All of high school math in one year?

In my previous post, I responded to Michael Thayer’s comments about my Mathematics Overview. In this post I will respond to Mike’s proposal for a one-year course to replace all of high school math.

Mike and I largely agree about the failings and shortcomings of traditional curriculum and pedagogy, but I don’t agree with his solution. Here is a quote from his blog:

If we honestly boiled down our algebra 1, algebra 2, and trigonometry/pre-calculus curricula, we’d find that: 1) most of it is repeated without much notable increase in difficulty (only the numbers become bigger or more decimal-laden, or the word problems become well, more wordy) from course to course, and 2) most of it deals with techniques for solving problems (or worse, doing calculations by hand) that are as antiquated as the steam engine and the corset. Example: we spend tons of time on factoring polynomials (17th-19th century), even though the vast majority of them can’t be factored by hand, but usually fairly little time on matrices and vectors (19th – 21st century). Another issue is: we assume “mathematical literacy” to equate to “readiness for calculus”, which I have claimed is incorrect.

There is indeed substantial redundancy in the traditional curriculum. Some of that is due to the fact that each topic is taught superficially and in a rote manner, with the goal of being ready for the next quiz (or standardized test), and with little attention for the depth of understanding that would make for lasting mastery. Another reason for review of previously-studied material is because in reality students learn at different rates, and for difficult and important topics, review is actually necessary if your goal is to reach the whole spectrum of students. This remains true even if you have an excellent student-centered program.

And yes, teaching the same content the same way as it was done 50 years ago is absurd, given all the changes in society and technology, and given the much broader range of students we are trying to reach. (Remember that 50 years ago, only some students went on to high school math, with the rest condemned to eternal “remediation” and mind-numbing arithmetic drill.) Alas, how to make the needed changes is far from obvious. Mike suggests as examples that we can do less factoring and more matrices, and I agree with those recommendations.

But it’s not so easy to figure out how to implement this. Yes, as I explain in the introduction to my Mathematics Overview course outline and surely elsewhere on my Web site, it is both misguided and unrealistic to expect students to compete in accuracy or speed with Wolfram Alpha or the many other available electronic tools, from basic calculators to electronic graphers, spreadsheets, and CAS. What’s not so clear is what level of understanding and manipulative fluency is required to effectively use those tools. As an early adopter of educational technology, I have decades of experience struggling with this issue, and I can tell you unhesitatingly that merely teaching less algebra is insufficient to prepare students for intelligent use of electronic tools. The answer is to teach better, not less, and it will take much classroom time by all of us to sort out how to do that. Thought experiments and online discussions alone won’t do the job.

Given all this, I am certain that I don’t support Mike’s proposal for a one-year, 100-hour course to cover all the high school math most students need. Some additional reasons why I’m skeptical:

1. He is of course aware that which topics should be deleted from the list is tricky to sort out. I had to pick some for my Math Overview course, and I have some justification for those choices, but in the end I don’t think it’s easy to know whether those were the right choices. For example, both Mike and I would agree that some grasp of probability and statistics is essential for a well-educated, politically aware citizen, but neither his nor my one-year course includes any of that. Moreover, it’s all well and good to have one’s own theory about what people need, and it’s another to make sure they are ready to meet the expectations that today’s society has about what is essential. (For example, meeting graduation or college acceptance requirements, or being ready for calculus. Whether or not we agree with those expectations, surely we are letting our students down if we close those options for them in order to comply with our own philosophical leanings.)
2. Mike figures that the missing topics can be learned by students as they pursue courses in other subjects, adding the math as they need it. I’m sorry to say, but that is completely unrealistic. It’s hard enough for most students to learn Algebra 2 / Precalculus / Calculus topics with the help, support, and guidance of a competent math teacher. To think they can teach it to themselves, perhaps with the help of the Khan Academy or such is pure fantasy. Even if we manage to restructure high schools enough to involve math teachers in the teaching of other subjects, so as to pursue the mirage of strictly teaching math in the context of its applications, the curriculum development effort and teacher training needed to pull this off would be vastly greater than the already gigantic struggle to merely rethink the math curriculum. Besides, interdisciplinary learning has been a darling of reform-minded educators as far back as I can remember, but the practice does not measure up to the theory: the math in most of those programs gets diluted beyond recognition, and in any case those programs have very short lifespans.
3. Mike’s goal is for students to be mathematically literate. Amen! That is much more ambitious than today’s memorize-and-practice-in-silence pedagogy, which aims very low. I do think it is a realistic goal, but teaching for understanding requires the sort of pedagogy I promote in my Mathematics Overview: exploratory, hands-on, constructivist, tool-rich, discussion-provoking approaches. Those take more time, not less, than the traditional approaches. They can only be implemented by doing the sort of trimming of topics Mike recommends (though not as drastically), but in the end you don’t end up with a reduction in total time.  Deleting enough topics to have a program that would fit in a one-year 100-hour course without changing the pedagogy would only work with the small number of students who are good at rapid rote reproduction of teacher-taught techniques.

Perhaps the fundamental flaw in Mike’s proposal is that it is based on the widely held belief that the math we should teach to most students is different from the math we should teach to future mathematicians. I disagree, but that is a big enough topic to deserve its own post: stay tuned.

[next post in this series]

–Henri