Keys to Success: What Students Should Learn in Mathematics
Summary: In these brief remarks I will broadly outline the mathematical background that is required for a student intending to follow a career in science, engineering, economics or other technical fields. I will also discuss to what degree we are successful in providing this knowledge, and how this impacts on the learning and teaching of mathematics at the university level. These comments are based on my own experiences as an educator, past department chairman, and research mathematician at the University of Wisconsin, but they reflect widespread views among the mathematical community.
It is well known that prosperity in our highly competitive global community is fueled by a steady stream of technological and scientific advances, all of which inevitably contain a mathematical core. Likewise, successful careers in science and engineering all require a thorough and rigorous understanding of basic mathematics. The level of sophistication of the mathematical tools used say in a Wall Street firm has increased dramatically over time, and similarly many biological disciplines are becoming increasingly quantitative in nature. It seems clear that if our undergraduates are to compete successfully in this highly technological society, then they must be provided with adequate mathematical skills.
A high school student aspiring to follow a successful career in science, engineering, economics or any other technical field must obtain a solid mathematical training before she or he enrolls in college. For this to be realistic, a student has to enter college with at least a solid high school pre-calculus background, or, even better, a course in calculus as preparation. Obtaining a realistic comprehension of the depth and complexity required in the mathematics of science and engineering is a critical goal at this stage. A key objective must be to ensure that the student understands mathematical concepts, which is very different from memorization. This involves both knowing the concepts as well as being able to apply them in a variety of settings. The following is a brief summary of the main topics which are required as background for learning higher mathematics such as calculus:
Arithmetic, Algebra and Symbolic Manipulation: the ability to handle mathematical expressions and correctly apply rules of arithmetic and algebra is fundamental, and requires extensive practice. Students should know how to solve systems of linear equations, the general solution of a quadratic equation, polynomial identities, rules of exponents , etc.
Geometry and Trigonometry: the elements of plane geometry, including congruent and similar triangles, ratios, the Pythagorean Theorem, angles, sines, cosines, trigonometric identities, areas and volumes; understanding lines, circles, ellipses, parabolas, etc. using their defining equations and geometric representations.
Functions and Sets: understanding the notion of a function between two sets; extensive familiarity with examples including polynomials, trigonometric functions, logarithms, exponentials; ability to graph and interpret functions.
This list is of course not an exhaustive one, there are other important (but perhaps less fundamental) topics to consider, such as for example: geometric series and their application to computing interest rates, elements of probability theory, basic notions of statistics, etc.
The broad topics outlined above should be taught in a manner which makes clear that mathematical proofs necessarily underlie the whole subject, but at the same time providing abundant `real world' examples. At each stage, problem-solving skills should be nurtured, and students should be constantly challenged. The use of calculators and computers is of course an important component in solving mathematical problems, but they should simply be considered educational tools, as they are no substitute for mathematical reasoning. More generally the student should acquire a level of comfort in manipulating mathematics at an appropriate level. This means being able to apply the concepts to solving problems in a consistent, systematic and accurate manner, thus obtaining an appropriate measure of self-confidence which will be indispensable when moving on to courses at a higher level. Clinging on to sheets of formulas without a sound theoretical knowledge is counterproductive. The student should be taught to fully comprehend the intrinsic value of a precise mathematical definition, and to appreciate the logical steps behind mathematical thinking.
I have outlined the basic mathematical background required for taking more advanced mathematics courses in college. Of course it would be ideal for a student to cover calculus in his senior year at high school. However, this subject can be successfully taught to college students within their first two years, provided they have an adequate preparation along the lines we have described above. One should keep in mind that calculus and sometimes even more advanced mathematics courses are a prerequisite for practically every technical major offered at a university; indeed mathematics plays the role of a fundamental language for an enormous number of diverse disciplines.
Unfortunately, the current situation seems to be that too many incoming university students have not obtained an adequate mathematical background in high school, and they are poorly prepared to learn mathematics at a higher level. Essential skills in basic subjects such as arithmetic and algebra are often weak, and due to this poor preparation every year, nationwide, hundreds of thousands of incoming undergraduates must take remedial math courses, which are comparable to courses in early high school or even middle school. Unfortunately, this huge remediation effort does not provide much help in steering students towards careers in science or engineering. Once they have fallen behind, the vast majority of these students will opt for a non-technical major.
The reasons for this situation are not hard to find, and the fault does not lie with the students. Far too many of them simply do not receive a proper mathematical education in K-12, regardless of the curriculum that may have been used in their schools. If essential skills in arithmetic, algebra and geometry are not adequately developed early on, these deficiencies will build up over time, and when the student enrolls at a university, they are usually hard to remedy. A lack of mathematical background becomes increasingly difficult to overcome as courses reach higher levels. Once students have fallen behind it becomes almost impossible to catch up. As a consequence of a deficient K-12 background, these mathematically deprived students are denied the opportunity to unleash their natural talents in areas such as science, engineering and economics.
Hence we have a paradox: while society now requires an increasingly sophisticated level of mathematical training, it seems that average students have declining mathematical skills. Of particular concern are the often deficient math skills of prospective elementary school teachers that we see in the college classroom; this most certainly has a direct impact on the quality of their instruction, thus perpetuating the vicious circle of low performance. One of the key objectives of the administration's Math and Science Initiative is to improve the math skills of prospective and in-service K-8 teachers. To make a difference this will require the involvement of research level mathematicians in structuring better courses for current and future teachers.
As most of us know, US doctoral programs are the best in the world, and a large share of the world's leading scientists and mathematicians hold American degrees. However the situation is not as good as it sounds. The truth is that most math graduate programs rely very heavily on the influx of bright young scholars from abroad. Students from Asia, Latin America and Eastern Europe are eager to seize the opportunity of a higher education in the United States (indeed I am such an example, as I received a college degree in Mexico, a Ph.D. at Princeton University and today I am a United States citizen). In contrast, the pool of native born American talent is insufficient, and many domestic students are simply too poorly equipped to compete with foreign students. This is often due to an undergraduate preparation which is substantially weaker than the international standard, caused by an inadequate K-12 preparation for college; catching up can be a formidable challenge. Once again this is a result of a less than optimal preparation at a lower level, which telescopes into deficiencies at each subsequent step in an individual's education.
Looking at the situation one step further we see that when hiring mathematics and science faculty, foreign applicants will very often have a much stronger record than their American counterparts. Hence our universities hire large numbers of foreign scientists and engineers. These people are welcome additions to our talent pool, but it is a sad commentary that it comes at the expense of our own children.
To summarize, one can identify two basic problems: (1) attracting more students into mathematics, engineering, science and other technical areas, and (2) providing them with the necessary skills for success. These problems arise early on in our educational system, but as we have seen they can telescope out to the highest levels, impacting the entire mathematical enterprise in our country. Although solutions are not easy to find, at the very least we should make a concerted effort to significantly increase the involvement of research mathematicians in K-12 education issues. Given their knowledge of higher mathematics and extensive teaching experience, mathematicians (as well as other scientists and engineers) can surely make critical contributions in helping to address the problems we have discussed.
We cannot afford to ignore the deplorable consequences of the current state of affairs. In order to ensure the continued success of American science and industry we must develop a strategy to significantly raise the mathematical standards in our educational system. In the absence of decisive action, the problems which we are witnessing are likely to cascade into a disastrous avalanche.