| Unless students have basic skills, they cannot develop enough understanding of mathematics to succeed in college or on the job. |
The revised National Council of Teachers of Mathematics standards are a sign that a new consensus has emerged about effective mathematics instruction. Now school districts throughout the country will have to re-evaluate the “content lite” mathematics programs they purchased over the past 10 years—programs that mirrored the council’s 1989 standards in slighting mathematical substance.
On April 12 of this year, when NCTM officials unveiled the 2000 version of the math standards, they emphasized repeatedly that basic skills matter. (“Revised Mathematics Standards Provide More Guidance,” April 19, 2000.) Those officials and curriculum and instruction specialists around the country now recognize that unless students have basic skills, they cannot develop enough understanding of mathematics to succeed in college or on the job.
So it’s back to the drawing board for programs like the Connected Mathematics Project that go to extremes to avoid teaching long division. (To that end, Connected Mathematics presents material that is, in fact, mathematically incorrect.) Likewise, we can only wonder how the trend-conscious officials in Minnesota will explain to parents and taxpayers the recent decree that only programs like Connected Math can be used in their state.
We commend the NCTM for acknowledging that its last try at writing national standards had serious flaws. The council has learned from actual teaching experience and has put the interests of students above saving face. The new standards represent a significant improvement over the 1989 standards in many areas:
(1) The level of student work expected is better aligned with the international benchmarks provided by such successful countries as Japan, Singapore, and Hungary.
(2) The illustrative examples and the content discussion focus on:
- Mathematical issues (for example, on page 59 of the standards document, a child solves the problem 6 + 7 by calculating 6 + 6 and adding 1; the child, the document points out, is drawing on “her knowledge of adding pairs, of adding 1, and of associativity”);
- Possible content to reinforce the desired standards (for example, the picture on page 159 gives teachers a clear and intuitively understandable explanation for students of the reason the sum of the first K successive odd whole number is the square of K);
- What to emphasize and how;
- Classroom methods for handling intricate problems; and
- Clearing up frequent misunderstandings by students.
(3) The 2000 document contains far fewer mathematical errors than the 1989 one.
The message throughout is that mathematics instruction needs to be balanced. Students today certainly need calculation and symbolic-manipulation skills that go beyond the merely mechanical. They must understand concepts sufficiently well to be able to handle new situations flexibly and confidently, to be able to recognize where mathematics can be applied to problems, and to devise strategies to solve the problems that arise.
In the new standards, however, as distinct from the 1989 standards, it is no longer an article of faith that such deep understanding can occur even if students lack basic skills. Indeed, the previous standards led many curriculum planners to think that the availability of calculators and computers had replaced the need to teach basic skills. Such thinking is rejected in the new standards.
Although the 1989 standards said that calculators should be available to students “at all times,” the new standards say that often such availability is not appropriate.
There are still vestiges of the utopian dream of calculator as deus ex machina in the new standards, but on pages 25 and 26, the new standards explicitly say that technology should “not be used as a replacement for basic understandings.” It is “not a panacea,” and, as with any teaching aid, it can be used “well or poorly.”
Proficiency with operations is given prominent place. For example, on page 32 of the revised standards, the NCTM makes it clear that students are expected to develop real “proficiency” in arithmetic computation as well as “deep and fundamental” understanding of number systems.
Likewise, in algebra, symbolic manipulation is “critical in mathematical work.”
Furthermore, students must come to understand “the concepts of algebra, the structures and principles that govern the manipulation of the symbols” (pages 37 and 297). Thus do skills and conceptual understanding march hand in hand in the 2000 NCTM standards.
The 2000 standards seek to have teaching interweave conceptual understanding with basic skills. |
This represents a shift in emphasis from the 1989 standards. Indeed, it was precisely when the 1989 standards disparaged symbolic manipulation that those standards were most severely criticized by professional mathematicians and concerned parents. The 2000 standards recognize the critical role that skill with symbolic manipulation plays in developing mathematical competency.
Here is an extended passage that illustrates how the 2000 standards seek to have teaching interweave conceptual understanding with basic skills (operational fluency):
“In high school, students’ understanding of number is the foundation for their understanding of algebra, and their fluency with number operations is the basis for learning to operate fluently with symbols. Students should enter high school with an understanding of the basic operations and fluency in using them on integers, fractions, and decimals” (page 291-92).
In the 1989 standards, mathematical proof was depreciated. In pleasant contrast, the geometry section of the new standards includes a strong emphasis on teaching students “careful reasoning and proof,” so that students learn to think logically and learn “to see the axiomatic structure of mathematics.”
The revised standards say that students should be required to understand how to prove mathematical statements (and, of necessity, also learn to recognize incorrect arguments). That is a big improvement on what the earlier standards said on this point. The new consensus is that mathematical competence has to include ability to demonstrate proofs.
One of the central issues in what the news media have called the “math wars” has been whether students are expected to learn the standard algorithms for addition, subtraction, multiplication, and, above all, long division. The 1989 NCTM standards opposed teaching any standard algorithms; any methods used to solve problems had to be made up by the students themselves.
But the new standards (for example, on page 79) say that students need to learn “efficient and accurate strategies,” that is, correct, general algorithms, “that they understand” for the basic operations of addition and subtraction.
Again (on page 155), the 2000 standards say that “the conventional algorithms for multiplication and division” should be learned “as one efficient way to calculate.” This is an endorsement of requiring students to learn the standard algorithms.
This endorsement of teaching algorithms emphasizes (correctly) the importance of students’ learning to understand why the algorithms work. But the 2000 standards go on at unnecessarily great length to point out that minor variations exist on the standard algorithms and that there are other correct, but less efficient, methods. Nonetheless, by unequivocally endorsing “efficient and accurate methods,” the NCTM has unequivocally endorsed the standard algorithms.
The mathematical point is that, using the base-10 representation of numbers, there is only one efficient (and correct) algorithm each for addition, subtraction, multiplication, and division, when using multidigit numbers. There may be minor variations in execution, but at the conceptual level they are all the same.The 2000 standards, quite reasonably, recommend (on page 220) that students devise methods for solving problems as one way of enhancing understanding. They suggest that students develop their own methods and share them with one another. Students should explain why their methods work and are reasonable to use. They should then compare their student-devised methods with “the algorithms traditionally taught in school.” In this way, students can comprehend “the power and efficiency of the traditional algorithms” and also understand the connection between them and student-constructed methods “that may sometimes be less powerful or efficient but are often easier to understand."Ultimately, students should apprehend that algorithms are “tools for solving problems,” rather than think that learning them is “the goal of mathematics study” (page 144).
We would add that the study of algorithms has become even more important to students today than it was in the past. Constructing computer programs is nothing but writing down sequences of algorithms—some of which can be looked up in books, but most of which have to be constructed by the programmer. The new math standards say:
“Mathematics topics such as recursion, iteration, and the comparison of algorithms are receiving more attention in school mathematics because of their increasing relevance and utility in a technological world” (page 16).
| As in any work of this size and complexity, the revised standards contain errors. |
In truth, algorithms have to be a crucial component of K-12 mathematics education today. Students must learn how to construct algorithms for searching, ordering, formatting, and so on, and they must begin to get some idea of how to recognize correct and incorrect algorithms. In particular, they must learn that checking an algorithm against examples is not a proof of correctness. One could even hope that part of the development of proof techniques in geometry could be extended to discuss issues like these.
Having said all these positive things about the 2000 version of the NCTM standards, we have to add some cautionary comments about the standards’ imperfections. As in any work of this size and complexity, the revised standards contain errors.
An error on page 161 reflects the NCTM’s deficient understanding of real-world applications of mathematics:
“Historically, much of the mathematics used today was developed to model real-world situations, with the goal of making predictions about those situations. As patterns are identified, they can be expressed numerically, graphically, or symbolically and used to predict how the pattern will continue. Students in grades 3-5 develop the idea that a mathematical model has both descriptive and predictive power.”
In fact, much of mathematics was developed not to predict but to operate machinery and systems. In these cases (such as cars, amplifiers, robot arms, and the registers in computer chips), we know what material things will do when small changes are made here and there. Mathematics then provides the tools that enable people to make such devices actually do useful things in the real world. It is this interaction of mathematics with science, medicine, engineering, computer science, and other areas that provides the vast majority of applications of mathematics.
The techniques used in such applications include solving quadratic equations, factoring polynomials, and the use of exponential functions and logarithms. Other techniques are far beyond K-12 mathematics, but even then, preparation for learning them starts with mastering the K-12 techniques we are talking about here. The new standards mention these techniques, but do not give them the attention they deserve.
This deficiency is perhaps the gravest failing of the new document. (The 1989 standards also failed in this regard.) Students who might work in these applied-mathematics areas should receive K-12 instruction that will prepare them for such work. But the NCTM standards largely ignore real real-world applications in all too many of these areas.
Here is another basic error from page 188:
“Formulating conjectures and assessing them on the basis of evidence should become the norm. Students should learn that several examples are not sufficient to establish the truth of a conjecture and that counterexamples can be used to disprove a conjecture. They should learn that, by considering a range of examples, they can reason about the general properties and relationships they find.”
The difficulty here is not with the first two sentences; it lies in the last. The point is that it doesn’t matter how large a set of examples a student checks: Without a proof, the relationship that is being explored could still be incorrect. This weakness pervades these standards. It stems from the fact that it is unrealistic to expect young students to be able to construct proofs; but at the same time, those who wrote the NCTM standards are uncomfortable with a teacher’s ever telling students certain statements are, in fact, always true, and that students will just have to accept them without proof at this point. Unfortunately, it is probably impossible to do a reasonable job of teaching math in grades K-7 without, at some points, making such statements.
In addition, although we think the new NCTM standards go a long way toward embodying the emerging consensus on mathematics instruction, they reflect neither valid research nor the consensus of teaching practice in their premature enthusiasm for using technology (calculators and computers) in instruction.
The new standards say that “electronic technologies” are “essential tools” for teaching, learning, and doing mathematics. But the research cited in support of this statement mostly involved small-scale, controlled situations. (The data from the only somewhat large-scale study—involving about 5,000 students—were so ambiguous that two different researchers drew widely different conclusions from them.)
The reality is that curriculum-planning and teaching in school systems is a large-scale operation where close supervision is not going to be possible. As a result, the only studies that are valid are those that look at such large-scale operations.
To date, only the Educational Testing Service researcher Harold Wenglinsky’s 1998 study does this. (“Does It Compute?,” from ETS.) It is based on data from the National Assessment of Educational Progress, and the author’s conclusions are much more cautious and nuanced than the NCTM’s ardent embrace of technology.
This is a step forward for mathematics instruction in the United States. |
Moreover, the Third International Mathematics and Science Study found that, for the 8th grade TIMSS mathematics test, the majority of the students from three of the five nations with the highest scores (Belgium, Korea, and Japan) rarely or never used calculators in the mathematics classroom. In contrast, two-thirds of the students from 10 of the 11 nations (including the United States) with scores below the international average used calculators several times a week and sometimes nearly every day.
Should the new standards have been less vague? Yes. Is there room for improvement in aligning with top-performing countries? Yes. Could the proven value of drill and practice have been spelled out? Yes. Is long division given rather short shrift, considering its importance? Yes. There are disappointments as well as improvements in the new NCTM standards.
Yet this is a step forward for mathematics instruction in the United States. It marks out a large amount of common ground and reduces the amount of contested ground. Nonetheless, we hope that the National Council of Teachers of Mathematics will consider making some limited revisions to the revised document and be willing to correct the errors that readers will point out over the next few months. After all, the 2000 standards are primarily available electronically, and revision is quite a bit simpler in this format than is the case for publication in hard copy.
Further information and comments can be found on the World Wide Web site developed by the authors: math.stanford.edu/~milg ram/nctm2000-errata.html.
Bill Evers is a research fellow at Stanford University’s Hoover Institution in Stanford, Calif., and a member of its Koret Task Force on K-12 Education. He served on the California State Academic Standards Commission. Jim Milgram is a professor of mathematics at Stanford and a member of Achieve’s National Mathematics Standards and Assessment Panel.
