Mathematics Areas in Grades 9-12, Opinions of Bert Fristedt

The comments below concern a standard four-year mathematics program. In them I give no particular attention to either calculus for the accelerated students or appropriate omissions for those taking only three or fewer years of high school mathematics. All areas mentioned here appear in `Core-Plus' and `Interactive Mathematics Program' but with the emphases and numbers of pages so different from those of traditional programs that I feel compelled to say that the changes in subject matter from the traditional to these programs are massive.

Major Topics

It is a contradiction in terms for someone to say that they know algebra and trigonometry and to be unable to perform a wide variety of algebraic and trigonometric calculations in an efficient manner. Skill is the key for these two subjects, and to acquire a skill, one must practice extensively; this emphasis on practice is missing in `Core-Plus' and `Interactive Mathematics Program', or at least delayed far too long.

For story problems, skill is also the main feature. Here the skill is that of translating verbal descriptions into mathematics and then giving a verbal interpretation of a mathematical answer. Strange as it may seem to those who have glanced at `Interactive Mathematics Program' or `Core-Plus', I think these curricula shortchange the student in this area also. It seems to me that the attempt to treat very realistic issues in which mathematics might play a useful role has resulted in there being very few story problems that both involve a significant amount of mathematics and are sufficiently straightforward so that an average student could do the entire problem in a fully satisfactory manner working alone. Rather the complications of the problem are likely to lead to vague inconclusive discussions.

Somewhere in the 9-12 curriculum, students should work on writing proofs, not just for isolated theorems, but in the context of a coherent package of definitions and theorems. An axiomatic treatment of plane geometry is one place where the proof-writing can occur. When this approach is taken, one goal should be for all students to develop to the extent that they can prove that the diagonals of a parallelogram bisect each other and other theorems at the same level of difficulty, One should also help a large number of students develop to the point where they can write a proof requiring them to introduce lines not specified in the theorem.

Analytic geometry is a necessity in the high school curriculum. For this reason, there is an argument for excluding from the curriculum the axiomatic presentation of geometry described in the preceding paragraph. Two precautions in case of such an exclusion: (i) The geometry in `analytic geometry' must be given full weight; for instance many exercises involving triangles and circles should appear. (ii) Conceptual proofs should appear somewhere in the curriculum.

Certain aspects of algebra, trigonometry, and geometry that were in my high school curriculum a half-century ago should be eliminated in the curriculum of today. The use of tables and interpolation in connection with the trigonometric, logarithm, and exponential functions is no longer relevant. The half-year I had of solid geometry should be somewhat reduced. These eliminations leave more room for the following topics than was available when I went to high school. Indeed the last four of these topics were not in my K-12 curriculum, but at least three of them belong in the K-12 curriculum of today.

Further Topics

In my grade school and junior high school we spent considerable time organizing data, constructing pie charts, bar graphs, and so forth---and we calculated quartiles in connection with exercises on converting between fractions and decimals and percentages. My education was fine in this regard, and it would also be fine today. `Core-Plus' and `Interactive Mathematics Project' give considerable attention to organization of data and drawing conclusions from it---at the cost of sacrificing other aspects of mathematics.

Consider, for instance, problem 4 on page 58 from Course 1 of `Core-Plus. For each of the 50 states, the percentage of people speaking a language other than English at home is given. The students are not asked to calculate these percentages from population data. An extra, but important mathematical twist, could be added by giving populations in thousands of people, and asking students to calculate percentages. The problem focuses on two types of states: outside (or border) states and inside (or interior) states. There are data differences between the two types which the student is asked to illustrate in various ways. The student is also asked a non-mathematics question: ``Why do you think there are these differences?'' The teachers guide mentions immigration and also separates the border states into two categories, those where immigration is high and those where immigration is low. What isn't clear, even in the teachers guide, is whether the patterns of immigration are being obtained from the data or whether the patterns of immigration are taken as known and then used to explain the data. In any case, the teachers guide does not even mention the possibility of certain groups holding on to their languages over many generations in some states but not others.

I would not have included the preceding discussion of a particular problem did I not think it typical of how often there is a rather interesting story for which the mathematics is minimal or nonexistent. This problem belongs at an appropriate place in a social studies course, and even in such a course, the students could be expected to use their calculators to do the grade school arithmetic of calculating percentages from population data.

Discrete mathematics is a loosely defined term that encompasses combinatorics (which treats systematic methods of counting), number theory (which treats properties of whole numbers connected with factorization), graph theory (where `graph' has a meaning different from its use in phrases such as `graph of an equation' or `graph of a function'), and modular arithmetic (sometimes called clock arithmetic). My own grade school education gave considerable attention to number theory and some to graph theory (via puzzles). Discrete mathematics should get more attention in K-12 than it did a half-century ago. My view is that when traditional-type materials are chosen for high school and especially for middle school attention should be given to the sections on discrete mathematics. Let me emphasize however: In my opinion it is much better to use good traditional-type materials, together with a purchased discrete mathematics supplement, than it is use materials such as `Interactive Mathematics Program' or `Core-Plus'. Discrete mathematics can occupy a well-defined separate unit in the curriculum. A slight modification would be for combinatorics to be integrated with probability. It is not important that all aspects of discrete mathematics be treated; focus is more important than breadth. A further comment: Discrete mathematics is a good area for stressing conceptual proofs in case plane geometry is not used for that purpose.

Apart from whatever materials are used, a major challenge for the teacher is to integrate teaching about calculators with teaching about mathematics. For instance, it is important that students develop independent ways of checking answers for reasonableness. I cringe, only internally I hope, when someone in my class gives an answer larger than one for a probability or an answer giving the three side lengths of a triangle with the largest of the three numbers being larger than the sum of the other two. This type of student oversight seems to arise more often among students who rely heavily on a calculator than among students who use it more sparingly.

Some basic properties of probability are: (i) the probability that at least one of two events occurs equals the sum of the probabilities of the two events if the events are mutually exclusive; (ii) if they are not mutually exclusive a correction has to be made---say, by subtracting the probability that both events occur; (iii) the probability that both of two events occur is the product of their probabilities if the events are independent. Just using these basic facts one can do a rather wide variety of interesting story problems. The symbolism for writing these basic properties is simple and easy to remember, and can become easy to use in a variety of contexts with practice. Situations for which these properties are relevant are discussed rather extensively in Lesson 3 of Unit 5 of `Core-Plus 3'. Nevertheless, it is very difficult to find these properties clearly identified in a nice neat compact form. [In fact, it is only in the last problem of the lesson that the student is asked to write a symbolic rule for the probability that at least one of two events occurs in case the two events are not necessarily mutually exclusive. Then the student is asked to pose and solve one problem using this rule. While it is true that a few earlier problems lead towards this problem, there are no follow-up story problems (and there should be several) where the student has to use the symbolic formula.] And Venn diagrams, one of the good contributions to high school mathematics to come out of the `new math' of a few decades ago, can be used to make the first two of these properties transparent; but Venn diagrams do not seem to be used in this lesson.

I have seen a preliminary version of Unit 5 of `Core-Plus 4', a unit that treats binomial distributions and statistical inference. There is quite a nice sequence of a few pages on `Characteristics of Experiments' which introduces students to the types of errors that can occur in setting up experiments to be analyzed statistically. It describes precautions such as `double-blind experiments'. My opinion of the rest of the unit is unfavorable. For instance, the formulas for the mean and standard deviation of the binomial distribution appear as mere assertions, but there are many ways to connect these to general definitions and theorems in a very natural way. Some of these ways involve interesting algebraic manipulations involving factorials, and students in the fourth year should practice such manipulations.

Some introduction to vectors and matrices might be appropriate in high school, but only if connections to geometry are made or row operations of matrices are treated in conjunction with solving simultaneous linear equations. (Of course, `matrix', as defined in a standard dictionary, might be used in many courses where structure is involved.)

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