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). 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. 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 constituencys expertise is often entirely absent from the decision-making process.
As a member of that constituency and an ad-vocate for authentic re-form in mathematics ed-ucation, I was part of a group that decided to pre-pare a point-by-point refu-tation 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. The original and complete version of this chart can be found on the NYC HOLD website www.nychold.com. The direct link is: http://www.nychold.com/myths050504.html. The names of all those who prepared these Ten Myths can be found at the end of this chart.
Ten Myths About Math Education and Why You Shouldn't Believe Them
Myth # 1: Only what students discover for themselves is truly learned.
Reality: Students learn in a variety of ways. Basing most learning on student discovery is time-consuming, does not insure that students end up learning the right concepts, and can delay or prevent progression to the next level. Successful programs use discovery for only a few very carefully selected topics, never all topics.
References: Dixon, R., Carnine, D., Lee, D. Wallin, J., & Chard, D. (1998). Review of High Quality Experimental Math-ematical Research: Executive Summary. Eugene, OR: National Center to Improve the Tools of Educators, University of Oregon. http://idea.uoregon.edu/~ncite/documents/math/report.pdf Klahr, D. & Nigam, M. (2004). The Equivalence of Learning Paths in Early Science Instruction: Effects of Direct Instruction and Discovery Learning. Psychological Science, 15, 10, 661667.http://www.psy.cmu.edu/faculty/klahr/KlahrNigam.2-col.pdf John R. Anderson, Lynne M. Reder, & Herbert A. Simon. Applications and Misapplications of Cognitive Psychology to Mathematics Education http://act-r.psy.cmu.edu/papers/146/Applic.MisApp.pdf
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 algorithms for basic arithmetic operations and are not taught the reasons for the standard algorithms and how to use them.
Reality: Children who do not master the standard algorithms begin to have problems as early as algebra I. Long division is a skill that students must master to automaticity for algebra (polynomial long division), pre-calculus (finding roots and asymptotes), and calculus (e.g., integration of rational functions and Laplace transforms.) By demanding estimation and computation skills, it develops number sense and facility with the decimal system of notation as no other single arithmetic operation affords.
References: From 1998 issue of the Notices of the American Mathematical Society: We would like to emphasize that the standard algorithms of arithmetic are more than just ways to get the answer that is, they have theoretical as well as practical significance. For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials David Klein (California State University, Northridge) & R. James Milgram (Stanford University). The Role of Long Division in the K-12 Curriculum. http://www.csun.edu/%7Evcmth00m/longdivision.pdf
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 dont need to spend long hours practicing and reviewing basic arithmetic operations. Its the concept thats important.
Reality: The starting point for the development of childrens creativity and skills should be established concepts and algorithms Success in mathematics needs to be grounded in well-learned algorithms as well as understanding of the concepts. What is taught in math is the critical component of teaching it. How math is taught is important but must be dictated by the what.
Much understanding of math comes from mastery of basic skillsan idea backed by most professors of mathematics. The idea of having to make a choice between conceptual understanding and skills is essentially false a bogus dichotomy. That students will only remember what they have extensively practicedand that they will only remember for the long term that which they have practiced in a sustained way over many yearsare realities that cant be bypassed.
References: Kenneth Ross, Chair, Mathematical Association of America Presidents Task Force on the NCTM Standards. (June 1997). Response to NCTMs Commission on the Future of the Standards. http://www.maa.org/past/maa_nctm.html Hung- Hsi Wu (Department of Mathematics, University of California, Berkeley). (Fall 1999). Basic Skills vs. Conceptual Understanding: A Bogus Dichotomy. American Educator. http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdfDaniel Willingham. (Spring 2004). Practice Makes PerfectBut Only If You Practice Beyond the Point of Perfection. American Educator. http://www.aft.org/pubs-reports/american_educator/spring2004/cogsci.html Stanley Ocken. (September 2001). Algorithms, Algebra, and Access. http://www.nychold.com/ocken-aaa01.pdf Ethan Akin. (March 30, 2001). In Defense of Mindless Rote. http://www.nychold.com/akin-rote01.html Ralph Raimi. (2002). On the Algorithms of Arithmetic. http://www.nychold.com/raimi-algs0209.html
Myth # 4: The math programs based on NCTM standards are better for children with learning disabilities than other approaches.
Reality: Educators must resist the temptation to adopt the latest math movement, reform, or fad when data-based support is lacking Large-scale data from California and foreign countries show that children with learning disabilities do much better in more structured learning environments.
Miller, S.P. & Mercer, C.D., Educational Aspects of Mathematics Disabilities. January/February 1997, Journal of Learning Disabilities, Vol. 30, No. 1, pp. 47-56.Darch, C., Carnine, D., & Gersten, R. (1984). Explicit Instruction in Mathematics Problem Solving. The Journal of Educational Research, 77, 6, 351359.
Myth # 5:Urban teachers like using math programs based on NCTM standards.
Reality: Mere mention of [TERC] was enough to bring a collective groan from more than 100 Boston Teacher Union representatives
References: Editorial, Mathematical Unknowns, The Boston Globe, November 8, 2004.
Myth # 6:Calculator use has been shown to enhance cognitive gains in areas that include number sense, conceptual development, and visualization. Such gains can empower and motivate all teachers and students to engage in richer problem-solving activities.(NCTM Position Statement)
Reality: Children in most of the highest scoring countries in the Third International Mathematics and Science Survey (TIMMS) do not use calculators as part of their instruction before grade 6.A study of calculator usage among calculus students at Johns Hopkins University found a strong correlation between calculator usage in earlier grades and poorer performance in calculus.
References: Lance Izumi. (2000). Calculating the cost of calculators. Capitol Ideas. http://www.pacific research.org/pub/cap/2000/00-12-21.htmlW. Stephen Wilson, K-12 Calculator Usage and College Grades Educational Studies in Mathematics. http://www.math.jhu.edu/~wsw/ED/pubver.pdf
Myth #7:The reason other countries do better on international math tests like TIMSS and PISA is that they select test takers from only a group of the top performers.
Talk of the Nation program on education in
the U.S., Grover Whitehurst, Director of the Institute of
Education Sciences at the Department of Education, stated
that test takers are selected randomly in all countries
and not selected from the top performers.
Myth # 8Mathematical concepts are best understood when presented in context. When presented that way, the mathematical concept will be automatically understood and mastered.
Applications are important and story problems make good motivators, but under-standing should come from building mathematics for universal application. When story problems take center stage, the math it leads to is often not practiced or applied widely enough for students to learn how to apply the concept to other problems.
Hung-Hsi Wu. (1996). The Mathematician and Mathematics Education Reform. In Notices of the American Mathematical Society, 43, 1531-1537.http://math.berkeley.edu/~wu/reform2.pdf
Myth # 9 NCTM math reform reflects programs and practices in higher performing nations.
Reality: A recent study commissioned by the U.S. Department of Education, comparing Singapores math program and texts with U.S. math texts, found that Singapores approach is distinctly different from NCTM math reforms. A study reviewing videotaped math classes in Japan shows teacher-guided instruction, including a variety of hints and help from teachers while students are working on or presenting solutions.
American Institutes for Research. (January 28, 2005). What the United States Can Learn From Singapores World-Class Mathematics System (and what Singapore can learn from the United States). Washington, D.C.http://www.air.org/news/documents/Singapore%20Report%20(Bookmark%20Version).pdf Alan R. Siegel. (May 2004). Telling Lessons from the TIMSS Videotape: Remarkable Teaching Practices as Recorded from Eighth-Grade Mathematics Classes in Japan, Germany and the U.S. Chapter 5. In W. M. Evers & H. J. Walberg (Eds.), Testing Student Learning, Evaluating Teaching Effectiveness. Stanford, CA: Hoover Institution Press, pp. 161-194. http://www.cs.nyu.edu/faculty/siegel/ST11.pdf
Myth # 10:Research shows NCTM programs are effective.
Reality: There is no conclusive evidence of the efficacy of any math instructional program. Increases in test scores may reflect increased tutoring, enrollment in learning centers, or teachers who supplement with texts and other materials of their own choosing. Much of the research on NSF-supported programs has been conducted by the companies selling the programs. State exams increasingly address state math standards that reflect NCTM guidelines rather than the content recommended by math-ematicians.
National Research Council. (September 2004). On Evaluating Curricular Effectiveness; Judging the Quality of K-12 Mathematics Evaluations. Washington, DC: National Academies Press. http://www.nae.edu/NAE/naepcms.nsf/weblinks/MKEZ-5Z5PKX?OpenDocument The state tests in Maryland have a number of 3-pointProblems. Students are awarded 1 point for performing the math correctly and 2 points for explaining it. It is thus possible to do the math right but get half the credit that another student gets with the wrong answer.
The original document was prepared by Karen Budd, Member, Board of Directors Parents for Better Schools in Fairfax County, Elizabeth Carson, Co-Founder and Director NYC HOLD (Honest Open Logical Decisions on Mathematics Education Reform), Barry Garelick, Analyst U.S. Environmental Protection Agency, David Klein, Professor of Mathematics, California State University, Northridge, R James Milgram, Professor of Mathematics, Stanford University, Ralph A. Raimi, Professor of Mathematics, University of Rochester, Martha Schwartz, Paleomagnetism Laboratory, University of Southern California, Sandra Stotsky, Research Scholar, Northeastern University, Vern Williams, Mathematics Teacher, Longfellow Middle School, Fairfax County, Virginia, W. Stephen Wilson, Professor of Mathematics, Johns Hopkins University,
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