AMS Association Resource Group on NCTM Standards 2000

RESPONSE TO NCTM'S ROUND 4 QUESTIONS

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

• use inductive reasoning to recognize patterns and form conjectures;
• use reasoning to develop plausible arguments for mathematical statements;
• use proportional and spatial reasoning to solve problems;
• use deductive reasoning to verify conclusions, judge the validity of arguments, and construct valid arguments;
• analyze situations to determine common properties and structures;
• appreciate the axiomatic nature of mathematics.

ARG Summary response:

All members of the AMS ARG agree that mathematical reasoning, deduction, and formal proof are part of the nature of mathematics and should be part of the mathematics curriuclum. According to the accepted usage of the word "nature," this means that mathematics is not mathematics without these items. The question is not if, but how much and when.

The bottom line is that at every point mathematics should make sense to every child. From the beginning, kids should routinely be encouraged to ask questions about why things work, encouraged to disagree with what they think is faulty reasoning, and encouraged to explain why they think their assertions are true.

Simple examples of mathematical reasoning should be introduced in courses at the elementary school level, moving to more and more of it as the student advances. Too often in the past mathematical reasoning has been left to the geometry course. The use of the areas for basic geometric figures to compute the areas of more complicated figures in elementary and middle school is an example of mathematical reasoning as is the derivation of formulas in algebra. That fact should be explicitly understood by students. Examples of deductive reasoning should be part of every course at the high school level. Students should learn the fact that mathematics can be developed on an axiomatic basis during high school. Although it is easily possible to carry this too far, students should become aware of the coherent nature of mathematics.

Students should understand the difference between a proof and a collection of examples. They should understand the difference between a true mathematical statement (which can be proved, whether or not it is proved at this point in the curriculum) and a useful mathematical model for a "real world" phenomenon (which is always an idealization that does not catch all the details of the phenomenon).

Mathematical disposition and the items in Evaluation Standard 10 are really aspects of the understanding of mathematics. Of course it is the understanding of mathematics that we are trying to achieve, and so these are goals that we should aim for in all mathematics courses.

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

ARG Summary response:

The members of the AMS ARG mentioned the following topics:

Geometric objects: Lines, circles, triangles, three dimensional shapes, and their basic properties such as parallelism, constancy of angle sums in triangles, perpendicularity, the Pythagoream Theorem,

Measurement: Area, perimeter, volume, surface area

Transformations: A vital part of geometry, used to study symmetry, congruence, & similarity. Transformations should also be used in proofs. Some members of the group observed that the notion of transformation and invariance is the basic one in geometry, and that everything mentioned in this list are ways to describe transformations and what remains invariant under a given transformation.

Proofs: Geometry needs to be more than deductions from axioms, but that is not the problem now. Now we need to get a few more proofs back into mathematics classes. Students should understand the logical coherence of geometry. That understood, it should also be said that there needs to be proofs in algebra as well as geometry, and even informal ones in some arithmetic classes.

Analytic geometry: Start with arithmetic and the number line. Equations and graphs of lines, planes, and conics. Analytic geometry should be used as a method of proof.

There is divided option on the need to study what is sometimes called vector geometry in high school. However, this aspect of geometry, together with the use of matrices, is being introduced earlier and earlier in the college/university curriculum, simply because it is needed in so many application areas. It is only a matter of time before this urgency reaches our secondary schools.

Another strong recommendation we would like to make to NCTM is that geometry should be emphasized as early as possible, and developed in parallel to arithmetic. Too often, geometry before 9th grade involves merely learning the names of figures and applying formulas as exercises in arithmetic (or sometimes algebra). This is certainly necessary but geometry should receive even more early emphasis.

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

• represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations;
• represent and analyze finite graphs using matrices;
• develop and analyze algorithms;
• solve enumeration and finite probability problems;

ARG Summary response:

There was unanimity among the members that discrete mathematics should not be a separate standard. On the other hand, certain elements of discrete mathematics should be included in the precollege curriculum.

A list of those items would include a reasonable treatment of binomial coefficients, elementary counting techniques (permutations and combinations), Pascal's triangle, and the binomial theorem. A treatment of matrices, in particular matrices related to geometry, with rotation matrices related to trigonometry, and the determinant interpreted in terms of volume. Experience with proof by induction is desirable. A good discussion of algorithms, including recursion and iteration.

It is easy to get carried away in making up such a list. The AMS ARG members want to caution against making the K-12 Standards too much of a wish list. A careful delineation of a solid but realistic curriculum would be valuable. There is just so much one could wish for.