all agree that students should understand mathematical concepts.  All parents want that.  But what the parents worry about is that students who learn mathematics in this way will never perform mathematical operations efficiently and automatically.

Now, this is just one of many issues in contention, and the debate is now widespread.  Parents' groups from California to New York have organized to try to make sure that more traditional mathematical instruction is at least an option.  Our main presenter today, Mike McKeown, on my near left and your far left, was one of the organizers of Mathematically Correct, a California group that has had considerable impact on the instructional program for mathematics in California.

After his presentation, we will hear from Gail Burrill, a former president of the NCTM, the National Council of Teachers of Mathematics; David Klein, a professor of mathematics at Cal State University; Lee Stiff, currently president of the NCTM; and Tome Loveless, a senior fellow at the Brookings Institution.

Mike McKeown, our main presenter, is currently a professor of medical science at Brown University.  Mike, I invite you to take the podium, and we will remove this distraction here.  And I'll even move the little box I have to stand on.

[Laughter.]

DR. McKEOWN:  And I was going to leave your secret safe with me.

Good afternoon.  As Lynne said, I'm here, I am a parent in the guise of a parent activist, although in real life I am a research scientist and a faculty member at Brown, where I use genetics and molecular biology to study the development and function of complex organisms.

My wife, Erica, was a math major in college.  She was taught by a number of excellent mathematicians, at least one of whom is here in the audience today.  And after she graduated from college, she worked for a number of years as a math teacher, teaching from seventh grade through high school, and then as a math tutor, tutoring everything from elementary school through calculus.  So we have some knowledge of what's out there in various different math programs, and, in fact, we have a wall of math books at home covering back from the mid-'60s to the present.

The two of us, along with a number of others, are co-founders of Mathematically Correct, and we have a website, www.mathematicallycorrect.com, and like three-quarters of the founders, the two of us are Democrats, so this is an organization that spans the political spectrum.  And among the other founded, we have a chip designer who works in the aerospace industry, a research statistician who, among other things, does statistics for large medical research studies and for studies of neurobiology and how learning and memory work, and we have a person who is a lifelong learner, who first taught school, then went back to school, got a geology Ph.D. in paleomagnetism, and has also taught school mathematics and college mathematics.  So we have some experience in the kinds of things that are going on.

Now, just to give me some sense of the audience, I'd like to know how many of you have either children or grandchildren who are between the age of 6 and 16.  Okay.  Quite a few.  So I have a question for you.

It's the year 2002.  Have you looked at your math book today?  Because that's where the rubber hits the road.  It's what happens in classrooms when the programs hit the children and are they learning what you want.

So let me give you a brief history of some of the things, the developments in mathematics since I was a child.

In the early '60s, programs known under the general guise of new math were introduced.  There were some aspects of those that were probably quite helpful--I certainly found them so--but many things that did not work and led to a backlash.

In 1989, the National Council of Teachers of Mathematics introduced their first standards document, and as should happen with standards documents, curriculum followed the standards and a nationwide adoption of programs that might be called standards math.  And implementation of that came in California in 1992 with the adoption of the California Mathematics Framework of '92, which was a particularly radical interpretation of the standards.

In rebound from that, in 1997 California adopted standards which are strikingly different in some aspects of their content and style from the NCTM.  And in California since then, curriculum has followed the standards, as have tests and other aspects linked to these classroom activities.

Finally, in 2000, the NCTM introduced its new standards which go under the acronym of--or go under the name of the Principles and Standards of School Mathematics.  The acronym for that is PSSM.  I'll let you figure out how to pronounce it.

Now, the NCTM standards and all standards are based on an idea that is quite appealing, and it is a plan for generalized improvement, develop good standards that give us an idea of what students should know, when they should know it, the degree of depth to which they should know it, develop good tests that test the content of the standards and link to the standards, and develop good curriculum that will go with the standards and be linked with the standards.  So you have a triad of things that lead to improvement of student performance and student learning.  This is good.

Now, one of the things that is a danger in this triad, it sounds--it is very appealing in its structure.  It has an appropriate structure.  Unfortunately, there's a certain danger of bait and switch, meaning that you can have good talk, you can have good structure, but the implementation can be less than optimal.  So, for example, you could have standards that are bad, which will lead to tests that are bad, which will lead to a bad curriculum, and you were perhaps worse off than you were before.

Now, let me review the NCTM standards.  Others will speak today who can correct me if I make any major errors.  I'm sure they will.  I'm counting on it.

The NCTM standards of 1989 were deliberately revolutionary, self-consciously so.  They emphasized teaching method at least as much as they emphasized the content of the mathematics that should be taught.  They're generally low on the specifics of content, but they clearly and explicitly downgraded aspects of that content, specifically things such as mastery of skills, memory, standard algorithms, analytical methods, for example, the ones that you find and should learn in algebra.  And they emphasize inventive methods, generic problem solving, techniques such as guess and check, stressing conceptual understanding, perhaps differentiable from true understanding, statistics, and data analysis.  And, in addition, they abjure direct teaching and practice in favor of discovery learning and creation of ad hoc methods.

So to caricature the structure of a course using what we might call NCTM math, it's a group of children working together in heterogeneous groups, rediscovering the great principles of mathematics, and inventing new ways to do common things.  And then when they're done, they write long essays about it.

If you've been there, you know it's true.

Now, this isn't just something that I'm making up in my interpretation, and aspects of this have reached the common press.  So, for example, at least in the semi-popular press--and this is an article from the New Yorker by Christopher Buckley. It was intended to be humorous, but to many of us parents, it's not humorous at all.  It's just all too close to the truth.

Parents will find that the current new math bears no resemblance to one of their own day or, indeed, to any math.  Ptolemy and Euclid would not recognize it.  The multiplication table, for instance, has been replaced by a system whereby the digits of the right hand are interlaced with the toes of the left while the child hops backwards in 3/2 time.

Now, he was directing this particularly at Chicago math, but there are others that are worse.

Now, one of the characteristics I asserted about these was that they, in fact, de-emphasized standard methods and standard algorithms.  And, indeed, many people who have been associated one way or another with reform math and the NCTM have essentially said that standards methods are ready for the ash heap of history.

So, for example, Constance Kamii and Dominic wrote, "Algorithms not only are not helpful in learning arithmetic; they also hinder development of numerical reasoning."  And if that weren't enough--this is not a strange or unusual event--Steve Leinwand, who at one point was going to be a member of this panel when it was going to be in the fall and was involved in the exemplary and promising program study for the Education Department, wrote, "It's time to recognize that for many students real mathematical power on the one hand and facility with paper-and-pencil computation 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."

Imagine how smart we all could have been if we hadn't learned how to compute.

Now, luckily, these ideas are not necessarily the ones that are agreed by all.  Indeed, the NCTM, when they were in the process of making their revisions, solicited various mathematical organizations to comment on various different topics.  And the American Mathematical Society made the following suggestion about algorithms:  "We'd like to emphasize the standard algorithms 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 they 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."

Or as Hung-Hsi Wu, a math professor at Berkeley, said, "Could these authors possibly be unaware of the fact that the division algorithm, like other standard algorithms, contains mathematical reasoning that would ultimately enhance children's understanding of our decimal system?"

I have asserted then that many of the fundamental ideas that contribute at least on the "what do we teach" side of the NCTM standards are badly flawed and, in fact, in some cases are likely to at least leave children unprepared.  So I have suggested these ideas are quite bad.  But then why have they been adopted and why are they favored?

One of the possibilities is foolish romanticism.  There is a strain in American thought that likes to think that we are all natural blank slates who are just little learning machines and that if we only left children to be exposed to a print and spoken word rich environment, they would all automatically learn to read.  That's whole language.  If we allowed children to be in a math-rich environment and explore and discover for themselves, they will learn the basics of mathematics and they will all be mathematical geniuses.

These are wonderful, appealing ideas.  Unfortunately, I don't think they're at all right.  And there are also potentially less flattering reasons.  This is a quote from Jack Price, who was then president of the National Council of Teachers of Mathematics.  This was in a radio program in San Diego.  I've heard him make similar statements on television, and I have also heard other people from NCTM make similar statements.  "What we have now is good for the most part for relatively high socioeconomic Anglo males, and we have a great deal of research that is done showing that women, for example, and minority groups do not learn the same way.  Males, for example, learn better deductively, where we found with gender differences, for example, that women have a tendency to learn better in collaborative effort when they're doing inductive reasoning."  Or as my wife said when she heard this, "Oh, great, women's intuition."

Now, luckily--I actually want to make one more comment about that.  So this is an interesting attitude to have about math education, and one of the things that we have seen is that various people who support the NCTM style of math have suggested not only that we need math for reasons of ethnic or gender differences, but that if you oppose the NCTM standards, their content or their methods, that perhaps you are merely trying to keep one social group up, the group to which I happen to be in, while keeping other groups down.  And that is certainly something that we have seen in various places.

Luckily, Dr. Stiff has recently written in a document about whether the NCTM standards are a political document, what should really decide us is data.  And that's a good thing because there are data out there.  There are large data sets developing from movements related to standards.  For example, in California, with the largest school population in the country, they test essentially every second grader to eleventh grader every year for the last four years, so we have a huge data set.  There are search engines such as the one at the Ed. Trust that allow us to now search out high-minority, high-poverty schools, truly at-risk schools statistically and find those schools and ask what they're doing.

So I searched for schools that have greater than 90 percent African American and Hispanic students and have poverty levels at 70 percent or greater, asked for those schools that had been high-scoring schools for the last two years in a row, get the top four:  Kelso in Inglewood, Bennet-Kew in Inglewood, Robert Hill Lane in Los Angeles, and Paine, also in Inglewood.  All of those, greater than 90 percent African American and Hispanic, greater than 70 percent poverty.  At least two of them are greater than 90 on both of those.

The lowest ranking of those schools, the lowest relative rank of those schools ranks at 75, which means in this scale that they are better than or equal to 75 percent of all the other schools in the state.  The top one, which is Kelso, ranks at 83, better than 83 percent of the schools in the state.

This means that this is the result of from grades three to five, the average test score in those schools is between the 70th and the 76th percentile.  These are high-performing schools.

What do they do?  We can ask.  We can look at the data.  One thing I can say is that none of them use NCTM-style math as their main program.  They don't use Chicago, they don't use connected, they don't use MathLand, they don't use connections, they don't use number data in space.  What they use is what we might have called traditional math.

As Nancy Ichinaga, principal of Bennet-Kew at the time, said in 1995 when I called to ask how they could succeed so well, she said, "No new books."

Now, they have broken that precedent in recent years.  They have gotten new books recently.  Specifically they've gotten books that are--they've been using books that are published by Saxon Publishers, which are decidedly not a traditional, traditional book, different in their own particular way, but certainly quite effective.

Now, in the few minutes I have left, I just want to go over a few things that I think will be certainly objects for discussion, and these are what I would consider pernicious, appealing, and romantic fallacies.

The first is that we are hard-wired to learn math just as we are hard-wired to learn to speak, and that letting students discover the great truths of math is as natural as learning to speak.  And this is completely false.  We are hard-wired to learn spoken language, but we are not hard-wired to learn the alphabetic principle.  We are hard-wired to learn some things about math, small numbers, their relationships, and particularly who has more than I do.  So those kinds of things we're hard-wired to learn.  Relatively simple arithmetic and the techniques as complicated as algebra, we are not hard-wired to learn.  We learn them better when we're taught.

The second--and I have heard this statement so many times, "brain research shows," and then that's followed by something like "discovery learning and inventive methods work best."

Well, I don't think that's true either because the best of brain research and top-level cognitive psychology show that repetition and appropriate repeated stimulation are critical for long-term memory and mastery, and mastery and long-term memory free the short-term memory for concentration on the problems at hand.

You'll also hear, "Skills and knowledge, they're no longer important.  The world is changing so fast that mere facts"--they're just mere facts.  They're not facts, they're mere facts, little tiny facts--"will be outdated before we know it."  This is, of course, false for most--for 99 percent of what we do in every day.  Newton's laws still hold for almost all situations we'll encounter.  Algebra is still algebra.  Analytical methods of algebra are still useful and among the most powerful methods of problem solving available.

Then they will stress, "Since we're not teaching facts, we will teach problem-solving skills."  Now, the reason this is dangerous is because, in fact, attempts to teach generalized problem solving are largely ineffective.  Problem-solving skills, particularly high-level skills in problem solving, are domain-specific because they're based on mastery of domain-specific knowledge.  What you know in math, what you know in physics, what you know in history allows you to solve problems in those domains.  If you know nothing about history, you won't be able to solve problems about history in any deep way, similarly with math.

Finally, of course, we teach phonics or, of course, we teach skills or, of course, we teach computation to mastery.  And the embedded phonics that we see in various balanced literacy programs is ineffective in teaching children letter-sound correspondences and ineffective in teaching decoding skills.  Discovery learning and invention of some method for computation is not the same thing as teaching and practicing an effective, logically developed general method, and it is not--and burying these things at the back of the document or in an obligatory one-sentence thing--of course, all of us agree that students should master computational skills and then going on to describe a document--a program in which computational skills are unimportant is not the same as learning it.

Finally, some words that we have to nail down.  Remember, it's what gets to the classroom that matters, so some things are rather like Mom and apple pie and the flag these days.

Standards.  Everybody recognizes that good standards are good, but you have to look at the standards you've got.  Good standards, bad standards, indifferent standards, that matters.

Balance.  Who could be opposed to balance?  Well, it depends on where you balance the bar and how much weight you put on one side or the other, because those things do make a difference.  Look at what is meant by balance.

And, finally, skills.  As I just commented, we can stick things--we can say we're in favor of skills, but if we reduce algebra to one-sixth of all the strands that are in the program, when we finally end up with what hits the classroom, algebra is certainly going to be de-emphasized and replaced by something of less value.

And with that, I hope you enjoy the rest of your afternoon.  I hope we have an interesting discussion, and thank you for your time.

[Applause.]

MRS. CHENEY:  Our next presenter, Gail Burrill--our first commentator, really--is a past president of the National Council of Teachers of Mathematics and is currently in the science and mathematics education division of Michigan State University.

DR. BURRILL: Good afternoon.  I'm very pleased to be here, and I have to say that my remarks are based on my more than 25 years as a high school mathematics teacher, on my work in mathematics education, both national and international, and my most recent experiences working with pre-service teachers.

Before I kind of get into some of the points that I'd like to make, I actually just have to say that, as a math teacher, I believe that my students should master skills.  I believe that computation is important.  I believe that groups are good sometimes.  I believe my students need a good, sound lecture sometimes.

I clearly have lots of students who are not hard-wired to learn mathematics.  That's what made my job so interesting, to try and help those kids make the connections.

I believe that you need to have a sound understanding of knowledge and skills and conceptual understanding in order to do problem solving.  But I would like to be able to have my students be problem solvers in mathematics.  And I think that pretty much I was able to succeed as a teacher in preparing students to be problem solvers.

I think Mike and I and the rest of the panel probably all agree that having a mathematically literate society is important.  And I think we fundamentally have the same goal:  to provide every student with a challenging, high-quality mathematics education.

Given that we want students to understand mathematics, what does that mean?  I asked my pre-service students that, and their reply:  Students who understand mathematics will be able to solve problems--problem solving.  They'll be able to make connections between what they know to what they are learning.  They'll be able to make decisions using mathematical reasoning.  They can represent and analyze situations and communicate their thinking to others.

I think those are pretty worthwhile goals, and I've tried the same question on teachers.  I suspect, if I had time, I could try this with this group.  And I probably would get pretty much the same set of answers.

How do we make this happen with our students?  Over the last decade, we've begun to understand some things about what it means to learn.  According to the National Research Council's "How People Learn," we have a lot of things that we know about pedagogy, instruction, curriculum, and assessment that differ significantly from things that are common in today's schools.

Research provides us some pointers.  Consider, for example, these statements taken from a compilation of the literature:

Students need to see that they are learning something useful and relevant.  Students learn if they're actively involved in choosing and evaluating strategies, considering assumptions and receiving feedback.  Students learn by building or transferring knowledge from previous experiences.

For most students, learning is dependent on teaching.  But according to how people learn, classroom teaching has often focused too narrowly on the memorization of information, giving short shrift to critical thinking, conceptual understanding, and in-depth knowledge of subject matter.  And the development of intellectual competence requires more than the accumulation of discrete pieces of information.

Research about teaching suggests that learning may be hindered by presenting too many topics too quickly, presenting isolated sets of facts that are not organized or connected, and not helping students understand where, when, and why to use their knowledge.

Research does say that activities should be structured so that students are able to explain, explore, extend, and evaluate their progress.  And I would point out it doesn't mean--notice it didn't say "invent their own knowledge."

Research goes on to say that struggling at first with concepts enables students to benefit from a lecture that brings ideas together.

Many approaches to instruction look equivalent when the only measure of learning is based on memory, asking students to recall information masking as understanding, according to Bransford.  Instructional differences become apparent when measuring how well the knowledge transfers, procedural knowledge without underlying conceptual understanding does not transfer.  And I ask you to think about many of your colleagues and neighbors, the people that I sit next to on planes and buses who say, "Oh, math.  I never did get it."

How does research related to mathematics?  After our set of standards, which, in the words of Alan Schoenfeld, catalyzed a national standards movement, much to our surprise.  We put out, as Mike said, the principles and standards, a vision statement designed to reflect a decade's experience since the publication of the original standards.

The teaching and learning principles from the standards, the principles and standards, support the kind of teaching and learning advocated by the research that I talked about.  Students who have learned mathematics with understanding and are able to build new knowledge from their experience and prior knowledge, and teachers who understand what students know and need to learn and challenge and support them to learn it well.

"How People Learn" advocates creating knowledge-centered environments that include an emphasis on sense-making.  However, as Schiffler (ph) argues, many mathematics curricula emphasize not so much a form of thinking as a substitute for thinking.  The process of calculation or computation only involves the deployment of a set routine with no room for ingenuity or flair, no place for guesswork or surprise, no chance for discovery, no need for the human being, in fact.

According to the research, traditional curricula often fail to help students understand the structures within a discipline, and instead emphasize mastering procedural objectives, not helping students develop as learners of mathematics--as a matter of fact, as learners of anything.

In mathematics, we are fortunate to have new and promising curricula based on the standards that support the kinds of learning and teaching suggested by the research.  Evidence is mounting that these curricula, when implemented as intended--and there's lots of things that go wrong between intentions and what actually happens in classrooms, for many things that are even beyond our control.  But when these are implemented as part of a coherent system over a long enough time for cohorts of students to experience the complete program, we're producing students with a kind of mathematical understanding valued by my students.  And in contrast to traditional curricula, these new curricula have been field tested, revised, and the results have been monitored for continual improvements.

That's not to say we're perfect, but we're moving towards a better curriculum for all students.

The new book from the National Research Council, "Adding It Up," that talks about the research that looks at how elementary students learn mathematics asks for mathematically proficient students, where we have the development of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition to learn mathematics.  And it goes on to say that instruction should not be based on extreme positions that students learn on one hand solely by internalizing what a teacher or book says, or on the other hand solely by inventing mathematics on their own.

We have made gains over the last ten years.  The NAEP scores, the SAT, and the ACT scores have risen.  Students are taking more mathematics.  In 1998, 45 percent were enrolled in upper-level mathematics in high school.  More students have access to algebra and geometry.

We still have a long ways to go.  A Japanese colleague told me that teaching math is 80 percent certainty and 20 percent doubt.  That 20 percent leaves us a lot of room to think about improvement.

Some of our practices and actions are still responsible for leaving students behind.  My pre-service students did case studies of math-avoiding peers, university students who often chose careers because they didn't involve mathematics.  The case studies reported statements like:  Math's useless for most of us.  There was only one way to proceed, and I didn't get it.  I was lost.  There was nothing to think about, no place to discuss ideas.  The teacher didn't explain it.  It's just me and numbers, we don't mix.  We had all of these timed tests and I got nervous and couldn't do things fast and quit trying.

My pre-service students had excelled as these speed tests, at being first and doing well.  They had never considered what it must mean to those who are last.

We have a lot of challenges.  Too many students do not have the opportunities to learn.  We have not necessarily figured out how to use calculators and computers effectively.  Quality teaching and adequate teacher preparation and development are critical.  Unfortunately, while 38 states require professional development for recertification, only seven require that professional development have any content.

Teachers need good resources.  Thirty-four percent of the fourth grade students in NAEP had math teachers who did not have all of or most of the resources they need.

Accountability is important.  We have problems but we have promises.  As "Adding It Up" says, "For too long, students have been the victims of cross-currents from mathematics, as advocates of one learning goal or another have attempted to control the mathematics to be taught and tested."

I argue that we as a community have to work together to change the mathematics that needs to be changed so that we have students who can do what we want them to do.

Thank you.

[Applause.]

MRS. CHENEY: David Klein, our next commentator, is a professor of mathematics at California State University-Northridge.  One of the good things that has happened as a result of the debate surrounding mathematics education is a number of mathematics professors have begun to be involved in what we should teach in our schools, David Klein among them.

DR. KLEIN:  Good afternoon.  No single institution in the United States has caused more damage to the mathematical education of children than the National Science Foundation.

In saying this, I want to make it clear that I am not criticizing the NSF's admirable and important role in supporting fundamental scientific research.  I am talking about the education and human resources division of the NSF.  This is the division within the NSF that funds K-12 education projects.  It is responsible for systematically promoting the worst math education fads of the past decade.

Fuzzy math originated with the NCTM and the nation's colleges of education.  These negative programs have been aggressively funded and promoted not only by the National Science Foundation but also by numerous private foundations such as the Noyes Foundation, Bill and Melinda Gates Foundation, Flora Hewlett Foundation, to name just a few.  Perhaps later I will have time to provide some specific examples.

Parents are up in arms and are rebelling against the NSF and NCTM fuzzy math programs.  Hundreds of mathematicians are, too.  Let me give you an example.

In October 1999, the U.S. Department of Education released a list of ten so-called exemplary and promising math programs that it recommended to the nation's schools.  More than half of these exemplary and promising math programs were created with NSF money, and others were and are aggressively promoted with NSF funding.

Parents and mathematicians have opposed them for years.  Why would they do this?  The answer is that these so-called exemplary and promising programs are among the worst math books and programs in the country.  They radically de-emphasize basic skills in arithmetic and algebra.  Uncontrolled calculator use is rampant, and calculators are often introduced starting in kindergarten.

Fuzzy math books claim to teach conceptual understanding, but they don't.  Instead, they squander valuable class time on aimless projects with little or no intellectual content.  One can draw a parallel between the philosophy that underlies the failed whole language learning approach to reading and these NSF/NCTM programs.

Some of these math programs don't even have textbooks because books might interfere with children's creativity and the so-called discovery process.

Many of America's leading mathematicians were alarmed by the Federal Government's official endorsement of fuzzy math books.  In November 1999, I faxed an open letter to then-Education Secretary Richard Riley that was co-signed by more than 200 other mathematicians and scholars.  Our open letter urged the Department of Education to withdraw the entire list of exemplary and promising mathematics curricula and to announce that withdrawal to the public.

Among the endorsers of our open letter are many of the nation's most accomplished scientists and mathematicians.  Department heads at many universities including Cal Tech, Stanford, Harvard, and Yale, as well as two former presidents of the Mathematical Association of America, added their names in support.  Seven Nobel laureates and winners of the Fields Medal, the highest international award in mathematics, also endorsed.

The open letter was published as a full-page ad in the Washington Post thanks to the generosity of the Packard Humanities Institute.

Following its publication and press coverage, the NCTM denounced our open letter and expressed its complete support of the fuzzy math programs.  Specifically, in a letter dated November 30, 1999, the NCTM Board of Directors sent a letter to the Secretary of Education that said, "The Board of Directors of the National Council of Teachers of Mathematics wishes to inform you of their unconditional support for the work of the expert panel, the criteria used by the panel, the process employed by the panel, and the quality and appropriateness of their final recommendations," that is, for the 10 so-called exemplary and promising math programs.

The so-called exemplary and promising math programs and others like them continue to be aggressively promoted by the NSF.  Parents in New York City are now working with mathematicians from Courant Institute at NYU and other New York universities to find ways to resist the NSF/NCTM-imposed fuzzy math programs.  But so far the NSF is winning.

Some of the parent leaders and mathematicians in New York City are in attendance at this event.  Perhaps we'll hear from them later.  There are similar alliances between parents and mathematicians in other communities.

The NSF and NCTM fuzzy math programs cause problems for all schoolchildren, but they are particularly harmful to children with limited resources.  Upper-middle-class parents can afford tutoring to compensate for what the National Science Foundation has done to their schools.  The tutoring industry has skyrocketed as NSF/NCTM fuzzy math programs proliferated across the country.  Sylvan Learning Center, Kaplan's, Score Learning Program, and Kumon are among the examples.  But the greatest damage is to lower-income children who directly bear the brunt of these defective, anti-arithmetic, and watered-down algebra programs.

The NCTM has adopted the point of view that most girls and minority children have learning styles that are different from white males and Asians of both genders.  Professor McKeown has already commented on this.  While he was president of the NCTM, Jack Price said that minority groups and women do not learn math the same way as white males.  He said, "Women have a tendency to learn better in a collaborative effort when they are doing inductive reasoning."  This was in contrast to the way white males learn math.  According to Price, "Males learn better deductively in a competitive environment."

This attitude toward women and minorities is consistent with the NSF-funded math books.  They rely heavily on superficial repetitive patterns--a form of inductive reasoning--rather than logical deduction, which is the core of mathematics.

The NCTM has attempted to redefine mathematics itself in order to support a notion of learning styles in math associated with skin color and gender.  This is misguided in the extreme.  There can be no doubt that children of all races and backgrounds can excel in classical, content-rich educational environments.  Jaime Escalante, portrayed in the movie "Stand and Deliver," sent his low-income Hispanic calculus students to top universities in record numbers using traditional methods.

Nancy Ichinaga, Marjorie Thompson, and many other principals and teachers in Inglewood, California, described by Professor McKeown, have also proved beyond a shadow of a doubt that African American and Hispanic students not only learn but excel in traditional content-rich programs.

What do these education leaders say about the NCTM/NSF math reform agenda?  Nancy Ichinaga told me, "Reform is for the birds."

According to Escalante, "Whoever wrote the NCTM math standards must be a physical education teacher."

While I don't think Escalante's comment is quite fair to PE teachers--

[Laughter.]

DR. KLEIN:  --his point is clear, and the National Science Foundation would do well to listen to criticisms and change course.

If President Bush is listening now, I strongly urge him to find new leadership of the education and human resources division of the NSF.  The National Science Foundation needs leaders who can stop the damage that organization is causing to the mathematics education of America's schoolchildren.

Thank you.

[Applause.]

MRS. CHENEY:  Lee Stiff, our next commentator, is currently president of the National Council of Teachers of Mathematics and a professor of mathematics education at North Carolina State University in Raleigh.  Mr. Stiff?

DR. STIFF:  Thank you.  I will speak from here.  I have so many comments that I could make based on the presentations that preceded me in addition to the remarks that I had already prepared, but I must respond to several of the ones that have preceded me.

I want you to think about the mathematics that you took as a child, and I want you to think about the mathematics that your friends and neighbors have taken when they were children and their reaction or response to that.

I don't mind talking about fuzzy math because I know fuzzy math exists.  It existed in traditional mathematics.  Because if you didn't understand algebra, if you never understood geometry, if you didn't make sense out of the mathematics that was taught, yeah, you could add, subtract, multiply, and divide, but did you understand place value?  Did you understand grouping and regrouping, or what we used to call back in my little town in Murfreesboro "borrowing" and "carrying"?

If you didn't really understand it, then it was unclear to you.  And I maintain that something that is unclear to you is fuzzy to you.  The fuzzy mathematics existed long before some NCTM standards that were created in 1989.

Also, it's interesting to me that of the speakers that preceded me, they talk about what the presidents have said.  Why haven't they talked about what this president has said?  I'm the president that was in office when Principles and Standards of School Mathematics was released in the year 2000.  Why haven't they talked about all the commentaries that I have made in our news bulletin and all of the kinds of commentaries that I have made about principles and standards since?

We are not in that much disagreement.  We want young people to be able to add, subtract, multiply, and divide.  We just want them to do more.  We want them to understand the mathematics that they are learning and be able to delve deep enough so that when it comes to learning algebra, they will actually understand the structure of arithmetic so that they can understand the structure of algebra.

Really, it is all about change.  It's about change.  We want a high-quality mathematics education for every child.  That means that we have to support teachers.  We have to support teachers who are already in place.  We don't blame teachers at NCTM for the state of conditions per se because states around the nation have said if you want to be a middle grades teacher, you may not have to take more than six or nine hours of mathematics.  But the mathematics that middle-grade students are taking now requires teachers who have a major or minor in math or math education.  So that means that they have to have a fuller understanding of mathematics, and we support that.

We have said that in our commentaries.  Have you read what I said about that?  We want our middle grades teachers to be capable mathematics learners so that they can impart the vision of Principles and Standards of School Mathematics.  And we want to upgrade the mathematics preparation of our elementary teachers as well.

We recognize that those teachers in position need to have professional development that will make them stronger, and we support those pre-service teachers having the requirement of taking more mathematics before they ever go into the classroom.

When you look at TIMS, the Third International Math and Science study, when you look around the world and you see how we compete and compare, and you look at the teachers of those students in other countries around the world, their teachers are mathematicians by training, math educators by training.  We want that for our children.  Why don't we require it?  If we're serious about the mathematics, why don't we require it?

Standards show the way.  Forty-nine of the 50 states have endorsed the ideas that are found in Principles and Standards for School Mathematics.  Since 1920, NCTM has been leading the way.  It's not just these standards or the so-called new math that people might berate and the mathematics that was talked about that was good back in the '60s and '70s.  That was NCTM mathematics.  We are doing what we've always done.  We are doing what we have always done:  trying to create better mathematics for students.  Meeting the needs of students changes over time.  It doesn't stand still.  We don't look back and say it was the good old days and it should be this way forever because times have changed.  It's all about change.  We need students now who are flexible and resourceful problem solvers.  We need students who can take those basic facts, understand how they are developed, and then apply those basic facts.

And you talk to a teacher, talk to any teacher.  If you want a student to be able to do something, you have to teach for it.  People would have you believe that if you learn the basic facts alone and you can recite them adequately, you can recite them expertly, you can recite them without a flaw, that will make you a great problem solver, a great applier of mathematical knowledge.  But it doesn't work that way.

A teacher will tell you that if you want young people to use the knowledge, they have to be given opportunities in school to use the knowledge.  We give them those opportunities.  I want my child to be able to divide 21 into 1,344 using an algorithm.  I certainly do.  And the example that was shown earlier is not an adequate representation of what the standards say.

Now, if a program embraces the standards and does something different than what we talk about, is that our fault?  I am sure in that program they show that using the additive process for dividing--and that was what was going on--makes sense for smaller numbers, not like 1,344 but maybe like 25 divided by 7 or maybe even 128 divided by, you know, 4 or 5.  But the point is you can show the structure.  That's the point of that.

But then when it comes to larger numbers and bigger examples, young people have to use their algorithms that they've learned.  And how do they understand those algorithms?  Because they've had exposure to the simpler cases where the structure can be revealed.  And now they understand how it all fits together.  No one would advocate that a child should divide 1,344 divided by 21 in the manner that was shown.  No one would really advocate that.  But the key is for young people to begin to understand so that when the algorithms are put forth, they see how the structure of them all fit.

Standards have had an impact, and if it's all about change, we believe that not all change is for the better.  Just because it's about change doesn't make it better.  But we have to have evidence that what we are doing with standards is working.  And we have that evidence.  We have it in Pittsburgh.  We have it in Puerto Rico.  We have it in North Carolina.  We have it in L.A.  We have it in Philadelphia.  We have it in Massachusetts.  We have evidence that what we have embarked upon is beginning to make some headway.  The TIMS data, the NAEP data, the National Assessment of Educational Progress, it tells us that we're making some headway.

It's not a complicated problem that we're discussing here.  It's a complex one.  There are many facets.  But what to do is well understood.  We've got to get better teachers in the classroom.  That's understood.  We have to support them.  We have to provide professional development opportunities so that they can be successful in implementing better and more mathematics.

My father, who had a third grade education, used to say back in Murfreesboro, my little small town that I grew up in, "Weighing a pig several times over won't make it heavier."  We can assess and assess all we want to, but when we find out that the pig is too light, we got to feed it.  So that when we find out that young people are performing in a way that we would wish, then we've got to provide teachers who are prepared to teach mathematics.  We have got to go the urban settings in this nation to say you don't have the kind of teacher representation that will make your students be successful in mathematics.

Less than 50 percent of the students in urban settings have a chance of having a teacher with any credentials for teaching mathematics, whether it be middle grades or high school.  We've got to say we have a responsibility to provide our young people the opportunity to learn this mathematics.  They may not be hard-wired, but we can put the wiring in.  Good instruction can put the wiring in.

And so I'm going to close with a paraphrase of a quote from our distinguished moderator, that a system of education that fails to nourish students' understanding of mathematics denies the students a great deal:  the satisfaction of mature thought, an attachment to abiding concern, and a perspective of human existence.

Mathematics does all of that.  Mathematics helps us understand the world around us, and every child should have access to that understanding.  Mathematics helps us understand political races, so the statistics of an outcome will say there was no problem here, statistically speaking.  Mathematics informs us.  It gives us a citizenry that we would be proud of.  And the NCTM is trying to make that happen--not for a few students, but for all the students that we teach.

[Applause.]

MRS. CHENEY: Thank you very much.

Tome Loveless, our final commentator, is a senior fellow at the Brookings Institution where he directs the Brown Center on Education Policy.

DR. LOVELESS:  I want to thank Mrs. Cheney for inviting me here today.

President Stiff referred to evidence, and that's really what I want to direct my comments to this afternoon.

In your packet--I'm an old sixth grade teacher, so that is going to tell you two things about my presentation.  First of all, I prepared some graphs and charts, and in your packet you should have them.  If you'll follow along, my comments will make more sense to you.  And then the second thing you'll notice is that I'm going to focus quite a bit on arithmetic, and that's because I consider arithmetic to be essentially to mathematics what phonics is to reading.  It is simply a non-negotiable thing that children must learn.

Now, on the first slide that hopefully now you have located in your packet, there are two graphs.  President Stiff referred to the NAEP scores, and Gail Burrill did the same thing, the NAEP scores going up.  Actually, there are two different NAEP tests.  If you take a look at these two graphs, the graph at the top--this is the fourth graders--you can see that their math scores have been going up, in fact, very dramatically have been going up since 1990.

However, you need to know something about this main NAEP test.  It is loaded towards the NCTM.  It is based on the NCTM standards.  Children are allowed to use calculators on a portion of this fourth grade test.  And such concepts as geometry, data analysis, and problem solving are far more important on this NAEP test, and that's one of the reasons why children are scoring better at it.

But take a look at the graph at the bottom of that same page.  This is the other NAEP test.  It's called the Trend test.  It's been around since the early '70s.  Now, that line looks quite different.  It's basically flat.  There has been some growth, but only two scale score points.

Something is going on here.  In terms of the federal test of mathematics in the early grades, one test is saying there's tremendous growth.  The other test is saying that we're basically flatlining.

Now, what's holding down that graph at the bottom?  What's holding down our Trend NAEP scores?  Let's go to the second slide.

What we did in the Brown Center--and, by the way, you can't really answer this question very well about how well are kids doing arithmetic.  Unfortunately, the Federal Government has a flawed NAEP program.  It has a terrible math test that it's using because you can't answer basic questions like:  How are fourth graders doing in arithmetic?  How are fourth graders doing in adding, subtracting, multiplying, and dividing whole numbers?  How about eighth graders?  Do they know fractions?  Do they know about decimals?  Do they know how to compute percentages?  You cannot answer any of those questions using federal data because the Federal Government does not report--it doesn't break out and report trends on those very important domains in math.

In the Brown Center, what we did is we took the public release items and we broke them out, and we created what are called arithmetic clusters, and at the bottom you'll see we define exactly what we measured.

Now, let me explain something about this chart.  You cannot compare relative performance.  Thirteen-year-olds are not performing better than 17-year-olds, okay?  This chart is only valuable for a trend analysis, a time trend analysis.  So what I want you to do is take a look at the left side of that vertical line.  The vertical line is when NCTM math reform started coming into our schools.  You'll notice there was tremendous growth in arithmetic by all three age groupings in the 1980s.

But something happens.  Suddenly the brakes are slammed on beginning in 1990.  Both 13-year-olds and 9-year-olds flatline from 1990 through 1999.  And if you look at 17-year-olds, their arithmetic scores have actually declined.

So this is like--is this conclusive?  Absolutely not.  It's not conclusive.  But it is sort of like a canary in the coal mine, and the canary has dropped dead.  We need to take this seriously.

Let's go down now and dig under this data and look one step further.  What is it about arithmetic that seems to be a problem?  Go to Slide 3.  Again, this is an item analysis of fractions items.  Now, these are a small cluster of items.  There are only three items at age 17 and four items at age 13.  The value is we can track student performance, again, back to 1978.

Look at what happened to the 17-year-old scores beginning in 1990.  They fall right off a cliff.  Seventeen-year-olds are not learning fractions as well as they did in 1990.

Now, there is a point that I agree with both Gail Burrill and Lee Stiff on, and that is, we should not harken to some golden age.  We don't want to go backwards.  Arithmetic instruction and math instruction was not very good in 1990, and I don't want to go back to the conditions of 1990.  We need to move forward, and we need to make progress.  However, we haven't done that.  We actually, I suspect, are going backwards.

Take a look at the next slide, Slide No. 4.  Fortunately, there is one state out there, the State of Iowa, that has been giving essentially th