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Mathematics

Also see:

"I finally got around to examining your web site and I was amazed. What a marvelous job you have done! ... Best regards and again, congratulations on your web site."
-- Frank Allen, Emeritus Professor of Mathematics at Elmhurst College, National Advisor for Mathematically Correct, and former president, National Council of Teachers of Mathematics (NCTM)

Understanding the Battles Over Math

  • Math Education: An Inconvenient Truth, produced by Where's the Math?, Washington State.
    Meteorologist M. J. McDermott presents vivid illustrations of the problems with fuzzy math, using specific examples from Everyday Math and TERC Investigations:

  • Math Education: A University View, produced by Where's the Math?, Washington State.
    Prof. Cliff Mass, Department of Atmospheric Sciences, University of Washington notes that students are clearly less capable in math than 10-15 years ago, leading to the need to "dumb down" college courses accordingly. Prof. Mass lays the blame squarely on the impact of fuzzy math in K-12.

  • Math with Madeline, produced by Where's the Math?, Washington State.
    MUST-SEE VIEWING! Fifth grader Madeline shows us vivid examples of the differences between the fuzzy math program that she used in her old school (TERC Investigations) and the mastery math programs (Saxon Math and Singapore Math) she now uses.

  • Kevin Killion's statement to the Chicago session of the National Math Advisory Panel, April 20, 2007. Excerpts:
         "Another weapon is to blame lousy math performance on intractable, dusty old methods. Schools are urged to 'embrace change' and teachers are exhorted to be 'agents of change.' The reality couldn't be any more starkly different. Everything has already changed. On our Illinois Loop website, we provide extensive information about how math is taught in Illinois school districts, from Addison to Zion. This resource is well-used by parents in tracking what districts are doing. Here's what we've found:
         "In Chicago, some 290 schools use progressivist, constructivist math programs in early grades. On the flip side, we have been able to identify only 5 -- count 'em -- 5 conventional CPS schools that use practice-and-mastery math programs, plus another 5 schools that are charters offering Saxon Math.
         "Now the suburbs. The Illinois Loop has collected info on the math programs used in 118 suburban K-8 districts in five collar counties. We find that progressivist, constructivist products are the math foundation in 77 percent of those districts.
         "But even that only hints at the severity of the problem. On the North Shore, or in Lake County, it's almost impossible to find any schools with anything but constructivist math. And across the area, we identified only 6 -- 6! -- districts -- out of 118 -- that make use of those math programs most recommended by practice-and-mastery reformers, such as Singapore Math or Saxon Math. So much for the argument that parents in the suburbs already have the schools they want.
         "Here's a twist: We've all heard of the 'dance of the lemons.' Well, there is also the 'Dance of the Math Lemons' performed by districts unhappy with their math programs. Example: District 39 in Wilmette dumps Math Trailblazers and picks up Everyday Math, even while District 109 in Deerfield drops Everyday Math to take a chance on Math Trailblazers. Like Lois Lane who couldn't see the truth staring her in the face, these districts stick with constructivist math and merely substitute one program for another. We're sure not seeing any 'agents for change' there! These districts are firmly mired down with a philosophy they refuse to abandon."

  • Here are two excellent short articles that provide a terrific introduction to understanding the issues of math instruction. They are written by Paul Clopton (one of the co-founders of Mathematically Correct), and are from the August 2001 issue of Parent Power, a publication of the Center for Education Reform:

  • It Works for Me: An Exploration of Traditional Math, Part 1
    It Works for Me: An Exploration of Traditional Math, Part 2
    It Works for Me: An Exploration of Traditional Math, Part 3
    by Barry Garelick, November 12, 2007. "Anyone who has been involved in the debates surrounding math education [has] come across the arguments that 'traditional math doesn't work' or 'the old way of teaching math was a mass failure'."
         But what do the facts say?
    These three juicy articles provide background on the math texts used in the 1940s through 1960s, leading to revealing conclusions, particularly in light of the fact that math scores on Iowa tests steadily increased from the 40's through the mid-60's, after which they began a decline. Highly recommended!

  • Here's another excellent article. This one does a good job giving specific examples on the differences between "fuzzy math" and "traditional" methods:
    U.S. Math Woes Add Up to Big Trouble by Ken Gorrell, Concord [NH] Monitor, April 8, 2007. Excerpts:
       "A bit of context is important. The reformers, representing the education establishment, believe learning 'process' is more important than memorizing core knowledge. They see self-discovery as more important than getting the right answer. For them it's the journey, not the destination.
       "Traditionalists, consisting mainly of parent groups and mathematicians, advocate teaching the traditional algorithms. They advocate clear, concrete standards based on actually solving math problems. The destination - getting the right answer - is important to traditionalists.
       "Two examples will help to make the difference clear. ..."

  • "We sometimes lower the bar because we want to make sure everyone gets over it."
    -- Chicago CPS Exec
    Illinois Students Not Up To Test Mark: ACT Scores Indicator Of College Readiness by Stephanie Banchero, Chicago Tribune, August 17, 2005. Excerpt: "If performance on the ACT college entrance exam is any indicator, this year's graduating Illinois high school seniors lack the academic skills necessary to pass basic college-level math, science and reading courses, according to data released Tuesday by the testing company. Of the 136,000 Illinois students who took the ACT, only 25 percent posted a science composite score high enough to indicate they are likely to succeed in a first-year college science course. Only 38 percent met the standard in math. About half did so in reading."

  • A quote from the above article, reminiscent of President Bush's complaint about schools that fail due to their "soft bigotry of low expectations":
    "...I think we are sometimes guilty of not teaching to the rigor of those courses. ... We sometimes lower the bar because we want to make sure everyone gets over it."
    -- Donald Pittman, Chicago Public Schools chief officer for high schools

  • Cognitive Child Abuse in Our Math Classrooms by C. Bradley Thompson. Excerpts:
        "The test results are in: America's children are flunking math. ... As educators scramble to explain America's math meltdown ... few are willing to look at the fundamental cause: the new, 'whole-math' method for teaching. ...
        "In a typical whole-math classroom, children do multiplication not by learning the abstract multiplication table, but by using piles of marshmallows. They count a million birdseeds in order to understand the concept 'million.' They measure angles by stretching rubber bands across pegged boards. One whole-math program preposterously claims to foster a 'conceptual understanding' of math by asking fifth-graders the following stumper: 'If math were a color, it would be ______ , because ______.' Surely such exercises foster in children only conceptual stultification -- along with a bewildered sense of frustration and disgust.
        "Another whole-math program asks sixth-graders to address the following problem: 'I've just checked out a library book that is 1,344 pages long! The book is due in three weeks. How many pages will I need to read a day to finish the book in time?' The proper way to solve the problem would be to use the method for long division: 1,344 divided by 21. By contrast, the whole-math approach assigns students to a group, requires them to design their own problem-solving rules, and urges them to guess if all else fails. In other words, children are told that their random 'strategies' are just as good as the logically proven principles of long division. They are taught that the vote of the group, rather than the reasoning of the individual mind, is the means of arriving at the truth.
        "Now imagine flying on a plane designed by aeronautical engineers who have been trained to concoct their own math schemes and to use a 'guess-and-check' method."

  • Math Wars: lead editorial in the Wall Street Journal, January 4, 2000.
    "Reinventing math is an old tradition in this country. It has been around at least since the 1960's, when the inimitable Tom Lehrer mocked the New Math in Berkeley cafes. Even Beatniks understood that a method that highlights concepts at the expense of plain old calculation would add up to trouble. ... Today the original New Math is old hat, but many folks in the education world are hawking yet another reform. It is known by names like "Connected Math," or "Everyday Math." ... Not that all members of the Academy are joining the movement. ... 200 mathematicians and scientists, including four Nobel Prize recipients and two winners of a prestigious math prize, the Fields Medal, published a letter in the Washington Post deploring the reforms. ... And well they might. For programs of [this] sort ... turn out to be horrifyingly short on basics."


    Used with permission

  • How About That: Back To Basics
    Editorial, Chicago Tribune
    , September 25, 2006. "In recent years, any parent who has sat at the kitchen table with a child completing math homework has watched the pages grow fuzzier and fuzzier. The child busily begins to attack a math problem using a 'new math' method. The parent looks on and sighs, wistful for the days when a student could arrive at an answer in a much more concrete and direct way. Such is the heart of the so-called math wars, the conflict between math traditionalists who stress the basics and those who push the looser "constructivist" approach."

  • A powerful speech from a parent in an upscale suburb:
    Elizabeth Gnall's statement to the National Math Advisory Panel\, September 6, 2007.
    Excerpts:
        "I live in the affluent public school district of Ridgewood, New Jersey. But my district has a dirty little secret. Ridgewood Public School district is segregated -- on one side of town, elementary school-aged children are taught math following a logical sequencing of topics, honoring the scholarly body of mathematics.
        "In another part of town the math is not taught but instead it is left for the children to discover and to construct. The math where for grades beyond Kindergarten the use of scissors, glue, paperclips, and any other object now defined as a manipulative, are deemed acceptable and encouraged. Sadly, this is the side of town where my children attend school. ...
        "Across this nation, parents just like me, will ultimately triumph in the math wars because it is OUR children, not the children of the state. ... Give us a choice in math education and we would choose a math education that is rigorous, focuses on content, is not driven by constructivist pedagogy, emphasizes the learning of mathematical facts, principles, and algorithms, uses the proper language and symbolic notation of math, and defines mathematical reasoning as the interconnections within mathematics. It is the kind of math that is being taught in other parts of this nation, the world, and in other parts of my town of Ridgewood, New Jersey. It is the math I believe that will provide a solid foundation for my children so if they desire, if they dream, to become a scientist, an architect, or like their dad, a Wall Street finance executive, or like their mom, an engineer, they can."

  • Ten Myths About Math Education, And Why You Shouldn't Believe Them by Karen Budd, Elizabeth Carson, Barry Garelick, David Klein, R. James Milgram, Ralph A. Raimi, Martha Schwartz, Sandra Stotsky, Vern Williams, and W. Stephen Wilson, in association with New York City HOLD and Mathematically Correct, two education advocacy organizations of parents, mathematicians, and K-12 educators, May 4, 2005.
    You can make good use of this chart to dispel these destructive claims:
    • Myth #1: Only what students discover for themselves is truly learned.
    • Myth #2: Children develop a deeper understanding of mathematics and a greater sense of ownership when they are expected to invent and use their own methods for performing the basic arithmetical operations, rather than study, understand and practice the standard algorithms.
    • Myth #3: There are two separate and distinct ways to teach mathematics. The NCTM backed approach deepens conceptual understanding through a problem solving approach. The other teaches only arithmetic skills through drill and kill. Children don't need to spend long hours practicing and reviewing basic arithmetical operations. It's the concept that's important.
    • Myth #4: The math programs based on NCTM standards are better for children with learning disabilities than other approaches.
    • Myth #5: Urban teachers like using math programs based on NCTM standards.
    • Myth #6: Calculator use has been shown to enhance cognitive gains in areas that include number sense, conceptual development, and visualization.
    • Myth #7: The reason other countries do better on international math tests like TIMSS and PISA is that those countries select test takers only from a group of the top performers.
    • Myth #8: Math concepts are best understood and mastered when presented "in context"
    • Myth #9: NCTM math reform reflects the programs and practices in higher performing nations.
    • Myth #10: Research shows NCTM programs are effective.

  • The Myths and Realities about "Fuzzy Math" by Sandra Stotsky, July 4, 2005. For almost two decades, mathematics education in K-12 classrooms has been driven by unsupported pedagogical theories constructed in our schools of education and propagated by the National Council of Teachers of Mathematics (NCTM). ... But many parents, mathematics experts, and K-12 teachers of mathematics do not share this vision. They reject the NCTM doctrine and model for mathematics reform. The views of this diverse constituency, comprised of mathematicians, scientists, engineers, K-12 teachers of mathematics, educational researchers, and concerned parents across our nation have been regularly marginalized by the dominant voice of mathematics educators in our schools of education and of NCTM officials. This constituency's expertise is often entirely absent from the decision-making process. As a member of that constituency and an advocate for authentic reform in mathematics education, I was part of a group that decided to prepare a point-by-point refutation of a set of common myths spread nationally and internationally by mathematics educators in our schools of education and NCTM officials. These myths are often presented as facts to policy makers and the general public. I offer this slightly revised chart for possible use by a curriculum committee in a school or district appointed to revise its K-12 mathematics curriculum or to decide on new mathematics textbooks, and by candidates for school boards or committees in local elections.

  • Traditional Math Means Never Having to Say You're Sorry by Barry Garelick, November 30, 2007. "The difference between traditional and present-day teaching is striking. The emphasis is now on big concepts. These come at the expense of learning and mastering the basics. Getting the right answer no longer matters. In theory, it is student-centered inquiry-based learning. In practice it has become teacher-centered omission of instruction. With the educational zeitgeist having been planted and taken root, the development of the NCTM standards in 1989 were an extension of a long progression. To top it all off, the reform approach to teaching math is being taught in education schools, thus providing future teachers with 'work-arounds' to those few math textbooks that actually have merit."

  • The Math Wars, Conceptual Thinking, and Traditional Algorithms: This is a more complete review of the issue, with well-chosen examples of the differences between "traditional" math instruction and modern fuzzy math. You'll also find useful discussion of the intentions and underlying philosophy of fuzzy math.

  • An observer (who writes under the pseudonym "John Dewey") comments on a video (Edspresso, October 10, 2006) shown to ed school students as an example of best practices in math: "Another video showed a teacher with his students standing around a table in the center of the room while he explained that day's assignment. ... This lesson was about parabolas, how the various constants in the vertex form of the equation for a parabola governed its shape, location and direction. He had them split into four groups, each group exploring what happens when you vary one particular constant. They were to use colored pipe cleaners to show the various parabolas on a poster. When through, the students all convened around the central table again and the teacher asked many questions which the students answered, some correctly, some not. There was no 'That's right, that's wrong', just more questions. The teachers in both videos were extremely good at what they were doing, which brought home an unsettling realization to me: You can be very good at doing something that is absolutely horrible."

  • A Brief History of American K-12 Mathematics Education in the 20th Century by David Klein, Mathematical Cognition, 2003. Here is a solid, well-researched look at how math education came to its current dismal state, after a century of unrealistic philosophizing and tinkerings with how math is taught.

  • Highly recommended!
    An A-Maze-ing Approach to Math (HTML) or as a (formatted PDF doc) by Barry Garelick, Education Next, Spring 2005. This is a definitive review on fuzzy math, written by Barry Garelick, an analyst with a federal agency in Washington. He digs solidly into fuzzy math's roots in the depths of constructivism, and the educrat politics (and money!) behind its encroachment on our schools. If you're involving in a math battle, get this article to help your cause. Here are excerpts from two articles about this Garelick article:
    • An Amazing Article, by Andrew Wolf, The New York Sun, March 7, 2005. Excerpt: "Mr. Garelick was tutoring a ninth-grader who was learning geometry. ... Mr. Garelick was horrified to learn that what he called a 'mainstay of mathematics' was largely missing from his student's geometry text. ... So Mr. Garelick began looking at other textbooks and found the same truncated instruction. Students were being given a Reader's Digest version of math."
    • Math, New Math, Fuzzy Math, Anti-Fuzzy Math, by Bill Leonard, March 7, 2005. Excerpt: "We collectively groan when new test scores are released demonstrating just how poorly U.S. students perform in math compared to their international counterparts. If you have ever asked yourself how this sorry situation came to pass, then I encourage you to read the article 'An A-Maze-ing Approach to Math,' by Barry Garelick."

  • The Havok Wrought by Modern Math by Dr. Charles Ormsby. "If you are a parent of elementary school children, you've probably seen it: elaborate make-work homework assignments, cutting and pasting extravaganzas, overly complex and roundabout procedures to add or multiply numbers, estimation exercises that won't quit, and the use of calculators in place of traditional arithmetic methods. You thought: 'Of course, the educational professionals must know what they are doing. Once my children catch on to these clever techniques, they will develop into mathematics geniuses!' Unfortunately, what you discover is that they never learn the core facts and methods, their confusion grows, they lose their self-confidence, they decide they just can't do math, and you are stuck paying for tutoring. Even worse, children who might have become exceptional mathematicians, engineers, or scientists are denied their rightful future."

  • 2+2=5: Fuzzy Math Invades Wisconsin Schools (PDF file) by Leah Vukmir. This article is highly recommended, both for those new to the issue and those who have been battling for years. It provides a thorough and passionate review of what the math controversy is all about. (Note: Although the title refers to Wisconsin, the material covered would be useful nationwide.) When written, the author was the organizer of an extremely effective group of parents and teachers woprking for academic reform. She went on to become a visiting fellow with the Wisconsin Policy Research Institute, and she is now a State Representative.

  • Math Fluency by Mary Damer. Mary writes about four stages of learning and how this view applies to learning and mastering mathematics.

  • Is This Math Program Proven? by Mary Damer. -- Mary gives tips on how to dissect the publisher's presentation and sales material, and to determine whether evidence presented on behalf of a math program has any validity or merit.

  • How To Respond When Your School Announces a New-New Math Program by Kevin Killion. What do you say? How do you respond when your school tells you that your child's math program is going to be replaced? What is your reaction when the replacement's main advantages are a "Tokyo by Night" layout, fuzzy-headed but politically correct examples, oddball algorithms and methods (or no methods at all), and a big emphasis on writing essays and playing games? Here are some suggested responses!

  • NCTM and "Problem Solving" by Charles E. Breiling, December 30, 2003. "One thing the NCTM (National Council of Teachers of Mathematics) is good for is revisionist history. The idea that we used to teach students the multiplication table, and then never used that information in solving math problems is preposterous. ... In the real world, if you teach basic skills (facility with number facts, fractions, decimals, percents, algebra, etc.) and have students use these facts to solve problems (without using calculators) then what you end up with is really good problem solvers! But in the NCTM universe, if you depend on a calculator for your number facts, you can just jump in with your "problem solving." Sounds reasonable enough, but what you end up with is a student who sees "A man has 5 trucks, and each truck holds 10 cases"--immediately adding 5+10 (on the calculator, natch) for the answer of 15 cases. Students who don't know beans about number facts won't be able to solve problems, no matter how much you focus on problem solving! This isn't a case of putting the cart before the horse, this is a case of removing the horse entirely."

  • How NOT To Teach Math by Matthew Clavel, City Journal (NYC), Winter 2003. In New York City, a Whole Language reading program and the very fuzzy "Everyday Math" (a.k.a. "Chicago math") program have been mandated for use system-wide by central administrators. Here is one NYC teacher's view on what fuzzy math has done to his classroom and school.

  • The Math Wars by David Ross, Ph.D. This is a very lucid and compelling essay that does a balanced and fair job presenting the arguments of both reformers and "traditionalists". Ross (a mathematician at Kodak Research Labs) then firmly concludes in favor of the traditional point of view. An especially novel aspect of this report is the author's dissection of just what is meant by "conceptual thinking." Using examples in math and from elsewhere, Ross concludes that the basics are absolutely necessary, so that students can eventually move (as he has titled the last section) "From Addition to Wonder".

  • Independent Analysis of Mathematics Textbooks (PDF) by Chris Patterson, January 1999. This is an extremely helpful review of the "math wars" controversy. This paper, prepared by the Texas Public Policy Foundation and Education Connection of Texas, includes background on the opposing viewpoints in math instruction, gives examples of each, and shows how these views on math affect math textbooks. It concludes with a clear and attractively formatted summary of detailed curriculum reports conducted by Mathematically Correct

  • Uncivil War: A Bloodless Account of a Bitter Battle, by Ralph Raimi, Education Next, Spring 2004. The failure of the education establishment to support proven, effective math instruction has far more to do with raw political battles than it has to do with scholarly research. This article looks at the progress of improved math instruction in California and the efforts of reformers and the resistance of education bureaucrats. Prof. Raimi also provides an extended version (PDF) of the same article, which adds more details than could be accommodated in the printed version.

  • How Did It Ever Come to This? by Ralph Raimi, notes for a talk to meeting of National Association of Scholars, New York, New York, May 22, 2004. Outlines the genesis of fuzzy math, and gives examples of how fuzzy math precepts handicap mathematical learning.

  • The Math Wars - 1960's Revisited by Erica Carle, October 13, 2003. Debates over the merit of teaching basic math facts reminded this author of a memo exchange over the same topic -- 40 years ago.

  • "Applications and Misapplications of Cognitive Psychology to Mathematics Education", by John R. Anderson, Lynne M. Reder and Herbert A. Simon, Department of Psychology, Carnegie Mellon University, Excerpt: "...Some of the central educational recommendations of these [educational] movements have questionable psychological foundations. We wish to compare these recommendations with current empirical knowledge about effective and ineffective ways to facilitate learning in mathematics and to reach some conclusions about what are the effective ways. A number of the claims that have been advanced as insights from cognitive psychology are at best highly controversial and at worst directly contradict known research findings. As a consequence, some of the prescriptions for educational reform based on these claims are bound to lead to inferior educational outcomes and to block alternative methods for improvement that are superior."

    We have 25 boxes of paper clips, 100 per box. Take away 50. How many are left? Here's the fuzzy math way to do it in third grade -- and get the wrong answer.
    (Click to enlarge)

  • What is the role of "learning the basics" in mathematical operations and fundamental algorithms? How do repeated practice and mastery of computation lead to "higher-order thinking" later? Here's a wonderful essay that gets to the heart of these questions!
    In Defense of "Mindless Rote" by Ethan Akin, professor of mathematics, City College of New York.

  • A Quarter Century Of U.S. 'Math Wars' And Political Partisanship (preprint) by David Klein, California State University, Journal of the British Society for the History of Mathematics, Volume 22, Issue 1, p. 22-33 (2007). Is fuzzy math a left-wing plot? Is traditional math a far-right obsession? In this interesting paper, the author laments, "Why did disagreements about school mathematics books in the US diverge according to left and right politics? ... In the course of the math wars, parents of school children and mathematicians who objected to the dearth of content were dismissed as right wing, but there is nothing inherently left wing about the NCTM aligned mathematics programs. ... Progressive math is a purely capitalist phenomenon. Indeed, one of the promotional themes of the NCTM was to prepare students for the needs of business. Ultimately, the injection of left and right ideologies into mathematics education controversies is counterproductive. The math wars are unlikely to end until programs espoused by progressives incorporate the intellectual content demanded by parents of school children and mathematicians."

  • New-age Math Doesn't Add Up by Bruce Ramsey, editorial columnist, Seattle Times, April 22, 2007.
        "It's called reform math, discovery math, constructivist math, fuzzy math. I think of it as new-age math, and believe it is one reason why last year nearly half the 10th-graders in Washington public schools failed the mathematics portion of the high-school graduation test. It is also one reason American kids do so poorly when measured against kids from Europe and East Asia. ...
        "New-age math ... came packaged with a garden basket of fragrant thoughts. ... It tends to introduce topics in a roundabout way that aims for a eureka moment. That is the 'discovery' part. It introduces many subjects early, focusing on concepts rather than calculation. That is the 'constructivist' part. It sometimes wants the student to estimate an answer rather than find the right one. That is the 'fuzzy' part. It demands written explanations of how an answer was arrived at, often in 'math journals.' ... New-age math uses games, colored blocks, dice, poker chips and other manipulatives. It requires working in groups. 'The idea is that if you let them struggle and come up with their own solutions, they'll learn it better,' ... None of these things is necessarily bad. ... But there are drawbacks. ...
        "[A teacher] at Ballard High, says, 'Supposedly, reform math is heavier in concepts but weaker in skills. But in my experience, kids are weaker in both.' He says the weakness is most noticeable in 'B' and 'C' students. ... And after high school? At community colleges, half the students take remedial math. At the University of Washington, [a professor] says, 'I saw a profound drop in math skills starting in the mid-'90s.' New-age math, he says, has created 'a whole generation of students who can't do fractions.'"

  • Calculating the Effects of "Discovery" Math by Bruce Ramsey, editorial columnist, Seattle Times, May 16, 2007.
        "A mother in Everett wrote, 'I discovered this past year that my eighth-grader is calculator-dependent ... The math skills she lacks stem from the fact that she never learned her basic math facts. She doesn't know how to do long division. She relies on her calculator for simple math problems.'
        "Says a Bellevue mom with a son, 15: 'Students are told to create their own algorithms to solve addition and subtraction problems, and these algorithms are frequently incomplete and unreliable. [Students] are presented with a little probability, introduced to matrices, presented with a smattering of this and that, but never achieve mastery of a topic. They are not taught long division. They are taught to use calculators to do the most simple problems.'
        "Writes a Redmond mom: 'Our fifth-grader has not been taught how to multiply double-digit numbers without a calculator, or what the heck to do with long division.'
        "A Shoreline dad helping his seventh-grade daughter had forgotten the rule for solving a math problem. He discovered that the rule wasn't in the book. The kids were supposed to figure it out themselves. Math, he grumbled, was being taught 'like philosophy, with no set rules and right answers.' ...
        "A Boeing engineer says he is 'constantly amazed by the gap in math skills between our junior American engineers and those educated in any other country,' especially those with the British system of education. He and his wife are teaching their two kids at home, using Singapore Math.
        "Another e-mail was from a Marysville-Pilchuck High School graduate who scored so low on a math placement test that it shocked him. 'I went back to some older textbooks my mother had,' he writes. He worked with them, retook the test, and placed himself three levels higher. He is now graduating from the University of Washington, though, he writes, with 'no thanks to my high-school math.'"

  • The Demise of Basic Surveying Mathematics by Richard L. Elgin, Ph.D., American Surveyor, May 2007. "The entering students' knowledge of the basic subjects of algebra, trigonometry and geometry has sunk to such a low level in the past five to ten years, that for all practical purposes I can say they have virtually none when it comes to being prepared to attack surveying. Discussing such pre- surveying topics as triangle solutions, orthogonal vectors calculation (i.e., latitudes and departures), even doing something simple as recognizing similar triangles when trying to derive horizontal curve equations just draws blank stares. What's the problem? I believe we can lay most of the blame at the feet of high school math curricula."

  • How should we approach math education for children with disabilities? This paper tackles this question head-on:
    Educational Aspects of Mathematics Disabilities by Susan Peterson Miller and Cecil D. Mercer, Journal of Learning Disabilities, January/February 1997. Excerpt:
    "Numerous educators have expressed concern regarding the application of the [NCTM] Standards to students with disabilities (Carnine, 1992; Hofmeister, 1993; Hutchinson, 1993; Mercer, Harris, & Miller, 1993; Rivera, 1993). Among the concerns are the lack of references to students with disabilities in the [NCTM] Standards document, lack of research related to the [NCTM] Standards, and overall vagueness of the document. These issues need to be addressed if we are to avoid another failed reform movement, with students paying the greatest price."

  • Guide to Mathematics and Mathematicians on The Simpsons: Here's an astonishing collection of every morsel of math that has even been on the show, compiled by Dr. Andrew Nestler and Dr. Sarah J. Greenwald. It includes some terrific quotations from the Simpsons, including plenty from the legendary "Girls Just Want to Have Sums" episode, as well as some screen captures. The website even includes some lovingly-prepared classroom worksheets on math topics mentioned on the show!

Practice Leads to Insight and Mastery

  • Discovery Learning in Math: Exercises Versus Problems by Barry Garelick, Nonpartisan Education Review, Vol.5, No.2, 2009. Ed school theorists promote the notion that traditional math courses provide merely mechanical and algorithmic approaches that do not lead to "off the script" thinking. This article takes a close look at how so-called "exercises" do indeed lead to discovery and to unscriptlike thinking. Excerpt:
         "Whether in driving, math, or any other undertaking that requires knowledge and skill, the more expertise one accumulates, the more one can depart from the script and successfully take on novel problems. It's essential that at each step, students have the tools, guidance, and opportunities to practice what they learn. It is also essential that problems be well posed. Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems. As a result, the challenge is in figuring out what they mean -- not in figuring out the math. Well-posed problems that push students to apply their knowledge to novel situations would do much more to develop their mathematical thinking."
         "Students given well-defined problems that draw upon prior knowledge, as described in this article, are doing much more than simply memorizing algorithmic procedures. They are developing the procedural fluency and understanding that are so essential to mathematics; and they are developing the habits of mind that will continue to serve them well in more advanced, college level mathematics courses. Poorly-posed problems with multiple 'right' answers turn mathematics into a frustrating guessing game. Similarly, problems for which students are expected to discover what they need to know in the process of solving it do little more than confuse. But well-posed problems that lead students in manageable steps not only provide them the confidence and ability to succeed in math, they also reveal the logical, hierarchical nature of this powerful and rewarding discipline."

  • "This study shows that conceptual insights emerge unconsciously during practice"

    Math Discoveries Catch Kids Unawares by B. Bower, Science News, January 2, 1999. Many of those who have careers that depend on math generally feel they learned it not by talking about "ways" of solving problems, but rather by doing math, with lots and lots of practice. This article from Science News reports on an interesting study that suggests exactly that!

    Three amazing conclusions from this study:

    1. kids deduced an important mathematical algorithm simply from doing computation problems,
    2. nearly all of the kids in the study eventually did this, and
    3. most of the kids were not able to explain in words what they were doing, thus suggesting that being forced to convert mathematical understanding into words was a skill quite distinct from mathematical competency.

    Here are some excerpts: "Many educators and scientists assume that conscious knowledge is the engine that drives learning. A new study suggests instead that, at least among grade-schoolers, unconscious problem-solving insights often set the stage for academic advances. Second-graders who practice solving inversion problems -- such as 8+10-10 = 8 -- start out by computing the answers but frequently turn to a more efficient strategy unconsciously. ... However, after becoming aware of the shortcut, kids employ it only part of the time, returning at other times to more time-consuming calculations. In the long run, the child's nurturing of an array of problem-solving tactics allows for adjustments in tougher math challenges. ... 'This study shows that conceptual insights emerge unconsciously during practice,' remarks psychologist David C. Geary of the University of Missouri in Columbia."

  • How Can Learning Facts Make Thinking More Enjoyable -- and More Effective?
    by Daniel T. Willingham, American Educator, American Federation of Teachers, Spring 2009.
         "Data from the last 30 years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that's true not simply because you need something to think about. The very processes that teachers care about most -- critical thinking processes like reasoning and problem solving -- are intimately intertwined with factual knowledge that is in long-term memory (not just in the environment). ...
         "Take two algebra students -- one is still a little shaky on the distributive property, whereas the other knows it cold. When the first student is trying to solve a problem and sees a(b + c), he's unsure whether that's the same as ab + c or b + ac or ab + ac. So he stops working on the problem, and substitutes small numbers into a(b + c) to be sure that he's got it right. The second student recognizes a(b + c), and doesn't need to stop and occupy space in working memory with this subcomponent of the problem. Clearly, the second student is more likely to successfully complete the problem."

  • "Most mathematics textbooks follow precisely the approach that our studies find so ineffective"
    Temporal Spacing and Learning by Hal Pashler, Doug Rohrer, and Nicholas J. Cepeda, The Observer, Association for Psychological Science (APS), March 2006. "Studies going back a century and more have found that spacing learning episodes across time sometimes enhances memory. The so-called spacing effect is the topic of hundreds of articles, and one might assume that we know all we need to know about it. However, the subtitle of an article on spacing effects that Frank Dempster published in American Psychologist in 1988 -- "A case study in the failure to apply the results of psychological research" -- remains appropriate now. Whether one looks at classrooms, instructional design texts, or language learning software, there is little sign that people are paying attention to temporal spacing of learning.
         "[O]ur team ... has been teaching students abstract mathematics skills ... In a study ..., students learned to solve a type of permutation problem, and then worked two sets of practice problems. One-week spacing separating the practice sets drastically improved final test performance (which involved problems not previously encountered). In fact, when the two practice sets were back-to-back, final performance was scarcely better than if the second study session was deleted altogether. This fits with other research from our team showing that benefits of over-learning decline sharply with time ...
         "Interestingly, most mathematics textbooks follow precisely the approach that our studies find so ineffective: a brief lesson on a topic is followed by a practice set containing virtually every problem in the book relating to this topic. Far more useful, we suspect, is to intersperse problems related to older topics covered over past weeks and months."

  • "The ideal textbook also has a tremendous number of practice problems because practice, practice and more practice is demanded from each student. Finding enough practice problems has always been difficult, thus I am currently developing my own texts and practice workbooks for class and homework use that are consistent with the lesson plans of the program."
    -- legendary math teacher Jaime Escalante

Fuzzy Math Impedes Learning

       "This year's graduating Illinois high school seniors lack the academic skills necessary to pass basic college-level math, science and reading courses ...
        "Of the 136,000 Illinois students who took the ACT ... only 38 percent met the standard in math."

    Chicago Tribune
    August 17, 2005

  • An Illusory Math Reform; Let's Go To The Videotape by Linda Seebach, Rocky Mountain News, August 7, 2004. Excerpt: "American children come off badly in international comparisons of mathematics performance, and they do worse the longer they're in school. One such comparison, the Third International Mathematics and Science Study, tested more than 500,000 children in 41 countries, starting in 1995. As part of the study, researchers videotaped more than 200 eighth-grade math lessons. These lessons have been studied intensively in an effort to figure out why Japanese students do so well in math while American students do so badly. [But] Alan Siegel, a professor of computer science at New York University, ... believes that many of the [published articles about the] TIMSS studies contain 'serious errors and misunderstandings.' [The videotapes reveal] teaching in the traditional mode, beautifully designed and superbly executed, but nothing like the parody of instruction that goes by the term 'discovery learning' in math-reform circles in the United States. The videotape shows, Siegel says, that 'a master teacher can present every step of a solution without divulging the answer, and can, by so doing, help students learn to think deeply. In such circumstances, the notion that students might have discovered the ideas on their own becomes an enticing mix of illusion intertwined with threads of truth.' Illusion prevails in far too many American classrooms."

  • Trends in Math Achievement: The Importance of Basic Skills, presentation by Tom Loveless, Senior Fellow, Governance Studies, Brookings Institution, February 6, 2003. Excerpts: "Take a closer look at the scores for nine year olds. These skills comprise the basic arithmetic that all fourth graders are expected to master -- addition, subtraction, multiplication, and division of whole numbers. All four areas reversed direction in the 1990s, turning solid gains that were made in the 1980s into losses. ...
    "A similar concern can be raised about the performance of thirteen and seventeen year olds. Their level of proficiency on computation skills remains unacceptably low. Look closely at fractions. Proficiency with fractions is critical in preparation for algebra. In 1999, only about half of thirteen and seventeen year olds could compute accurately with fractions on the NAEP. Students who leave eighth grade not knowing how to compute with fractions enter high school as remedial matmanipuh students. Students who leave high school lacking proficiency with fractions are inadequately prepared for college mathematics. On the most recent trend NAEP, both age groups were less proficient at computing with fractions than in 1982, twenty years ago."

    Lattice Multiplication

     
     "Lattice multiplication"
  • New-Math Multiplies by Linda Schrock Taylor. "Yes, New-Math is multiplying, but I am sorry to report that too many children are not learning to multiply with New-Math. ... Multiplication is not all that difficult if one learns the multiplication tables and the logical, precise algorithm for the process. One day I was teaching traditional multiplication when one of the special education students wanted to show me the process she had been taught. Her problem even shocked me, and luckily I had my camera with me. This illustration should help the unaware to understand why so many children in special education, as well as most other children, are coming to believe that math is an alien life form. It is no wonder that, when such foolishness is passed off as an intelligent math procedure, math scores are dropping like stones, while confusion is rising to new heights. It is no wonder that our students grow up: seeing themselves as stupid, hating math, and actually mathematically incompetent!"

  • Posting on an education discussion board:
    "Towards the end of 3rd grade ... I wrote a letter to my son's teacher/principal telling them that unless they could provide me with compelling evidence that using the lattice method of multiplication was helpful (yes, Everyday Math), I expected my son to do all of his multiplication problems using the standard algorithm so that he would get very good at one method, rather than not-so-good at four methods. Shortly thereafter, he came home with a math assignment. The class had been given a worksheet and told to use the lattice method to solve the problems, but Eric's teacher told him he was supposed to use the standard algorithm. I asked him what he thought about that. His response? 'Mom, it took the other kids three times as long to get their work done.' Duh."

  • What Is 5536 Divided By 82?: Compare the standard method versus the "method" used in Everyday Math:

Do "Manipulatives" Make Math Harder?


    Used with permission
  • Don't Be Manipulated, Center for Education Reform. "The use of "manipulatives" has become a buzzword in education circles. The concept refers to kindergarten and elementary students' use of concrete objects - anything from blocks or magnetic letters to complete systems specially designed for use in the classroom - for, literally, hands-on learning of math and language concepts.
         "However, a recent study brings into question their effectiveness - in this specific case, the use of a particular set of manipulatives usually did not transfer into faster or greater proficiency in the symbolic, written worlds of math and language.
         "The lesson here for parents? Just because something's "hot" doesn't mean it's helpful. Check out the science behind the curricular buzzwords. Get specifics on what classroom methods are being used, if they require special teacher training (and whether your teacher has it), and whether their effectiveness has ever been independently assessed. And finally, make sure your school keeps the focus on results, and not just process."

  • Mindful of Symbols: Educational Ramifications by Judy S. DeLoache, Scientific American, August 2005. ( Full article here.) Excerpt: "Teachers in preschool and elementary school classrooms around the world use 'manipulatives' -- blocks, rods and other objects designed to represent numerical quantity. The idea is that these concrete objects help children appreciate abstract mathematical principles. But if children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive. And some research does suggest that children often have problems understanding and using manipulatives.
        "Meredith Amaya of Northwestern University, Uttal and I are now testing the effect of experience with symbolic objects on young children's learning about letters and numbers. Using blocks designed to help teach math to young children, we taught six- and seven-year-olds to do subtraction problems that require borrowing (a form of problem that often gives young children difficulty). We taught a comparison group to do the same but using pencil and paper. Both groups learned to solve the problems equally well--but the group using the blocks took three times as long to do so. A girl who used the blocks offered us some advice after the study: 'Have you ever thought of teaching kids to do these with paper and pencil? It's a lot easier.'"

  • Learning From Symbolic Objects by David H. Uttal and Judy S. DeLoache, the Observer, Association for Psychological Science (APS), May 2006. Note of local interest: Uttal is an associate professor of psychology at Northwestern University.
         "In the classroom, teachers sometimes use more formal manipulative systems composed of concrete symbolic objects, such as Cuisenaire Rods or Digi-Blocks, that have been explicitly designed to help young children learn mathematics. ... Based on the writings of scholars such as Piaget, Bruner, and Montessori, educators have suggested that young children learn best through the use of highly concrete objects. However, our prior research on a variety of symbol systems (e.g., scale models, pictures, and maps) leads us to think twice about the value of having young children play with objects that are intended to be used as symbols. ... letter and number toys as representations may have just the opposite effect than what is intended: making children focus more on them as objects and less on what they stand for.
         "... we investigated the effectiveness of concrete symbolic objects, known as manipulatives, in helping young elementary-school children learn the procedures associated with two-digit subtraction. We taught children using either using the traditional written method or a commercially available manipulatives set, per the manufactures instructions. ... We found that children initially performed equally well in both training conditions. However, those children who learned with the manipulatives had trouble transferring knowledge to written versions of the math problems; they did not use what they had learned using the manipulatives to solve written versions of the same or similar problems. Moreover, learning with the manipulatives took almost three times as long as learning with the written method. This result does not mean that manipulatives are never useful, but it does challenge the typical assumption that they are more effective than other teaching tools in all contexts."

  • Posting on an education discussion board:
    "My daughter was taught to use the Cuisinaire blocks, lining up yellows against greens and all that. ... In later years, when she was an adult, I mentioned the Cuisinaire blocks and asked her if they had been of any help. She said she had no idea, at the time, that they were trying to teach her arithmetic. It was just another 'activity.'"

Spiraling ("The Death Spiral")

     
    Many of the new fuzzy math programs embrace a concept called "spiraling" in which the same material is presented multiple times. The intentions is that brighter students get a look ahead, while providing slower students an extra opportunity to catch up. But unlike reinforcement strategies, spiraling does not seek mastery first, so successive presentations give children numerous wonderful opportunities to experience total, humilating, discouraging failure.

    Here are some articles about the "death spiral" of fuzzy math education:

  • In this paper, Spiraling Through UCSMP Everyday Mathematics (March 2003), Bas Braams of New York University shows how so-called "spiral learning" is applied in one notorious fuzzy math program. Braams identifies this spiral practice as being one of the most devastating elements of the program: "The Everyday Mathematics philosophical statement quoted earlier describes the rapid spiraling as a way to avoid student anxiety, in effect because it does not matter if students don't understand things the first time around. It strikes me as a very strange philosophy, and seeing it in practice does not make it any more attractive or convincing."

  • Educational Aspects of Mathematics Disabilities by Susan Peterson Miller and Cecil D. Mercer, Journal of Learning Disabilities, January/February 1997. Excerpt:
    "At the elementary and middle school levels, basal mathematics programs are frequently used to guide instruction. Basal programs typically include a sequential set of student math books with accompanying student workbooks. Placement and achievement tests are frequently included to determine whether students have mastered the material in one book that then allows them to move to the next book. Teacher guides with suggestions for the teacher are provided in basal series. The typical basal curriculum uses a spiraling approach to instruction; in other words, numerous skills are rapidly introduced in a single graded book. The same skills are reintroduced in subsequent graded books at higher skill levels. Basal instruction using this spiraling curriculum approach is supposed to add depth to the math topics taught, but in reality the result seems to be superficial coverage of many different skills. Skill mastery is unlikely, because new skills are introduced too quickly in an attempt to "get through the book." The primary concerns regarding basal programs are the lack of adequate practice and review, inadequate sequencing of problems, and an absence of strategy teaching and step-by-step procedures for teaching problem solving (Wilson & Sindelar, 1991). Research has demonstrated that the basal approach to teaching mathematics is particularly detrimental to students who have learning difficulties (Engelmann, Carnine, & Steely,1991; Silbert & Carnine, 1990; Woodward, 1991)."

  • Excerpt from interview with Mike Feinberg, co-founder of the Knowledge Is Power Program (KIPP), on PBS special, "Making Schools Work":
         "If I put in front of you a 5th, 6th, 7th, and 8th grade textbook in math and opened up to page 200 and I jumbled them up, and said, 'order them from fifth through eighth grade in order,' you'd have a very tough time because they all look the same. That's because, unfortunately, we have this national strategy of 'we're not really going to teach to master, we're going to teach to exposure and over lots and lots of years of kids seeing page 200 in the math book, eventually somehow they're going to learn it. We're going to teach them how to reduce fractions in fifth grade, in sixth grade, in seventh grade, in eighth grade, in ninth grade and continue until finally somehow magically they're going to get it.' -- instead of thinking, 'let's teach the kids how to reduce fractions at a mastery level in fifth grade, maybe spend a little time reviewing it in sixth grade but let's move on to pre-algebra and let's move on to algebra then.'"

  • Things Don't Add Up In B.C. Math Classes by Bill Hook and Karin Litzcke, Vancouver Sun, Issues & Ideas, March 04, 2005. "British Columbia's elementary math curriculum is crippling learning, especially among disadvantaged students. B.C. has used what is called a "spiral' curriculum since 1987, following a tradition of emulating U.S. educational practice. A spiral curriculum runs a smorgasbord of math topics by students each year, the idea being that they pick up a little more of each with every pass. In reality, the spin leaves many students and teachers in the dust. ... Presently, teachers face having Grade 4 classes who still cannot add 567 + 942 nor multiply 7 x 8 because the Grade 1, 2, and 3 teachers were forced to spend so much time on graphing, polygons and circles, estimating quantity and size, geometrical transformations, 2D and 3D geometry and other material not required to make the next step, which is 732 x 34. And because elementary math fails to provide a solid foundation, many basically capable students simply give up when faced with the shock of high school algebra, which would be the doorway to advanced technical training at all levels. ... [T]eachers cannot make up Grades 1 to 7 while teaching Grade 8."

  • "... the most barbaric form of confusion for a young mind ..."
    A mother posted this online: "My younger children have already suffered through first, second and third grades - being taught how to take the [state] test - rather than learning basic math, spelling, etc ... The elementary schools in our district have *spiraling* math, the most barbaric form of confusion for a young mind! Spiraling math never touches on a math concept long enough to cement it into the child's brain ... The next concept is quickly implemented in some absurd attempt to relate it to the first concept ... which fails miserably and causes complete confusion in the childs mind. Try introducing addition and subtraction for a few weeks - then changing to fractions - going on to multiplication then sneaking in some pre-algebra ... all within a 2 month period in the third grade! We have had to fill in the blanks at home with older math text books! Most parents are not aware of this crazy, low budget math curriculum. They just believe their child is not brilliant. Teachers are aware of the damage the spiraling math causes and are apologetic - yet can do nothing about it.

  • Another mother posted: "I don't blame the teachers for this...They are the first to agree with me. They are given a curriculum at the beginning of each school year and God forbid if they criticize it! I had a math teacher tell me he spent his own money on books that would help fill in the gaps that the spiraling math creates - only to have the school district come to his room and yank them out...Why would they do this?"

  • An elementary teacher posted: "I'm on a committee to rewrite our math curriculum. ... The lower grade teachers dislike most of the [proposed curriculum] choices as they feel they 'cover too much' (k - 3) and the upper grade teachers (5 - 6) feel that the kids come to them unprepared. The fourth grade teachers are too stressed about state testing to get involved in this conversation. So, after weeding through the responses from our teachers, it seems that the K -3 teachers feel that they are 'rushing through' the text, due to the spiral learning concept and the upper grade teachers feel that they spend too much time reteaching the basics for the same reason. The 1st and 2nd grade teachers are frustrated by the fact that they are expected to introduce algebra ideas and the 5th and 6th grade teachers are frustrated as they are reteaching multiplication facts instead of their core content standards. Are we the only ones who face this issue? Are other schools as frustrated as we are?"

  • Another math teacher replied to the above comment: "For those who are quick to pick up math concepts it is great. My third grade daughter is doing the same fraction concepts that I teach to most of my 5th graders and is doing very well with it. But for those that struggle with math, the pacing [of spiraling] is horrible. If you fail to get basic step one, you are then unprepared for basic step two, and so on and so forth. So, it appears that the kids fall further and further behind and by the time they appear in the upper grades they are lost and convinced they cannot understand math at all. And, so now, I have two distinct math groups: those that were successful in the 'fast paced program' and who are moving at a fast clip, and those who are struggling to get to grade level in math and who I've spent half the year to get them to the point where they no longer tell me they are too stupid to do math."

  • Another teacher: "Our curriculum coordinator says that [spiraling] presents a concept and then revisits it later on in the curriculum. Well, it does do that but at a higher level so students that didn't get it the first time really don't understand when it rolls around again at a more difficult level."

Illinois Standards for Mathematics

  • Here's a huge warning notice about the Illinois math standards: In a state FAQ on its standards, the ISBE explains the relevance of so-called "national" standards this way:
    Standards have been prepared by learning area groups to serve as a resource for the nation. For example, the National Council of Teachers of Mathematics (NCTM) developed widely acclaimed math standards in 1989. National work exists in many learning areas and was used as one of many resources in developing the Illinois Learning Standards.
    Illinois' claim that the 1989 NCTM standards were "widely acclaimed", without even a hint that those standards also ignited a firestorm of controversy and rebuttal, casts serious doubt on the intellectual honesty of the ILS proponents. It also helps to explain why independent reviews of the Illinois standards for math have been so negative.

  • In a special report in the Spring 2008 issue of American Educator (American Federation of Teachers), the AFT scored Illinois' Math standards:
    • The Illinois Math standards DID NOT MEET criteria at the Elementary level
    • The Illinois Math standards DID NOT MEET criteria at the Middle school level
    • The Illinois Math standards DID NOT MEET criteria at the High school level

  • Illinois given a grade of "C" in math:
    See the Fordham Foundation's "State of State Math Standards 2005", which gave that grade of "C" to Illinois.

    CriteriaScoreGrade
    Clarity:1.50D
    Content:2.00C
    Reason:1.00D
    Negative Qualities:2.50B
    Final Score, 2005: C
    Final Score, 2000: D
    Final Score, 1998: D

    Here are some excerpts from Fordham's comments on the Illinois standards:

    • "The element added to Illinois' standards since our last review -- the 2002 Performance Descriptors -- does add some specificity to the generally poor Learning Standards and thus helps to improve Illinois' grade. Unfortunately, it also adds confusion to Illinois' standards: When are students supposed to learn what?"
    • "The standards, taken alone, are terse and frequently indefinite, as illustrated by the early elementary standard, 'Select and perform computational procedures to solve problems with whole numbers.'"
    • "In the lower grades, there are serious deficiencies in the treatment of arithmetic; for example, students are not expected to memorize the basic number facts."
    • "...there is no mention of the standard algorithms of arithmetic in either the Standards or the Performance Descriptors."

  • States' Math Standards Don't Measure Up, Study Says by Brian L. Carpenter, School Reform News, March 2005. Excerpts: "In a Thomas B. Fordham Foundation study published in January [2005], states earned an average grade of a 'high D' for their mathematics content standards. Chester Finn Jr, president of the Fordham Foundation, writes in the foreword to the report, 'the essential finding of this study is that the overwhelming majority of states today have sorely inadequate math standards.' ... The study found ... major problem areas with math content standards in most states, [including] ...
    • 'excessive emphasis' on calculator use,
    • failure to require students to memorize 'basic number facts',
    • absence of 'standard algorithms of arithmetic for addition, subtraction, multiplication, and division',
    • inadequate standards for student understanding of fractions by late elementary and early middle school years,
    • a nearly 'obsessive' focus on requiring students to identify 'patterns',
    • 'the use of a dizzying array of manipulatives (physical teaching aids) in counterproductive ways',
    • 'a tendency to overemphasize estimation at the expense of exact arithmetic calculations'..."

Math Anxiety

    Parents with a child with severe math anxiety should find these articles extremely interesting!

  • Math Fears Subtract From Memory, Learning by Bruce Bower, Science News, June 30, 2001, Vol. 159, No. 26. Excerpt:
    "Some scientists have theorized that kids having little math aptitude in the first place justifiably dread grappling with numbers. That conclusion doesn't add up ... according to a study [that concludes] people's intrusive worries about math temporarily disrupt mental processes needed for doing arithmetic and drag down math competence. ...
    "Math anxiety exerts this effect by making it difficult to hold new information in mind while simultaneously manipulating it ... Psychologists regard this capacity, known as working memory, as crucial for dealing with numbers. 'Math anxiety soaks up working-memory resources and makes it harder to learn mathematics, probably beginning in middle school' [a researcher says]."

  • Researchers: Math Anxiety Saps Working Memory Needed To Do Math, Reuters, February 20, 2007. "Worrying about how you'll perform on a math test may actually contribute to a lower test score, U.S. researchers said on Saturday. Math anxiety -- feelings of dread and fear and avoiding math -- can sap the brain's limited amount of working capacity, a resource needed to compute difficult math problems, said Mark Ashcroft, a psychologist at the University of Nevada Las Vegas who studies the problem. 'It turns out that math anxiety occupies a person's working memory,' said Ashcroft ... Worrying about math takes up a large chunk of a person's working memory stores ... spelling disaster for the anxious student who is taking a high-stakes test."

NYC HOLD

Mathematically Correct

    An extensive source of information about the crisis in mathematics education is the website of Mathematically Correct. If you have a child in a school that uses a fuzzy math program, you owe it to your child to read up on this national controversy. Your first stop should be the spectacular website of Mathematically Correct. THAT is THE place to read about "fuzzy math," "new-new math," or "rainforest math." Find out what's in your child's math program at school, and what's behind it. This website is just loaded with information and links on the strategies used by educational theorists, and specific comments on a wide variety of specific math curricula. You can also learn how parents all over the country are organizing to fight dumbed-down math.

    Here are some items you will find at that website:

Calculators

     
    Used with permission
  • Calculating the Cost of Calculators by Lance T. Izumi, Capital Ideas, December 21, 2000. "A September 2000 Brookings Institution study found that calculator use decreases student math achievement. Analyzing national test data, Brookings concluded that students who used calculators every day scored lower than students who used the devices less frequently. Given this finding, it is disturbing that Brookings also found that while only 27 percent of white students used calculators daily, half of African-American students made daily use of calculators. Yet, despite such evidence, university schools of education, which place so much emphasis on the learning 'process,' actively promote the use of calculators, devices that eliminate the process of learning math. For instance, a book on math teaching methods required at San Francisco State University tells future teachers that 'there's no place for requiring students to practice tedious calculations that are more efficiently and accurately done by using calculators.' ... The evidence shows that calculator use and other trendy teaching methods harm rather than help students. These failed methods end up cheating children out of the basic knowledge and skills they will need to succeed. That is both a fraud and a tragedy."

  • TI Celebrates 40th Anniversary of Calculator by Andrew D. Smith, Dallas Morning News, September 30, 2007. Excerpt: "... they distressed math lovers, who said calculators diminished understanding. Math teachers, for example, spent years resisting calculators. Older engineers took a similarly skeptical view. [TI engineer Jerry] Merryman, who kept using his old slide rule long after he invented the calculator, says that engineers who learned their trade post-calculator have no 'feel for numbers.' TI eventually found itself in an odd place. It was a math and science company that made a fortune selling students a product that some said hurt their understanding of math and science."

  • "... lower grades ..."
    K-12 Calculator Usage and College Grades (PDF) by W. Stephen Wilson and Daniel Q. Naiman, Educational Studies in Mathematics 56: 119-122, 2004. This study concludes that students in the math courses at Johns Hopkins University who had been encouraged to use calculators in K-12 had lower grades than those who weren't.

  • Just Say No To Calculators by David Klein, professor of mathematics at California State University, American Teacher, American Federation of Teachers (AFT), March 2001. "One of the most debilitating fads to sweep American public schools in the last decade has been the heavy use of calculators, especially in elementary schools. According to the Third International Mathematics and Science Study, or TIMSS, use of calculators in U.S. fourth-grade mathematics classes is about twice the international average. In six of the seven top-scoring nations in the study, teachers of 85 percent or more of the students report that students never use calculators in class. ... Even universities are forced to run remedial math classes at unprecedented levels, including classes in arithmetic for entering freshmen. This intolerable state of affairs can be laid at the doorstep of the uncritical use of calculators in elementary schools."

  • "... doomed to solving only trivial mathematical problems ..."
    Computation Skills, Calculators, and Achievement Gaps: An Analysis of NAEP Items by Tom Loveless, Director, Brown Center on Education Policy, Senior Fellow, Governance Studies, April 15, 2004. This valuable paper examines national trends in computation skills, investigates whether allowing calculators on NAEP items produces significantly different results compared to not allowing calculators, and analyzes the impact of allowing calculators on the performance gaps among black, white, and Hispanic students. It concludes: "If students are only able to compute accurately with calculators -- or if their computational skills are so weak that only the simplest of calculations can be made -- then students are doomed to solving only trivial mathematical problems."

  • Paper-and-Pencil Math by Richard H. Escobales Jr., Canisius College, Notices of the AMS, American Mathematical Society, August 2004 -- a mathematician takes on those who give lip service to the learning of algorithms while in reality favoring calculator use over the steps necessary to actually attain competency. "Mastery of addition and the other algorithms of basic arithmetic act as a flashlight, allowing the young student to move freely about in the world of numbers and basic numeric operations. Without such mastery a young student is condemned to move about blindly in this intriguing unknown world of numbers."

  • Calculator Dependence by William Kohl, June 7, 2006. The author begins by describing a conversation he has with a student he is tutoring. An algebra problem asks, in the equation y=x2+5, if the constant 5 is changed to 1, how does the plotted curve change? "I said, 'What would you do to find the answer?' He said, 'I have to get my calculator.' I said, 'Why?' He said, 'I need it to work the problem.' I said, 'Couldn't we just think about the problem first?'" The author concludes, "dependence on the calculator has short circuited the learning of math and the development of analytical skills ... younger people are not comfortable with numbers."

  • "The calculator subtly undermines the whole math curriculum"
    Kick Calculators out of Class by David Gelernter, professor of computer science at Yale University, New York Post, May 21, 1998. "They should be banned from American elementary schools. ... The calculator subtly undermines the whole math curriculum."

  • Trash the Calculator, It's Back to Basics in Britain by Alexander MacLeod, Christian Science Monitor, January 26, 1998

  • Of Culture, Calculators, Math Anxiety And How We Learn, UniSci News, May 14, 2001: "'Better performance on the complex arithmetic [problems in a study] was associated with lower reported calculator use in elementary and secondary school,' says [Jamie] Campbell [of the University of Saskatchewan]. Complex arithmetic places special demands on short-term memory skills that simple arithmetic usually does not, because complex arithmetic involves operations such as carrying, borrowing and place-keeping. 'This is demanding mental juggling for most people's short-term working memory processes,' says Campbell. 'Using a calculator might restrict the level of expertise achieved with respect to short-term memory skills for complex arithmetic.'"

  • Calculators Are Like Bicycles by Charles E. Breiling, April 21, 2004. "We could go on for days about what we really think of calculators in K-12 math classes (we think they're great for science classes, especially stochiometry in Chemistry), but suffice to say it just might be one of the signs of the Apocalypse. Personally, we think calculators ... are great, and they sure do save a lot of time. But that's not the point behind teaching children arithmetic in school. The point of doing paper-and-pencil arithmetic (for example, finding the product of 47 and 9) isn't to actually find the product (like we didn't know what the answer was, and that's why we have kids do these problems). The whole purpose of this exercise is to practice a skill accurately to the point of automaticity."

  • A discussion about calculators on Joanne Jacobs' website provided some fascinating comments. Here are some excerpts:

    • "I've taught calculus for a couple summers ... I was frustrated beyond belief that students couldn't give a rough sketch of a function without those darn graphing calculators... I actually forbade their use in my classroom and on tests ... but then I was further shocked when I heard the A.P. Calculus exam requires these darn graphing calculators ... when did they do that? WHY did they do that???"
    • "The reason the students are fooling around with calculators instead of learning calculus, of course, is that the students *can't* learn calculus because they never mastered algebra."
    • "my daughter was utterly baffled by some of her Algebra II homework before I shoved the (required) graphing calculator aside and showed her how to do the work the old-fashioned way. She's since shown some other students in her class who were similarly stumped."
    • "By analogy, carpentry is not about operating a nail gun or a power saw. Carpentry is knowing how to cut the wood to fit and where to put the nails. A skilled carpenter can produce more and better work with power tools, but without the underlying skill, he'll only make more sawdust and noise."

  • "speed up the process of not learning anything"
    Calculators in High School Linear Algebra, by Prof. Ralph Raimi. The author describes this article as telling "how recent NCTM-approved advances in mechanized pedagogy can speed up the process of not learning anything."

  • Calculated Controversy: Do High-Tech Calculators Take the Challenge Out of Learning Math? Excerpt: "Among the leading opponents is Hung-Hsi Wu, a math professor at the University of California, Berkeley, who worries that calculators may deprive students of a critical tool. 'I want them to learn how to think,' he says. 'That's the greatest virtue in having a good mathematics education. ... At some point, there's no point in using a calculator,' he reasons, 'because if you can't think anyway, then what's the point of using a calculator?'"

  • A Horizon without Calculators, August 31, 1996. Excerpt: "There has been much concern about student use of calculators. Many parents feel that their children exercise their fingers too much and their brains too little by the over-use of calculators. Examples include algebra students using calculators to solve 300/3 or 63/9. These should be immediately solved by mental math, virtually without stopping to think about it. When students reach for calculators to do simple problems something is wrong. Such behavior is learned by having calculators present at all times, a policy that is common in many 'reform' math classes."

  • Ditch the Calculators: Letting Children Punch Numbers Into a Machine Does Not Add Up To Learning Math by Diane Hunsaker, Newsweek, November 3, 1997. "Some teachers argue that calculators let students concentrate on how to solve problems instead of getting bogged down with tedious computations. ... Some of my elementary-school children look at a word problem and instantly guess that adding is the correct approach. When I suggest that they solve the problem this way without a calculator, they usually pause and think before continuing. A student is much more likely to minimize his work by reflecting on the problem first if he doesn't have a calculator in his hand. ... A student who learns to manipulate numbers mentally can focus on how to attack a problem and then complete the actual computations easily. He will also have a much better idea of what the answer should be, since experience has taught him 'number sense,' or the relationship between numbers. A student who has grown up with a calculator will struggle with both strategies and computations."

  • The Great Calculator Debate: Concerns About Calculator Use in Elementary Schools by Thomas W. Cowdery


  • Used with permission
    Calculators in Class: Freedom From Scratch Paper Or 'Crutch'? (also here) by Mark Clayton, Christian Science Monitor, May 23, 2000. Excerpts:
    "Critics ... say calculators are overused in US middle and high schools. They warn that a wave of 'new new math' programs that employ calculators much more than traditional approaches are entering grade schools, threatening basic math skills. David Klein, a math professor at California State University in Northridge, says calculators should 'not be used at all in grades K-5, and only sparingly in higher grades.' That's not where America's schools are headed, however. ...
    "And as calculators become cheaper and more powerful, even some college students question their impact on learning. 'I feel as though three years of math at high school were lost,' says Amir Emami, a freshman at Kalamazoo College. Even though he graduated with a 3.4 grade point average, he says he has a weak understanding of math. 'The answers [in high school] were written paragraphs, not equations or number crunching. You learn to depend on our TI-82 calculator.'
    "Of course, math performance is tied to many factors, Klein notes, but the highest-performing countries on international math tests used calculators less. At the eighth-grade level, students from three of the top five performing nations in math (Japan, Korea, Belgium) rarely or never used calculators. But in the US and 10 of 11 nations with scores below the international average, many used calculators every day.
    "Down the highway from Okemos is the city of Portland, where Portland Middle School is a pilot for [new-new math program] Connected Math. A sixth-grader recalls that at his elementary school kids used calculators in every grade. 'I can't even do division without a calculator,' he says. 'Last year we did this weird type of division, I think they called it 'long' ... I didn't really get it. The teacher told us 'don't worry, you're doing the work. The calculator is just showing you the answer.''"

  • "a dramatic loss of insight"
    Dangers of the Paradigm Shift by Jose Miro-Julia: This very interesting paper on math education considers the positive role of calculators, but worries about a profound "loss of knowledge" from their misuse. Among the author's concerns are students thinking of all operations as being of equivalent utility (repeated addition on a calculator requiires little more effort than multiplication), a dramatic loss of insight into the subtleties of a mathematical problem, and the blind faith in the "sorcery" of the calculator. The author surmises that calculators inhibit rather than encourage critical thinking.

  • A group of mathematicians at Northwestern University offers this lovely item for sale:

  • Mathematicians agree 93-to-0:
    Basic math, not calculators

  • Math professors 93-to-0 in supporting basic math: Stephen Wilson, professor of mathematics at Johns Hopkins University, asked a number of mathematicians their thoughts on the following statement:

    Statement:

    "In order to succeed at freshmen mathematics at my college/university, it is important to have knowledge of and facility with basic arithmetic algorithms, e.g. multiplication, division, fractions, decimals, and algebra, (without having to rely on a calculator)."

    The response was overwhelming and unanimous: 93 mathematicians agreed with the statement, and NONE disagreed! Prof. Wilson comments on these results, "This is particularly remarkable because if you ask this same bunch what mathematics is, then no two of them will agree. There are at least a couple of dozen who would normally disagree just to be disagreeable."

    Click the link to see their affiliations and many of their extended comments, but here are some of the highlights:

    • Professor, Wayne State University: "That it is even slightly in doubt is strong evidence of very distorted curriculum decisions. I do not know even one university-level teacher of mathematics who would disagree with it. I would be truly astonished to meet a person who disagrees."
    • Professor, University of Calgary: "Last winter at this time I was visiting Chennai (Madras), and one of my talks was at a Jr H S (called a Secondary School there). I was amazed to see the facility these young students had with arithmetic - of course they never use calculators at all in school."
    • Math Ph.D., Union College: "It is essential to have knowledge of and facility with basic arithmetic algorithms, e.g. multiplication, division, fractions, decimals, and algebra, (without having to rely on a calculator). Students without this ability typically do not make it successfully through their introductory calculus courses, and are often forced to repeat courses or to drop out of engineering/science programs."
    • Professor, University of Illinois: "Here is a new twist. My daughter got a terrible score on her SAT II or whatever, because her calculator's battery was dead, so she borrowed her brother's, but it was set to give exact answers, and she didn't know how to convert them to decimals, and she was too stressed to figure it out, so she couldn't do multiple-choice answers that required knowing whether pi or e (say) was closer to 3."
    • Professor, Northern Illinois University: "No doubt you are aware that the US educational system is releasing increasing numbers of students who fail to meet even the minimal standards imposed by state boards of review. ... It's a bad situation and I'm glad you're standing up for a minimal competence."
    • Associate Dean and Professor of math, Queen's University [Ontario]: "It makes an enormous difference to have students who can add fractions, for example. This last skill is a key indicator of success - if you cannot add fractions, you do not belong in university - we do not have the time nor the resources to teach you that which you should already know on being accepted. If you have graduated from high school without such skills, your high school has cheated you. In my view, the teaching of basic arithmetic skills is not an option for schools, but rather an important part of their mandate. I'd be very unhappy to send my children to a school that thought otherwise."
    • Assistant professor math, Penn State: "Having taught 20 years in public school mathematics and now 16 years at Penn State Altoona in the Mathematics Dept. I heartily agree that public school students must learn the basic arithmetic algorithms to be successful in college mathematics courses. Calculators are a good thing and are being used extensively in my engineering math classes, but successful students know the basics without a calculator."
    • Professor, Fordham University: "I more than agree with your statement about the need for children to learn arithmetic; and the necessity of being able to do simple arithmetic without a calculator. ... I see many students who, when confronted with an expression like (64)^(-2/3) will hit their calculators to find out the value; but, because they have been raised with the calculator, have no idea what the expression means; how it has been defined; what are the algebraic properties of exponents."
    • Ph.D., University of Goettingen [Germany]: "What a question: the answer is of course 'yes, obviously'!!!!"
    • Professor, Univesite Louis Pasteur [Strasbourg, France]: "I completely agree with your statement ... It is sad that such things which ought to be completely obvious re controversial!"

    (Click here for the full list of math professionals and their comments.):

    Equity and Calculators

  • Calculators May Be the Wrong Answer As a 'Digital Divide' Widens in Schools by Daniel Golden, Wall Street Journal, Friday, December 15, 2000. "Teachers like Mr. Martin favor calculators as motivational tools. ... But more calculator use in inner-city schools generally hasn't added up to higher test scores. The majority of experts on elementary-school learning maintain that, for students who lack basic number proficiency, calculators may provide only the illusion of progress. 'Kids get to use calculators as a substitute for practice, and they never really understand arithmetic,' says Sandra Stotsky, deputy education commissioner in Massachusetts, a state that has taken a back-to-basics approach."

  • Do Other Schools Have an Unfair Advantage on SATs and AP Exams? by Sarah Brown, Co-Editor-in-Chief, Cougar Crier, John F. Kennedy High School, Bellmore, New York, December 2002. This is a remarkable article, not just for the depth of its research and quality of its writing, but also for the fact that it was written by a high school student for her school newspaper. Excerpts: "Most Kennedy students are unaware that upper-level math classes in Port Washington, Jericho, Great Neck, and other high schools across Long Island and the country are using the Texas Instruments-89 calculator, a far more advanced calculator than that which is used at Kennedy. ... Many students, parents, and teachers complain about schools' usage of the TI-89, claiming that the calculator inhibits learning. 'The kids don't have to know as much,' admitted [Great Neck North High School's Mathematics Chairperson Helen] Kramer."

  • Pricey Calculators Give Some Edge In Math Tests by Nedra Rhone, Newsday, November 23, 2003. "Staples office supply store was a madhouse during the late summer back-to-school shopping crush. ... Mothers were screaming, 'I didn't know the calculator was $90!' The calculator, a graphing handheld listed among the necessary class materials for Deer Park High School and others on Long Island, earned a place in the great Regents Math A debate. While the issues of content and grading dominated educators' discussions after an estimated one-third of students in the state failed the exam last June, the use of the pricey calculator that sent some parents into a frenzy raised questions of equity and stirred age-old debates about the use of technology in math instruction. Students who are able to afford graphing calculators, or who live in districts that provide the tools, have a distinct advantage over other students if permitted to use the calculator on a Regents exam, reported an independent panel of educators last month."

  • Math Teachers Debate The Use Of High-Powered Calculators by Lisa Black, Chicago Tribune, November 3, 2003. "A hand-held calculator that can solve brain-numbing algebra equations within seconds has high school math teachers divided over whether it will make algebra more accessible or rob students of basic skills."

  • Knock-Knock: Its the C.E.O., New York Times, April 12, 2009, interview with Terry J. Lundgren, chief executive of Macy's. Excerpt:
    Q. Anything you would like business schools to teach more? Less?
    A. In our business, there's not enough emphasis on math. Coming out of college, we really like to have kids who like math, study math and get it. And so I'd like to make sure that there is an emphasis on math. I think there is a strong emphasis on marketing already, and we want that and we need that. But to me, the math piece is weak in most business school educations, and I'd like to have more emphasis on that.
    Q. But somebody might say, "That's what calculators are for."
    A. And that's exactly the problem. ... I think there's logic that has to go into this. And I don't think you should actually have to have a calculator for every decision that you make that has numbers attached to it. Some of that should just come to you quickly, and you should be able to quickly move to your instincts about that being a good or not good decision."

    Comments on Calculators

Algebra

    "Guess-and-Check" Is Not Algebra!

  • Parents love it when they see their kids assigned "pre-algebra" problems like this: Two numbers add up to 15, and the difference between them is 3. What are they? This is a simple problem -- with algebra. But what a mess it turns out to be when educrats use this problem to introduce "Guess and Check" to grade schoolers as a "problem solving strategy".

  • Pseudo-Education Marches On by Domenico Rosa, December 4, 2004. "Guess-and-check appears to be the 'strategy' of choice that is being taught for solving simple problems. ... These 'strategies' are becoming more and more widespread under the guise of 'solving algebra problems' and 'algebra for all.' In my opinion, far from teaching any meaningful concepts, these mechanical calculations are doing little more than enhancing the pseudo-education of American students. This type of pseudo-education is being promoted -- at conferences, workshops, minicourses, and training sessions -- by assorted 'experts' who promise to boost scores on assorted 'mastery tests' and other standardized tests. These promotions are being adopted mindlessly by administrators and teachers, whose bonuses and other financial rewards are based on the results of these tests. As long as this rampant pseudo-education continues to be promoted, the situation in the U.S. will only worsen."

    Basic Math is a Prerequisite for Algebra

  • California's Algebra Crisis by Paul Clopton and Bill Evers, October 6, 2003. "The strongest predictor of failure to learn algebra is not race or income; it is a lack of adequate academic preparation. The problem begins before students get to their first algebra class. Many school districts have watered down the content of pre-algebra courses, removing important but difficult material. The districts want more students to pass math classes, and they want to guarantee high pass rates by making the classes easy. But classes without content set students up for later failure in algebra. ... Admirably, California embraces learning algebra by the end of eighth grade as a long-term goal. But strengthening academics from kindergarten on is necessary before this goal can fully be met. Algebra placement rates ought to depend on student readiness."

  • The Dumbing Down of Algebra by C.F. Navarro, Ph.D. Excerpts:
         "At the George Washington Middle school where I taught eight-grade math in 1998, only a few of my math students were at grade level. The rest were at a fourth-grade level, or lower. ... Most had not yet learned their multiplication tables and were still counting with their fingers. By the end of the year some had progressed to about a fifth-grade level, a substantial improvement, but far short of the comprehension and skills required for algebra. Nonetheless, all were required to register for algebra the following year.
         "More troublesome still was my algebra class. ... with few exceptions, they didn't know how to work with fractions, decimals or integers. They lacked the power of concentration to set up and solve multiple-step problems. They were incapable of manipulating symbols and reasoning in abstract terms. Like most of my general math students, some had not yet learned their multiplication tables and were still counting with their fingers. All had been issued graphing calculators (a terrible mistake) and led to believe that algebra consisted simply of pushing buttons and getting the right answers.
         "Given another year or two to mature and learn their basic math, most would have mastered algebra and gone on to higher mathematics without much trouble. But as it turned out, all they got from their premature exposure to algebra was a lot of stress. Some, I suspect, will hate math as long as they live.
         "The education establishment, however, is not wont to give up a bad idea. If it cannot bring the kids up to algebra, then it will bring algebra down to the kids. ... But no matter how much the subject is fragmented and, in the process, dumbed down; no mattter how many how-to-teach-algebra workshops high school teachers are forced to take, students unprepared for the subject are not going to learn it. ... The early algebra and algebra-for-all program in our public schools looks great on paper. It gives the impression that our local kids have finally caught up with their counterparts in Japan and Norway. But in truth, they are just as far behind as ever."

  • What Does It Take To Learn Algebra? First You Have To Master The Fundamentals by Karin Klein, Los Angeles Times, February 4, 2006. "Things looked pretty hopeless to both of us those first couple of sessions, as Johnny stumbled through algebra problems while I tried to figure out exactly what he didn't understand. Then, as we took it down to each step of each little calculation, the trouble became clear: Johnny somehow had reached ninth grade without learning the multiplication tables. Because he was shaky on those, his long multiplication was error prone and his long division a mess. As Johnny tried to work algebraic equations, his arithmetic kept bringing up weird results. He'd figure he was on the wrong track and make up an answer. This discovery should have made us feel worse. How could we possibly make up for a dearth of third-grade skills and cover algebra too? But at least we knew where to start. ... in all these decades, the same school structure that failed Johnny goes on, dragging kids through the grades even if they don't master the material from the year before. This especially makes no sense for math, which is almost entirely sequential."

  • Things Don't Add Up In B.C. Math Classes by Bill Hook and Karin Litzcke, Vancouver Sun, Issues & Ideas, March 04, 2005. "... because elementary math fails to provide a solid foundation, many basically capable students simply give up when faced with the shock of high school algebra, which would be the doorway to advanced technical training at all levels. ... [T]eachers cannot make up Grades 1 to 7 while teaching Grade 8."

    Is Algebra Needed?

  • Our own Dave Ziffer defends the teaching of algebra, in this open letter to Education Week magazine. Excerpt: "Mr. Bracey contends that algebra is an essentially useless skill, malevolently imposed upon our students for the purpose of sorting out which children will attend college. But wait a minute, don't all of us in the scientific and mathematical communities emphatically claim that algebra is the very foundation of the work that we do every single day? Not so, according to Mr. Bracey. Apparently he thinks that people can build bridges, produce electricity, design cars, and fly airplanes without any use of algebra at all."

  • Algebra Benefits All Students, Study Finds by Kathleen Kennedy Manzo, Education Week, November 15, 2000.

  • Algebra and Its Enemies by Kenneth Silber, May 8, 2006

    Learning Algebra

  • The Effects of Cumulative Practice on Mathematics Problem Solving by Kristin H. Mayfield And Philip N. Chase, Journal of Applied Behavior Analysis, Summer 2002. "According to the latest international study, the average score of U.S. students was below the international average, and the top 10% of U.S. students performed at the level of the average student in Singapore, the world leader. In addition, recent tests administered by the U.S. National Assessment of Educational Progress revealed that 70% of fourth graders could not do arithmetic with whole numbers and solve problems that required one manipulation. Moreover, 79% of eighth graders and 40% of 12th graders could not compute with decimals, fractions, and percentages, could not recognize geometric figures, and could not solve simple equations; and 93% of 12th graders failed to perform basic algebra manipulations and solve problems that required multiple manipulations. These statistics reveal students' deficits in the fundamental skills of mathematics as well as mathematical reasoning and problem solving."

Quotes on Math Education

    From our extensive page on education quotations:

    "I recommend you to question all your beliefs, except that two and two make four."
    -- Voltaire (L'homme aux quarante écus)

    "Strange as it sounds, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations."
    -- Ernst Mach, physicist

    "A mastered algorithm in the hands of a student is an incomparable tool laying bare the conceptual structure of the mathematical problems that the algorithm solves. With such tools, and with the guidance of good teachers in their use, a student can grasp and integrate in twelve years a body of mathematics that it has taken hundreds of geniuses thousands of years to devise."
    -- David Ross, Ph.D., a mathematician at Kodak Research Labs

    "Piaget's constructivism and Bourbaki's austere rigor have left their marks on our schools. Will such trenchant educational theories ever give way to more serene and better optimized teaching methods, based on a genuine understanding of how the human mind does mathematics?"
    -- Stanislas Dehaene, "The Number Sense: How The Mind Creates Mathematics"

    "Discovery lessons, students writing to learn mathematics, the teaching of so-called general problem-solving concepts, field trips, math lab lessons, alternative assessments, collaborative partner tests, student presentations, and open-ended problems should all be used sparingly. I use some of them, but they have limited value. Pencil-and -paper analytic solutions are the heart of mathematics education."
    -- Michael Stueben, Twenty Years Before the Blackboard

    "These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward."
    -- Albert Einstein
    Dr. Einstein would have done lousy on "performance assessment" tests with that attitude! -- Editor

    "It's been my experience that students in secondary math education ... are generally among the worst students in my class. The background in math of prospective elementary school teachers is even worse, in many cases nonexistent."
    -- John Allen Paulos, Innumeracy

    "The mathematics background that elementary school teachers typically receive is atrocious -- little or none"
    -- Paul Sally Jr., professor in mathematics, University of Chicago

    "Nothing flies more in the face of the last 20 years of research than the assertion that practice is bad. All evidence, from the laboratory and from extensive case studies of professionals, indicates that real competence only comes with extensive practice. ... In denying the critical role of practice one is denying children the very thing they need to achieve real competence."
    -- John R. Anderson, Lynne M. Reder and Herbert A. Simon, Carnegie Mellon University, in Applications and Misapplications of Cognitive Psychology to Mathematics Education

    "...varied and repeated practice leading to rapid recall and automaticity is necessary to higher-order problem-solving skills in both mathematics and the sciences. ... lack of automaticity places limits on the mind's channel capacity for higher-order problem-solving skills. ... only intelligently directed and repeated practice, leading to fast, automatic recall of math facts, and facility in computation and algebraic manipulation can one lead to effective real-world problem solving. ... [These conclusions are based on] reliable facts, figures, and documentation ... not just from isolated lab experiments, but also from large-scale classroom results."
    -- E. D. Hirsch

    "Computational algorithms, the manipulation of expressions, and paper-and-pencil drill must no longer dominate school mathematics."
    -- National Council of Teachers of Mathematics (NCTM), Professional Standards for Teaching Mathematics, 1991

    "The NCTM denigrates the idea of practice, which is thoughtful, considered repetition, and confuses it with drill, which is blind, mindless repetition."
    -John Saxon

    "Never underestimate the joy people derive from hearing something they already know."
    -- Enrico Fermi (1901-1954)

    "In the new approach, as you know, the important thing is to understand what you're doing rather than to get the right answer."
    -- Tom Lehrer, New Math

    "So you've got thirteen ...
    And you take away seven,
    And that leaves five...
    well, six actually, but the idea is the important thing."
    -- Tom Lehrer, New Math

    "Presumably no one would argue that the conservative view on the sum of 14 and 27 differs from the liberal view."
    -- Carl Sagan, The Demon-Haunted World, page 257

    "Mathematical discoveries, small or great are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious."
    -- Jules Henri Poincaré, mathematician (1854-1912)

    "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in."
    -- Richard Feynman

    "Mathematics is the language in which God has written the universe."
    -- Galileo Galilei

    "This notion that one has to 'interest' students in mathematics in order to make them do it has gone much too far, to the point where real mathematics in many cases has just disappeared entirely from the courses. They're just a discussion of what mathematics does and beautiful pictures and imprecise ideas."
    -- Paul Sally Jr., professor in mathematics, University of Chicago

    "Belief is no substitute for arithmetic."
    -- Henry Spencer

    "Newsrooms are full of English majors who acknowledge that they are not good at math, but still rush to make confident pronouncements about a global-warming 'crisis' and the coming of bird flu."
    -- John Stossel

    "My whole experience in math the last few years has been a struggle against the program. Whatever I've achieved, I've achieved in spite of it. Kids do not do better learning math themselves. There's a reason we go to school, which is that there's someone smarter than us with something to teach us."
    -- High school student Jim Munch, relating his battle against his school's constructivist math program

    "It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity."
    -- Pierre-Simon de Laplace (1749-1827)

    "Well, the math stuff I was fine with up until 7th grade. But Malia is now a freshman in High School and Iım pretty lost. Itıs tough."
    -- Barack Obama, The Tonight Show, 2012

More Excellent Resources on Math

  • Weapons of Math Destruction: Cartoons about fuzzy math!

  • Resources on mathematics education, assembled by Dr. Martin Kozloff

  • Math Reform Issues: an outstanding collection of articles and reports on the math controversy, presented by the Oregon Education Coalition.

  • Kids Do Count: Seeking Excellence in Math Education is an extensive (and beautifully designed) website organized by a group in Utah protesting the use of fuzzy math there. They provide excellent commentary and personal histories on problems, and a very rich set of links for additional information.

  • How well are American students prepared in math? Decide for yourself: take a look at these examples of math problems from Japan's University Entrance Center Examination (UECE) (PDF doc).

  • The Math Forum at Drexel University: An extensive and fascinating collection of articles and links on math teaching issues, assessment, computers and calculators, research, education reform, societal issues, public policy, and public understanding of math.

  • From our page on books on education, see this section on math.

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