Response to Fourth Set of Questions from NCTM Commission on the Future of the Standards

MAA President's Task Force on the NCTM Standards

Reported by Ken Ross

Contents

Question 1. In previous responses, ARGs have highlighted the importance of mathematical reasoning, of deduction and formal proof, and of mathematical disposition. These topics relate to what might be called the "nature of mathematics." What should K-12 students learn about the *nature of mathematics*? At what grade levels? You might find it useful to consider Evaluation Standards 7 and 10 from the Curriculum and Evaluation Standards, which follow.

Evaluation Standard 7: Reasoning

The assessment of students' ability to reason mathematically should provide evidence that they can-

Evaluation Standard 10: Mathematical Disposition

The assessment of students' mathematical disposition should seek information about their-

Response

If, as seems to be the case, "mathematical disposition" is the assessment of students' mathematical disposition to seek information about a variety of things, we do not understand the inclusion of Evaluation Standard 10 here. Nevertheless, we want to concur most emphatically with the importance of the second and third bullets.

The remainder of this question deals with the "essence" of mathematics which we believe to be its abstraction. From it, the discipline derives its great power for application to a great number of seemingly unrelated areas. A singular feature of mathematics which gives it power is that it approaches a problem by stripping it down to the essentials and throwing away all irrelevant data. In this way, widely dissimilar problems are seen to come under a single heading while a simple idea turns out to be applicable to diverse situations. Often this process will transform what seems to be a hard problem into an easier problem. Mathematics teachers at all grade levels should be so well versed in this concept of the field that they transmit a sense of excitement, interest, and joy in having students learn an important fundamental discipline.

As students proceed through the grades, they should gain an understanding and appreciation of the exactness, elegance, and beauty of the field. They should be aware that precise logical thinking dominates in mathematics, so that terms must be clear and well defined. Hypotheses, whether implicitly understood or stated explicitly, must be taken into account in the solution of any problem. Precise logical thinking is the only way to communicate mathematics reliably and gain consensus about the workings of complex phenomena, make dependable predictions, and achieve universal applicability. We also stress precise logical thinking in mathematics, since otherwise many students find it difficult because of the sense of alienation they experience in trying to learn the subject. They perceive that it is a game whose rules have been set randomly and arbitrarily in ways that they cannot hope to understand. Consequently, most players (students) view mathematics as something that does not concern them. Under the circumstances, it would be a miracle if they could be persuaded to work hard to achieve its mastery.

Studies from around the world have shown that students' serious problems with mathematics begin when they study properties of fractions. We believe that the introduction of fractions in the elementary grades is often so lacking in logic and precision that the rules for adding, multiplying and dividing fractions are presented in a random and arbitrary manner. As a result, many students do not take what they learn seriously. It should be no surprise that when asked to add fractions a/b and c/d, many students opt to write (a+c)/(b+d). Similar difficulties occur with the rules for working with signed numbers and the laws of exponents.

  1. Students should be invited to make mathematical conjectures as soon as they have enough arithmetical knowledge to do so, and thereafter students should realize when they are making a conjecture and when they are firmly establishing a statement. For example, early in their elementary school careers students can be asked to start with a number that is a multiple of three, add together its digits, then add together the digits of the resulting number if it has more than one digit, and continue until the result has only one digit. The students can be invited to do this with a collection of different numbers they select themselves that are all multiples of three, and then make a conjecture based on the observed outcomes. The teacher should congratulate students on correct conjectures, but should stress right from the beginning that since the students have not shown this for every integer that is a multiple of three, they cannot yet be certain that this really is true for all such numbers. They can also explore the inverse of the conjecture by trying the same process on numbers that are not multiples of three. The advent of technology has made the numerical checking of many simple conjectures fashionable and painless. Unfortunately it may increase the likelihood of confusing--in the students' minds--the distinction between checking a general statement for a few special cases and knowing the reasoning that underlies that statement. We believe NCTM2000 would do well to underscore this distinction.

  2. Students should be given a gentle introduction to proof early in the elementary grades. For example, the teacher can ask students how many digits there will be in the product of two two-digit numbers. After the students experiment awhile and decide that there can be either three or four, the teacher can ask them how they can be sure there are no other possibilities. Perhaps spontaneously, and almost certainly with a little prodding, the students should be able to come up with an acceptable argument based on the two extremal cases. (While this depends on some order properties that the students will almost certainly not have studied rigorously, their proof will be appropriate and acceptably rigorous for their grade level.) This would also give the teacher an opportunity to point out that to answer a question like this, it is not enough for students to give examples, because they could give lots of examples to support the erroneous conclusion that the answer is always three.

    While they are in high school, students' experiences of proof should include other mathematics besides geometry. In elementary algebra classes they can be asked to give counterexamples showing that commonly used, but erroneous, formulas such as (a+b)2=a2+b2 are incorrect. Or they could be asked to find values of a and b for which such a formula is true (the difficulty of which might help bolster a sense for the general falseness of the formula). In grades 11 and 12, students could be asked to verify basic divisibility properties of integers, such as: if a, b, and c are integers and a is a factor of both b and c, then a is a factor of b+c.

  3. Students should be given an introduction to the principles of deductive reasoning, beginning in the early elementary grades with the notion that the truth of a statement does not imply the truth of its converse. They should also explore the truth and falsity of statements involving the quantifiers "some," "all" and "no." As the students progress through the grades, other items can be added to their store of knowledge about the principles of deductive reasoning. Before they are ready to take high school mathematics courses, they should know that one counterexample disproves a statement, but no finite number of examples that illustrate a general result is sufficient for a proof. By the time they reach the end of their high school mathematical career they should have learned all of the basic principles of deductive reasoning used in mathematical arguments, including argument by contraposition. Given a concrete argument within their scope, they should be able to identify the hypotheses and conclusions. They should also be able to pick up glaring logical errors and to identify simple reasoning from assumptions to conclusions. There should be some evidence that some of this can be formalized. But formalization should be on examples that are understood, not memorized, and for which the need to generalize using symbolic notation is recognized.

  4. Students should be given enough exposure to deductive reasoning from a system of axioms that they understand something of the axiomatic nature of mathematics. This is important, not just for mathematics, but also to make sure students understand that the end products of deductive reasoning in any situation are usually based on the underlying system of axioms that a person accepts as obviously true. It would seem that the appropriate place for this is the traditional one, namely somewhere in the middle of the high school years.

  5. The curriculum should avoid the inclusion of topics if they are not actually needed at a certain grade level and if they cannot be satisfactorily explained at that grade level. As a specific example, a detailed investigation about the decimal expansions of rational and irrational numbers should be deferred to grade 11 or 12. At that time, teachers should discuss how to take a repeating decimal and find its fractional representation and how it happens that the decimal expansion of a fraction is bound to terminate or repeat. This is also the appropriate time to prove results such as the irrationality of the square root of 2. When a number such as pi comes up in the earlier grades, it can simply be noted (to be discussed in greater detail later) that no matter how accurate one's calculator, it will never display an exact decimal representation for the number. Thus, when one substitutes a decimal version into an expression, one will always have to round it to a certain number of decimal places and use an approximately-equal sign. Practice working with such numbers in grades 6-9 can turn a discussion of irrationality in the 11th or 12th grade into a meaningful experience.

Question 2. What are the four or five most important geometric concepts or themes that should be included in the K-12 curriculum?

Response

First, we would like to express our hope that this question, and answers to it, will not limit the coverage of geometry to four or five concepts and hence lead to the continued neglect of geometry in the K-12 curriculum. Rather, we interpret the question as asking for help in identifying some key themes that can be used to strengthen geometry in the K-12 curriculum. We shall identify five possible themes below, but would first like to call attention to the current failure to devote at least a full semester to proofs in Euclidean geometry. We are concerned that many of the usual presentations of Euclidean geometry, with an emphasis on the foundational and obvious materials, do not appeal to most students. But by being selective in presenting the foundations, and by adopting the method of local axiomatics, students can be led quickly to beautiful and nontrivial theorems such as the nine-point circle and the Simson line. *These* should appeal to all students. We strongly advocate emphasizing Euclidean- geometry-with-proofs in the curriculum because NCTM's call for developing higher order thinking skills resonates with us. It seems to us that Euclidean-geometry-with-proofs is the ideal part of the high school curriculum to develop the ability to formulate conjectures (by simple drawings) and to check their correctness, often using multi-step reasoning. The visual appeal of the theorems provides additional incentive.

  1. Coordinate geometry, which reflects the relationship between geometry and number that arises from the fact that real numbers can be represented as points on a line.

  2. Similarity and its relationship to proportional reasoning. Similarity is a geometric idea that has many applications, one of them being a proof of the Pythagorean theorem.

  3. Measures of geometric quantities, such as length, area, volume, and angle, and the relationships between them in specific situations. This provides a connection between number and shape and includes the Pythagorean theorem and the constant ratio between a circle's circumference and its diameter.

  4. Visualization of geometric objects, particularly three-dimensional ones. (This is one place where computers can play a useful role.) In particular, students should be able to visualize and abstract various plane geometric theorems.

  5. Symmetry, in its broadest sense, is a very rich theme in geometry. From it one can develop the ideas of congruence, and it provides a natural way to introduce functions (transformations) into geometry.

We strongly recommend that students receive one full year of mathematics in grades 8-12 that requires them to deal directly with geometry and the visualization of geometric objects, in both two and three dimensions. This one-year coverage of geometry need not necessarily all take place in the same grade; it could be spread over two or more grades or with portions of other appropriate courses satisfying this requirement.

Question 3. In Round 3 responses, ARGs identified a number of areas within discrete mathematics that should be considered in Standards 2000, including iteration and recursion, graph theory, the binomial theorem, and combinatorics. Should there be a separate standard addressing discrete mathematics across grades K-12, or should these topics be dispersed among the other proposed Standards (number, algebra and functions, geometry, measurement, and probability and statistics)? You might find it useful to consider Standard 12 for grades 9-12 from the Curriculum and Evaluation Standards; which follows.

Standard 12: Discrete Math

In grades 9-12, the mathematics curriculum should include topics from discrete mathematics so that all students can-

and so that, in addition, college-intending students can-

Response

We favor dispersing these items among the other areas covered by the proposed standards. While discrete mathematics is an important part of mathematics, the K-12 curriculum is already crowded. We are concerned that if a great deal of emphasis is put on discrete mathematics as a separate topic, then it is likely to crowd out other things that may be more basic to a student's success in later mathematical studies.

We suggest that, of the areas within discrete mathematics listed in question 3, only the binomial theorem and basic combinatorics be included in the "core" high school curriculum. Moreover, they should be included among the topics for algebra II, though they will reappear in probability. In particular, we recommend that students in algebra II use fundamental counting principles to compute combinations and permutations; use combinations and permutations to compute probabilities; and use the binomial theorem to expand binomial expressions which are raised to positive integer powers.

Finally, we note that algorithms should be a part of just about every mathematics course. Sequences and recurrence relations would be appropriate topics in an elective course. On the other hand, there is probably no room for linear programming in the K-12 mathematics curriculum, not even for college-intending students.

OTHER AREAS WE WOULD LIKE TO ADDRESS

 4. The first question asked about the "nature of mathematics," which we have addressed without discussing statistics. However, statistics is finding a place in the curriculum. We strongly support the continued inclusion of statistics and probability in the K-12 curriculum. It should not be an optional topic.

5. We want to reiterate the importance of teachers raising their expectations of ALL students.

6. We have been advised that we should avoid discussing teacher preparation, because a different committee will be considering that issue. That committee will need to take account of our emphasis on the importance of students developing a gradually increasing ability to reason logically and deductively starting before high school. In particular, we are suggesting that middle school teachers should actually prove certain things for their students. We acknowledge here that we are not currently adequately preparing teachers for these tasks and recommend that curricular materials be written so that the mathematical facts and their rationales are clearly and explicitly stated. In particular, this means that, if a topic is introduced through a discovery activity, the students should receive follow-up materials giving a clear summary of the main insights to which they were supposed to be led by doing the activity.

7. The MAA Task Force on the NCTM Standards recommends that the revised Standards include a brief glossary of those terms in the revised Standards that have been the subject of misinterpretation or misunderstanding or are likely to be given different interpretations by different people. Such a glossary should be brief, only include terms actually appearing in the revised Standards, and not include the definitions of mathematical terms whose meanings are unambiguous across the various mathematical communities. Inclusion of such a glossary would go far to address one of the most frequently heard criticisms of the present NCTM Standards, namely that its readers have interpreted terms used in these Standards in entirely different ways. Giving clear definitions of such terms would be a very important contribution to clarity in the revised standards.

Here are examples of some terms which, if they appear in the Revised Standards, should be considered for inclusion in the glossary.

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