In my last post, I complained about the shrinkage of geometry, a decades-long trend in US math education. Some of the reasons I suggested for this state of affairs is the offering of a substantial amount of algebra to a much broader population, the growth of calculus as a high school subject, and the increasing place given to statistics in math department offerings. I also pointed out that the Common Core State Standards for Math (CCSSM) are bloated at the high school level, in spite of the abandonment of some customarily taught theorems in geometry. All these arguments revolve around the fact that the amount of instructional time devoted to math is finite: if you have more of this, you perforce have less of that.
Stepping back a bit, we may guess at cultural reasons for geometry’s decline. A friend who trains students for math competitions tells me that in those circles, it is well known that American students have much less grounding in geometry than their peers in other countries. If it is true, this does not surprise me. There may be a self-reinforcing mechanism at play: the very people who thrived in an educational program that privileges the manipulation of symbols become the educators of the next generation. The multi-generational retreat of geometry may also be symptomatic of a priority given to the pragmatic over the beautiful, which the students reflect back to us: “when will I ever use this?” Note that in Catalyzing Change in High School Mathematics (CCHSM) NCTM correctly argues that our goals in teaching math are many — not just preparation for college and career, which was the big emphasis of CCSSM. In any case, if it is pragmatism that led to the shrinking of geometry, it is a short-sighted pragmatism. In fact, thinking visually is important in further mathematics (including calculus!), in the sciences, in architecture, in the building trades, in design, in the arts, and no doubt in other fields.
In addition to the importance of visual thinking, there are more reasons to value geometry. It is the part of math where instead of breaking problems up into pieces, we look at the whole picture, literally and figuratively. It is a part of math with contributions from many cultures. It is a part of math that appeals to a different type of student (and, frankly, a different type of teacher!) It is the part of math where the connections to art and culture are the greatest. Thus, it is the part of math where the beauty of the subject is most apparent. It is the one part of math that “made sense” to so many of the people who tell us they are “not a math person”.
It is also the part of school math where students are introduced to proof, which of course is a core value in mathematics. However, in my view that is a less convincing argument for geometry. Reasoning and reflection about reasoning should occur at an appropriate level throughout K-12 math, and the CCSSM authors quite correctly embedded that in the excellent mathematical practices standards. In any case, the teaching of proof is an important topic which I hope to return to, but it is not the topic of this post.
Instead, I will suggest some of the geometric content I’d like to see.
- In elementary and middle school, I mostly like the content suggested by the CCSSM, but I would add geometric puzzles to the mix. When I was a beginning teacher in the 1970’s, there were excellent geometry materials for the lower grades, some of which were developed by EDC in the Elementary Science Study (ESS) program. Those included activities with tangrams, pattern blocks, mirrors, and probably other things I did not see or do not remember. Alas, the ESS materials are out of print, and are not likely to see the light of day any time soon. In the 1980’s, I added to that genre with my pentomino and supertangram activities. (See my Geometric Puzzles pages for more on this.) Doing some work on geometric puzzles at an appropriate level is possible all the way from kindergarten to 12th grade, sometimes just to develop visual sense, and sometimes with implicit or explicit curricular connections. For example, in the lower grades, it ties in with the standards about “composing and decomposing shapes”. I was going to say that in high school puzzles could be used to explore the areas of similar figures, but I can’t find a CCSSM standard about this. (Could it be that this too is a topic that was a victim of geometry shrinkage?)
- There are several mentions of symmetry in the K-8 standards, with a specific standard about bilateral symmetry in 4th grade. There is plenty more that could be done in that arena: students could be introduced to rotational symmetry, to figures with multiple lines of symmetry, and to frieze and wallpaper symmetry. This can be explored for example by using pattern blocks, or geometry templates. I suggest some activities along these lines in my Geometry Labs book (free download), and I went into greater depth in a post-Algebra 2 elective (Space.) Symmetry reveals the deep connections between math and design in many cultures. It is highly interesting to many students, and not always the same students who enjoy symbol manipulation. The CCSSM emphasis on geometric transformations could have provided a unique opportunity to expand the place of symmetry in the secondary curriculum.
- Tiling the plane (tessellation) offers another geometry-rich context for exploration. It allows connections with symmetry, and can be used to introduce standard foundational geometry topics such as vertical angles, angles determined by parallels and transversals, the sum of the angles in a triangle or quadrilateral, angles in regular polygons, and isometries (known as rigid transformations in the CCSSM.) Find some tiling activities on my Web site.
- In high school, I like much of the content of the traditional course, though not always how it is taught. I will return to this in a future post. Today, I will just say that the existence of GeoGebra and other interactive geometry software makes it possible to teach the traditional content much better. This technology enhances learning in a variety of ways. Direct manipulation on a screen is an excellent environment to distinguish the essential properties of a figure from the particular way it appears on a static page. It makes it easy to generate conjectures, and to find counter-examples. It is what finally might help fulfill the promise of geometry as an appropriate arena for the teaching of proof. Finally, this technology is more accessible than ever, as it is now available in free software that will run on tablets as well as computers.
Even given the current version of math standards, the suggestions I have made up to this point are realistic, because they can be interpreted as ways to teach the existing standards effectively, and because most do not require much extra time. At least, they are realistic if you agree with the points I made in my last post. I will now go on to propose possible topics for 11th and 12th grade math electives. These topics probably should not be considered to be really core, even though I personally like them a lot more than much of what we usually teach in precalculus classes.
- School math has long underemphasized solid (3D) geometry, a field with a very long history (Platonic and Archimedean solids,) beautiful and accessible results (Descartes’ theorem, Euler’s formula,) and multiple applications (chemistry, architecture, 3D printing.) This is another part of geometry made dramatically more accessible by technology. Absurdly, the CCSSM restricts 3D work to grades K-8. In high school, it appears only as a topic leading to the standard calculus problems.
- We could do more advanced work with geometric transformations: theorems about composition of isometries, connections with complex number arithmetic and very basic group theory, a more complete coverage of transformation matrices, and some introductory work with geometric transformations in three dimensions. (I taught all this in my Space elective and it provided an excellent opportunity for well-motivated more advanced work on proof.)
- Fractals, self-similarity, and fractional dimension. This ties in with similarity, obviously, as well as geometric sequences and series, logarithms, and computer programming. (I taught all this in my Infinity elective, and it was extremely popular with students.)
OK, I should stop. This turned out to be a longer post than I expected. You probably think it’s mostly a reflection of my own mathematical preferences, and of course you’re right. But that does not make it wrong! And in any case, I’m not the only one. I hope some day you’ll join me!
PS: In this post, I not only quoted John Lennon at the end, but I re-used various bits from my article on the CCSSM, re-ordered, tweaked, expanded, and in some cases verbatim.