1. Focusing on changes in mathematics and in undergraduate education:

a. What current advances in the field of mathematics might (or should) influence the updated Standards?

b. 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?

We shall address these two questions together, since both ask about how progress in mathematics might affect the content areas presented in pre-college mathematics. One of the factors that must be kept in mind as we consider current advances is that the field of mathematics is dependent on standard basic skills that must be acquired and mastered before much substantive progress can be made.

One of the biggest changes in mathematics education in recent years is that we are beginning to teach students to recognize the features it shares with the experimental sciences. Not only do mathematicians solve problems and prove theorems, they have to FIND the problem to solve or the theorem to prove. This is especially true of applied mathematicians, though mathematicians in all areas of mathematics use experimentation, some making substantial use of the computer. One of the strengths of the Standards is that they make clear that mathematical inquiry has an experimental aspect. However, there needs to be a general warning that technology is often abused in the hands of less well-informed people. For example, the data furnished by a computer cannot serve as replacement of a proof.

The questions about what new areas or topics should be treated in the updated Standards stimulated a great deal of discussion among members of the Task Force, with little that we all agreed on. For one thing, the question is multi-faceted. At what grade levels should the new topics be treated, and to what depth? What would be taken out of the curriculum to make room for the new topics? How would they be integrated into the curriculum? There was concern that we were unable to identify current *standard* components of the curriculum that could be replaced, which led to a reluctance to encourage the introduction of new topics. There also seemed to be concern that we could easily end up with a curriculum of even greater breadth and less depth than we have now. In view of all this, the Task Force is not making any recommendations at this time concerning new areas for the updated Standards. We will continue our discussion and may write a supplementary report on this topic. Or individual members of the Task Force may submit their own reports.

The term "discrete mathematics" means different things to different people, and its meaning should be clarified in any serious discussion. In this response, we use the term in the rather narrow sense of including graph theory, a little set theory, elementary combinatorics and the principle of induction, since most of the other topics that are sometimes viewed as discrete mathematics (logic, algorithms, sequences and recursion, and probability) are referred to explicitly in this or previous responses. There are also topics, such as Boolean algebra, automata and semigroups, that are covered in discrete mathematics courses in college, but which we assume are not relevant here. For the record, areas that were discussed in response to question 1b included discrete mathematics, exploratory data analysis, probability, optimization and recursively defined sequences. In addition, in our earlier responses we recommended a greater emphasis on logical reasoning and proof.

We interpret this question as follows: There have been changes in pedagogy in many post secondary programs. How should these changes reflect the recommendations made by the Standards for pedagogy in school mathematics?

We are not convinced that the pedagogical models to which the students have been exposed in pre-college courses have a major impact on their ability to deal with the pedagogical models that they encounter in college. The one (increasingly rare exception) to this statement is that some students, who have had a K-12 experience within a single learning style, react quite negatively to instructional methods involving a mixture of lecture, individual discovery, cooperative activities, and extended oral and written feedback. Thus, we are more concerned that the pedagogical models used in K-12 are appropriate for students at their grade levels than we are about how pedagogy in post secondary programs will affect that in K-12 education.

The Standards should emphasize that students learn in a variety of ways, and good pedagogical models should reflect that. They should also note that teachers teach in a variety of ways and need to have a good balance of approaches so that they can appeal to the way that is most effective for them and their students in a particular situation. The Standards should explicitly caution against any doctrinaire adoption of one particular pedagogy exclusively.

First and foremost, our mathematics expectations of all students must be higher than in the past so that whether they intend to continue their formal education or not, they will strive to perform at a reasonable level based on their potential.

The Standards should strongly encourage all students to study mathematics throughout their school years. Students who are not college bound should have as sound and solid an education in mathematics as possible since they may not be exposed to the discipline again. With a strong mathematical background, they will be in a good position to enter the work force and be acceptable for the more challenging positions. On the other hand, students who are college bound who have a firm understanding of mathematics will have the opportunity to enroll in a broad range of courses that are mathematically based! In either case, the result will be adults who are more flexible citizens, able to cope with a continually changing world as they enter the ever increasing information age.

The obvious benefits of providing differentiated instruction are that students who are genuinely prepared for more challenging mathematical experiences will be able to have them. They are more likely to remain interested in mathematics and mathematically-related fields (and able to pursue that interest) at the point in their lives when they are making genuine career decisions.

One obvious cost is the visible one involving the resources needed to provide differentiated instruction. Another, more subtle and serious societal cost comes from students "leaking out of the mathematical pipeline" when they are misplaced into less challenging mathematical tracks even though they are capable of handling the more rigorous ones. We are not saying that the "less challenging" track is automatically inferior to the common track that would be there if the "more challenging" one were not available. But this does happen in some cases where, for example, "business math" programs have been put in place in high schools as an alternative to more traditional sequences of courses.

Concerning differentiated instruction, there is substantial agreement that, at some point, there has to be some branching of the curriculum to allow each student to be appropriately challenged, while achieving the highest feasible level of mathematical proficiency for that student. One issue is how early in the curriculum such branching should take place. An argument for relatively early branching is to assure that many students are not under-challenged. However, in this case certain essential safeguards must be put into place to deal with the worst abuses. These would include:

(a) More instruction for teachers and counselors on proper evaluation and counseling of students that is less influenced by the students' socioeconomic and ethnic backgrounds and gender. We recognize that this is an apple-pie-and-motherhood type of recommendation, but the issue is too important to ignore. We are optimistic that, with suitable preparation, most teachers and counselors can avoid actions and decisions based on unwarranted group-wide assumptions. Preparedness for mathematics courses can and should be measured, as already done in many schools, by the use of diagnostic examinations which are regionally and nationally available. Of course, the results need to be carefully interpreted and used wisely.

(b) Differentiated paths through the curriculum are designed carefully so that if a student is misplaced into a less challenging path, either by the student's own underestimation of his or her preparedness or by that of others, then it is possible to correct this within a reasonable period of time and get the student back into the proper path. This design should also allow for less traumatic transitions for students who move from one school district to another.

It is especially important that these safeguards be in place if the curriculum branching takes place before the last year of high school. This is one of the arguments for keeping all students together as long as we can, in which case we must assure that students are not under-challenged. This would require commitment and resources on the same scale as those described in (a) and (b).

Some members of the Task Force are prepared to make a concrete proposal concerning the curriculum branching. They propose "Starting with Algebra 2", which in most cases would be the 10th grade. We suggest that courses including Algebra 2, trigonometry, pre-calculus and calculus be available to future college science-math students, while courses useful to the general citizenry, like statistics and discrete mathematics, be available to the others. This would assume the ability to offer courses of the latter kind which are less technically oriented than, but equally educational as the other courses.

a. 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?

One activity in which mathematicians and scientists engage, while in the pursuit of solving problems, is the making of conjectures. Such conjectures are usually based on the accumulation of evidence and the intuition that comes from experience. As is consistent with the Standards, students across the K-12 curriculum should have opportunities to engage in exploratory mathematical activities and to make mathematical conjectures. They should discover, however, that the ultimate truth of a reasonable conjecture cannot be based *solely* on the accumulation of evidence and intuition. Indeed, they must realize that there is a substantial difference between a proof and a conjecture no matter how many examples they can demonstrate that satisfy a given general statement. Students can appreciate what is involved here best if they have a solid grounding in logical reasoning to an extent appropriate for their grade level. We have already addressed this in point 5 of the "Reservations and Concerns about the Standards" in our first report to the NCTM Commission. The AMS ARG also addressed this in their statement that:

**While the notion of logical deduction is not completely lacking in the description of K-8 education given in the Standards, the ARG discussion suggests that this strand could be made more prominent and more coherent. In particular, there is a need, once filled by the standard geometry class, for students to learn basic syllogistic logic, including notions such as converse, inverse, and contrapositive.**

Logical reasoning and deduction are the most important activity, in which mathematicians engage, that should be mirrored in the K-12 curriculum. The Standards should make it clear that careful logical reasoning and deduction are crucial not only for evaluating the results of the investigative and exploratory activities which it encourages, but also for the actual conduct of such activities in all but the most trivial cases. In addition, the Standards should encourage awareness about the explicit and implicit assumptions on which conclusions rest.

b. 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?

This question seems to be designed to address one way or another the objections of some skeptics, in which case it would be helpful to know the nature of this skepticism, and in particular to have some description of the "precursors of algebra" whose introduction in the elementary grades is raising the hackles of these skeptics.

We suspect that this question is really about *symbolic* algebra. Given that, we favor the gentle introduction of the notion of variable quantities early in students' mathematical career in K-12. The use of variables are pervasive in our society and can be introduced in a natural way in the earliest grades. Indeed, the use of letters to represent numerical quantities shouldn't be treated as any more unusual than the use of words to represent objects and ideas.

Another interpretation is that this question is about algebra as viewed by the public, i.e., the laws governing addition and multiplication and their relationship, the negative numbers, and word problems. These, together with variables, are part of the "popular" conception about algebra. With this interpretation, our answer is the same: The ideas should be introduced early. Using a box or a circle or an "x" to hold the place for the name of a number we don't know yet, but hope to find out, one can start that with 6-year olds! The distributive law and the rest of the algebraic structure pack a lot more punch in algebra when their usefulness has been seen in earlier years. For example, the distributive law is what makes the standard algorithm for multiplication work. A good way to motivate the product of negative numbers, with proper preparation, is to assure that the "laws" that govern the arithmetic of nonnegative numbers also work for negative numbers. Also, the number line should be drawn so that it does NOT end at the left with a point labeled "0", but has points to the left of 0 which just have not yet been named.

In general, we advocate starting to sensitize students to mathematical language, especially with truth and falsity of quantified statements, at a very young age. Once variables have been introduced, questions requiring more and more logical sophistication can be posed in grades 7-12. One can start with ones like "For what values of x is 10x = x, or xx = 1 or 6x = x6?" which only require imagining what happens when x takes different values. Next, because it's easier for students to grasp the idea of counterexample than proof, one can go on to ask things like "True or false? For all real numbers a and b, (a+b)^2 = a^2+b^2. Justify your answer." Then one can ask questions like "Is the equation x(x+5) = x^2+5x true for all real numbers x? Justify your answer" which students can answer merely by citing a known property, such as the distributive law. Finally, one can ask students questions that require them to prove simple general statements. For instance: "Indicate whether the equation below is true for all values of a and b, for some values of a and b but not for others, or for no values of a and b. Justify your answer. a^2 + b^2 = -1." Besides those found in geometry courses, other appropriate problems for high school involve divisibility properties of integers, properties of rational numbers, properties of logarithms, one-to-one and onto functions, increasing and decreasing functions, and properties of trigonometric functions.