Teaching Geometry

I taught two geometry sections my first year at a different school, and while I didn’t do a particularly good job (the classroom management problems were horrible for a different reason, and the curriculum was CPM–ick), I came away with useful insights that have really improved my execution this year.

Geometry Then and Now

Back in the dark ages, we used to say “There are two sorts of people in the world: those who prefer algebra and those who prefer geometry.” This mindset comes from a time when advanced students took algebra in 8th grade, most of everyone else college bound took it in 9th or 10th grade, and then followed it up with geometry, Algebra II, and precalc if there were enough time. Students who were really bad at math took Basic Math or Business Math and maybe took algebra their senior year. If this sounds familiar to you, fine–but it’s not like that anymore.

“Students who were really bad at math” were not representing our nation’s racial balance, and research unsurprisingly showed that students who went beyond Algebra II in high school had higher college completion rates. Naturally, this meant that everyone should take algebra as early as possible, cognitive ability or readiness be damned. The resulting carnage of this policy did not lead to re-evaluation, but rather to the determination that pre-algebra preparation should start earlier–and, of course, kids who fail algebra need to take it again.

Consider the effect of this policy on the average, “okay at math” kid today. Starting in sixth grade, it’s All Algebra, All the Time. By the time they get to geometry, “math instruction” and “algebra” are virtually synonymous–and they don’t even realize it. Kids have spent three, four, or even five years with algebra preparation or instruction. Specifically, using processes to solve for an unknown.

And then: geometry. Good god, what fresh hell is this? Facts. Vocabulary. Relationships. And then, in some weird way, you use these facts and vocabulary and relationships to come up with more facts and vocabulary and relationships. There’s no solving. There’s not even an answer. Half the time the book gives you the answer but then expects you to explain it using, god help us all, facts and vocabulary and relationships.

This is a whole galaxy away from “I like algebra better than geometry.” First off, all but 10-15% of my students found algebra completely unmanageable, so they aren’t looking back fondly at an easier subject. They’re trying hard not to curl up in a fetal position at the realization that math gets worse than the horror of the past three years.

Geometry teachers would do well, I think, to acknowledge this confusion. I tell my students some version of what I’ve just explained above and I see the light dawn. They get it. They might not get geometry, yet, but they get why they feel so lost. And that helps them move forward.

Try, try again
Two years ago, I could see that many of my students weren’t getting it. I retaught, thought of other ways to explain things, but I didn’t understand the degree of their lostness until relatively late in the first semester. I adjusted my teaching more, but I still hadn’t figured out why they were so lost.

This year, I was teaching parallel lines and transversals in week 2 or so and I suddenly realized that most of my students didn’t get it. They weren’t complaining, they weren’t acting out, they were just lost. I recognized the look from two years ago, and was now able to distinguish a furrowed brow of mild confusion from a blank look of utter nihilistic despair.

So at the end of day 2, I did a thumb check. “Okay, guys, my sense is a lot of you are feeling lost. Thumbs up if you feel confident, sideways or down if you’re kind of or totally lost.” And sure enough, most of the class was sideways or down.

I told the class I would come up with a different way to explain it. The next day, I used Geoboards, rubber bands, and little wooden geometric shapes to create a visual image of corresponding, alternate interior, and so on. (I’ll write that up some time). The lesson was very effective in helping students understand the angle relationship. But more important, the students recognized that I had stopped everything, rethought the lesson, came up with a radically different way of explaining the concepts–and had gone through this effort because I could see they were lost. The feedback from the new lesson was very enthusiastic, kids felt much less lost–but more importantly, they felt like I understood their confusion and was willing to spend time and effort helping them out.

This created a lot of good will, and since then they’ve been very trusting of my oftentimes bizarre way of building visual images to help them grasp geometric concepts.

It’s easier to do this in geometry than algebra, since geometry is new to everyone. Even my top students appreciate the occasional visual exercise, and I always have extra challenges for them. In algebra, some kids are lost right from the beginning, and it’s impossible to reteach everyone. (Which means, now that I think about it, that if I differentiate immediately after my assessment test in algebra, I might have an easier time. Hmmm.)

But fundamentally, it’s important to understand that time spent at the beginning, pacing be damned, will really pay off in student investment. I now realize that many of my geometry students two years ago had checked out because none of it made sense, and I didn’t pick up on that early enough to intervene. I took five days to explain parallel lines and transversals rather than two, but every minute of it was well spent.

De-emphasize what they won’t use

Most college graduates think proofs, logic, and construction as quintessential geometry subjects. That’s because we never use them again. We don’t spend any time in formal logic, never do formal proofs, and as for construction, forget it.

So I mostly dump them (which is how I make up the five days on transversals). Not completely. My Holt text starts with these five chapters:

  1. Foundations (Undefined Terms, Segments, Angles, area/perimeter formulas, coordinate geometry, transformations)
  2. Geometry Reasoning (logic and proofs)
  3. Parallel and Perpendicular Lines
  4. Triangle Congruence
  5. Triangle Properties

I dumped transformations entirely. I then took coordinate geometry, proofs, and logic and broke them up into tiny digestible chunks (coordinate geometry was review), rather than cover them all at one time, and covered about 20% of the material. So rather than an entire section on proofs, I introduced algebraic proofs at a natural pause point, when I had a day or two between major sections. I just introduced it; my goal was familiarity and recognition but not competence. Then, after introducing congruent triangles, I introduced two column proofs, and the students used congruence shortcuts to create two column proofs. This was much more successful than introducing a whole chapter on proofs when they were still in the WTF stage.

Yes, I know, the purists out there, assuming anyone is reading, is shocked. What? Proofs and logic introduce an invaluable way of thinking logically and methodically! Yep. But ask geometry teachers in heterogeneous classrooms if their kids understand proofs, and they will sigh. There’s just no way to get the lower ability half of the population to understand proofs and they’ll never use it again. I could spend lots of time trying, but I have better things to do with their time.

Ideally, I’d love to make my top students go through rigorous proofs, but it would take more instruction time than I can manage in differentiation. I hope to figure this out at some point, but I’m not as practiced at teaching geometry as I am at algebra.

So you’re thinking my class is too easy, right? Well, we just had our semester final and the geometry teachers agreed to start with a common assessment, built by a traditional geometry teacher who had covered far more material than I had in the first semester. I considered the test a little too easy and more picayune than I would build, so I substituted some harder questions. I didn’t dump more than two or three of the questions that we didn’t cover, because I felt pretty confident the students could figure it out–and, for the most part, they did.

Always remember where they are going
Geometry is just a brief respite. The next year, it’s back to algebra II, another course that causes a lot of carnage. Half of my class has extremely weak algebra skills, half of the rest are adequate, and the top students were rarely challenged with tough material. They need the practice. My sophomores will be taking the algebra and pre-algebra intensive state graduation test and my juniors are taking the SAT. Algebra is a big part of their testing load this year.

So I teach my geometry course as Applying Algebra with Geometry Facts. My students will never again need to prove that triangle ABC is congruent to triangle XYZ, but they will always need to know how to find the angle measures of a triangle whose angle ratio is 2:3:5. They will never use a compass again, but they will need to know what to do if Angle A and Angle B are supplementary and Angle A = 4x+ 13 and Angle B = 2x + 17 and they need to solve for x.

The problem is that the state tests tend to emphasis more traditional geometry. Aggravating, really, given that the state has clearly de-emphasized traditional geometry in its overall curriculum, but so be it.

Results

I had told all my students this year that if they showed up and worked, they’d pass with a D-. In my Algebra II course, several students did not in any way demonstrate understanding of the material we covered that year (not for lack of trying, in most cases), but I kept my promise.

But in my Geometry classes, my D students were genuinely Ds. They struggled, but got Ds or “respectable Fs” (50% or higher) on all the tests and quizzes. On the 100 question final (40 correct is a D-, 15 point grade scale instead of 10), 15 students failed. All but two had “respectable” Fs (answered 30 or more questions correctly, and those two were just below 30. The distribution was pretty close to normal, the average score and the mode were C. So far, so good.

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9 responses to “Teaching Geometry

  • Dave

    Lots of good stuff here. Two specific items stand out for me.

    1) As you know, but many, many do not: correlation is not causation – “and research unsurprisingly showed that students who went beyond Algebra II in high school had higher college completion rates.” I cannot believe how this argument is used to justify placing more students into classes for which they are unprepared. I am all for access and equity, as long as one possesses the needed skills to succeed. However, forcing students into classes in which they are lost does more to harm their self-esteem and future earning potential than placing them into a class where they can experience success, assuming they are so driven, or encouraged. Unless, and until, this default placement process is improved, the students will continue to suffer through school-induced self-doubt and self-limiting thinking and behavior.

    2) I believe the old-school adage “There are two sorts of people in the world: those who prefer algebra and those who prefer geometry,” presaged what we now call “differences in learning styles.” People were just not conversant in that verbiage then. Fortunately, through the efforts of many teachers, via the NCTM, other entities, and the “new math” craze, which did spawn some beneficial pedagogical thinking in spite of missing, and obliterating, many of the finer details of mathematics, the NCTM content and process standards provided a framework for improving mathematics instruction, and making mathematics more accessible to more students. However, there are two-sides to every coin, or sword for that matter, so positive and negative impacts result. The implementation of the content and process standards vary amongst the millions of thousands of teachers instructing students in mathematics, especially at the primary school level. So, while students are provided a broader perspective, and experience, of mathematics, those who previously excelled at one or the other, due to a closer alignment of the content with their learning style, are prone to perform slightly less well if they are not adept at integrating multiple learning styles. [I acknowledge this is a fairly dense, and perhaps incomplete, comment, which may require further explanation later.]

  • Geometry: Starting Off « educationrealist

    […] We never really use it again. Geometry has mostly been subordinated to algebra in high school, as I’ve written before, and my geometry class is best thought of as algebra applications with geometry. Or is it the other […]

    • John Beach

      ” That’s because we never use them again “. Thanks for the article and comments. After 39 years as a design engineer, I’ve started teaching 7-12 math, Algebra through Calculus.
      As a mechanical design engineer, I used geometry most of the time. It is very valuable in electromagnetic fields as well. (That’s what allows your TV and cell phone to work). I understand lawyers appreciate the logic of proofs; I’ve heard an aspiring lawyer went back and studied geometrical proofs as she started her legal studies.
      The subject is worth understanding. My surprise, as a first year teacher, is how difficult it is for some students to learn, while for others it is as easy as inhaling.

      • educationrealist

        Relatively few find it easy. Most find it impossible. I wouldn’t argue it’s not important, but rather that it should wait until more students can understand it. Trig or later. Glad you liked the essay.

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    […] hang on. As I’ve mentioned before, I don’t teach proofs and rarely prove theorems. I also typically dump transformations, most […]

  • Dan_Kurt

    I took Geometry in 10th Grade 1957-1958 at a Catholic all boys HS. We were sorted by ability first to get in the school and then by IQ and grade school records into classes A through H. Everyone took Algebra 1, Geometry, Algebra 2, and the top three groups took Trig, Solid Geometry and Analytic Geometry the senior year. Geometry was all proofs all the time for a year as I remember it. As I recall the course, the teacher expected each student to work out Euclid himself. BTW, my Geometry teacher also taught me 2nd year Latin (Caesar) that same year.

    Geometry was a wonderful experience compared to the other math courses which were drudgery. The method used to teach those other math courses was Strict Drill. One had to do circa 25 to 30 problems five days a week. The teacher minimally taught so the class time was used to start on the problems which were in the books pretty much as I recall, books designed for drill. A quiz a week was given and a final exam each semester. For three years the total of math problems done had to be over 15,000. It was a painful system but those of us who weathered the storm found College Math was a breeze. our brains were wired for math.

    When my son took geometry in HS, early 1990s, proofs had been shunted aside. I talked to his teacher at the fall parent-teacher meeting and was told that the course was now relevant. I told him that the course was gutted and dumbed down. My wife was scandalized by my bluntness placing me in the dog house for a while. My response was to tutor my son in math for the next two and a half years so he was prepared for college. It worked for him as he is now a Ph. D. in Mechanical Engineering.

    Dan Kurt

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  • Core Meltdown Coming | educationrealist

    […] proofs and constructions. I’ve written about this extensively (see the above link, here, and here). Geometry teachers quickly learn that, with extensive, patient, instruction over […]

  • surfer

    Interesting post. I had a very strong public school experience, but intrinsically geometry was always weird. You don’t use it the way you do algebra 1, 2, trig, functions, analytical geometry, calc 1-3, ODE, or even engineering math (PDEs, etc.) or linear algebra. It was explained that this was our exposure to proofs and fine. But it always seemed different.

    A course that was more algebraic geometry would be better. Not analytical geometry, that is too hard. But perimeters and areas (and surface area and volumes for 3D) seem like good practice of algebra.

    You never finished off the story and said how your guys did versus the other classes on the final.

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