On 06 May 2000, Nancy Buell wrote: >Today I sent the Harvard Crimson the following response to Schmid's >op-ed piece. Don't know if it will get printed. Nancy
Dr. Wilfried Schmid [New Battles in the Math Wars; 5/04] is concerned >that two third of Massachusetts fourth graders could not correctly >solve 256 x 98 on last SpringÂs MCAS tests. In both the traditional >and reform mathematics curricula that IÂve used over the past 30 >years, this would be a fifth grade expectation, not fourth grade. Of >course one wonders why such a problem appeared on a test that is >supposed to measure what all students should know and be able to do by >the end of fourth grade.
In fourth grade students spend a great deal of time developing >important mathematical ideas related to multiplication: * understanding what multiplication represents and how it relates to >real experiences. * understanding the multiplication operation itself where all parts >of one factor operate on all parts of the other. This is different >from adding ones with ones and tens with tens. * recognizing the reasonableness of an answer in the thousands when >multiplying two 2-digit numbers. This is unreasonable when adding two >2-digit numbers. * extending number sense into the thousands. * recognizing multiplication as repeated addition, but also >recognizing the effeciency of moving beyond addition as a strategy for >solving multiplicative problems. * developing an understanding of the distributive property and how it >works in partitioning multiplication problems. * mastering the basic multiplication facts.
These are mostly complex ideas that develop over time and with varied >experiences. These are the ideas that lay the foundation for >flexibility, efficiency, and accuracy in multiplication work in fifth >grade. Dr. Schmid states, ÂAmerican school children rank near the >bottom in international comparisons of mathematical knowledge.Â In >fact, fourth graders were about average in the recent TIMSS study, but >are weakest at estimation and number sense, the very areas where the >reform curricula, that Dr. Schmid decries, are putting more emphasis. As we work to develop studentsÂ conceptual and computational >understandings, we do need to be mindful of what is known about >childrenÂs cognitive developmental levels and about how children >learn, as well as the mathematical understandings and skills we hope >they will develop. Pushing more complex computation to lower grades >does not necessarily raise the bar. It may mean many students will >resort to memorizing procedures rather than working for understanding >of those procedures.
My experience with reform curricula is that students are doing far >more mathematical reasoning than with any traditional program IÂve >used in the past. In addition they have more practice, although not >in the familiar form of worksheet drills. Their ability to estimate >and to calculate accurately have improved.
Nancy Buell Brookline 4th Grade Teacher
On May 4, Anne Wheelock wrote: >FYI - This op-ed may be of interest. It's posted at
What is 256 times 98? Can you do the multiplication without using a calculator? Two thirds of Massachusetts fourth-graders could not when they were asked this question on the statewide MCAS assessment test last year.
Math education reformers have a prescription for raising the mathematical knowledge of schoolchildren. Do not teach the standard algorithms of arithmetic, such as long addition and multiplication, they say; let the children find >>their own methods for adding and multiplying two-digit numbers, and for larger numbers, let them use calculators. One determined reformer puts it decisively: "It's time to acknowledge that continuing to teach these skills (i.e., pencil-and-paper computational algorithms) to our students is not only unnecessary, but counterproductive and downright dangerous."
Mathematicians are perplexed, and the proverbial man on the street, when hearing the argument, appears to be perplexed as well: improve mathematical literacy by downgrading computational skills?
Yes, precisely, say the reformers. The old ways of teaching mathematics have failed. Too many children are scared of mathematics for life. Let's teach them mathematical thinking, not routine skills. Understanding is the key, not computations.
Mathematicians are not convinced. By all means, liven up the textbooks, make the subject engaging and include interesting problems. But don't give up on basic skills! Conceptual understanding can and must coexist with computational facility--we do not need to choose between them.
The disagreement extends over the entire mathematics curriculum, kindergarten through high school. It runs right through the National Council of Teachers of Mathematics (NCTM), the professional organization of mathematics teachers. The new NCTM curriculum guidelines, presented with great fanfare on April 12, represent an earnest effort at finding common ground, but barely manage to paper-over the differences.
Among teachers and mathematics educators, the avant-garde reformers are the most energetic, and their voices drown out those skeptical of extreme reforms. On the other side, among academic mathematicians and scientists who have reflected on these questions, a clear majority oppose the new trends in math education. The academics, mostly unfamiliar with education issues, have been reluctant to join the debate. But finally, some of them >>are speaking up.
Parents, for the most part, have also been silent, trusting the experts--the teachers' organizations and math educators. Several reform curricula do not provide textbooks in the usual sense, and this deprives parents of one important source of information. Yet, also among parents, attitudes may be changing. A recent front-page headline in the New York Times declares that "The New, Flexible Math Meets Parental Rebellion."
The stakes are high in this argument. State curriculum frameworks need to be written, and these serve as basis for assessment tests; some of the reformers receive substantial educational research grants, consulting fees or textbook royalties. For now, the reformers have lost the battle in California. They are redoubling their efforts in Massachusetts, where the curriculum framework is being revised. The struggle is fierce, by academic standards.
Both sides cite statistical studies and anecdotal evidence >to support their case. Unfortunately, statistical studies in education are notoriously unreliable--blind studies, for example, are difficult to construct. And for every charismatic teacher who succeeds with a "progressive" approach in the classroom, there are other teachers who manage to raise test scores dramatically by "going back to basics."
The current fight echoes an earlier argument, over the "New Math" of the '60s and '70s. Then, as now, the old ways were thought to have failed. A small band of mathematicians proposed shifting the emphasis towards a deeper understanding of mathematical concepts, though on a much more abstract level than today's reformers. Math educators took up the cause, but over time, most mathematicians and parents became unhappy with the results. What had gone wrong? Preoccupied with "understanding," the "New Math" reformers had neglected computational skills. Mathematical understanding, it turned out, did not develop well without sufficient computational practice. Understanding and skills grow best in tandem, each supporting the other. In most areas of human endeavor, mastery cannot be attained without technique. Why should mathematics be different?
American schoolchildren rank near the bottom in international comparisons of mathematical knowledge. Our reformers see this as an argument for their ideas. But look at Singapore, the undisputed leader in these comparisons: their math textbooks try hard to engage the students and to stimulate their interest. In early grades, they present mathematical problems playfully, often in the guise of puzzles. Yet the textbooks are coherent, systematic, efficient, and cover all the basics--worlds apart from the reform curricula in this country. How I wish Singapore's approach were adopted in my daughter's school!
The curriculum, of course, is not the only reason for Singapore's success, nor is it even the most important reason. The teachers' grasp and feeling for mathematics: that is the crucial issue, already for teachers in the early >>grades. Here, it turns out, many of the reformers agree with the critics. Teacher training in America has traditionally and grossly stressed pedagogy over content. The implicit message to the teachers is: If you know how to teach, you can teach anything! It will take a heroic effort--by mathematicians and math educators--to change the entrenched culture of teacher training.
Mathematicians do not want to invade the educators' turf. We are not qualified to do their work. Yet we are qualified as critics of reforms in math education. We should call attention to reforms we see as well meaning, but hectic and harmful. Most music critics would not do well as orchestra musicians. They do have acute hearing for shrill sounds from the orchestra.
Wilfried Schmid is Dwight Parker Robinson Professor of Mathematics. Earlier this year, he served as a mathematics advisor to the Massachusetts Department of Education.