A friend and colleague writes:
I have a question for you. U.S. education seems to require extensive higher math in high school and college, even for kids going into fields like journalism where calculus won’t be used much. I would prefer to see more balance where we instead require all kids to learn HTML and basic coding in high school and college. That would seem a might wiser route to go. I would appreciate your thinking on educational priorities.
This is an excellent question. The traditional high school math curriculum (geometry, algebra, trigonometry, and in modern times, functions) was designed to prepare students with the skills necessary for problem solving in calculus. Although calculus is arguably the greatest engineering achievement of humans and is also a thing of great beauty and elegance, and although it used to be the standard minimum requirement in mathematics for a high-quality liberal arts bachelor degree, the fact is most students now don’t take calculus in college. For students not preparing to take calculus we ought to be teaching other things in high school math.
The most important aspect of curriculum in math and computer science is mathematical reasoning and problem solving, which are central to critical thinking in general. Anything we do as educators to change curriculum must insure this focus as a core thread. Certainly learning algebra provides a vehicle for the development of mathematical reasoning and problem solving. But what content, other than algebra, could provide a focus for this development? (Contrary to popular belief, understanding basic beginning algebra-now usually taught in the eighth grade-is in itself an important development toward a well-educated adult. It should not be optional, though its timing should be flexible.) The traditional curriculum comprises years of algebra beyond basic algebra. What could we teach instead?
Certainly statistics and probability are much more useful content areas for the average educated person. Sadly, these topics are ignored by the traditional curriculum, so we’ve had generations of non-calculus learners forced to study boring and useless algebra while failing to learn to read critically a newspaper article about risk of disease. This is explained strikingly and humorously in Innumeracy, by John Allen Paulos. Many high schools now have modernized their college-bound curriculum by replacing pure algebra or trigonometry with “FST,” functions, statistics, and trigonometry. Some colleges and universities have developed “Finite Math” courses that count toward a degree for non-technical majors. These are steps in the right direction.
Computer science is another pertinent area amenable to development of mathematical reasoning and problem solving. However, we have to be careful; practically useful content and computer literacy in the usual sense do not necessarily have the depth and rigor required for this development. Learning to program (using Java, say) and studying algorithms and data structures are great in this regard. But, for example, basic HTML is not. Now, a rudimentary understanding of basic HTML is essential to being computer literate, and I think it ought to be part of the curriculum for that reason. But true programming in the computer science sense is different. An interesting problem with adopting programming as a vehicle for learning mathematical reasoning and problem solving is that the skills are esoteric and most students find it boring. Hmm…that sounds similar to algebra and calculus. But certainly programming has tremendous appeal to some and should be supported as an option.
Some further thoughts:
Content and critical thinking are both important. If you ask an individual Calculus II teacher whether they’d prefer a student who had learned all the basic secondary topics (content) in Calculus I or a student who missed those but was capable of basic problem solving, the teacher will always choose the problem solver. But if you ask the same question of the calculus teacher regarding basic algebraic symbolic manipulation skills versus problem solving, they will probably prioritize that basic content over problem solving. So the importance of content depends on the content’s connections with current endeavors. Interestingly, a committee of calculus teachers typically cannot agree on what secondary content to remove from the syllabus.
As an aside, I think it’s important to consider that what seems like extensive higher math in college is actually not that extensive, or high, or out of balance in the context of a liberal education. We study Shakespeare, which is at least as esoteric as calculus, and certainly literature has many obvious alternative curricula available. Though the detailed answers to “why” vary dramatically from field to field, the answers are analogously the same: we believe earnestly that a well-rounded liberal education stimulates a person to acquire motivation, discipline, and background content necessary as a life-long independently thinking learner in an open and democratic society.
The devaluation of this philosophy due to prioritizing the capitalist need for skilled workers, the commoditization of education itself in a pseudo free market, and the development of students as “customers” is all concomitant with relaxing degree requirements, grade inflation, requiring performance based learning objectives, and other deplorable developments in higher education. (The relationships between a free market and education standards and pricing are well explained in Howard Hotson’s essay, “Don’t Look to the Ivy League.”) Though I do not suggest giving up gains in popular access to higher education, we should not lose sight of the benefits of a liberal education. I think that to get a four-year piece of paper from an accredited university, one should read some Shakespeare and learn a little calculus, though the cows are already out of that barn. One result of the commoditization of higher education is that high-quality college degrees are becoming once again the privilege of the rich. The rest of us can get degrees in “Packaging” or “International Business” at the lower-tier state university. What we really needed was more public money for less wealthy college applicants while maintaining intellectual quality standards of grades and degrees.
I happened upon this due to a search. I don’t mean to plug my own efforts, but when I started on a (possibly ill-advised) project to write my own calculus textbook, I thought about one that might actually have a liberal-artsy philosophy behind it, so I started with symbolic logic. We can get our math-for-nontechnical-majors students to understand truth tables and much symbolic logic, and it comes in handy when giving epsilon-delta definitions and proofs, and getting them to understand what the theorem that a convergent series’ terms must go approach zero in the limit says and does not say. My book can be found in preliminary form a couple links from my site, and I once wrote a paper arguing that we should teach symbolic logic in high school to get the critical thinking juices going earlier, and help students understand the logical flow of even simple algebra problems (like when you do and don’t have to “check” the answers that are spit out of the equation solving). I do agree about the critical thinking, and the idea that the math we teach is arguably less esoteric than Shakespeare. Anyhow, I’m not a lurker here but found this and saw I could leave a comment, so I did. Thanks. -Mike Dougherty, Weatherford, OK (USA)
Teaching symbolic logic (or formal logic) in high school is an interesting idea. I would be very interested to see your paper on this. Some community colleges have experience with formal logic as a course or as part of a critical thinking course.
I think a true liberal arts calculus course would start with Galileo (leading to the derivative) and with Kepler (leading to Newton and the need for the integral).