The major problem with the 1989 National Council of Teachers of Mathematics (NCTM) Standards is their lack of balance. One can easily accept many separate statements and sentiments expressed in the Standards, but it is the whole that exhibits a strongly one-sided slant. The danger is that each state and organization that is using the Standards as a guiding document will take what it believes is the essence of the Standards and ignore the caveats and admonitions. HOLD is based in California, where we have our own 1992 Mathematics Framework that is supposedly derived from the Standards. When one reviews that framework, one sees how easily the Standards can be misconstrued. If the Standards propose both individual and group learning, but the stress is on group work, then the Calif. Framework talks about group learning as the only approved way to teach. If the Standards call for decreased attention on computation skills, then they are completely gone from the Calif. Framework. If the Standards call for stressing mathematical communication, and mentions only in passing that mathematical symbolism is preferred, then the Calif. Framework is already using English as the language of mathematical communication, and mathematical symbolism is gone.
Another danger occurs when the Standards are careless and make statements that are not supported by research, or when they ignore teaching and school realities. Statement like “There is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations” (p. 8) is unsupported by research, and is inconsistent with the reality we see in California. Focus on alternative assessment, to the near exclusion of standardized assessment (p. 201) ignores the reality in the classroom and in the society, where standardized assessment is often the only kind available, and certainly the only kind that has a proven track record of predicting achievement. The Standards are taken by many teachers as gospel - they certainly read like one sometimes - and unproved theories and statements should not be casually presented to such an audience.
Therefore, HOLD requests that the new standards will include the following clarifications:
1. New Goals for Students.
“(1) that they learn to value mathematics, (2) that they become confident in their ability to do mathematics, (3) that they become mathematical problem solvers, (4) that they learn to communicate mathematically, and (5) that they learn to reason mathematically” (pp. 5,16).
HOLD suggests the order be reversed. Putting the most important aspect of mathematics, often the only one that lasts—“learning to reason”—as fifth is a mistake. Surely someone was confused when “learning to value mathematics” was deemed more important than knowing mathematics. The most important thing is learning mathematics, not appreciating mathematics.
2. Reasoning vs. Results.
The Standards say “In fact, a demonstration of good reasoning should be rewarded even more than students’ ability to find correct answers" (p. 6). While nobody would argue with this principle, good reasoning is notably difficult to assess. Many teachers are unable to fairly assess reasoning, both because of lack of time and lack of training, but this has not stopped them from reading the above as a license to totally ignore the value of correct math, preferring “student creativity” instead, whether mathematically correct or not. It must be clarified that “good reasoning” means “mathematically correct reasoning,” and that correct math has intrinsic value too.
3. The goals of learning mathematics should be clarified
“Change has been particularly great in the social and life sciences. In fact, quantitative techniques have permeated almost all intellectual disciplines. However, the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus-calculus sequence, a sequence designed with engineering and physical science applications in mind. Because mathematics is a foundation discipline for other disciplines and grows in direct proportion to its utility, we believe that the curriculum for all students must provide opportunities to develop an understanding of mathematical models, structures, and simulations applicable to many disciplines” (p. 7).
The average teacher will think that the passage above says that the traditional math curriculum is obsolete. This is patently incorrect. While an added focus on statistics and data representation in the curriculum is welcome, this passage sounds like an attack on classical mathematics per se and needs to be justified or struck out.
4. The role of the teacher, and the importance of “closure” in the class
The Standards suggest that “Instructional approaches should engage students in the process of learning rather than transmit information for them to receive” (p. 67). It suggests less reliance on “outside authority (teacher or an answer key)” (p. 71). It promotes methods that “require the teacher’s role to shift from dispensing information to facilitating learning, from that of director to that of catalyst and coach” (p. 128).
Many teachers interpret those exhortations to mean that they should never teach in the class, but instead only facilitate. This is especially dangerous when coupled with the poor training in math and insufficient knowledge of math of all too many teachers. These teachers are not able to give correct mathematical guidance to students. In other cases we have seen students left with erroneous results as the teacher was reluctant to tell them that they were wrong.
HOLD strongly believes that the role of the teacher should be first and foremost to teach, which includes direct instruction and immediately responsive constructive criticism, in addition to coaching. The new Standards must make the point that it is crucial for the teacher to achieve closure of ideas in the class and not let students leave the class full of erroneous math as a result of student “brainstorming.”
5. Problem Solving Curriculum
The Standards promote a curriculum that is centered on problem solving. Unfortunately, many teachers interpret that to mean that there is to be no explicit teaching of any mathematical subject, unless it is embedded in a “problem.” Oftentimes the result is that students are never taught any subject matter in depth, as subjects tend to pop up sporadically but never systematically. This issue should be clarified, and the importance of clear scope and sequence stressed.
6. Mathematics Communication and the Language of Mathematics
The Standards correctly identify mathematical symbolism as the language of mathematics. Unfortunately, often mathematical communication is misinterpreted to mean only or mostly English prose.
The results are that teachers in middle and high schools are actively discouraging the use of mathematical symbolism and penalizing students who write correct but short mathematical arguments using such notation. The teachers believe that “the more English prose, the better.” The Standards should clarify this issue, in particular because it penalizes immigrants, minorities and language-impaired students.
7. Whole Math philosophy
Many sections of the current Standards are permeated by a “Whole Math” philosophy. Like “Whole Language,” it lacks a research base and has only limited merit. The belief that students will learn math by just immersing themselves in doing math and discussing math, without explicit expository teaching is without merit. If math were so simple, it would not have required centuries for mathematical giants and professional practitioners to develop its ideas. Doing math and arguing math with one’s peers, with the teacher as a “facilitator” will not make the students see far, as they “will be standing on the shoulders of midgets.”
Statements such as “This suggests that instead of the expectation that skill in computation should precede word problems, experience with problems helps develop the ability to compute. Thus, present strategies for teaching may need to be reversed; knowledge often should emerge from experience with problems” (pp. 8-9) are without solid research behind them. Claiming that an approach “capitalizes on children’s intuitive insights and language in selecting and teaching mathematical ideas and skills” (p. 16) sounds nice, but it ignores the fact that mathematics in not “natural”, but rather a highly complex acquired skill. The “intuitive insights” of young children are more often than not a result of previous parental direct teaching, rather than some innate ability. One would not expect an average child to learn to play the piano by telling him to listen to a concert or play Bach - why would one then expect a child to learn mathematics this way?
The Standards say “All mathematics should be studied in contexts that give the ideas and concepts meaning” (p. 67). This is a naive and unproved approach that is often interpreted by teachers to mean that they should never teach any skills directly. Some amount of drill is important for the math student, just as the piano student needs to practice scales. This need should be clarified in the Standards.
8. Use of Calculators
The Standards send a mixed message. Statements such as “appropriate calculators should be available to all students at all times” (p. 8) are tempered by others like “We recognize, however, that access to this technology is no guarantee that any student will become mathematically literate” (p. 8).
Still, saying that “There is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations” (p. 8) and “Classroom experience indicates that young children take a commonsense view about calculators and recognize the importance of not relying on them when it is more appropriate to compute in other ways” (p. 19) borders on the irresponsible. If one needs evidence that calculator usage actively destroys the ability to calculate, one has just to step into one of many California classrooms. Even in affluent places like Palo Alto, students cannot mentally divide 300 by 3 (a procedure many eight graders failed to do properly in a recent test here).
9. Clarify the applicability of the examples to specific grade levels
Currently it is difficult to relate the examples to a specific grade level. K-4 or 5-8 spans too broad a spectrum of abilities, and the examples should be specific.
10. Add specific performance standards for each grade level
The Standards would be much more useful and helpful with such performance standards.
11. Objective, standardized, assessment
The Standards are replete with admonitions that standardized assessment is not sufficient and not indicative of real mathematical achievement. While there is nothing wrong with promoting alternative assessment, the importance of standardized assessment cannot be overstated, despite rhetoric to the contrary. Low results on standardized testing is what created the call for reform in the first place. In claiming now that these tests do not represent reality, the Standards are undermining the reform movement itself. Universities and the public are looking at standardized results, and if they do not show improvement, the reform movement is doomed.
12. Focus on content, not form
“The focus is on what happens in the classroom as students and teachers interact” (p. 189). Wrong. The focus should be on what students are to learn or have learned, and not how they learned it. The Standards should shy away from preaching a single philosophy of teaching, and then—even worse—promoting an assessment that evaluates not content but style.
13. Delete or modify the need for the assessment to be aligned with the curriculum
“Alignment is a critical issue in the development or selection of assessment instruments or in the use of assessment data. Teachers, test developers, and administrators all must be concerned about the alignment of curricula and assessment, although their interpretations of assessment and its scope may vary’ (p. 193). HOLD does not believe that this is a true statement. Assessment can be different from curriculum, especially if the curriculum claims that it is centered on problem solving. Isn’t an assessment a form of problem solving? Students should be able to cope with any form of assessment, if the Standards are right that the new way of teaching math does indeed produce mathematically powerful problem solvers.
In summary, the Standards’ call for simultaneous change of both the curriculum and assessment is unwise. No one who values the scientific method, would change the experiment, while changing the measurement criteria at the same time. But that is exactly what the Standards are supporting. Shouldn’t the Standards measure up to their own criteria? “standards often are used to ensure that the public is protected from shoddy products. For example, a druggist is not allowed to sell a drug unless it ...[includes] evidence of its effectiveness” (p. 2, emphasis added). The Standards must prove their effectiveness based on the original assessment that called them into existence, before trying to espouse new assessment.
HOLD Steering Committee
- Bill Evers
- Hanna Hoffman
- Michal Anne Plume
- Marina Polyak
- Ze’ev Wurman
