Restructuring the Math Pyramid?
A friend recently shared with me the following video from TED (see below). In it, mathematician (or, in this case, mathemagician) Arthur Benjamin gives a brief argument for eliminating calculus as the top of the "mathematical pyramid" in high school education, and replacing it probability and statistics. The main reason for this shift is that unless you are planning to have a career in a technical field, it's unlikely you'll find a use for calculus in your everyday life, but an understanding of statistics can benefit you no matter what you do. For example, it can help you to build an intuition about day to day decision making when risk and uncertainty are involved. Here's the video (it's short, only a couple of minutes):
A noble goal, to be sure, and it's certainly a solution that wouldn't cost a whole lot. There is an argument to be made for such a change lurking in here somewhere, but coming in at under 3 minutes, Benjamin's argument barely scratches the surface. In no particular order, here are some of the problems I have with his proposal:
1) Arguing that students shouldn't learn calculus because they may not use it in their everyday life is specious. By this reasoning, I should never have taken any courses in history, biology, or chemistry. The purpose of high school education in this country seems to be not only determining what educational avenues students want to pursue further, but also what avenues they don't want to pursue. If you want to argue that students should only be learning things that they can apply to their everyday lives, then you are arguing for a much more sweeping reform of education.
I do acknowledge that there is an opportunity cost at work when we spend a year teaching a student calculus rather than statistics, and certainly the average student will find more use later in life for the latter. But there's also an opportunity cost at work when we spend a year teaching a student statistics rather than calculus, especially for students who aren't sure in what direction their academic future will head. If anything, this seems to be an argument for offering both statistics and calculus for students, rather than forcing them into one option or the other.
2) About 2/3rds of the way through the talk, Benjamin asserts that "if our students, if our high school students, if all of the American citizens knew about probability and statistics, we wouldn't be in the economic mess we're in today." This is met with some cheers from the audience, but is it actually true?
The answer depends on your definition of "knowing" probability and statistics. I agree that having some knowledge of statistics is a good thing for the population at large, and there are no doubt many fundamental principles that could be taught at a high school level - for example, the idea that correlation does not imply causation, or the ways in which one can manipulate data or graphs of data. These topics, among others, are discussed in the book How to Lie with Statistics, which would be a great required reading book for any teacher trying to impart intuition and a healthy dose of skepticism onto his or her students, and is written for a general audience.
However, even if everyone in America had this basic level of knowledge, it's not at all clear that this would have somehow saved us from economic catastrophe. If you work in finance, odds are pretty good that you already have a knowledge of statistics that goes beyond a high school level, but this didn't stop the economy from tanking.
Nassim Nicholas Taleb wrote an excellent article last year on the limits of statistics, which is well worth a read if you can spare the time. One of the arguments he makes is that part of the reason the financial models caused such an economic implosion was that these models are necessarily unable to predict black swan events, which can have a tremendously negative impact, but are also tremendously rare. In fact, he argues that statistics is actually quite poor at trying to predict what will happen in extremely complex systems where rare extreme events can have a profound effect on the system.
However, this is not what one learns in a high school or even undergraduate class on statistics. Most problems at this level involve simple systems (games of chance, for example). In other words, studying statistics at a low level does not expose one to the subtleties and limitations of the subject - in particular, I don't think it's feasible to say that if every high school graduate had taken a course in statistics, somehow we would have prevented the current economic catastrpohe. To do so would have required a much deeper understanding of statistics among those applying the financial models than can be supplied at the high school level.
This brings me to my third point...
3) To have a good understanding of statistics, one must already have a working knowledge of calculus. There is a limit to the amount of depth a probability or statistics course can explore when calculus is not a prerequisite, and because of this many results (such as the Central Limit Theorem) are stated without proof. This is fine if you are simply trying to expose students to some of the standard tools in the subject, but if you can't go deeper, there really is a limit to the level of understanding a student can achieve.
I agree that there is a great deal of value in teaching statistics to high school students, even at the level of pre-calculus. One can still impart a significant amount of intuition at this level. However, for students who plan to use statistics in any significant capacity, it's important that they develop a working knowledge of calculus as well.
If you're not planning on going into a technical field, certainly you'll get more value out of a basic statistics class than you will a calculus class. But students who dislike math in general will probably still dislike statistics, even though there's more to like for someone who's not interested in math than there is in a calculus course.
This, in turn, brings me to my next point...
4) In the larger debate over the failings of math education in America, the choice of whether to teach statistics or calculus in secondary school misses the point entirely. By the time students reach the later years of their high school career, most already have a pretty well developed sense of their relationship to mathematics - either they had good teachers and enjoyed the subject, or through a series of misfortunes which may have been out of the student's control, they feel like math is a subject they will never understand, and will struggle with until they have the freedom to not take a math class, and are finally free from its iron grip.
Sadly, the problems with math education in this country run much deeper, and swapping out calculus for stats at higher levels won't alleviate the fundamental problems students have with mathematics. When I grade papers in a calculus class, students make just as many (if not more) algebra mistakes as they do calculus mistakes. In other words, many students leave the year without having mastered the math concepts presented to them during that year. Compound this over several years, and it doesn't matter if you give them a calculus book or a statistics book - they will have trouble because they haven't mastered the prerequisites.
Certainly one can argue that there are fewer prerequisites in a statistics class, but prerequisites are still present, and algebra is certainly one of them. If a student has a poor understanding of algebra, it's reasonable to assume he will have significant gaps in his understanding of statistics, and if the goal is to give students an intuition for randomness and understanding data that can help them in their everyday lives, gaps in statistical understanding are significant problems. Therefore, achieving this goal isn't as simple as making sure every high school senior has taken a statistics class - we really need to insist that every student first has a working knowledge of algebra. This is a problem we have already, and is not resolved by Benjamin's proposal.
5) This is a small point, but important. What really bothers me about this talk is when Benjamin makes the statement that, "If [probability and statistics] is taught properly, it can be a lot of fun!" Well, yes, but this is true of any subject. Implicit here seems to be the idea that calculus cannot be fun, even if taught well. I'm sure this isn't Benjamin's intention, but it's easy to misinterpret, especially if you are someone who has never taken a calculus class, or has fallen victim to the commonly held opinion that calculus is some kind of black magic whose secrets only a chosen few can hope to unravel.
The truth (and one that Benjamin knows) is that any math class can be fun if taught properly. A more accurate statement might be "it's easier to make probability and statistics fun for students," because of the vast applicability to everyday life, from games of chance to calculating the probability that someone in a family is colorblind. But to suggest that statistics is inherently more fun for students than calculus does a disservice to all the great teachers of calculus. Either class can be fun and valuable if taught well, or traumatizing if taught poorly.
I understand what Benjamin is saying, and I also understand its appeal. The argument works well as a 3 minute sound clip, but upon further reflection, there are some significant questions that need to be addressed. There are many problems with math education in this country, and I'm not sure which, if any, are solved by this proposal.
From my own experience, no students in my high school were forced to take either calculus or statistics, although both courses were offered. Preparing students exclusively for either one or the other will of course do a disservice to some, so perhaps putting both on the table is the best compromise, although this becomes a problem for schools with limited resources. I am confident, however, that simply putting statistics on the pedestal currently occupied by calculus doesn't do a whole lot in terms of fixing everything that's broken.
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