In Defense of Geometry: Part II

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! 

— Henri

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. 

8 thoughts on “In Defense of Geometry: Part II”

  1. “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.”
    I believe this was an unintended oversight. (That and the corollary about volume are the only ones I know about.)

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  2. The idea that algebra is particularly “practical” is, in my opinion, true nonsense. There is nothing stranger to me than the idea that there is some sort of economic or even social value in more people becoming proficient in algebra. The only argument is one of equal access to higher education and fields that require algebra — in nearly every case this is just a chance to use algebra as a proxy for something else. And we should find better proxies and signals.

    It might be true, I guess, that algebra is more practical, but algebra is not that practical and geometry isn’t that much less so. We’re playing a crazy game in math education.

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    1. Algebra is more practical in the sense that it is pretty much a prerequisite to further work in math, science, and statistics. And it is not just as a proxy. For example, I don’t think you can understand high school physics without some facility in algebra. We can’t know in advance who will want to or need to study science, engineering, or statistics, which is why we do need to teach algebra to all students.

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      1. I was going to respond that we might as well prepare students for other careers — why should math/science careers get privileged in k-12? — but maybe it really is that math/science is widely used in careers. I guess I don’t know.

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    1. But there is limited space in the K-12 curriculum and therefore a limited number of doors that we can leave open.

      To take an extreme example: it is much harder to become a oboe player if one has not had the chance to learn oboe proficiently as a student. A student who decides in college to pursue a career in oboes will find themselves with an insurmountable gap between themselves and their peers. Ditto for someone who decides to wants to be a professional painter or dancer or poet. It’s not too late to start — neither is it too late to start studying algebra in college, forgetting the gatekeeping systems we’ve created — but it will be very difficult to catch up to one’s peers.

      So we really are privileging math/science careers over others. Whether we should or not is a different question. Maybe there really are more careers that algebra prepares you for, and in some utilitarian calculus algebra keeps more professional doors open than music/dance/art. Or maybe it’s more importantly to train kids in math/science to keep up our military advantage over other countries. But the idea that we’re keeping all doors open to all kids by focusing on algebra…I don’t buy it.

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      1. True. We can’t keep _all_ doors equally open, because K-12 time is finite. But for the same reason, we do need to figure out some societal priorities. To respond to your example: the oboe cannot be deemed more important than reading and writing. The latter are necessary for further learning in any field. Not everyone becomes a journalist or a playwright, but everyone needs to be literate if they’re going to learn almost anything. Playing the oboe is wonderful, but it’s mostly useful for learning to play music, which can be done through other instruments as well. Thus we should have band, or orchestra, or music one way or another in the curriculum. In the visual arts, drawing would probably take priority over stone-carving for similar reasons: it’s useful for all the visual arts, and in fact for any field where confidence in sketching an idea matters.

        Anyway, I’ve written about the question of literacies here:
        https://blog.mathed.page/2015/10/15/literacies/
        If I remember correctly, it was part of a not unrelated conversation with you about “coding for all”.

        Yes, if one didn’t learn algebra in high school, one can pick it up later, but not as easily, and a lot of time has been wasted.

        Bottom line: we should discuss these questions, and not leave them to politicians. (Which is why I appreciate your comments!) When the importance of math is used to oppose music and the arts, we should fight back, but we need not throw up our hands and say all subjects deserve absolutely equal time, or just be taught in whatever amount they have been taught before. (For example four years of high school math for the well-off, two years of arithmetic review for the poor, which is how it used to be when I was new to the profession.) These are all legitimate areas for discussion. Thus my argument in defense of geometry, which flies in the face of a long-running trend, but hey, I don’t need to defer to the powers that be. All of us need to think about what is important, and critique the reasoning of others!

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