10 Myths (Maybe) About Learning Math
By Jay Mathews
I love debates, as frequent readers of this column know. I learn the most when I am listening to two well-informed advocates of opposite positions going at each other.
I have held several debates here, although not all of them have worked because the debaters lose focus. One will make a telling point, and the other, instead of responding, will slide off into a digression.
So when I found a new attack on the National Council of Teachers of Mathematics (NCTM), the nation's leading association for math teachers, by a group of smart advocates, I saw a chance to bring some clarity to what we call the Math Wars. For several years, loosely allied groups of activist teachers and parents with math backgrounds have argued that we are teaching math all wrong. We should make sure that children know their math facts -- can multiply quickly in their heads and do long division without calculators, among other things -- or algebra is going to kill them, they say. They blame the NCTM, based in Reston, Va., for encouraging loose teaching that leaves students to try to discover principles themselves and relies too much on calculators.
The NCTM people, on the other hand, said this was a gross misstatement of what they were doing.
The advocates call their new assault "Ten Myths About Math Education and Why You Shouldn't Believe Them." I took the myths, and their explanation of each, and asked the NCTM to respond to each one. Here is the result. There are some quotes that are not attributed, but are found in sources cited on the myth Web page, and some technical language, but I think this provides a good quick review of what this raging argument is all about.
Feel free to send your comments to one of the people who came up with the list of 10, Elizabeth Carson at nycmathforum@yahoo.com or to the NCTM at president@nctm.org. The NCTM Web site is http://www.nctm.org/about/position_statements/, and the names of the dissident group are on the myth Web page.
Myth #1 -- Only what students discover for themselves is truly learned.
Advocates: 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.
NCTM: NCTM has never advocated discovery learning as an exclusive or even primary method of instruction. In fact, we agree that students do learn in a variety of ways, and effective learning depends on a variety of strategies at appropriate times. The goal is not just to know math facts and procedures but also to be able to think, reason and apply mathematics. Students must build their skills on a strong foundation of understanding.
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 being taught the standard arithmetic algorithms and their rationale, and given practice in using them.
Advocates: Children who do not master the standard algorithms begin to have problems as early as algebra I.
The snubbing or outright omission of the long division algorithm by NCTM- based curricula can be singularly responsible for the mathematical demise of its students. Long division is a pre-skill that all 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.) Its demand for estimation and computation skills during the procedure develops number sense and facility with the decimal system of notation as no other single arithmetic operation affords.
NCTM: NCTM has never advocated abandoning the use of standard algorithms. The notion that NCTM omits long division is nonsense. NCTM believes strongly that all students must become proficient with computation (adding, subtracting, multiplying, and dividing), using efficient and accurate methods.
Regardless of the particular method used, students must be able to explain their method, understand that other methods may exist, and see the usefulness of algorithms that are efficient and accurate. This is a foundational skill for algebra and higher math.
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.
Advocates: "The starting point for the development of children's 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 most critical component of teaching math. How math is taught is important as well, but is dictated by the "what." Much of understanding comes from mastery of basic skills -- an approach backed by most professors of mathematics. It succeeds through systematically empowering children with the pre-skills they need to succeed in all areas of mathematics. The myth of conceptual understanding versus skills is essentially a false choice -- a bogus dichotomy. The NCTM standards suggested "less 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."
NCTM: Math teaching does not fall into two extremes. There are several ways to teach effectively. Even a single teacher isn't likely to use the same method every day. Good teachers blend the best methods to help students develop a solid understanding of mathematics and proficiency with mathematical procedures.
It's worth noting that standard algorithms are not standard throughout the world. What is most important is that an algorithm works and that the student understands the math underlying why it works.
Every day teachers make decisions that shape the nature of the instructional tasks selected for students to learn, the questions asked, how long teachers wait for a response, how and how much encouragement is provided, the quality and level of practice needed -- in short, all the elements that together become the opportunities students have to learn. There is no one-size-fits-all model.
Myth #4 -- The math programs based on NCTM standards are better for children with learning disabilities than other approaches.
Advocates: "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.
NCTM: Most of the math programs published in this country claim to be based on the NCTM Standards. More important than the materials we use is how we teach. Students, all students, are entitled to instruction that involves important mathematics and challenges them to think.
Myth #5 -- Urban teachers like using math programs based on NCTM standards.
Advocates: Mere mention of [TERC, a program emphasizing hands-on teaching of math that this group doesn't believe demands enough paper and pencil work] was enough to bring a collective groan from more than 100 Boston Teacher Union representatives. ..... "
NCTM: Curricular improvement is hard, takes a lot of work, and demands support -- for the teacher, for students, and for parents. It should be noted that Boston students using the TERC-developed curriculum seem to be thriving. The percentage of failing students on the Massachusetts state assessment decreased from 46 to 30 percent and students scoring at the Proficient and Advanced categories increased from 14 to 22 percent between 2000-2004 (Boston Globe, December 14, 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)
Advocates: Children in almost all of the highest scoring countries in the Third International Mathematics and Science Survey (TIMMS) do not use calculators as part of mathematics 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.
NCTM: The TIMSS 1999 study of videotaped lessons of eighth-grade mathematics teachers revealed that U.S. classrooms used calculators significantly less often than the Netherlands (a higher achieving country) and not significantly differently from four of the five other higher-achieving countries in the analysis. When calculators are used well in the classroom, they can enhance students' understanding without limiting skill development. Technology (calculator or computer) should never be a replacement for basic understanding and development of proficiency, including skills like the basic multiplication facts.
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.
Advocates: On NPR's "Talk of the Nation" program on education in the United States (Feb. 15, 2005), 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.
NCTM: This is a myth. We know that students from other countries are doing better than many U.S. students, but certainly not all U.S. students. One reason U.S. students have not done well is that the way we have taught math just doesn't work well for enough of our students, and we have the responsibility to teach them all.
Myth #8 -- Math concepts are best understood and mastered when presented "in context"; in that way, the underlying math concept will follow automatically.
Advocates: Applications are important and story problems make good motivators, but understanding should come from building the math 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.
"[S]olutions of problems ..... need to be rounded off with a mathematical discussion of the underlying mathematics. If new tools are fashioned to solve a problem, then these tools have to be put in the proper mathematical perspective. ..... Otherwise the curriculum lacks mathematical cohesion.
NCTM: For generations, mathematics was taught as an isolated topic with its own categories of word problems. It didn't work. Adults groan when they hear "If a train leaves Boston at 2 o'clock traveling at 80 mph, and at the same time a train leaves New York ..... " Whatever problems and contexts are used, they need to engage students and be relevant to today's demanding and rapidly changing world.
An effective program lets students see where math is used and helps students learn by providing them a chance to struggle with challenging problems. The teacher's most important job in this setting is to guide student work through carefully designed questions and to help students make explicit connections between the problems they solve and the mathematics they are learning.
Myth #9 -- NCTM math reform reflects the programs and practices in higher performing nations.
Advocates: A recent study commissioned by the U.S. Department of Education, comparing Singapore's math program and texts with U.S. math texts, found that Singapore's approach is distinctly different from NCTM math "reforms."
Also, a paper that reviews videotaped math classes in Japan shows that there is teacher-guided instruction (including a wide variety of hints and helps from teachers while students are working on or presenting solutions).
NCTM: The study commissioned by the U.S. Department of Education comparing Singapore's mathematics program and texts with U.S. math texts also found that the U.S. program "gives greater emphasis than Singapore's to developing important 21st-century mathematical skills such as representation, reasoning, making connections, and communication. The U.S. frameworks and textbooks also place greater emphasis on applied mathematics, including statistics and probability."
NCTM's standards call for doing more challenging mathematics problems, as do programs in Singapore, Japan and elsewhere, but they also recognize the needs of 21st-century learners.
Myth #10 -- Research shows NCTM programs are effective.
Advocates: 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. Also, much of the "research" touted by some of the NSF programs has been conducted by the same companies selling the programs. State exams are increasingly being revised to address state math standards that reflect NCTM guidelines rather than the content recommended by mathematicians.
NCTM: True, there is no compelling evidence that any curriculum is effective in every setting, nor are there data to show exactly what causes improvement in student learning when many factors are involved. There is evidence that some of the more recently developed curricula are effective in some settings. However, the effectiveness with which a program, any program, is implemented is critical to its success, as are teacher quality, ongoing professional development, continuing administrative support, and the commitment of resources. Again, the issue of effectiveness is more likely to be attributable to instruction than to any specific curriculum.
Contrary to what is stated in some of these myths, there is no such thing as an "NCTM program." NCTM does not endorse or make recommendations for any programs, curricula, textbooks, or instructional materials. NCTM supports local communities using Principles and Standards for School Mathematics as a focal point in the dialogue to create a curriculum that meets their needs.
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