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 (affiliations and more), in association with New York City HOLD and , two education advocacy organizations of parents, mathematicians, and K-12 educators.

May 4, 2005

For almost two decades, 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). Their curricular and pedagogical "vision" for mathematics education reform, articulated in the two NCTM standards documents (1989 and 2000), has dominated local, state and federal education decision-making and policies, as well as public discussions, and press coverage. But many parents, mathematics experts, and K-12 teachers of mathematics do not share this vision.

A well-informed group of education stakeholders rejects the NCTM doctrine and model for mathematics reform. The expertise and viewpoints of this diverse group, comprised of mathematicians and scientists, K-12 teachers of mathematics, educational researchers, and concerned parents across our nation has been regularly eclipsed and 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. We are members of that constituency, and are part of an informal bipartisan grassroots coalition of advocates for authentic reforms in mathematics education.

The chart below offers our point by point refutation of a set of common myths propagated by mathematics educators in our schools of education and NCTM officials that are often presented as fact to policy makers and the general public.

"NCTM" (Fuzzy) Reality References Myth

Myth #1 Students learn in a variety of Dixon, R., Carnine, D., Lee, D. Wallin, ways. Basing most learning on J., & Chard, D. (1998). Review of High Only what students student discovery is time- Quality Experimental Mathematical discover for themselves consuming, does not insure that Research: Executive Summary. Eugene, is truly learned. students end up learning the right OR: National Center to Improve the concepts, and can delay or prevent Tools of Educators, University of progression to the next level. Oregon. Successful programs use discovery for only a few very Klahr, D. & Nigam, M. (2004). The carefully selected topics, never all Equivalence of Learning Paths in Early topics. Science Instruction: Effects of Direct Instruction and Discovery Learning. Psychological Science, 15, 10, 661-667. Becker, W. C. and Engelmann, S.; Sponsor Findings From Project Follow Through. University of Oregon.

John R. Anderson, Lynne M. Reder, Herbert A. Simon. Applications and Misapplications of Cognitive Psychology to Mathematics Education.

R. James Milgram, "What is Mathematical Proficiency?," March, 2004. Invited address, First Workshop on Mathematics Education. Mathematics and Science Research Institute, Berkeley, CA.

Myth #2 Children who do not master the General reference: The algebra, pre- standard algorithms begin to have calculus and calculus teachers and Children develop a problems as early as algebra I. professors who must remediate or flunk deeper understanding these children. of mathematics and a The snubbing or outright omission greater sense of of the long division algorithm by From 1998 issue of the Notices of the ownership when they NCTM- based curricula can be American Mathematical Society: are expected to invent singularly responsible for the "We would like to emphasize that the and use their own mathematical demise of its methods for performing students. Long division is a pre- standard algorithms of arithmetic are more than just 'ways to get the answer' -- the basic arithmetical skill that all students must master operations, rather than to automaticity for algebra that is, they have theoretical as well as practical significance. For one thing, all study, understand and (polynomial long division), pre- the algorithms of arithmetic are practice the standard calculus (finding roots and algorithms. asymptotes), and calculus (e.g., preparatory for algebra, since there are integration of rational functions (again, not by accident, but by virtue of and Laplace transforms.) Its the construction of the decimal system) demand for estimation and strong analogies between arithmetic of computation skills during the ordinary numbers and arithmetic of procedure develops number sense polynomials." (The above was quoted in and facility with the decimal an open letter to Secretary of Education system of notation as no other Richard Riley in 1999, which was signed single arithmetic operation by 200 prominent U.S. mathematicians.) affords. The Role of Long Division in the K-12 Curriculum; David Klein (California State University, Northridge), R. James Milgram (Stanford University)

Myth #3 "The starting point for the Kenneth Ross, Chair, Mathematical development of children's Association of America President's Task There are two separate creativity and skills should be Force on the NCTM Standards. (June and distinct ways to established concepts and 1997). Response to NCTM's Commission teach mathematics. The algorithms... Success in on the Future of the Standards. NCTM backed mathematics needs to be grounded approach deepens in well-learned algorithms as well Basic Skills vs Conceptual conceptual as understanding of the concepts." Understanding; a Bogus Dichotomy; understanding through Hung-Hsi Wu, Department of a problem solving What is taught in math is the most Mathematics, University of California, approach. The other critical component of teaching Berkeley (American Educator, Fall, teaches only arithmetic math. How math is taught is 1999). skills through drill and important as well, but is dictated kill. Children don't by the "what". Much of Willingham, D. (Spring 2004). Practice need to spend long understanding comes from Makes Perfect-But Only If You Practice hours practicing and mastery of basic skills - an Beyond the Point of Perfection. reviewing basic approach backed by most American Educator. arithmetical professors of mathematics. It operations. It's the succeeds through systematically Algorithms, Algebra, and Access, by concept that's empowering children with the pre- Stanley Ocken (Sep 2001). important. skills they need to succeed in all areas of mathematics. The myth of In Defense of "Mindless Rote", by Ethan conceptual understanding versus Akin (Mar 30, 2001). skills is essentially a false choice - a bogus dichotomy. The NCTM On the Algorithms of Arithmetic, by standards suggested "less Ralph Raimi (2002). emphasis" on topics needed for higher math, such as many basic skills of arithmetic and algebra.

"That students will only remember what they have extensively practiced - and that they will only remember for the long term that which they have practiced in a sustained way over many years - are realities that can't be bypassed."

Myth #4 "Educators must resist the Miller, S.P. and Mercer, C.D., temptation to adopt the latest math "Educational Aspects of Mathematics The math programs movement, reform, or fad when Disabilities." January/February 1997, based on NCTM data-based support is lacking..." Journal of Learning Disabilities, Vol. 30, standards are better for No. 1, pp. 47-56. children with learning Large-scale data from California disabilities than other and foreign countries show that Darch, C., Carnine, D., & Gersten, R. approaches. children with learning disabilities (1984). "Explicit Instruction in do much better in more structured Mathematics Problem Solving." The learning environments. Journal of Educational Research, 77, 6, 351-359.

Myth #5 "Mere mention of [TERC] was Editorial, "Mathematical Unknowns," enough to bring a collective groan The Boston Globe, November 8, 2004, Urban teachers like from more than 100 Boston A10. using math programs Teacher Union representatives..." based on NCTM standards. Myth #6 Children in almost all of the Calculating the cost of calculators, Lance highest scoring countries in the Izumi, Capitol Ideas, Pacific Research "Calculator use has Third International Mathematics Institute, Vol. 5, No. 51, December 21, been shown to enhance and Science Survey (TIMMS) do 2000. cognitive gains in areas not use calculators as part of that include number mathematics instruction before W. Stephen Wilson, K-12 Calculator sense, conceptual grade 6. Usage and College Grades Educational development, and Studies in Mathematics. visualization. Such A study of calculator usage among gains can empower and calculus students at Johns Hopkins motivate all teachers University found a strong and students to engage correlation between calculator in richer problem- usage in earlier grades and poorer solving activities." performance in calculus. (NCTM Position Statement)

Myth #7 On NPR's "Talk of the Nation" Grover Whitehurst, Director, Institute of program on education in the U.S. Education Sciences; on NPR Talk of the The reason other (Feb. 15, 2005), Grover Nation, February 15, 2005; countries do better on Whitehurst, Director of the international math tests Institute of Education Sciences at like TIMSS and PISA is the Department of Education, that those countries stated that test takers are selected select test takers only randomly in all countries and not from a group of the top selected from the top performers. performers.

Myth #8 Applications are important and The Mathematician and Mathematics story problems make good Education Reform; Hung-Hsi Wu, Math concepts are best motivators, but understanding University of California, Berkeley; in understood and should come from building the Notices of the American Mathematical mastered when math for universal application. Society, 43(1996), 1531-1537). presented "in context"; When story problems take center in that way, the stage, the math it leads to is often underlying math not practiced or applied widely concept will follow enough for students to learn how automatically. to apply the concept to other problems.

"[S]olutions of problems ... need to be rounded off with a mathematical discus-sion of the underlying mathematics. If new tools are fashioned to solve a problem, then these tools have to be put in the proper mathematical perspec-tive. ... Otherwise the curriculum lacks mathematical cohesion."

Myth #9 A recent study commissioned by What the United States Can Learn From the U.S.Department of Education, 's World-Class Mathematics NCTM math reform comparing Singapore's math System (and what Singapore can learn reflects the programs program and texts with U.S. math from the United States); American and practices in higher texts, found that Singapore's Institutes for Research; for U.S. performing nations. approach is distinctly different Department of Education; January 28, from NCTM math "reforms." 2005; Washington, D.C.

Also, a paper that reviews Siegel, Alan R. Telling Lessons from the videotaped math classes in Japan TIMSS Videotape: remarkable teaching shows that there is teacher-guided practices as recorded from eighth-grade instruction (including a wide mathematics classes in Japan, Germany variety of hints and helps from and the US. Chapter 5 in ``Testing teachers while students are Student Learning, Evaluating Teaching working on or presenting Effectiveness,'' Williamson M. Evers and solutions). Herbert J. Walberg, Eds., Hoover Institution Press, May, 2004, pp. 161- 194.

Myth #10 There is no conclusive evidence of On Evaluating Curricular Effectiveness; the efficacy of any math Judging the Quality of K-12 Mathematics Research shows NCTM instructional program. Evaluations; National Research Council, programs are effective. the National Academies Press; Increases in test scores may reflect September, 2004. increased tutoring, enrollment in learning centers, or teachers who The state tests in Maryland have a supplement with texts and other number of 3 point problems in which materials of their own choosing. students are awarded 1 point for Also, much of the "research" performing the math correctly and 2 touted by some of the NSF points for explaining it. It is thus possible programs has been conducted by to do the math right but get half the credit the same companies selling the that another student gets with the wrong programs. State exams are answer. increasingly being revised to address state math standards that reflect NCTM guidelines rather than the content recommended by mathematicians.

This document was prepared by: Additional web resources for mathematics education advocacy Karen Budd Member, Board of Directors Parents for Better Schools in Fairfax County National Organizations

Elizabeth Carson New York City HOLD # Mathematically Co-Founder and Director Correct NYC HOLD Honest Open Logical Decisions on News Mathematics Education Reform Email: [email protected] NYC HOLD News Page Barry Garelick Local Activism Analyst U.S. Environmental Protection Agency Email: [email protected] Illinois Loop # Parents for Evidence Based Education (IA) # Teach Us Math, Parents David Klein Concerned With Penfield (NY)'s Math Professor of Mathematics Programs # Plano (TX) Parental Rights California State University, Northridge Council # PBSfx: Parents for Better Schools (Fairfax, VA) # Save Our R James Milgram Children from Mediocre Math (Conejo Professor of Mathematics Valley, CA) # Simsbury (CT) Stanford University # Teach Utah Kids # Kids Do Count (UT)

Ralph A. Raimi Curriculum Reviews Professor of Mathematics University of Rochester Mathematically Correct Program Reviews # NYC HOLD Curriculum Reviews # Martha Schwartz Illinois Loop Mathematics Reviews and Paleomagnetism Lab More # Earlier MC Program Reviews for University of Southern California Grades 2, 5, and 7 Sandra Stotsky Research Scholar Reviews of Mathematics Standards Northeastern University The State of State Math Standards 2005 # Vern Williams State Mathematics Standards 1998 # Math Teacher NCTM Principles and Standards for Longfellow Middle School, Fairfax County Virginia School Mathematics # NYC HOLD on Email: [email protected] Standards and Assessments

W. Stephen Wilson Personal Pages and Weblogs Professor of Mathematics Johns Hopkins University Bas Braams # Ralph Raimi # David Klein Email: [email protected] # Bill Quirk # Hung-Hsi Wu # Toby Earl's Teach Math # Kitchen Table Math # Jeff * Affiliations or positions are provided for identification Lindsay # Brian D. Rude # Bert Fristedt # purposes only and do not imply institutional or Lawrence Gray organizational support.

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