## Recent Trends in High School Mathematics, Opinions of Bert Fristedt

### Introduction

• How to succeed in calculus
• How to succeed in a college level liberal-arts-type mathematics course
• Mathematics for the workplace

I regard as unsatisfactory new materials like `Core-Plus' and `Interactive Mathematics Program'. By contrast I think curricula based on traditional mathematics topics are appropriate, although for any such curriculum, there would be places where I would recommend some changes. Much of what is written below is in support of the position described in this paragraph.

### How to succeed in calculus (in order of decreasing importance)

• Enter the course with the ability to do a wide variety of algebraic manipulations quickly and accurately, even when the constants in expressions are denoted by letters rather than specific numerals.
• Ditto for trigonometric manipulations (unless the calculus course is a short course which does not include the calculus of the trigonometric functions, in which case trigonometry need not even be a prerequisite).
• For each credit of a college calculus course give 3 hours of concentrated effort per week, counting both in-class and out-of-class effort, and distribute this effort rather evenly among the weeks and within each week.
• Enter the course knowing the concepts of `function', `inverse function', and `set'.
• Enter the course with the ability to do simple analytic geometry problems with accuracy and speed; for instance, be able to find the intersection of two given lines and to immediately write down the equation of a circle with a given center and radius, even when some of the given information involves arbitrary constants rather than specific numerals.
• Have the habit of writing mathematics in a very organized manner so that in a somewhat lengthy problem you can refer back to earlier steps accurately for use in later steps, and also so that you can easily check your work.
• Enter the course with the ability to turn precisely stated story problems into mathematics---I use the term `story problems' to emphasize that I am not thinking of problems involving significant technical information from some field outside of mathematics.
• Be able to quickly analyze simple situations involving similar or congruent triangles.
• Enter the course with the ability to use a hand-held calculator well. Some understanding of recursive definitions and their relation to making computer calculations would also be helpful.

Notice that none of the items concerns high mathematical aptitude; my 35+ years of experience tells me that hardly anyone fails in college calculus who has study habits appropriate for a college level course and adequate prerequisite knowledge and skills.

With respect to the top two items in the above list, I observe that it is true that machines can do many very complicated algebraic and trigonometric manipulations. However, these calculations come in the middle of many calculus solutions and one needs to automatically do these manipulations in order to keep one's focus on the major aspect of the problem; and it might even be that one wants to try a variety of manipulations in order to see what will make completion of the solution possible. As one simple example for those with a calculus background: In evaluating limits of quotients involving radicals, sometimes one wants to rationalize the denominator and other times the numerator.

### How to succeed in a college level liberal-arts-type mathematics course

• Enter the course with an understanding of the laws of exponents, the skill to smoothly perform intermediate algebraic manipulations, and the ability to solve quadratic equations expressing answers in simplest form with square roots if necessary.
• Enter the course with the ability to do simple arithmetic in your head---so, for instance, if weighted voting systems are discussed, the arithmetic involving one- or two-digit weights is so automatic that you can focus on the concepts being introduced, not the arithmetic.
• Be prepared to do calculations involving simple geometrical figures.
• Enter the course with the ability to use a hand-held calculator for elementary arithmetic, including expressions involving exponents.
• Enter the course with the ability to turn simple story problems into precise mathematics, because a college level liberal-arts-type mathematics course will likely have story problems involving several sentences.
• For each credit of a college course give 3 hours of concentrated effort per week, counting both in-class and out-of-class effort, and distribute this effort rather evenly among the weeks and within each week.
• Have the habit of writing mathematics in a very organized manner so that in a somewhat lengthy problem you can refer back to earlier steps accurately for use in later steps, and also so that you can easily check your work.

### Mathematics for the workplace

The mathematics that is used in manufacturing to decide if too many defective parts are being produced is the same mathematics that is used to construct good public opinion polls. The mathematics that governs the motion of a weight hanging on a spring is that same as that governing current in an electrical circuit. Whether or not an application is used as motivation for a mathematical topic, the end goal should be understanding and skill with the mathematics itself, separate from any particular application---because only then can the mathematics be used in a variety of applications.

The major aspects of algebra, geometry, and trigonometry as taught in the nineteenth century will find uses in the twenty-first century, even though many of the things to which they were applied in the nineteenth and twentieth century have become obsolete. Of course, mathematics, like all fields of knowledge, grows over time, but this growth does not make previously developed mathematics obsolete to the same extent that growth in other areas renders previous developments obsolete. Therefore, mathematics as a focused intellectual discipline combined with skills for carrying out its various algorithms and creativity in choosing from among its many concepts is especially useful for those whose working careers undergo many changes.

Mathematics is centered around its definitions and theorems, and these are often useful in the workplace. A carpenter uses the Pythagorean Theorem to create a right angle. I myself, in a summer job while still a student, used the precise definition of `definite integral' that I had just learned in advanced calculus. I was able to help engineers decide exactly which definite integrals they needed to calculate---despite the fact that at the beginning of a 2-hour conversation with them I knew nothing about the project on which they were working.

Many `real-world' situations are introduced in `Core-Plus' and `Interactive Mathematics Program'. Nevertheless, my opinion is that these materials, as compared with traditional materials, lessen the chance that the student will be able to use the learned mathematics in the workplace---because these new materials give insufficient practice with calculational algorithms and too little emphasis on definitions and theorems.

Some applications of mathematics should definitely appear in high school mathematics courses. For instance, compound interest should be taught in algebra. Three reasons: (i) It is a widely used application of mathematics that will not become obsolete. (ii) It does not require extensive knowledge outside of mathematics. (iii) It is intimately connected with the laws of exponents. Applications which do not have property (i) are better taught on the job than in the school room. Those that have property (i), but not property (ii), might belong in school classes such as physics, chemistry, biology, agriculture, and social science classes but not usually in mathematics classes.

This section has been designed to address mathematics-to-job concerns. But I do not accept the view that preparation for the workplace should be the only goal of education.

In my opinion, the most significant changes in new materials such as `Interactive Mathematics program' and `Core-Plus' occur in the subject matter itself, not in a delivery method such as an `integrated viewpoint'---and these changes in subject matter are very far-reaching, so far-reaching that even if I were in favor of such changes I would want them to be made in small steps---a reasonable precaution since so many people would be affected.

To advocate such changes in subject matter at the same time as advocating changes in the delivery system is to advocate a total package that cannot be carefully evaluated in a time-span of a few years.

Detailed comments related to these concerns can be found on separate pages via the following links:

In particular, the first of the above four links will clarify my assertion that the changes in subject matter from the traditional to `Core-Plus' and `Interactive Mathematics Program' are very far reaching.

Four items, 120B.02, 120B.031, 120B.10, and 120B.31 of Minnesota Statues 2000, Chapter 120B, which concern graduation rules and Profiles of Learning, focus more on strengthening requirements than changing them. They seem to anticipate more involvement of parents than I suspect has been the case. One item in these documents with which I strongly disagree is Subdivision 12 of 120B.031. It specifies who should be on the Examination and Evaluation Panel, and the specifications do not include faculty from the technical colleges, community colleges, state colleges, private colleges, and the University of Minnesota who will teach courses for which the graduating high school students must be prepared. The input of such people is especially crucial for mathematics where skill with prerequisite material is often the most important factor in how well a student does in these courses.