# An A-Maze-ing Approach To Math

### A mathematician with a child learns some politics

By BARRY GARELICK

Spring 2005 / Vol. 5, No. 2

I am not a mathematics teacher, but I have a degree in mathematics and an intense interest in how the subject is taught. When I retire, I would like to teach math, which is why I started tutoring high school students in my spare time three years ago. My first student was a 9th grader having difficulty with geometry. He stated his problem succinctly: “I don’t know how to do proofs.” Confronted with what I thought could be a common problem, I was still unaware that what I was really seeing was a national crisis in mathematics education.

Here’s some of what I would soon learn:

–Only 55 percent of 8th graders taking the National Assessment of Educational Progress (NAEP) exam in math correctly answered the question, “How many pieces of string will you have if you divide 3/4 yard of string into pieces each 1/8 yard long?”

–In an international math test taken by students worldwide in 1995 (the Third International Mathematics and Science Study, or TIMSS), U.S. student math proficiency for 8th graders fell below the international average (28th out of 41 countries). For 12th grade, U.S. math performance was among the lowest (18th of the 21 countries participating).

–In 1989 the National Council of Teachers of Mathematics (NCTM) published its Curriculum and Evaluation Standards for School Mathematics–an extensive set of mathematics standards for grades K-12 which de-emphasized memorization of number facts, the learning of proofs, and algebraic skills, but encouraged the use of calculators and “discovery learning.”

–The National Science Foundation (NSF) promoted the NCTM standards beginning in 1991 and awarded millions of dollars in grant money for the writing of math texts that embraced them and to state boards of education whose math standards aligned with them (see Figure 1).

Not knowing about these significant events (all of them before last December’s release of the Program for International Student Assessment report ranking American 15-year-olds 24th out of 29 in math among industrial countries (see Figure 2), I had no idea that our children were being deprived of a math education, thanks in no small part to a dubious education theory, watered-down standards, and a well-meaning but intellectually bankrupt federally subsidized program of math illiteracy.

What I did know was that my 9th grader didn’t know how to do proofs.
I looked through his textbook, one of whose authors was a recent president of NCTM, and I was surprised to find very few proofs of anything. More troubling, most theorems in the book were stated as postulates–that is, propositions stated without proof–and students were told to memorize them. The problems at the end of the chapter required students to do only a few simple proofs.

Proofs in geometry class have been a mainstay of mathematics. In fact, proofs were always considered an essential part of high school geometry, not only because of their importance in higher math, but because learning the rules of logical argument and reasoning has applications in science, law, political science, and writing. To see proofs being shortchanged in a geometry textbook was shocking.

Algebra texts were in no better condition, in terms of presentation and content–or, rather, their lack of content. Even if you accept the argument that geometry in general, and proofs in particular, are unnecessary for students to learn, at least algebra should be taught properly, since algebra is the common language of, and gateway to, all of higher math. The absence of clear explanation and logical development left students I later tutored in algebra as lost as my geometry student. Their textbooks (and, probably, their teachers too) encouraged them to use a graphing calculator. Operations with algebraic fractions, like a/b + c/d, were given little attention, to say nothing of quadratic equations, once the pinnacle of any first-year algebra course. Instead, the quadratic formula is presented for the students to memorize and apply–if it is even mentioned at all.

At the time I started tutoring, my daughter was in 2nd grade. I was concerned that she was not learning her addition and subtraction facts. Other parents we knew were saying the same thing. Teachers told them not to worry because kids eventually “get it.”

One teacher told me her understanding of the new method. “It used to be that if you missed a concept or method in math, then you were lost for the rest of the year. But the way we do it now, kids have a lot of ways to do things, like adding and subtracting, so that math topics from day to day aren’t dependent on kids’ mastering a previous lesson.”

In a world where it doesn’t matter when you learn something, because you’ll get it eventually, there seem to be few if any critical junctures, no mastery of procedure, no building on what you’ve learned–no learning.

The Hell-Hath-No-Fury Postulate

Coincidentally, I had the opportunity to find out how my experiences related to the politics of math education. I work for the federal government, which has a program that gives employees a chance to work on Capitol Hill to gain experience and knowledge of legislative and congressional procedures, which is valuable information when working in government. I applied for and received a six-month detail to work in a Democratic senator’s office. Senator X (so called, in keeping with mathematical convention to describe a class of variables, because, as I was also to learn, both the good intentions and the shortcomings of Congress are institutional) was interested in establishing a science project to nurture a “homegrown” breed of scientists and engineers who would then support that state’s burgeoning technology industry. Since I thought a likely place to start would be math education, the staffers working the education issue asked me to see what I could come up with.

I compiled a list of questions that I sent by e-mail to various mathematicians involved with the math education issue. The questions focused on the quality of textbooks and teaching, with emphasis on algebra and geometry. I also wanted to know whether K-6 texts taught arithmetic well enough to prepare students to learn algebra.

The nice thing about working on the Hill is that you almost always get responses to e-mails and phone calls. Fifteen minutes after I sent an e-mail to Harvard mathematics professor Wilfried Schmid, he called. I found out that his initiation into the world of K-12 math education was similar to mine–through his daughter. He explained how she was not being taught her multiplication tables. He was shocked at the math instruction she was receiving in the 3rd grade. Its substance was shallow, memorization was discouraged, students were kept dependent on mental crutches (her teacher made her work with blocks or count on her fingers), and the intellectual level was well below the capability of most of the kids in his daughter’s class.

Schmid’s reaction to the problems of math texts and teaching was similar to that of other mathematicians I talked to in the course of my Capitol Hill assignment, particularly those with children. Those dialogues led me to develop an ad hoc theory that I will postulate (this means I don’t have to prove it): Hell hath no fury like a mathematician whose child has been scorned by an education system that refuses to know better. In Schmid’s case, he talked to parents, school boards, and ultimately with the Massachusetts commissioner of education. Along with others, he succeeded in revamping Massachusetts’s math standards, much to the dislike of the education establishment and textbook publishers.

Framing the Debate

The controversy over K-12 math education has come to be known as the “math wars.” Like Schmid, mathematicians have been active in this debate, as has the “mathematics community” at large, including not only mathematicians at the university level, but teachers and others involved in the education establishment. They believe that students must master basic skills (the number facts, standard algorithms for adding, subtracting, multiplying, and dividing) in tandem with larger concepts about mathematics.

On the other side of the debate are the followers of an education theory that promotes discovery learning, minimization of both teacher instruction and repetitive drills, and a disdain for standard procedures (algorithms). The math being protested–by the mathematics community–is called a variety of things: “reform math,” “standards-based math,” “new new math,” and, most commonly, “fuzzy math.”

Although the education theories on which much of fuzzy math is based are promoted in many education schools in the United States, it is not accurate to say that everyone in the education arena buys into these ideas. Therefore, for purposes of clarity, and to be consistent with a vocabulary used by others describing the math wars, I will use the term “educationist” to refer to those who promote the contested theory of math education known generally as discovery learning.

Early Skirmishes

The math wars revolve around a four-part problem: A disputed theory of education that informs NCTM’s standards; state boards of education that base their standards of learning for mathematics on the NCTM standards; textbooks written to incorporate these standards; and teachers and others in the education establishment who are indoctrinated in the disputed education theory and who may not possess enough knowledge of mathematics to overcome the first three factors.

The education theory at the heart of the dispute can be traced to John Dewey, an early proponent of learning through discovery. But for all practical purposes, the story begins on October 4, 1957, when the Soviet Union launched Sputnik. This event signaled the American malaise in science and mathematics education and the need to overcome it if we weren’t to fall further behind the Soviet Union and lose the cold war. Thus the U.S. Congress was motivated to pass appropriations that triggered the development of the “new math,” a national effort, spurred by the National Science Foundation, that eventually found its way into most of our schools.

The curriculum, designed primarily by mathematicians, had problems, but it introduced to algebra, geometry, and trigonometry a long-missing formalism, logic, and consistency, and it resulted in calculus being taught in high school. The problem, however, was that a similar formalism was introduced into K-6 texts and curricula, with the result that students (and elementary school teachers, who were caught totally off guard) were exposed to number bases, set theory, and axioms long before they were ready for them. And soon enough the new math was being blamed for not teaching basic arithmetic: it was often said that new-math students could tell you that 5 + 3 = 3 + 5, but didn’t know that it was equal to 8.

Mathematicians have agreed for years that emphasizing sets and number bases in math programs designed for the lower grades was a horrendous mistake. Notwithstanding these errors, however, the difference between the current slew of textbooks and those from the new-math days of the 1960s is definitely worth noting: Accomplished mathematicians wrote many of the texts used in that earlier era , and the math–though misguided and inappropriate for the lower grades and too formal for the high school grades–was at least mathematically correct. Some of the high school texts were absolutely first-rate, and new-math-era textbooks like Mary Dolciani’s “Structure and Method” series for algebra and geometry continue to be used by math teachers who understand mathematics and how it is to be taught. (They usually use them on the sly, since most teachers are required to use the books that the schools have adopted.)

During the new-math era, which spanned the period between Sputnik to the early 1970s, mathematicians dominated the design of math texts and curricula for the first, and almost last, time. Up to that point mathematicians had been kept out of the math education picture, and K-12 mathematics tended not to include any examination of the logical structure of mathematics itself, with the single exception of Euclidean geometry. Students in the first half of the 20th century had instruction in practical matters: consumer buying, insurance, taxation–everything but algebra, geometry, or trigonometry, which, when they were taught, were frequently lacking in depth.

Significantly, the new-math era was one of the only times that mathematicians were given an opportunity to make proper math education available to the masses. (Not until the past few years, working with several state education departments, would they be allowed back into math education decisions.) And some believe that had certain prominent mathematicians who had started working with the development of the new-math programs managed to maintain their influence on those programs, the math education that would have emerged from new math–both lower grades and high school–would have been on par with the best of the math programs overseas.

Eventually, however, the problems with K-6 formalism and the logic and formalism of the program in general doomed new math. The general public, the education community, and even mathematicians themselves judged the new-math programs a failure. Mathematicians were assigned the blame, and the education establishment took back the reins. That establishment received an inadvertent boost in 1983 with the publication of A Nation at Risk, the shockingly pessimistic assessment of the nation’s schools by the National Commission on Excellence in Education. The report sounded another alarm about student math performance, and the NCTM, increasingly dominated by educationists, took advantage of this new education crisis to write revised math standards. The Curriculum and Evaluation Standards for School Mathematics, published in 1989, purported to put the country back on the math track. But because it was, in part, a reaction to the new math and those believed responsible for it, NCTM did not, as mathematicians point out, promote a lively public debate, as had the creators of the new math, but suppressed it.

Some Secrets about Discovery Learning

The NCTM standards were a brew of progressivism–a nod to the 1920s when math was supposed to be practical–and constructivism, which was progressivism that adapted research from cognitive psychology to the task of teaching and called it discovery learning. The standards were based on theories of learning that assumed that children had an innate ability to understand math. The group’s math curricula were thus structured to allow children to discover math concepts rather than to be given them, through direct instruction. The standards also expanded their reach to include, in addition to basic arithmetic, algebra, geometry, and trigonometry.

The NCTM’s view was that traditional teaching techniques, known as “drill and kill,” numbed student minds, turned them off math, and taught them nothing. And so the new standards recommended that students learn “strategies” for learning number facts rather than memorize those facts. It emphasized the use of calculators in all grades. Most important, however, the standards recommended certain areas that should receive “decreased attention” in grades K-4, including “complex paper-and-pencil computations,” “long division,” “paper-and-pencil fraction computation,” “use of rounding to estimate,” “rote practice,” “rote memorization of rules,” and “teaching by telling.” This last item, teaching by telling, is a reference to direct instruction (telling students what they need to know), which NCTM believed should be replaced by “discovery.”

Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned. There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students’ not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering–and taking ownership of–the wrong answer. Additionally, by working in groups and talking with other students (which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride.

Texts that are based on NCTM’s standards focus on concepts and problem solving, but provide a minimum of exercises to build the skills necessary to understand concepts or solve the problems. Thus students are presented with real-life problems in the belief that they will learn what is needed to solve them. While adherents believe that such an approach teaches “mathematical thinking” rather than dull routine skills, some mathematicians have likened it to teaching someone to play water polo without first teaching him to swim.

The Standards were revised in 2000, due in large part to the complaints and criticisms expressed about them. Mathematicians felt that the revised standards, called The Principles and Standards for School Mathematics (PSSM 2000), were an improvement over the 1989 version, but they had reservations. The revised standards still emphasize learning strategies over mathematical facts, for example, and discovery over drill and kill.

Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I’ve seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student’s going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method–one that is held in disdain because it uses rote memorization and is not discovered by the student–the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.

Enter Stage Left: The National Science Foundation

The NCTM standards received a boost of credibility in 1991 when the Education and Human Resources Division of the National Science Foundation funded two grant programs related to math education. The first was for state education departments that aligned their math standards with NCTM’s and school districts that adopted constructivist math programs aligned with NCTM’s standards; the second, for the development of commercial mathematics texts that also followed the NCTM party line, a spending program that now took NSF into the textbook business.

Eventually, NSF supplemented grants to school districts by funding “distribution centers” that promoted the very programs NSF had helped to create. This “Invasion of the Body Snatchers” technique has worked well in enhancing the NCTM constructivist ethic and the texts that NSF helped develop. Meanwhile, NSF continues to fund revisions to some of those texts.

The imprimatur of the National Science Foundation on both NCTM’s standards and the various textbooks they funded and NSF’s grant program caused states to revise their standards of learning for mathematics. Case in point: In 1992, the California State Board of Education adopted the California Mathematics Framework–a set of standards based prominently on NCTM’s standards of 1989.

And Stage Right: the Right

By this time, people were beginning to notice. NSF’s embrace of NCTM’s philosophy of education, among other problems with the teaching of mathematics, became grist for Lynne Cheney, then an active senior fellow at the American Enterprise Institute. With well-articulated essays in leading media, Cheney took out after the educationists and won the respect of mathematicians and scientists as she helped raise awareness among a wider audience across the United States. Isolated math revolts began to occur. One of the first was in Silicon Valley, where parents are engineers, scientists, programmers, and mathematicians–and they didn’t like the way their children were being taught mathematics. Two local mathematicians, Jim Milgram of Stanford and Hung-Hsi Wu of Berkeley, became key players in the rewriting of California’s math standards and the elimination of NSF-funded books from the state’s curriculum. Several years later, as I mentioned earlier, Wilfried Schmid from Harvard became active and helped rewrite Massachusetts’s standards. Two other states, Minnesota and Michigan (Milgram and Wu went to the Wolverine State as well), also just recently revised their math standards. Minnesota’s standards were changed due in large part to the efforts of Dr. Larry Gray, chair of the mathematics department at the University of Minnesota, and of concerned and outraged parents.

But the education bureaucracy did not roll over, and in the fall of 1999 the U.S. Department of Education released a list of ten recommended math programs, designated as “exemplary” or “promising,” all of them aligned with the NCTM standards and based on texts funded by NSF.

The reaction was swift. More than two hundred university professors–including Milgram and Wu, Schmid, and several winners of the Fields Medal, the highest international award in mathematics–wrote an open letter to Secretary of Education Richard Riley, calling on the Department of Education to withdraw the recommendations. The open letter was also published as a full-page ad in the Washington Post, paid for by the Packard Humanities Institute, long a critic of constructivist education. The Department of Education did not withdraw the recommendations, but instead added two more NCTM-aligned books to the list.
The Education Department’s 1999 list, as it represented a more visible–and seemingly overtly political–assertion of political will, was a critical point in the new-math wars. And when Lynne Cheney slammed the Clinton administration as she criticized math textbooks and the NSF in her editorials, she helped ensure that the math wars would become (as they remain) partisan. Republicans tended to be sympathetic to the issues, and some hearings were held. But nothing came of them, and no investigations into NSF and its funding practices were launched. No questions were asked about why the Department of Education didn’t rely on mathematicians in the review of proposals for these programs, nor was anyone in the department ever questioned about the NCTM’s education philosophy and the millions of tax dollars spent on texts that were the subject of fierce objections from 200 prominent mathematicians and scholars.

An Evidentiary Interlude

In the various arguments about how best to teach math, educationists make the point that research shows that their approaches work best. I tend to be suspicious of that research, and apparently others are as well. Ironically, a recent study by the National Academy of Science’s Mathematical Sciences Education Board, sponsored by the NSF itself, was also skeptical. “Evaluations of mathematics curricula provide important information for educators, parents, students and curriculum developers,” concluded the NAS about 19 specific mathematics curricula, including all 13 NSF-sponsored curricula, that it analyzed. They all “fall short of the scientific standards necessary to gauge overall effectiveness.”

While the proponents of these NSF-sponsored math programs may be able to claim that the research shows no evidence that the programs are “ineffective,” the mathematics community, and parents who are protesting to the various school boards across the United States, can now claim that the research cannot be used to support claims of superior effectiveness–or any effectiveness at all.

Is there a common ground in this war? It seems that for now both sides agree that subject matter should be as important as pedagogy and that improving the math education of teachers is probably the most important weapon in the battle. What constitutes a proper math education for prospective teachers, however, is subject to debate. A mathematician I know who teaches at a small college told me that a graduate of his education school caused much embarrassment when, during the interview for a job in an elementary school, the job seeker was unable to add two fractions when asked to do so. The college’s education department subsequently decided to put its math content courses under control of the math department.

Meanwhile, Back on the Hill

Though academic debate about mathematics curricula will no doubt continue, the field of argument is increasingly muddied by politics. It was in this context that I began my investigation into math education in 2002. I recall meeting with Senator X’s deputy chief of staff and two other staffers not long after completing my research on math curricula and the battles that had shaped–often, misshaped–them. “So what are your ideas on how math and science education can be enhanced?” they asked. My answer was something like, “You can enhance a car by painting it, but if the car has no engine, it’s not going to do much good.” This was not what they were expecting to hear. Nor were they expecting to hear that Lynne Cheney had also taken up the cause of anti-fuzzy math. At that point, the discussion took a decidedly troubling turn. These staffers–Democrats–now worried that they could not support policies that were also advocated by the wife of a powerful Republican.

I told them about the open letter from the two hundred mathematicians and urged them not to confuse the message with the messenger. “This is a real issue,” I said. “Kids aren’t learning the math they need to learn.”

I had discussions and sent e-mails in the hopes that I would at least get a chance to brief Senator X on the issue and, perhaps, persuade him to ask some tough questions of NSF when it came time to fund their programs. But I felt that at any moment everything was going to be whisked away.

And one day it was. The staffers in my office talked with other Democratic staffers on the Hill, who told them that it would be wise to stay away from the “fuzzy math/Lynne Cheney/Bush agenda” issue. Ultimately the staffers I was working with told me they couldn’t take a chance on having Senator X “come off like Lynne Cheney.”

This development was not surprising to any of the mathematicians with whom I had been working–most of them Democrats, like me. The senator was never briefed, and no investigation into NSF was launched. I was thanked for my hard work. I went back to my regular job and started tutoring middle school students in math at a school in D.C. while continuing to work with high school students in my neighborhood. That year a 9th-grade girl was having problems in geometry and came to me for help. “What seems to be the problem?” I asked. “I don’t know how to do proofs,” she said.

“I know,” I replied. “Don’t worry. It isn’t you.”

All politics is local, I decided.

Barry Garelick is an analyst with a federal government agency in Washington, D.C. The opinions expressed in this article are those of the author.

• […] This is creepily similar to the idiotic “lattice multiplication” lessons in Everyday Math that justify using incoherent, inefficient methods of multiplying because that’s the way the ancient Egyptians did it. […]

• Amy B says:

You don’t have to be a teacher or a math guru to know that the children are being cheated of a decent education. I was called on the carpet at my son’s school because I made him memorize the multiplication table. After all, if the school wasn’t teaching him, I would. You would have thought I was offering sacrifices to pagan idols. However, my son, now 29, has called me several times to thank me for the basic math background.

• Ann A. says:

Fascinating. How does this intersect with the standardized math tests required to get into college (ACT, SAT)? That is, don’t those tests force the teachers to teach actual stuff, rather than concepts? Just wondering how anyone gets into college otherwise.

(BTW, I homeschool math for all of my children, no matter where they are currently going to school.)

• JLSeattle says:

Re: “At the time I started tutoring, my daughter …was not learning her addition and subtraction facts. … Teachers told them not to worry because kids eventually “get it.”
“The authors of Everyday Math do not believe it is worth the time and effort to develop highly efficient paper-and-pencil algorithms for all possible whole number, fractions and decimal division problems….It is simply counterproductive to invest hours of precious class time on such algorithms. The math payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator.”

An unemployed man I was tutoring last month to take the qualifying exam for the 2010 Census – a good-paying though temporary job – could not add decimals or do long division. He wouldn’t be permitted the use of a calculator during the exam, of course. Even if he could use one, it wouldn’t help him decide whether in May 2009 a 78-year-old woman with a September birthday was born in 1930, 1931, or 1932. He couldn’t even begin to set up that problem on paper.

• Jac L says:

I teach 5th grade math. But, it is very difficult for the students to master division, simplifying (reducing) fractions, etc. due to the fact that they haven’t memorized their math facts. It’s so bad that in 5th grade some of them use their fingers to find the answer to 9 + 1. While some of the discovery methods are fun and can be enlightening, if they don’t learn the old fashioned algorithms, junior high and beyond will be incredibly difficult….

• Fact checker says:

While this is a good history lesson (with an agenda), it draws a completely false conclusion that reformed inquiry books has caused the failures of America students on international tests. Independent market research shows that only 20 to 30% at most of America’s students are taught from reformed textbooks. But Garelick wants you to believe that 100% of the problems stem from reformed methods. This mathematician needs a statistics lesson. But then again, he probably wants statistics removed from the K-8 curricula anyway.

• Barry Garelick says:

Fact Checker:

You have been leaving this message on various blogs and online journals, and despite many people having corrected your assumptions, you still persist. Please note: In my article, I talk about my daughter in second grade not being taught the math facts. This was prior to her school adopting Everyday Math. The explanation that the teacher gave me was consistent with the teaching methods that were and are still going on today, and which are heavily influenced by NCTM’s standards, as well as the use of fuzzy math texts.

• Debate says:

No one has corrected any assumptions. Some have provided opinions but without data, The facts are very clear as is the research. It is can be found at http://www.educationmarketresearch.com/ (for a small fee).
And I challenge anyone to show that across America, no more than 20 to 30% of students are taught from reform texts. BTW, your writings are very professional and insightful although your premise is flawed. It is unfortunate, though, that the majority of these blogs are filled with hateful and personal attacks on educators. Glad, though, we (my class and I) have your attention as it is a very good learning experience for young teachers.

• Barry Garelick says:

Debate:

I have been following the information flow being provided at the Scientific American online journal. My article in Education Next talks about reform texts, to be sure, but also highlights how non-NSF funded texts are also weak–case in point, the de-emphasis of proofs in geometry texts, and the general dumbing down of algebra texts. I also stand by my previous comment above, that the influence of ed school and other entities on how math should be taught contributes to the decline in performance that we are seeing nationwide, in addition to textbooks and programs that are content-deficient. I have been taking courses in ed school and have heard first-hand the philosophies of those teaching the courses about how math problems, even in Asian countries, are “inauthentic” and do not promote proper mathematical thinking. If there is any premise that is flawed, it is that one, and I have written about it elsewhere. One can also not overlook the approach to education in general–not just math. Group and collaborative learning is emphasized, to stimulate discussion and “construction of knowledge”. Process trumps content, so that getting the right answer is no longer important, and in some cases, even frowned upon.

I am happy that you find my writings professional and insightful, even though you disagree. I appreciate civil discourse, and like you, I am also dismayed at the level of personal attacks one finds on the internet–not just toward educators.

• Laura says:

Let us not take our eye off the target here. Mr. Garelick appears to be pointing to the downside of inquiry-based learning. This philosophy permeates ed schools in math, but also language arts and other subjects.

Students never actually acquire skills. They are left unable to perform arithmetic. They cannot spell. They have no grammar skills. Just step into a typical 8th grade classroom and observe. Observe the students and the frustrated teachers who wonder how Johnny got to 8th grade.

Textbooks do not teach our children. Teachers do. And, many teachers indoctrinated with the philosophy that inquiry-based learning is the only way students acquire knowledge, are being used as vehicles to perpetuate the very problems we have in our educational system today rather than using teachers as mediators of learning.

It is not the teacher on the frontline who is mediating this. It is not the textbook company, persay. It is the educational philosophy that is setting the agenda.

• Katharine Beals says:

Fact checker,

Everyday Math alone is mandated districtwide in two of the five largest U.S. cities: New York City and Philadelphia.

Everyday Math author Andy Isaacs touts its presence in Virginia Beach, VA; Kent, WA; Fayetteville, AR; Citrus County, FL; and Chattanooga, TN.

Other large Everyday Math districts that I am aware of include St Paul, MN; Montgomery County MD; Orange County, FL; and Washington DC.

Quite a number of suburban schools in the greater Delaware Valley area also use Everyday Math, in PA and NJ alike.

That’s just Everyday Math.

As far as other Reform Math programs go, Boston mandates Investigations; and I’ve heard reports of Investigations in Amherst and Trailblazers in Kansas City. We all know about Seattle, of course.

How many large districts that use “traditional math” can you cite?

Katharine Beals

• FactChecker says:

While Beals lists districts she knows or thinks use a reformed curriculum, the nationwide statistics tell the whole story. In grades 3-5, Everyday math has a 20% market share and investigations has a 10% market share. No other reformed books have a market share above 2%. Thus nearly 70% of the country learns from traditional books with traditional instruction. These facts can be verified at http://www.educationmarketresearch.com/ for a small fee.

Certainly in some areas of the country like Washington state, the reformed market share approaches 50% but in other states (California), the share is much much lower than the national average.

Moreover, in middle schools nearly 90% of students nationwide learn with traditional materials. Connected math is the only inquiry book with a market share above 2%. CMP has a 11% market share nationwide.

The facts are very clear and can be verified if you truly want the facts. But do you?

• Katharine Beals says:

Yes, as long as I don’t have to spend money on them.

• FactChecker says:

Unfortunately you do (have to spend money). The market research is copyright protected and I cannot distribute it electronically nor copy it. I have leaked enough data that can verified by anyone who wants to learn the complete truth about the market share of inquiry programs or any program for that matter.

It is unfortunate that many continue to perpetuate the myth that inquiry based programs are to blame for the low performance of America’s students when the truth is an overwhelming majority of students learn in traditional texts.

• Eric Fleming says:

I enjoyed reading this article and thought that it spoke a lot of truth about the climate of our nation’s attempts to educate k-12 students in the area of mathematics. I will point out my obvious bias and sympathy to agree with the author because I am as well a math major. If the previous statement does not jade you to my opinion, then I will continue.

I believe that it would not be outrageous to claim that the best people suited to teach a subject are the ones that have spent the most time learning it, granted they know how to simply explain what they know. Discovery learning can be great, but often inefficient in terms of time spent learning per profound realization in class. From my experiences, the best math teachers chose to incorporate discovery learning into homework assignments rather than force us to regurgitate mindless exercises from class, and used traditional methods during class.

Mathematics like anything else is no stranger to the basic blue-print for progress: build a strong foundation and then start working from there. If students cannot recite multiplication tables without hesitation by the time they leave elementary school, they are truly fighting an uphill battle with what is now probably their least favorite subject.

Memorizing basic arithmetic is not only necessary, but vital to the furtherance of mathematical knowledge. When we first teach children to read we do not first ask them to discover how to pronounce the alphabet, we make them memorize it! This is no different.

*As a side note, your “ad hoc theory” was hilarious.

• Rosalind Childs says:

Here in Detroit, at The Homework Center, we have experienced similar trouble. Ninety percent of the students we tutored this school year could not recite the multiplication table without hesitating. These students range from 3rd to 11th graders!

The school system is on a “Literacy Freight Train” which is leaving math in the background. I think we, the parents, should step up our game. Stop leaving it all to the teachers and start pushing our students to memorize the multiplication table (for starters). As time goes on, they will be able to memorize formulas as well. We, the parents, have to require more out of our “future leaders”.

• Epicurus says:

The reform math programs are generally poorly designed and will not help students acquire math skills, but there has been no significant decline in math skills nationwide since the implementation of reform math. It would therefore appear that they are not much worse than the old programs.

It isn’t difficult to teach computation, and no competent math teacher needs a textbook to do it. A teacher who fails to teach basic math skills because he is issued a copy of Everyday Math is not going to succeed because he goes back to a normal textbooks.

So, although I’m against reform math, it’s more important to say what you are for. The reformers have good ideas for clever story problems and applications. Praise them for that and then implement Saxon or Primary Mathematics (“Singapore Math”). But most importantly, make sure that the math teacher A) knows math B) knows how to get active responses from students and use them to shape his teaching.

• Taxpayer says:

Everyday Math absolutely STINKS. Because of Everyday Math, our daughter developed a serious case of math anxiety, which persists to this day (she’s now in 6th grade). She could barely understand the bizarre assignments and would get extremely frustrated. My husband is a scientist, and he had a heckuva time trying to figure out all the mumbo-jumbo so he could help our daughter with her homework!

We had to teach her how to add and subtract in columns and do traditional long division because Everyday Math didn’t teach it!

This kid can figure out probabilities in her head while playing Texas Hold ’em and whoop everyone’s hiney. Yet she has a near panic attack when doing long division or fractions.

• Barry Garelick says:

The rates of students who are taking remedial level math courses at community colleges and universities has been steadily increasing over the past twenty years. If you are basing your statement that there has been no significant decline in math skills since the introduction of reform math on NAEP scores, I point out that such questions are not exactly challenging and for many questions they allow students to use a calculator. I also point out that it is impossible to parse such data to ascertain the degree to which students have received instruction from tutors, learning centers such as Kumon and Sylvan, or help from parents.

If you want me to say what I am for, I am for students learning the basic skills in a sequential and logical manner that builds upon itself. To that end, I agreewith you that Saxon and Singapore do this nicely. I am also for letting competent teachers do this, and not dictate policies of reform which is happening in schools. The doctrine that learning basic skills and attaining procedural fluency prevents students from solving new or unfamiliar problems is one that is taught in ed schools.

It may not be difficult to teach computation, but if teachers are forced to stick to the reform agenda, they are not going to do well. How do you know that going back to the old textbooks would not work? That’s what many teachers who know how to teach have actually done. And yes, for those teachers who have many years under their belts, they don’t really need the textbooks, because they know what must be taught.

I agree with your last statement, but I fear that teachers coming out of ed school are not taught how to do this, and are actually taught that such practices are harmful.

• concerned taxpayer says:

Dear Taxpayer,

You are EXACTLY RIGHT!!

These mediocre programs have made their way into our schools because OUR TAX DOLLARS keep funding their development and implementation through NSF. Contact the House Science Committee to voice your disgust!

Check out this evaluation: http://books.nap.edu/openbook.php?record_id=11025&page=R1

• Niki Hayes says:

As a retired middle and high school math teacher and K-12 principal, I need to explain that MANY teachers are forbidden to use any materials other than the reform ones that have been adopted by the district. It’s called “fidelity of implementation.” This supposedly will prevent the pollution of the reformists’ equity-based, literary-designed, verbally-organized classrooms that are to appeal to girls and minorities (except Asians). (Yes, I consider such thinking as racist and sexist.) Of course, many teachers still manage to sneak traditional instruction to their students but that is a heck of a way to have to work each day. For one thing, the students must learn two “languages” of mathematics–the reformists’ methods and the tradtional/internationally-based methodologies that will prepare them for higher education. That duplicity adds undesired stress on both teacher and the student. (This also happens when students must be taught basic skills with tutors.)

Next, I will be glad to debate anyone at any time about the serious decline in math ability–as well as math scores, even though some of the scores have been manipulated to show growth–with the full-force introduction of reform math 1989. I watched it in my classroom over a 20-year period. As a principal, I saw how math achievement could be turned around with the use of Saxon Math as the K-12 curriculum on an Indian reservation and then in an all-white elementary school in Seattle, WA. It works on all kids from all “subgroups” and genders. It is not designed for “certain” groups while ignoring others.

I heard a speaker at a school board meeting ask once, when the reformists were demanding Everyday Math be adopted but, at the same time, saying that curriculum didn’t matter. It was the teacher who makes ALL of the difference, they said. The speaker asked, if that’s the case, why the district didn’t just issue teachers a copy of the Yellow Pages from which to teach mathematics. No one offered an answer.

Curriculum makes a huge difference for students if it is vertically implemented through grade levels because it is the one thing that can remain constant in a child’s learning. The adults change, both at school and in the home (sadly). But it must be user-friendly and proven in its results. None of the reform products can make that claim, at least not HONESTLY.

• Catherine says:

The claim that math skills today are the equal of math skills yesterday is not born out by enrollment figures in math/science graduate programs, where the ratio of foreign-born students to U.S. citizens is now very high.

Two weeks ago, I attended a talk given by David Steiner, New York’s new Commissioner of Education. During the question and comment portion of the evening, a local school board member told us that only 20% of the students in her daughter’s class in Yale School of Medicine are American citizens.

• Math Advocate says:

Revisiting this article gives me greater appreciation for the information provided and brought many things to mind. In a recent geometry tutoring session, I complimented the student for his ability to quickly and accurately make connections and solve problems. Like the author’s geometry students, this student struggled with proofs. He eventually did well with them. His response to my compliment was very insightful—he placed a lot of value on having worked proofs and told me that is why he is able to do so well now. He fortunately has a good example based text. Without that, this student would be totally lost.

Mention in the article of the new-math era caught my attention. The new-math texts are ones I would prefer to use over the new-new math or reform math programs prevalent in so many schools today. I have first hand experience teaching with both kinds of programs. While teaching grade 6 in an elementary school, I had to use the CMP program. I was fortunate that I could supplement it—that ability to supplement has since changed. I watched as Everyday Math (EM) was adopted for K-5 and the mandate to teachers was to only use EM. They were told they could not use any other materials or supplement in any way. Good teachers who might appropriate supplement a math program are not being allowed to do so. In places like this school, teachers fear being brought up on insubordination charges if they supplement the program. Still, some I know close the classroom door and try to fill in the gaps in an interest of providing their students with an adequate math education.

Epicurus—You may not see the decline in math skills and some test results may not appear to show a decline, yet our country does not fare well when comparing our students’ math abilities with those of other countries. Closer to home for me— only 25% of my incoming six grade students were fluent with basic multiplication facts through 10. Community colleges are offering more and more remedial math classes. Go visit some of those classes and talk with students. I have. Many of the students I talked to passed algebra 1 and 2, graduated from high school, and were admitted to college without ever developing fundamental math skills. They were calculator dependent (even for problems like 8+3).

Here is what I am for. I am for providing our students with a quality education. For me, I have found my students do better when I provide direct instruction and use a good example based textbook. While it may sound really good, I have not found discovery learning to be an effective way for me to help students learn and develop fundamental skills.

The article listed below gives some background on minimal guided instructional practices. These practices sound reasonable but there is no empirical evidence available to show they are more effective than explicit instruction.

Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching
Paul A. Kirschner Educational Technology Expertise Center Open University of the Netherlands Research Centre Learning in Interaction Utrecht University, The Netherlands
John Sweller School of Education University of New South Wales
Richard E. Clark Rossier School of Education University of Southern California
http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf

More information on Inquiry-Based Instruction versus Example-Based Instruction can be found at
Explicit Instruction or Reform.
http://soundmath.wetpaint.com/page/Explicit+Instruction+or+Reform

Many teacher preparation programs promote the reform programs and approach. Professional development offerings in my area also promote the reform programs and approach. I talked with a young man last year that had just completed his teacher preparation program. He was getting ready to take the test the state requires to obtain certification to teach elementary school. He was anxious and told me he hoped they would allow him to use a calculator. He admitted to not knowing how to do division without one. Today, this person may be teaching math to your elementary school child.

• JM O'Dell says:

As another poster pointed out, no one is accounting for the number of parents using tutors or teaching their children at home using a traditional method to make up for the disaster that is fuzzy math. Go ahead and ask your reform math advocates in your district how many parents use tutors or a traditional textbook at home to teach what they know their kids need to know. You will get silence or stammering, because it is all a dirty little secret.

• Brad Spencer says:

Niki Hayes said:

“This supposedly will prevent the pollution of the reformists’ equity-based, literary-designed, verbally-organized classrooms that are to appeal to girls and minorities (except Asians).”

I graduated from high school in 1957 so I doubt I was ever exposed to reform math. In the 3rd grade the teacher would regularly have contests at the blackboard to find who did best (I first tasted a 3 Musketeers bar when I won.) As I recall most of the consistent top contenders were girls. Similarly, in high school, most of the top students in math classes were girls.

I think I was exposed to reform math a couple of times when visiting my home town and tried to help children of my niece and nephew with their homework. It seemed that the goal was to ill-present material and to have the student re-discover millenia of mathematics on their own (which I find absurd in intent and in execution.) There was one problem for which all the assembled adults were perplexed & incapable to figure out. This group included one high school math teacher. What it appeared to me was that the author of the text and the problem had experienced what he regarded as an intellectual breakthrough and was trying to force students (who did not have his same set of experiences) to have that same breakthrough. Plus it was a trivial breakthrough (if his intent was as suspected. the entire chapter was massively unclear and unfocused.)

You appear to suspect that the purveyors of reform math have a non-math-related agenda which they are following. I think you are correct. I concur that what they are attempting and what they are doing is not beneficial to students and not only doesn’t enhance learning, it retards it.

Perhaps this is a manifestation of the general tendency of liberal arts people to minimize and deprecate all non-liberal-arts knowledge (or to try to twist such knowledge into the forms they know.) If so, that’s bad.

• TC Rogers says:

I am an elementary teacher who has taught in both private and public schools. In the private school, we used Saxon math which has a traditional, structured approach. I watched as students were able to succeed, even though they had previously struggled with math.

Later, I watched my public school students get bogged down in basic math topics. There were several differences in the two groups. Many of my public school students had NOT memorized their multiplication facts by 5th grade. The public school texts were full of beautiful, full color photos but very little instruction and example. Many support personnel and resources were available to my students, so they often said “I don’t get it”, knowing that help was on the way. It was easier to get help than to think about how to solve the problem.

I often found that they had very little spatial and number awareness as well. Concepts such as more/less, up/down, forward/backward, left/right were fuzzy for many of my students. It makes it hard to do math if the tools they need aren’t in their heads, and if they don’t know where they are in space in relation to a number line.

Math education is a two-way street. Teachers must teach both content and process in a scaffolding way so that students can confidently move to higher levels of difficulty. Teachers must be equipped with the knowledge and proven materials to accomplish this. However, students and parents must also be prepared to shut off the television, get off the couch, and work hard to own the knowledge being presented.

• Hiroko Jobst says:

Long time viewer / 1st time poster. Really enjoy reading the blog, keep up the good work. Will definitely start posting more in the future.

• Barry Garelick says:

“However, students and parents must also be prepared to shut off the television, get off the couch, and work hard to own the knowledge being presented.”

I agree. I would point out however that there are many parents who work hard with their children to teach what is not being taught in the schools. Parents have been told by schools to not contradict or go against the approaches being taught ins chools. This advice is also given in Dept of Education guidance; see http://www2.ed.gov/parents/academic/involve/homework/part_pg4.html#4 which advises ” don’t teach strategies and shortcuts that conflict with the approach the teacher is using”. So if the teacher is using one of the ridiculous strategies or alternative algorithms of Everyday Math, parents are advised not to teach them the standard algorithm–nor are they to teach them long division for that matter.

If students are saying they don’t get it, it could be that the math program is not effective. Did they say this a lot when you used Saxon? And if the teachers provide the help the students need, why is that viewed necessarily as “handing it to the student”? If students are given proper information and instruction and problems are scaffolded appropriately (as they are in Saxon’s program), there will likely be less of the “I don’t get it” question.

• Tan Hui Ning says:

I am a student from Singapore and would like to thank you for posting this article as it was a great and informative read.
Just wondering do students in America learn how to draw graphs?

• Barry Garelick says:

Tan Hui Ning:

Thanks for your comment. Yes, students in the US learn how to draw graphs. In the lower grades it is basic line and bar graphs, though there is a trend to teach “box and whisker plots” in the lower grades. In grades 7-12, graphing is taught for a variety of different functions.

• Michele Premeaux-Johnson says:

This may seam silly, but this article is an answer to a prayer. NO JOKE…I have been wondering what ‘fuzzy math’ was. I now have a 2nd grader who is clueless. I plan on keeping her out of school a year JUST so I can teach her her tables…I will drill and kill (OK, not kill but that was your words correct?)

I have a degree in Psy and a minor in Math. I didn’t want to teach and NOW I am so glad that I’m not dependant on my salary. The state of Texas is at a very sad place right now. I to believe all school districts have to let the teachers who WILL have a job next year know by the 15th if they get a contract. POINT is, I feel so bad taking my funding – I mean my child out of school and yet I am so upset that she’s been in school three years and is SO CLUELESS.

I can relate in so many ways to your article. I went to Catholic school till 7th grade (in Texas). I took Algebra in 8th and Gemometry in 9th (we had to show proofs, which I wasn’t great at but learne how to do it) and by the time I moved back to Texas I was a whole grade level AHEAD of my new small town High School. They even made up a class for me call “computer math” and offered it to those student who did NOT want to take Calculus!

Anyway, the reason I even found this article is politcally based. I do not understand how we can keep going down in ranks compared to other countries AND YET we keep spending so much on “bad education”. I don’t blame the teachers. I almost don’t blame the Adminstration (meaning, those that went and spent more money on more education to get a Master’s degree so they could make sure a student’s class schedule was going to work = aka what my High School Councilor did for me?) I found in 2007 we were ranked 10 in Math and 9th in Science…in 2010 we’ve gone down in both of these. What MORE DO PEOPLE NEED TO SEE. How can I get on the band wagon for “change” here?

• Christine DAmico says:

Barry,
You are a strong person and bright! It’s sad that Lynne Cheney who only wanted to do good, has polarized, it seems this issue! This is NOT about republicans and democrats, it’s about our kids and our future. Both parties and any other party should get out of the way and make way for common sense and reason.
The human brain MUST be challenged with rigor, memorization and stimulation in order to grow neurons. This fact is rarely if ever played into curriculum development. Read, “The Brain That Changes Itself” pages 41 and 42, the authors point out that the frontal cortex of people exposed to a Classical Education in the early 1900s was bigger than it is now! Amazingly, a Classical Education which requires memorization of facts and important documents, handwriting precision and much more, prepares the brain to THINK…… Go figure!!! :-)

• Lynn S. says:

I have a BS in mechanical engineering (currently a Federal Employee) with an additional BA in English with Secondary Ed emphasis and a Spanish minor. I’m also the mother of four very bright children: 9th grade girl, 7th grade girl, 5th grade boy and 3-year-old boy. I’ve been very puzzled since the beginning of this school year, when my former whiz kids are all in tears over their math homework. I Googled “why is my 9th grade daughter struggling with Geometry?” and found this article. Very enlightening; and I appreciate everyone’s comments, as well. Yes, our school district went to Everyday Math one year ago. We received letters from the teachers explaining the new curriculum and that we were not to teach our children any other methods, as it would “confuse” them. No kidding! I plan on doing some more research on this curriculum and launching a protest to our school district. Meanwhile, I’m going to make sure to teach my son long division this year, and any other missing concept. He wants to be an engineer like Mom, so I guess he’d better know the basics!

• Barry Garelick says:

I appreciate Lynn’s and Christine’s comments. The “math wars” continue as they always have. Parents continue saying the things that many of the commenters have above, and which I espoused in my article. And schools and school districts maintain their position that parents do not like the new methods because it’s different than how they learned math. Parents (many of whom are scientists, engineers and mathematicians) are given short shrift and their arguments are dismissed because, after all, they are “just parents”.

Please continue the dialogues with your schools and administrations. But unfortunately, change comes about slowly if at all. Plan on teaching your child real math after school (using a variety of good texts available, such as the Singapore series, Saxon, Sadlier Oxford and others), or enroll your child in after school programs like Sylvan, Huntington, Kumon, Mathnasium and the like. It is unfair to parents who pay their taxes to have to pay even more money (or spend more time) teaching their children what they should have learned in school. It is also unfair to the child who has to spend extra hours beyond those he or she spends in school. And it is particularly unfair to those students from poor households whose parents cannot afford such after school programs, or may not be able to help their children themselves.

Please check out the website for U.S. Coalition for World Class Math at http://usworldclassmath.webs.com/commoncorestandards.htm.

Also, read Laurie Rogers excellent book “Betrayed: How the Education Establishment has Betrayed America”. She has a blog that is well worth reading at http://betrayed-whyeducationisfailing.blogspot.com/.

• Debra Sheffield says:

I am about to retire with almost thirty years of teaching in the middle schools of the midwest. I can speak from my own experience that EveryDay Math, MathThematics, and Connected Math are mainly hurting students more than it is helping them. Inqiry can be fun, but time consuming and for every A HA moment, there is a class time wasted by having to go over the material again for greater understanding. Truly, better prepared elementary teachers in math would help the situation or more time spent on math would help. My students cannot do simple multiplying/dividing decimals by powers of ten and moving decimals, convert mixed numbers to improper, or do even basic divide or subtract correctly. I find them mixing up every different way they have been exposed to and have no idea that their answers make no sense.

Because of the lack of time students spend at home on math curriculum (or made to do quality work by a care giver at home), students do not know their basic skills and have a complete lack of confidence in their own ability to solve a problem on their own. Yet, I cannot get my so called Curriculum Superintendent to make any changes. Our administration believes that with technology, students will have tools to do their calculations for them. Our math scores on local state tests have dropped so dramatically that we are now in line to lose state funding. I have not been able to take any students to any math contests such at MATHCOUNTS years because none of my students can do mental math quickly or at all. They are capable of learning some Pre-algebra, but not at the pace I need to get ready for state testing which determines our funding.

I have to tell you that my own health has been affected by the frustation of teaching unprepared students and dealing with ridiculous administration. I have definitely done whatever I thought necessary to deliver a quality education in my classroom. I have tried to get parents to rally, but most are not aware or do not go to school board meetings as a group for more impact. My districts’ math program is failing our students. And, I cannot get anyone to listen.

• Barry Garelick says:

Thank you for this comment. It is especially pertinent given an article I have written on reform math vs traditional: http://www.educationnews.org/education-policy-and-politics/barry-garelick-math-education-being-outwitted-by-stupidity/

Your experience in a math reform classroom as you’ve documented above is important and would be a helpful comment to this article if you’d care to do so.

• Dan Dilworth says:

Interesting. I found this while searching for “constructivist teaching opportunities.” I learned math in parochial school in the 60s. I have an electrical engineering degree. I taught high school mathematics 3 years during a hiatus from my technical profession a few years ago. I think inquiry and constructivism are wonderful, but only when math-facts are know and basic skill developed. You have to learn how to read before you can read literature or history. One has to know how to manipulate numbers and symbols mentally before one can understand basic linear equations or derive the quadratic equation. When the facts are known, then the guided inquiry and construction must begin. Even though I loved and was a natural with math, my education was mediocre and rote, I think because the teaching was mediocre at best. I expect more than anything, good mathematics education takes social adept and mathematically talented teachers. But lacking that, then the basic facts have to be taught (adding, subtracting, multiplying and dividing, by hand and head).

• Igor says:

The Math tutoring in NYC public school is terrible. Children did not build a strong calculus foundation. Study of probability in 3rd/4th grade is a waste of time. The math nonsense program needs to be stopped as soon as possible.

• Jem Corraggio says:

Unlike most of the people who have responded to this article, I do not think that fixing the curriculum alone will fix the problem. Yes I dodged a bullet by having traditional math taught at home while Everyday Math was taught at school. I actually found Everyday Math intriguing and amusing, and did find some value in knowing both. For example I still break up products such as 23 * 16 into 20*10 + 3*10 + 20*6 + 3*6 when doing mental math, and it was helpful when learning the distributive property and FOIL to already have some idea that you could break up multiplication like that. I do agree that without the traditional method coming first, those methods would not have served me very well. However, no matter how good the curriculum is, we need the teachers to teach it.

“If you can’t do, teach.” This is the disturbing philosophy that I feel needs to be corrected in the United States. I taught English for a summer in China. The students arrived an hour early, stood when I entered the room, and hung on my every word. The entire community respected me, and that was passed to their children. The American community does not respect their teachers. I am not referring to my students, but their parents and the larger adult community. I have a math degree from RPI. My fellow students in college mocked my choice of profession. I mean, why teach when I could be an actuary, a programmer, an engineer, etc. etc. and earn triple the salary. I am not kidding, my intellectual peers earn triple my salary and I actually have a relatively high paying teaching job at the moment. At my last job I would have had to say they earn 4 times as much.

We cannot have the people who can’t do mathematics teaching mathematics. But the people who can do mathematics are doing other jobs that require mathematical knowledge because they are jobs that come with more respect and a higher salary. I’m not saying that there aren’t many qualified teachers out there, but only because they, like myself, love the job so much it is worth the dismal pay in their eyes. But there are far too many teachers out there who do not have basic mathematical knowledge, as some other responses above have mentioned.

The math Praxis (both I and II) is a joke, easier in my opinion than the math portion of the SATs. How can the required knowledge for high school teachers be less than the required knowledge for all high school students entering college?

Our success in math and science as a nation depends on our country’s math and science teachers. Instead of spending billions in curriculum reform after curriculum reform, we need to spend that money attracting and training a new breed of teachers. If we do that well, the teachers will be able to pick good curriculum for themselves instead of being force fed it, whether they agree with it or not. I think that we need to make being a teacher as difficult and prestigious a position to attain as being a doctor, lawyer, or engineer, with the salary to match. Only then will we attract the brightest minds to solving the problem of our children’s education.

• Barry Garelick says:

Jem, thanks for your comment. Teacher knowledge is indeed important, but there are also cases in which teachers are told explicitly what and how to teach. Currently, the Common Core math standards are being interpreted by various schools, school districts and PD vendors as requiring math to be taught “differently” than before. That is translating to inquiry-based, student-centered classes, with “understanding” trumping content. For more info on this see my article on Common Core math standards at http://www.theatlantic.com/national/archive/2012/11/why-the-new-common-core-math-standards-dont-add-up/265444/.

• Jem Corraggio says:

Barry, thanks for responding. I know all too well the frustration of being forced to teach in a manner that isn’t conducive to learning. In addition to having to align my lessons with standards that are often vague and grade-level inappropriate, I also am forced to teach students who have, for example, never passed Pre-Algebra, Algebra I during the same semester they are taking Pre-Algebra over again. Administrators force students to double up so that they will graduate on time, even though it is completely illogical to expect a student without the prior knowledge of the first year of math to pass the second. What I am proposing is a drastic and complete turn around from the current education system, but one that I believe is necessary. Changing the curriculum alone will not fix this. If we put teachers we trust to know math and how to teach math in the schools, then those individuals should also be able to be trusted with choosing their own curriculum. I agree with changing the curriculum, but that will only solve half the problem, a band-aid, if you will. To have a system that truly works, we need to address the entire problem. Put math education into the correct hands and only those hands. The only questions then are whose those hands should be, and how to attract and train those hands.

To be completely honest, Barry, I don’t think you took my original argument seriously. It makes me wonder if you too discount my mental faculties because of my chosen profession. We both have math degrees. I am at the disadvantage of not knowing where yours is from to completely compare our backgrounds. But you seem to be under the impression that I need someone to explain how my profession works. Would you trust me to choose your child’s curriculum? Would you prefer to leave it in the hands of an administrator that doesn’t have a math degree? That doesn’t know your child, their strengths, their weaknesses, how to interest them in the knowledge, how to bolster their confidence?

I guess what I am saying is the argument isn’t just about what to teach, but who gets to answer that question. I would not say it would be wise to leave it in the hands of teachers now, I don’t think it would be wise to leave it to administrators ever, but with the right teachers… :)

• Liz B. says:

My family is spending 30K a year sending our 2 bright sons to a private, independent school. Every night, my MD husband and I wade through the Everyday Math work with them, wondering why they are doing so poorly at it. Every grading period brings teachers’ comments about all the things they have failed to accomplish, and we wonder what to do to get them back on the spiraling track. This week we are being called in for a conference. I think the recommendation will be that our 5th grader needs to join the throngs of his classmates who are being tutored. They are going to suggest that we use an older child (8th grader) who can explain things to him in a way that the teacher cannot. Something is wrong. I am trying to compose my response.

I found this article in desperation, when I googled “Why is math so much worse than it used to be?” I still don’t know what to do about my own kids, but at least I don’t feel so alone. Thank you.

• Barry Garelick says:

Liz,

Sorry to hear what you’re going through. It sounds like the school can’t admit that EM is a failure so as is usually the case, they blame the student. Having another student tutor him is ludicrous!

Many parents either help their kids at home (using traditional math) or hire tutors, or send their kids to Sylvan, Kumon and the like. When the student comes around and does well on standardized tests, the school/school district will take credit for it, saying that EM is a good choice.

I would look into the Learning Center option; Mathnasium, Sylvan, Huntington and Kumon are all very effective–and they are used to tutoring EM casualty cases.

• FreezyPop says:

Everyday Math is one of the primary reasons we chose to home-school our sons (via a K12-provided online public school). It also makes scheduling their skating lessons more manageable…if no more affordable, considering we’ve accepted a change in income to home school. :)

I was flabbergasted to read the “don’t teach your children other methods” comments above. The day after I got that letter would be my kids’ last day in any such a district, and parents and children alike in such districts have my sympathy.

• Richard M says:

I’m a math major at the University of Washington and would consider myself as having a strong mathematical background. While I was taught, and learned, basic arithmetic and on through calculus largely through memorization and an integrative approach to Geometry and Algebra, but probably the most significant reason why I have been successful in Math has been because I loved it, I was TAUGHT to love it, and I saw it as an ART.

When I say “art” I want you to think about something very physical, constructive, and aesthetic (like drawing and painting). Knowledge of the “basic facts” (45 * 24 = 1080) really shouldn’t be nearly as important as WHY THEY ARE TRUE. Moreover, there are way cooler artifacts in Math that children should be exposed to early to stimulate interest.

That said, arithmetic can be fun too. It isn’t taught very well, and it’s treated as a chore (or even a punishment) that only some kids can do–this is utter ****. Check out Arthur Benjamin’s book. The left-to-right method (combined with other methods) are by far superior for teaching arithmetic. For some odd reason, long division is the only operation we taught left-to-right.
http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401

(An aside: 45 * 24 = (45 * 20) + (45 * 4) = 900 + 180 = 1080. The Egyptian lattice method is actually pretty close to being something useful–you just need to see it as left-to-right instead. We read words that way, why don’t we read numbers that way too? Oh right we do…but we don’t.

In this blog, Witten unpacks and explains complex analysis beautifully and so simply I should think a freshmen in high school could follow it.

http://acko.net/blog/how-to-fold-a-julia-fractal/

Why can’t all Math be taught this way?

Paul Lockhart provides a great analogy to all of this and there’s an interview with him that addresses a lot of these questions too:

http://www.maa.org/devlin/lockhartslament.pdf

Thanks, R

• Kristin says:

Barry,

Thanks so much for your work on this and for your article. I’m a homeschooling mother of two, 9 and almost 5, and am researching the best way to teach math to my children. Wow, it seems I lucked out, going through school in the 70’s & 80’s, and learning traditional math, at least in the early years. I took Geometry in 9th grade, the Algebra, pre-Calculus, then Calculus in 12th and went on to be a computer science major in college and I don’t know what you mean by proofs, so maybe I did miss something in those later years.

Thanks again.

Sincerely,
Kristin

• Sandra Warren, MsEd says:

Memorization and comprehension are distinctive activities and both must be taught. We need a solid fact base (learn only through memorization) in order to take the material to higher level learning. Without the fact base what exactly are we going to analyze, synthesize or

It makes no sense that the Priciple & Standards for School Mathematics recommend “ ..students use understanding of multiplication to develop quick recall”

How about we use songs, rhymes, illustratios and proven

• Joshua says:

I’m in 7th grade. And the answer to: How many pieces of string will you have if you divide 3/4 yard of string into pieces each 1/8 yard long? The Answer is obviously 6! And only 55% of 8TH GRADERS got it right? Wow…

• Barry Garelick says:

Joshua,

Well done. Yes, it is disturbing that so many 8th graders answered it incorrectly.

Thanks for your comment

Name ()

*