In Defense of Geometry: Part I

In 2016, I wrote In Defense of Algebra 2, a blog post addressed to math educators who do not see that Algebra 2 can be engaging and worth teaching to all students. In this post and the next, I defend geometry. This is a different sort of argument. Algebra 2 has been offered to more and more students in recent decades, while geometry’s place in the curriculum has gradually shrunk in the same time period. The argument about Algebra 2 was largely about curriculum and pedagogy, and teaching fewer topics. For geometry, before getting to the subject itself, I need to answer the question: if I want more geometry, what do I want less of? This is what I’ll focus on in this post.

Algebra

There was a time in the distant past where solid geometry was a standard part of high school math in the US. That was before I became a teacher, perhaps even before I was born. (I myself was exposed to the basics of 3D geometry as a high school student in the French system back in the 1960’s.) There was a time more recently where the geometry of the conic sections was part of high school math in precalculus classes or in late chapters of geometry textbooks. As far as I can tell, that is no longer the case.

When I first started teaching high school, in the 1980’s, geometry was typically a one-year course, wedged between Algebra 1 and Algebra 2, with enough content to fill a fat textbook. Back then, algebra was taught to only some students, the ones who could pick it up quickly, and few students took calculus in high school. Thus there was plenty of geometry time for the college-bound. Still, as a department chair, I wanted all students to get a solid grounding in algebra, as it is necessary for any work in science, and of course in further math. That led me to move some Algebra 1 content into our Geometry course, which led to a reduction in the amount of geometry we offered. In other words, I participated in the very shrinkage of geometry that I decry. The importance of algebra is in part what is going on at the societal level, especially once you take into account the increasing numbers of students who take calculus in high school.

However, some algebra topics which are still widely taught should be eliminated. For example, the authors of the Common Core State Standards for Math (CCSSM) wisely removed solving absolute value equations and inequalities from the high school curriculum. Likewise, they eliminated arcane topics like synthetic division, Descartes’ rule of signs, and the rational roots theorem from the CCSSM. They probably should have gone further (see below!) but that’s a solid start.

Statistics

But algebra is not the only time hog! There has been a substantial increase in the amount of statistics taught in high school math classes. Thus, something had to go, and geometry shrunk…

Teaching a lot of statistics in math class is not a good idea. Some work with data can enhance math education (for example binomial distribution, expected value, basic modeling) and should be taught. But concepts like standard deviation, regression, and correlation coefficients are based on a mathematical foundation that is completely outside the reach of most high school students, and in fact most high school teachers. (I certainly don’t understand the underlying mathematics.) This not only does not further our goals as math teachers, it actually works against what we usually try to do, which is teach for understanding. Instead, it forces reliance on black-box magical technology. (Using technology for line-fitting is quite different from simply using an electronic grapher, because we can and do thoroughly teach the mathematics of graphing — not so much the mathematics of regression.) Not only that, but to teach statistics effectively, students should collect their own data, which takes more time away from actual math. And finally, statistics is best taught with context and content, and therefore belongs in social studies and science classes.

Too Many Standards

Still, reducing the amount of statistics does not create enough space for more geometry. The reason is that there are just too many high school math standards. Most states have adopted the CCSSM, or something close to that. Given the governmental obsession with standardized testing, this forces a continuation of the mile-wide-inch-deep approach that has historically plagued high school math in the US. I know this, because I spent a supremely boring few weeks trying to sequence the high school CCSSM standards into a reasonable four-year sequence, and found that the only way to do it would be to race through everything.

When I compare the results of that thought experiment with the curriculum I developed and taught in my 30+ years as department chair, the biggest difference is that what worked in an actual classroom was to revisit the most important topics at greater and greater depth, in multiple representations, using a variety of learning tools. That can only be done if you focus on what I will call really core standards, and skip or deemphasize other topics. Which standards are really core is a good question. There are some obvious candidates: the distributive law, the Pythagorean theorem, the basic trig ratios, and so on. But figuring out where to draw the line between really core and merely interesting would require input from the whole community of math educators. As far as I know such input was not sought in the writing of the high school standards, and the standards are bloated. (Read my 20-page analysis of the high school CCSSM if you have time. I got nothing but rave reviews about it.)

Please don’t take this as an attack on the CCSSM authors. They made a huge and important contribution which moved the math education conversation forward on many fronts. Among other things, they brought focus and coherence to K-8 math, and got started on some good ideas for high school. But some time has passed, and I believe I’m within the professional consensus when I say there are too many standards in the high school program they recommend. Here is what NCTM has to say in Catalyzing Change in High School Mathematics:

Focusing on Essential Concepts is consistent with some concrete advice I offered on Pruning the Curriculum back in 2015. It would open up some time for better teaching, and yes, for more geometry. I will not give a complete list of standards I would remove — compiling such a list would take too much time I don’t have, and in any case no one would heed my suggestions.

Still, I hope I convinced you that it is possible to make some space for more geometry. In my next post, I explain why that would be desirable, and suggest what geometry I would like to see in grades K-12.

— Henri