Here are two complaints often heard in any math department. The first is that people do not understand any mathematics; this complaint is typically inspired by some horrendous misuse of statistics or basic misunderstanding about numbers. The second complaint is about students who have generally succeeded at basic mathematics but who do not truly understand the advanced tools that they’re using. These complaints are about very different aspects of math education, and we need to keep them straight.
Ok, secret educator speak decoding time!
- “math literacy” == understanding basic mathematics to get through life and interpret your world
- “workforce development” == preparing people for jobs in STEM fields (Science, Technology, Engineering, and Math)
People often don’t give enough thoughts to their goals. Say, for example, that you’re debating if calculus should be the pinnacle of high-school learning. If you think the purpose of math education is math literacy, then it clearly shouldn’t be: statistics would be much more valuable. If you think the purpose of math education is workforce development, then calculus probably should be the goal: you need calculus for engineering, for math research, for any kind of science, and without it you’re sunk. (Leaving aside other goals such as exposing students to beauty or to deeper thinking, which might culminate a discrete math course.)
Schools basically push everyone to a workforce development track until they get to college. Even assuming that this track adequately prepares people to study STEM fields (it doesn’t, judging from most incoming college freshmen), it’s pretty bad for anyone who doesn’t want to pursue a STEM field: they don’t get a lot of the skills they need for life (interpreting mortgages, research studies, and politicians) and they’ve learned to thoroughly hate math.
Yet we shouldn’t put everyone into a math literacy track, either, because it’s quite helpful to build up good abstract thinking skills early-on if you do plan to pursue a STEM career.
Naturally, you think, we should just let students choose or perhaps sort them somehow into different tracks. But this opens up its own host of problems. For example, a student might actually be quite good at math but lack the self-motivation to pursue it seriously, so it is a disservice not to push them harder. There are students who develop mathematical skill later in life, but it is very hard to catch up in the workforce development track if you haven’t been given the opportunity to learn advanced algebra or trigonometry. Let alone that low-income and minority students will track disproportionately into math literacy, and you’ll also create student self-images of being “not good” at math in the literacy track.
Which brings me to the recent New York Times op-ed, How to Fix Our Math Education. This is probably the best proposal I’ve seen for addressing the situation, but it’s still not good enough. The authors come up with credible ideas for courses that would demonstrate the connection between the “real world” and mathematics. However, I don’t believe that the resulting courses would cover enough for workforce development. Also, the plan feels too much like a retreat: since we can’t actually get most students to deeply understand calculus, let’s find a way to learn topics that don’t involve calculus. But we know that calculus is possible from the results of other countries!
I do not know how to resolve this. I don’t know how to make calculus and advanced, abstract mathematics accessible to those who are ready for it without the collateral damage of convincing other people (many of whom could excel in mathematics) that they are not good at mathematics. Any ideas?
Consider Learning 
Instead of one math course, why not make two courses part of the standard curriculum? Students could take both a math literacy and a workforce development course. Ideally they would be accelerated in the course where they demonstrated the most ability and passion.
Upon graduation students would have some basis in both and those who thrived in the workforce development course would have a clear signal and preparation to decide whether or not to pursue a STEM career. But both sets of students could at least understand their mortgage.
Whenever I see these debates over workforce development verses “real world skills” I always feel like an important aspect is being ignored, and it is just this aspect which leads people to hate math. What is it? Well, I have never seen anybody enunciate it better than Paul Lockhart, so I will leave the explanation to him:
Click to access LockhartsLament.pdf
If the results of other countries give you confidence that we should be able to teach calculus, then shouldn’t we be able to answer your final question by looking at how other countries do it?
I ask because it’s not entirely clear to me how much the results of one society tell us about what should be possible in another society.
I suppose that it depends on what you consider as fair game for the solution. Successful calculus instruction on a large scale just indicates that there’s nothing inherent about the mental development of an 18-year-old that prevents their learning calculus. However, there are certainly big cultural factors so just adopting what is done in China/Finland/whatever won’t necessarily produce the same results here. I’d like to think about what might produce this kind of success without trying to completely adopt another nation’s culture.
Dan, have you read Ian Westbury’s article “Comparing American and Japanese Achievement: Is the United States Really a Low Achiever?” (from the Educational Researcher, Vol. 21, No. 5)? When accounting for curriculum alignment with international assessments (IEA and SIMS) he found that the achievement gap between the US and Japan disappeared. This would seem to suggest that curriculum alignment considerations trump cultural differences (ignoring, of course, the possibility that cultural differences account for curricular differences). This, if true, would be encouraging from the point of view of the US.