The MAA and the New NCTM Standards
by Kenneth A. Ross
In 1989 the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics (the "1989 Standards"). This document had considerable national influence on textbook development and both pre-service and in-service teacher training, but it also stirred up a good deal of controversy. In 1996 the MAA and several other professional organizations were each asked to designate or appoint a committee to help the NCTM reassess the 1989 Standards and aid in revising them. The MAA appointed the President's Task Force on the NCTM Standards to "serve as a review group that will provide sustained advice and information concerning K-12 mathematics and the NCTM Standards." From 1997 through early 1999, this Task Force responded to four sets of questions supplied by the NCTM and also to a draft of the proposed revision of the 1989 Standards. The final version of the revision was published in April 2000 as Principles and Standards for School Mathematics ("PSSM").
Members of the PSSM writing groups have indicated that input from the MAA Task Force was very helpful in their work. Part of the reason may be that our Task Force was able to make consensus reports despite our sometimes diverse views. This article is an attempt to indicate to what extent the concerns of our Task Force are reflected in PSSM. The short answer is that our concerns were heard and that many of them are reflected in PSSM. In what follows, we will focus on a few of our main concerns and will comment on how they have been addressed in the relevant portions of PSSM.
We believe that PSSM outlines an ambitious, challenging and idealized program whose implementation would be a vast improvement over the current state of mathematics education. We hope that the mathematical community will now focus its energy on helping this program achieve its potential, especially by improving teacher education and by becoming more involved with producing high quality precollege texts and teachers' supplements.
CONCERN 1: Most members of the Task Force wanted the revision of the 1989 Standards to be more specific with regard to both skills and expectations for students' intellectual growth from one grade level to the next. They believed that many classroom teachers share this desire and also felt that performance expectations by grade would help states formulate their own standards and would ease the problem of student movement from one region to another. A few members felt, however, that expecting specific performance standards by grade is unrealistic since neither the United States nor Canada has a national curriculum and the variability across states and provinces is great.
COMMENT: Most Task Force members were disappointed that PSSM does not specify performance expectations for each grade.
CONCERN 2: Task Force members were concerned that mastery of basic skills was not sufficiently addressed in the 1989 Standards. None advocated "mindless drill," but they agreed that drills of important algorithms, which enable students to master topics while at the same time learning the mathematical reasoning behind them, can be used to great advantage by a knowledgeable teacher.
COMMENT: PSSM addresses this concern as follows: "Knowing basic number combinations -- the single-digit addition and multiplication pairs and their counterparts for subtraction and division -- is essential. Equally essential is computational fluency -- having and using efficient and accurate methods for computing. Fluency might be manifested in using a combination of mental strategies and jottings on paper or using an algorithm with paper and pencil, particularly when the numbers are large, to produce accurate results quickly. Regardless of the particular method used, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate and general." (page 32) "And experience suggests that in classes focused on the development and discussion of strategies, various 'standard' algorithms either arise naturally or can be introduced by the teacher as appropriate. The point is that students must become fluent in arithmetic computation -- they must have efficient and accurate methods that are supported by an understanding of numbers and operations. 'Standard' algorithms for arithmetic computation are one means of achieving this fluency." (page 35)
"Part of being able to compute fluently means making smart choices about which tools to use and when." (page 36) "Teachers must maintain a balance, helping students develop both conceptual understanding and procedural facility (skill)." (page 77)
CONCERN 3: Task Force members expressed the belief that the 1989 Standards did not sufficiently address issues of mathematical reasoning, the need for precision in mathematical discourse, and the role of proof throughout the curriculum. The growing appreciation of the important role of experimentation and conjecture in mathematical thinking may have obscured the fact that reasoning is the foundation of mathematics. While science verifies through observation, mathematics verifies through logical, deductive reasoning. Students need to be consciously aware of the distinction between exploring topics and providing more rigorous arguments. In particular, they need to use terminology in a precise manner and be able to specify their hypotheses, particularly if these are implicit.
COMMENT: In PSSM there is significantly more balance of emphasis than in the 1989 Standards between pure mathematics, where reasoning and proof are so important, and exploration and applications. It is notable that the 1989 Standards avoided the use of the word "proof" whereas this word is used throughout PSSM.
The general Reasoning and Proof Standard in PSSM does a good job of addressing the importance of proofs and logical reasoning throughout the curriculum. Here are some quotes: "By the end of secondary school, students should be able to understand and produce mathematical proofs -- arguments consisting of logically rigorous deductions of conclusions from hypotheses -- and should appreciate the value of such arguments." "Reasoning and proof cannot simply be taught in a single unit on logic, for example, or by 'doing proofs' in geometry. . . . Reasoning and proof should be a consistent part of students' mathematical experience in prekindergarten through grade 12." (page 56) "Beginning in the elementary grades, children can learn to disprove conjectures by finding counterexamples." (page 59)
Among the content standards, proofs are given the most attention in the Geometry Standard which "includes a strong focus on the development of careful reasoning and proof, using definitions and established facts." (page 41) "Through the middle grades and into high school, as they study such topics as similarity and congruence, students should learn to use deductive reasoning and more-formal proof techniques to solve problems and to prove conjectures. At all levels, students should learn to formulate convincing explanations for their conjectures and solutions. Eventually, they should be able to describe, represent, and investigate relationships within a geometric system and to express and justify them in logical chains. They should also be able to understand the role of definitions, axioms, and theorems and be able to construct their own proofs." (page 42)
Despite the emphasis on rigorous thinking, it should be noted that the proposed use of reasoning in geometry, while rich, is quite different from the classical presentation in which the geometry is developed as a logical sequence of theorems based on axioms. The following summary is from the Grades 9-12 Geometry Standard. "Judging, constructing, and communicating mathematically appropriate arguments, however, remain central to the study of geometry. Students should see the power of deductive proof in establishing the validity of general results from given conditions. The focus should be on producing logical arguments and presenting them effectively with careful explanation of the reasoning, rather than on the form of proof used (e.g., paragraph proof or two-column proof). A particular challenge for high school teachers is to integrate technology in their teaching as a way of encouraging students to explore ideas and develop conjectures while continuing to help them understand the needs for proofs or counterexamples of conjectures." (page 310)
Later, in the Reasoning and Proof Standard for Grades 9-12, it is pointed out that "students should understand that having many examples consistent with a conjecture may suggest that the conjecture is true but does not prove it, whereas one counterexample demonstrates that a conjecture is false. Students should see the power of deductive proofs in establishing results. They should be able to produce logical arguments and present formal proofs that effectively explain their reasoning . . ." (page 345)
Additional specific aspects of proofs and reasoning are addressed in the Communication Standard. "In order for a mathematical result to be recognized as correct, the proposed proof must be accepted by the community of professional mathematicians." (page 61) "For some purposes, it will be appropriate for students to describe their thinking informally, using ordinary language and sketches, but they should also learn to communicate in more-formal mathematical ways, using conventional mathematical terminology, through the middle grades and into high school. By the end of the high school years, students should be able to write well-constructed mathematical arguments using formal vocabulary." "The process of learning to write mathematically is similar to that of learning to write in any genre. Practice, with guidance, is important. So is attention to the specifics of mathematical argument, including the use and special meanings of mathematical language and the representations and standards of explanation and proof. . . . As students mature, their communication should reflect an increasing array of ways to justify their procedures and results. In the lower grades, providing empirical evidence or a few examples may be enough. Later, short deductive chains of reasoning based on previously accepted facts should become expected. In the middle grades and high school, explanations should become more mathematically rigorous and students should increasingly state in their supporting arguments the mathematical properties they used." (page 62)
The Communication Standard also includes a good section emphasizing the importance of using the language of mathematics to express mathematical ideas precisely. "Beginning in the middle grades, students should understand the role of mathematical definitions and should use them in mathematical work. Doing so should become pervasive in high school." (page 63)
"Students in grades 3-5 . . . need to know that posing conjectures and trying to justify them is an expected part of students' mathematical activity." (page 191) "This involves creating classroom environments in which intellectual risks and sense making are expected." (page 197)
Here are the statements pointing out that students need to be consciously aware of the distinction between exploring mathematical topics and providing rigorous arguments: In grades 3-5, "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." (page 188) "Using technology, students can generate many examples as a way of forming and exploring conjectures, but it is important for them to recognize that generating many examples of a particular phenomenon does not constitute a proof." (page 41) The issue is touched upon in the Reasoning and Proof Standard: "Students may not always have the mathematical knowledge and tools they need to find a justification for a conjecture or a counterexample to refute it. . . . Teachers can point out that a rigorous proof requires more knowledge than most high school students have." (page 57)
CONCERN 4: Task Force members were concerned that the 1989 Standards' focus on "mathematics for all" appeared to neglect the needs of "some." They felt that, while there are good reasons to emphasize that all students can learn mathematics, the revision of the 1989 Standards should address the needs, especially at the high school level, both of those students who will eventually become significant users of mathematics and also of the majority of students who will not.
COMMENT: At the outset, PSSM claims that "there is no conflict between equity and excellence." It goes on to say: "PSSM calls for a common foundation of mathematics to be learned by all students. This approach, however, does not imply that all students are alike. Students exhibit different talents, abilities, achievements, needs, and interests in mathematics. Nevertheless, all students must have access to the highest-quality mathematics instructional programs." (page 5) "Students with exceptional promise in mathematics and deep interest in advanced mathematical study need appropriate opportunities to pursue their interests." (page 369) Unfortunately, there is little follow-up on these statements nor any clear indication as to how the different needs of students should be addressed.
The first principle in PSSM is the Equity Principle: "Excellence in mathematics education requires equity---high expectations and strong support for all students." This is clarified as follows. "Equity does not mean that every student should receive identical instructions; instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students. . . . Likewise, students with special interests or exceptional talent in mathematics may need enrichment programs or additional resources to challenge and engage them." (pages 12-13) Again, there is little indication of how these goals should be accomplished.
The Grades 9-12 Standards emphasize that "all students are expected to study mathematics each of the four years that they are enrolled in high school." "These Standards describe an ambitious foundation of mathematical ideas and applications intended for all students." (page 287) "A great deal is demanded of students in the program proposed here, but no more than is necessary for full quantitative literacy." (page 288) We are not sure what the last sentence means, but it is hard to believe that it can apply to all students.
Here is the most complete statement in PSSM concerning the different needs of students, but it also seems vague. "High school students with particular interests could study mathematics that extends beyond what is recommended here in various ways. One approach is to include in the program material that extends these ideas in depth or sophistication. Students who encounter these kinds of enriched curricula in heterogeneous classes will tend to seek different levels of understanding. They will, over time, learn new ways of thinking from their peers. Other approaches make use of supplementary courses. For instance, students could enroll in additional courses concurrent with the program. Or the material proposed in these Standards could be included in a three-year program that allows students to take supplementary courses in the fourth year. In any of these approaches, the curriculum can be designed so that students can complete the foundation proposed here and choose from additional courses such as computer science, technical mathematics, statistics, and calculus. Whatever the approach taken, all students learn the same core material while some, if they wish, can study additional mathematics consistent with their interests and career directions." (page 289)
CONCERN 5: Task Force members agreed that in order to grow intellectually, students must have significant intellectual demands placed upon them. The 1989 Standards emphasized the responsibility of our profession to stimulate and "mathematically empower" students, but they did not simultaneously emphasize the necessity for students to work hard and stretch their attention spans. Students need to be aware that mathematics is not an inborn inherited trait and that everyone finds it difficult at some stage. They need to realize that in order to succeed they must not stop even when discouraged or frustrated.
COMMENT: This concern is addressed in PSSM in several places. "Regardless of the context, worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work." (page 19) "When challenged with appropriately chosen tasks, students become confident in their ability to tackle difficult problems, eager to figure things out on their own, flexible in exploring mathematical ideas and trying alternative solution paths, and willing to persevere. . . . Students should view the difficulty of complex mathematical investigations as a worthwhile challenge rather than as an excuse to give up. Even when a mathematical task is difficult, it can be engaging and rewarding." (page 21) "Students respond to the challenge of high expectations, and mathematics should be taught for understanding rather than around preconceptions about children's limitations." (page 77)
In the Problem Solving Standard, it is emphasized that students "should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and should then be encouraged to reflect on their thinking." (page 52) It is unfortunate that the importance of verifying or proving that their solutions are valid was not mentioned here.
In the Grades 6-8 Standard, after pointing out students' misconceptions about the nature of mathematics, it is proposed that "to counteract negative dispositions, teachers can help students develop a tendency to contemplate and analyze problems before attempting a solution and then persevere in finding a solution." (page 259)
There are a few comments about the importance of independent work, as distinguished from group work, but no indication of the relative emphases on these two approaches. I was able to find only one statement about homework: "All middle-grades and high school students should be expected to spend a substantial amount of time every day working on mathematics outside of class, in activities ranging from typical homework assignments and projects to problem solving in the workplace." (page 371)
CONCERN 6: Task Force members were concerned that the 1989 Standards recommended the inclusion of more topics (including statistics and discrete mathematics) and types of activities than the traditional curriculum without giving sufficient guidance about what material should be reduced or eliminated and how to achieve this. The result was a serious risk of superficiality, i.e., a curriculum that is a "mile wide and an inch deep."
COMMENT: PSSM contains five Content Standards: Number, Algebra, Geometry, Measurement, and Data Analysis and Probability. While the 1989 Standards contained an explicit Discrete Mathematics Standard for grades 9-12, PSSM incorporates the main topics of discrete mathematics into the other Standards spanning the K-12 years.
PSSM does not explicitly address the issue of how to deal with the increased number of topics and activities proposed for inclusion in the K-12 curriculum. Unlike the 1989 Standards, however, it suggests that "in the middle grades, the majority of instructional time would address algebra and geometry." (page 30) This would ease the pressure on the high school curriculum but may be difficult to implement.
As PSSM readily admits, the middle school geometry standards will not be easy to implement. ". . . significantly more geometry is recommended in these Standards for the middle grades than has been the norm. The recommendations are ambitious -- they call for students to learn many topics in algebra and geometry and also in other content areas. To guard against fragmentation of the curriculum, therefore, middle-grades mathematics curriculum and instruction must also be focused and integrated." (page 212)
PSSM rarely specifies topics than can be dropped or de-emphasized. But in connection with the customary U.S. and metric systems studied in grades 3-5, PSSM states that "they do not need to make formal conversions between the two systems at this level."
CONCERN 7: The Task Force focused on content rather than pedagogy, so it had little influence on the more pedagogical standards. Task Force members agreed that the Standards should acknowledge that students learn in a variety of ways and that good pedagogical models should reflect that reality. They also wanted it noted that teachers teach in a variety of effective ways and should be encouraged to develop a good balance of approaches. They recommended that the revision of the 1989 Standards should explicitly caution against any doctrinaire adoption of one particular pedagogy exclusively.
COMMENT: PSSM does not contain the kind of explicit caution recommended by the Task Force, but the general concern which led to the recommendation is addressed: "Teachers have different styles and strategies for helping students learn particular mathematical ideas, and there is no one 'right way' to teach." (page 18)
When PSSM discusses pedagogy, however, it tends to focus on student self-discovery, probably to encourage traditionally-trained teachers to experiment with new approaches. But some of its statements may be construed as minimizing the value of correcting misconceptions before they become entrenched. "A pattern of building new learning on prior learning and experience is established early and repeatedly, . . ., throughout the school years." (page 21) "To maximize the instructional value of assessment, teachers need to move beyond a superficial 'right or wrong' analysis of tasks to a focus on how students are thinking about the tasks. Efforts should be made to identify valuable student insights on which further progress can be based rather than to concentrate solely on errors or misconceptions." (page 24) The pedagogical vignettes seem convincing through grade 5 but are less so later on. Given the large number of topics proposed for the middle and high school curricula, it would appear necessary that the percentage of time devoted to group-discovery activities decrease in later grades, but this issue is not addressed.
Here is a sample of statements in the Representation Standard that try to balance self-discovery with learning conventional forms of mathematics. "Students should understand that written representations of mathematical ideas are an essential part of learning and doing mathematics. It is important to encourage students to represent their ideas in ways that make sense to them, even if their first representations are not conventional ones. It is also important that they learn conventional forms of representation to facilitate both their learning of mathematics and their communication with others about mathematical ideas." (page 67) "Teachers can gain valuable insights into students' ways of interpreting and thinking about mathematics by looking at their representations. They can build bridges from students' personal representations to more-conventional ones, when appropriate." (page 68)
CONCERN 8: Task Force members expressed considerable concern about the recruitment and education of future teachers, especially since the 1989 Standards appeared to demand much more of teachers than the traditional curriculum.
COMMENT: The NCTM has explicitly deferred this important issue to other documents and groups, but the authors of PSSM do make some relevant comments. A full page (page 146) of the Grades 3-5 Standards is devoted to this issue. This page includes references to "mathematics teacher-leaders" (persons who have particular interest and expertise in mathematics) and to "mathematics specialists" in the upper elementary grades. The conclusion is: "Ensuring that the mathematics outlined in this chapter is learned by all students in grades 3-5 requires a commitment of effort by teachers to continue to be mathematical learners. It also implies that districts, schools, and teacher preparation programs will develop strategies to identify current and prospective elementary school teachers for specialized mathematics preparation and assignment."
In connection with grades 6-8, it is observed that the "capacity of schools and middle-grades teachers to provide the kind of mathematics education envisioned needs to be built. Special attention must be given to the preparation and ongoing professional support of teachers in the middle grades. . . . Professional development is especially important in the middle grades because so little attention has been given in most states and provinces to the special preparation that may be required for mathematics teachers at these grade levels. . . . . In order to accomplish the ambitious goals for the middle grades that are presented here, special teacher-preparation programs must be developed." (page 213)
And in connection with grades 9-12, we are reminded that "these Standards are demanding. It will take time, patience, and skill to implement the vision they represent. The content and pedagogical demands of curricula aligned with these Standards will require extended and sustained professional development of teachers and a large degree of administrative support. Such efforts are essential." (page 289)
The crucial role of teacher education is recognized in the last chapter, Working Together to Achieve the Vision. "The reality is simple: unless teachers are able to take part in ongoing, sustained professional development, they will be handicapped in providing high-quality mathematics education. The current practice of offering occasional workshops and in-service days does not and will not suffice. Most mathematics teachers work in relative isolation, with little support for innovation and few incentives to improve their practice." (page 370) "The typical structures of teachers' workdays often inhibit community building, but structures can be changed. In some cultures, shared discussions of students and teaching are the norm. In Japan and China, the workdays of teachers include time for meeting together to analyze recent lessons and to plan for upcoming lessons. . . . Finding ways to establish such communities should be a primary goal for schools and districts that are serious about improving mathematics education." (page 371) A good case is made in support of the following statement: "There is an urgent and growing need for mathematics teacher-leaders -- specialists positioned between classroom teachers and administrators who can assist with the improvement of mathematics education." (page 375)
CONCERN 9: The task force did not address the issue of technology per se, but it did express general concerns about how technology may be used in mathematics education.
COMMENT: In The Technology Principle, the authors of PSSM express the belief that technology is an important and powerful tool. "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning." (page 24) The following is asserted. "Students' engagement with, and ownership of, abstract mathematical ideas can be fostered through technology. Technology enriches the range and quality of investigations by providing a means of viewing mathematical ideas from multiple perspectives. Students' learning is assisted by feedback, which technology can supply, . . ." Nevertheless, there are cautions. "Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions. . . . The effective use of technology in the mathematics classroom depends on the teacher. Technology is not a panacea. As with any teaching tool, it can be used well or poorly." (page 25) In the Grades 6-8 Standards, it is stated that "students should consider the features of [a] problem and the likely use of an answer to a calculation in deciding whether an exact answer or an estimate is needed, and then select an appropriate mode of calculation from among mental calculation, paper-and-pencil methods, or calculator use." (page 220)
Technology is encouraged from the beginning. "The mathematics program in prekindergarten through grade 2 should take advantage of technology. Guided work with calculators can enable students to explore number and pattern, focus on problem-solving processes, and investigate realistic applications. Through their experiences and with the teacher's guidance, students should recognize when using a calculator is appropriate and when it is more efficient to compute mentally." (page 77) "This set of Standards reinforces the dual goals that mathematics learning is both about making sense of mathematical ideas and about acquiring skills and insights to solve problems. The calculator is an important tool in reaching these goals in grades 3-5. However, calculators do not replace fluency with basic number combinations, conceptual understanding, or the ability to formulate and use efficient and accurate methods for computing. Rather, the calculator should support these goals by enhancing and stimulating learning. . . . Students at this age should begin to develop good decision-making habits about when it is useful and appropriate to use other computational methods, rather than reach for a calculator. Teachers should create opportunities for these decisions as well as make judgments about when and how calculators can be used to support learning." (pages 144-145)
Kenneth A. Ross is Professor of Mathematics at the University of Oregon. Among may other important MAA activities, he is chair of the President's Task Force on the NCTM Standards.
The various reports and responses from the President's Task Force on the NCTM Standards are all available online.
MAA Online is edited by Fernando Q. GouvÃªa (firstname.lastname@example.org).