AMS Association Resource Group on NCTM Standards 2000

RESPONSE TO NCTM'S ROUND 3 QUESTIONS

Summarized by Mark Saul
January 1998

Question 1a.
What current advances in the field of mathematics might influence the updated Standards?

While there were some very specific answers to this question, a few general points of consensus were also agreed on:

a) The current changes in mathematics education focus more on pedagogical style and on deeper questions of what students get out of studying mathematics, (such as the nature and importance of mathematical reasoning)than on specific new content.

b) While some "new" content items are being emphasized (many concepts in combinatorics, transformational geometry, matrix algebra), these items are not part of twentieth century mathematics. They were, for the most part, developed in the last century. (An exception is the body of statistical techniques studied in K-12, some of which are quite recent developments.) As historians have noted, the 'revolutions' that some writers find in the progress of the physical sciences are absent in mathematics. So the basic ideas behind these "new" curricular items have not changed, and recent use made of them, even when appropriate to discuss on the pre-college level, would not be central to curricular or pedagogical decisions.

c) Two topics new to the curriculum are matrices and transformational geometry. The point was made that both should be integrated into other curricular areas more than is the case in current practice. For example, transformational geometry should be applied to produce results familiar from traditional geometry, and matrix algebra should be connected to the representation of geometric transformations or of other objects already studied in the curriculum. Some specific remarks are included in the appendix to this report.

d) The answer to this question is closely related to the next question, about the effects of developing technology on the curriculum.

Question 1b:
Are there areas of mathematics that will be especially important for the beginning decades of the 21st Century? In particular, how should discrete mathematics be treated in the updated Standards?

Several members noted that new topics in this field provide a context for students to reason mathematically with only a minimal mastery of procedural skills needed.

As for specific topics, transformational geometry and matrix algebra have already been mentioned. Another:

"A particular aspect of finite mathematics that is missing is facility with iterative schema. This is one of those overarching notions that appears in many fields, from computer science to nonlinear dynamics. Its classical appearance in school mathematics in its incarnation of inductive proof (for the advanced student only) is no longer sufficient (and was seldom done well)."

There was some interest in the idea of adding some graph theory to the curriculum. One member mentions the binomial theorem as a useful way to bridge the gap between discrete and continuous mathematics.

As for technology:

"1. Students should make appropriate use of technology, eventually emulating the way things are in the real world: details are always automated.

2. But: exactly because details are always automated, we must stress what isn't yet automated: formulating problems (that is, translating them into a form that requires only computation to complete); conceptual understanding of mathematical ideas that enables their actual use; ability to use several skills in sequence."

The point was raised that large parts of the frontier of mathematical research proceed without reference to computers.

It is suggested that the final document include examples of appropriate and also inappropriate uses of technology, or at least an indication of the benefits and pitfalls in using graphing calculators, symbolic manipulation software, and the like. Several members gave examples.

Question 1c:
Higher and technical education, both generally and in mathematics departments, have undergone and continue to undergo much change. How should these changes influence the perspective of the updated Standards?

There was not much reaction to this issue. One member noted, "There hasn't been that much. Calculus reform is not a major shift in content, rather more in pedagogy." Another says, "If I thought there was consensus on a vision at this point in time, it would be useful to communicate - but unfortunately I don't sense there is."

Question 2:
Focusing on the role of mathematics both for intelligent citizenry and for careers in the mathematical sciences and related fields:

The current NCTM Standards focus on providing quantitative literacy for the broad range of students in schools today. Historically, school mathematics has functioned to identify students who are mathematically able and channel them into higher-level mathematics courses. How might the updated Standards balance these two purposes of school mathematics? What are the benefits and costs of providing differentiated instruction, given the ever-increasing level of mathematical literacy needed by society?

There was consensus that differentiated instruction is necessary. Students differ, so curriculum and instruction must differ. There was consensus, with two dissenting opinions, that differentiated instruction takes place most effectively in homogeneously grouped environments, at least in the upper levels. Several members expressed the view that that as children grow, their needs differentiate, and the need for more time in homogeneous groupings grows. There was recognition that this is a politically hot issue.

Several members thought that the question's phrasing was misleading, since it assumes a conflict between teaching all and teaching the talented. The comment came up several times, from many members, that how we group students is not so important as the curriculum and pedagogy they're exposed to.

One comment: "We have to teach all kids. That necessitates differentiation: different kids learn differently. There should be enough flexibility in the curriculum so that every kid can find something interesting --- "interesting" does not mean "trivial"; trivial is seldom interesting, and kids have contempt for the trivial. The real issue is what you do with students, however you group them. This gets into curricular and pedagogic issues, and this is where the discussion belongs."

Some members took issue with the premise of the question. One member pointed out that the TIMSS data does not support the statement it makes:

"If you look at the 8th grade TIMSS results, we have 5% of our students in the top 10% internationally, and according to the English report on TIMSS, we have 25% of our students in the lower 25% internationally. If our mathematics program has been designed to identify mathematically able students and channel them into higher-level courses, we have failed miserably, and done a better job of keeping the poorer students to international levels than we have with our better students."

Question 3a:
Assuming that having students who are able to think analytically and flexibly is a desirable goal, what kinds of activity should students be engaged in across the K-12 curriculum to produce those ends? Is it important that students be engaged in inquiry and investigation mirroring at a lower level the kinds of activity in which mathematicians engage?

The group members expressed a consensus that a balance must be sought between technique and process. Certain algorithms should be internalized and made automatic, but students should be able to re-think them, or apply them in new contexts, when necessary. Discovery learning can play an important role, but is not always appropriate. Cooperative learning or small group work is a specific pedagogical technique which may or may not relate to the goal of establishing the correctly balanced classroom environment.

Question 3b:
Some people have been skeptical of the notion that precursors of algebra should be included in the elementary grades. However, some activities in the elementary grades may form an important basis for later understanding of algebra. Is it reasonable to include algebraic concepts in the elementary school? Which concepts are important? What activities might be useful in helping students develop initial ideas about algebra?

The group expressed an enthusiastic consensus that the mathematics of the early grades should be connected to what the students will learn later on. Whether this be called vertical integration or early introduction of advanced ideas is not important. Likewise, some members expressed doubt that we have a sufficiently sharp definition of algebraic thinking to distinguish it from other types of mathematical thought, but that the precise distinction is not so important as the early introduction of good mathematical ideas in general. In particular, geometric concepts should be part of the curriculum from the very beginning.

The concern came up more than once that elementary school teachers need support in implementing this idea, since many are not comfortable with more advanced mathematical concepts, and won't see the connections with what they are teaching.

A member (with agreement from others) gave some examples of concepts which can be introduced early:

"Primary grades (K-2): patterns, filling empty boxes in equations, t-charts and other precursors of functions, some explicitness about, e.g., commutativity, additive and multiplicative identities, etc. (but of course without the multisyllabic words).

Upper elementary grades (3-5): explicit use of variables in formulas (e.g., for area) and functions (e.g., distance as a function of time, cost as a function of quantity); more complex patterns and precursors of functions, e.g., sequences."

A member mentions, in addition, the importance of an early introduction of "spatial visualization and connections to areas other than geometry and how visual approaches help students access certain concepts (including number)."

APPENDIX

I. Some specific connections between traditional curriculum and new items

1) Showing that transformations can be used to prove theorems that one recognizes as geometry in the traditional sense. This is done to some extent in "College Geometry" by Howard Eves. But, to give an idea of the relatively primitive state things are in, let me mention that the nine-point circle, one of the elegant staples of the subject, though usually considered beyond high school level I believe, is almost totally attributable to the fact that the medial triangle (the triangle formed by the midpoints of the sides) and the original triangle are similar by a dilation by -2 around the centroid, but almost no one seems to recognize this.

2) Matrices are notation for transformations. What I think of as some key facts: a) matrix multiplication is the algebraic parallel to composition of transformations. b) the transformations given by matrices are linear transformations. c) Geometrically, this means: they preserve lines (and the origin). This is probably too deep to really treat in high school, but it can be told to students. d) Rotations are given by rotation matrices. This makes sense out of the addition formulas for sines and cosines, in terms of the group law for rotation matrices. e) The complex numbers can be thought of as transformations of the Euclidean plane (orientation-preserving, origin preserving similarities).

II. A specific bit of 20th-century mathematics was provided by a member.

In a paper written with Gerry Alexanderson, George Polya wrote the following: "The binomial coefficients belong to the curriculum of the secondary school, their connection with combinatorics is known since the days of Leibniz, Pascal and Jacob Bernoulli. The `Gaussian binomial coefficients' are much less widely known, their connection with combinatorics is of a more recent date. We thought that an exposition of some of the relations between Gaussian and ordinary binomial coefficients may add some zest to a traditional secondary school subject."

This paper appears in Elemente der Mathematik, and is an extended version of a paper written by Polya which appeared in the proceedings of a meeting on Combinatorical Mathematics and Its Applications. Both are reprinted in volume 4 of Polya's Collected Papers. Binomial coefficients count the number of minimal lattice paths from (0,0) to (n-k,k) with unit length for each step. They thus provide a refined counting of the number of such paths of length n. The Gaussian binomial coefficients prove a further refined counting, dividing the paths from (0,0) to (n- k,k) into equivalence classes according to the area under the paths, or by the number of inversions. There is an easier way to do this than Polya found, which was discovered by Schutzenberger in 1953. Consider the two paths from (0,0) to (1,1), and call them xy when the path starts to the right, and yx when it starts up. There is one unit of area in the second case and none in the first. This led Schutzenberger to define a noncommutative multiplication, yx=qxy, with xq=qx and yq=qy. With this multiplication, the q-binomial theorem in the noncommutative form is

sum from 0 to n of G(n,k)*x^(n-k)*y^k
where G(n,k) is g(n)/g(k)g(n-k) and g(n)=1*(1+q)* ...*(1+q+q^2+...+q^(n-1))

The proof that Polya gave continues to work in this setting, and is easier than a proof of the binomial theorem since one can use the Pascal triangle type identities (there are now two) to derive what the coefficients in the expansion are.

This is very important late 20th century mathematics, since it is a major part of what lies behind the quantum group SU(2,q), which is starting to play a role in certain areas of physics. In fact, the discovery of quantum groups really came from the desire to understand some questions in mathematical physics. There are also a number of uses of quantum groups in different areas of mathematics.

Let me strongly suggest that all of you read the paper by Polya and Alexanderson, and then I will show how the noncommutative version helps make Polya's arguments easier. If Polya thought his version was suitable for high school when the only applications were in statistics and number theory, how much more he would have felt it was appropriate now that there is a more conceptual way of looking at these problems, and the result is a foundation for some very important contemporary work. Students are introduced to noncommutative multiplication when dealing with matrices. Here is a completely different way of seeing how mathematics changes and becomes richer when some of the old parts are extended by dropping or changing something that once seemed fixed and unchangeable.

III. A comment on new developments in statistics.

Current advances in statistics (and, I suspect, in most fields that employ math heavily) are largely driven by the continuing revolution in technology. New classes of models that (unlike the classical general linear model) require computationally-intensive iterative fitting are now standard. New ways of attacking even quite simple problems (think of the bootstrap and related resampling methods) require fast/cheap computing. Topics such as data mining, pattern recognition, nonparametric function estimation and the like are the hot areas of current research. Rarely in the history of the discipline has so much happened so quickly (even as the boundaries between statistics and information science in general are blurring).

IV. A specific picture of students "imitating" mathematicians:

Yes, it is important that students be engaged in inquiry and investigation mirroring at a lower level the kinds of activity in which mathematicians engage. Based on observations in classrooms serious about aligning themselves with the 1989 Standards, on some of the NSF-supported curriculum material, and on articles in and preprints for NCTM journals, the following picture of best practice emerges:

a) Starting in kindergarten, the mathematics classroom needs to be collegial, with students explaining what they mean, questioning each other, coming up with conjectures (which may be settled or may be left as conjectures) and fairly frequently encouraged to work together.

b) Teachers need to reflect student dialogue as well as direct it.

c) Somewhere around middle school, kids need to be introduced gently to more formal reasoning: we know this fact, what can we conclude? At this point the reasoning does not need to be directed towards a goal. For example: "If I know that a triangle has one right angle, what else do I know about it?"

d) At some point kids start proving theorems. Let's define our terms: a proof is a convincing argument from generally accepted mathematical principles by generally accepted logical rules, i.e., I'm not proposing a complete axiomatic development of, say geometry. Some sense of what the basic mathematical principles are is needed (e.g., when giving an algebraic argument we can assume arithmetic), and this can be tricky.

e) At some point the logical rules need to be made explicit: modus ponens, contrapositive, quantifiers, proof by contradiction, induction -- that kind of thing.

f) Modelling. Kids have to get some sense of what it means to use mathematics to model the so-called real world, whenever the mathematics lends itself to modelling, starting in K and going all the way through 12.