Arthur Benjamin thinks that the current model of the mathematics curriculum — leading from arithmetic to algebra and ultimately to calculus — is flawed and needs to be changed. Watch this 3-minute TED talk for what he thinks ought to be the real summit of the mathematics curriculum:
I am in a great deal of agreement with Prof. Benjamin here. The secondary school math curriculum does indeed seem poised to point students towards calculus. While that is appropriate for some, it is not appropriate for all; while on the other hand, a better knowledge of discrete math, especially probability and statistics, would be appropriate for everybody.
Moreover, Prof. Benjamin did not stress one of the most important selling points for refocusing on discrete math: The mathematical background requirements are a lot lower than they are for calculus. Students currently have to take two years of algebra, at least a semester of trigonometry, and often an entire course in Precalculus on top of all that just to have a fighting change in calculus. And even then it doesn’t always work. Probability and statistics, on the other hand, gets to good ideas, deep ones at that, without nearly so much training.
On the other hand, what about those students who do end up going into science, math, engineering, economics, or other fields requiring calculus? If probability and statistics becomes the summit of the secondary curriculum, then at what point do those kids get the precalculus training they need in order to complete calculus (which I interpret to mean a year of calculus) by the end of their freshman year in college? Would they be having to double up on math courses — statistics on the one hand and precalculus on the other? Would they need to decide that they wanted a STEM-related career early on in high school, and if so, is that good for them?
Casting Out Nines 
Hi, I am new to wordpress, but I came across your blog. Never truely being an enthusiast of mathmatics, I have always admired those who understood its nature. In college however, when I took statistics, I found it particularly useful and, as Dr. Benjamin so thoughtfully exlamined, fun. A child of social science, it appears in almost every Libral Arts class. I certainly agree with you and other who support that the educational pyramid in mathmatics should conclude with statistics and *cough* a foundation to micro- and macro-economics.
The problems are arithmetic and algebra, the bases, not which direction we go on from.
The appeal of stats and probability is that they need less foundation. Unfortunately, they are, while procedurally easier, conceptually difficult.
And, as you point out, calculus is a reasonable summit for many…. Do we cheat them?
Jonathan
I agree that probability and discrete math involve less foundation but require more conceptual understanding. As someone who teaches future high school teachers and occasionally gives workshops for in service hs teachers, I have noticed that their conceptual understanding of many mathematical ideas is very poor. The existing hs curriculum, including calculus, can be taught by anyone with procedural fluency but minimal conceptual understanding. I am not saying this is a good thing – it’s just reality.
To teach a solid intro stats /probability/ discrete math course would involve retraining hs teachers to appreciate and understand the conceptual underpinnings of mathematics. This is of course a good thing, but who is going to retrain these folks? Our students who aspire to be hs teachers are math majors, but a look at their transcripts reveals weak grades in theoretical math courses such as abstract algebra or advanced calculus.
Unless teachers are retrained, stats and discrete math in high school will simply degenerate to some sort of busy work. A better approach would be a senior course for non STEM types focusing on problem solving/modeling using algebra 1,2 and geometry and using graphing calculators or spreadsheets to aid in the process. This may involve less of a learning curve for the teachers.
I like the idea of focusing on discrete math because, as you say, the background requirements are less. High schools could, for example, offer a rigorous number theory class. Such a class could focus on quality rather than quantity, covering little material but covering it well.
But I’m concerned about the statistics proposal. I find it very difficult to work with people who have had a hack statistics class and think they understand more than they do. I bet not one person in a hundred who knows how to calculate a p-value actually knows what it means. I’m afraid that a high school statistics requirement would increase the ranks of those who know enough to be dangerous.
I agree with John above about his concerned of statistics proposal. I am more incline to Polya proposal that math should be taught as problem solving. This skill will be useful although the student won’t take any math course anymore.
I think it is very hard to teach statistics with anything more than a “black-box” approach if students do not have a solid foundation in calculus. I also find it quite odd that Dr Benjamin puts statistics in the discrete maths camp (I think maybe he means combinatorial probability, rather than statistics). Many of the great statistical thinkers of the 20th century (eg Kolmogorov) were actually analysts, after all.
Give the students instruction which fits helping them to make mathematical sense of what they will study IN Mathematics and WITH Mathematics. Start with Algebra 1 and Algebra 2, and Geometry. With those, and with further mathematical maturity which comes from using Mathematics and study of Trigonometry and Calculus, students will be better able to make sense of at least “Elementary” Statistics. There is a good reason why many “Elementary” Statistics courses at community colleges show “Intermediate Algebra” as a prerequisite. Beginning level Statistics uses many features of the intermediate level of Algebra (as well as some ideas typically outside of regular Intermediate Algebra).