Didymath

Closing the Teach For America Blogging Gap
Jul 08 2013

#73: Rethinking the Geometry scope and sequence

Last year, our district was in thrall to the unit exam schedule of CSCOPE, a very expensive curriculum/unit exam regime not at all appropriate for students who were behind grade level. Our district math coordinator at the time insisted at the time that fealty to this schedule would ultimately lead our students to the promised land.

Fast forward one year and our new district math coordinator has asked “What do you want Geometry to look like?” I am salivating at the opportunity to gut this bloated mess of a course. When Texas changed state tests from TAKS and STAAR, one of the philosophical shifts was to move away from lower-level skills like factual recall and mere identification to a test measuring “depth of knowledge.” I’m okay with that so far. You’d hope a test could measure critical thinking skills. Problem: your state standards cover a billion things(1). What in the name of high school football am I supposed to do with these billion topics if I want to cover each of them deeply(2)?

My answer is to say we can’t possibly cover everything! This is not some low expectations game I’m playing, this is just reality. I get 37 weeks of school time and a whole mess of that is testing, fire drills, students out for field trips, doctor’s visits, badly forged notes from fake doctor visits, court dates, paid employment, baby problems, baby mama problems, Fridays, Mondays, 7th periods, 1st period, periods adjacent to lunch, terminal explosive diarrhea and overactive bladders. I mean, once you factor all of that, I have about  four days to teach all this.

Last year, the units went a little something like this:

  • Unit 1: Foundations of Geometry. Introduction-type stuff like points, lines, and planes. Very heady, existential stuff and neither sitars nor analog synthesizers provided. Seriously, how am I supposed to do my job in these conditions? Conditional statements and their converses, inverses, and contrapositives make an appearance, albeit far too briefly (as do other logical novelties).
  • Unit 2: Geometry on the coordinate plane. In other words, how many ways can we bastardize the most beautiful mathematics course most people will take with algebraic nonsense? Get that slope outta here.
  • Unit 3: Parallel lines and transversals. Now we’re getting to the good stuff. Angle congruence, the beginning of proofs. Teach them transitive property, create a better world for all to live.
  • Unit 4: Triangles. The best unit, all kinds of proofs, triangle congruence postulates. Teach them how to argue logically and make sense deductively.
  • Unit 5: Right triangles. Special right triangles, Pythagorean theorem, and trig. Because they had done some in Algebra and 8th grade, Pythagorean theorem is enjoyed by many. They get special right triangle ratios only superficially, and struggle with the fraction values that often occur when hypotenuses have whole number values. Radicals are just…whoa. Don’t get me started on trig, though they dig the mnemonic device to remember SOHCAHTOA(3).
  • Unit 6: Quadrilaterals. Inexplicably short unit, don’t really get an opportunity to extend proofs this way. We talk about rhombi, rectangles, parallelograms, squares, and their properties.
  • Unit 7: Properties of two-dimensional figures and Unit 8 Measurement of two-dimensional figures. Combining these because, hey, two-dimensional figures! Lots of polygons, little bit of circles. Blow minds with the apothem. Regular hexagons got a little too much hot sauce. Lots of formulas and practice with the formulas (read: calculators required). This is when things start to get tedious.
  • Unit 9: Properties of two-dimensional figures and Unit 10 Measurement of two-dimensional figures. Combining these because, hey, three-dimensional figures! Lots of plastic models. Make a parody of the song “Bugatti” about polyhedrons. Blow any unblown minds with your acceptable rap skills. More formulas, more tedious calculations. We have become the robots. No Kraftwerk music in included but you play some on Spotify anyway before school.
  • Testing happens.
  • You do other things because testing happened.
  • Blog more because it is summer now.
I’m looking at this and I know I can’t just cut the calculations out of Geometry. I’d be totally screwing them over in Algebra 2 if I did. But I want more opportunities for thoughtful exercises and fewer “Label the base 1 and 2 and then plug those into the formula on your formula chart. CLEAR JAM IN TRAY 2. SERVICE ENGINE SOON. EXACT CHANGE ONLY.”
Some more specific thoughts:
  • Constructions were an afterthought last year, particularly because we lacked class sets of compasses until about November. No longer! I plan on integrating constructions as often as possible.
  • Where does that hideous Unit 2 go? It’s mostly midpoint, slope, and distance formula. I’m thinking dissolve it and integrate those elements where they’re more useful. Distance formula for right triangles and measurements of two-dimensional figures. Midpoint goes to triangles, perhaps. Slope with parallel lines and transversals. I haven;t thought about this one enough, but congealing them into one unit feels like a rote skills practice racket and not as meaningful as applying them in their “natural habitats.”
  • Whither Euclid, Pythagoras, or Riemann? So little room to discuss the history and development of this field. I didn’t understand it last year but via my summer reading, this is really interesting stuff! I don’t care if there is only one question on the end-of-course test, I want to talk more about this stuff! Have you looked at all 13 books of the Elements? Euclid was a gangster!
  • The poor circle is getting the short shrift. But where does it fit among all this angular hullabaloo? Seriously, where do secants and tangents and chords
  • A part of me wants to stop at quadrilaterals and get really deep with triangles before I even get there. How can we get to solids? You wanna talk about, like prisms, but you don’t even get how beautiful an equilateral triangle is yet? Objective: Student will be able to get how beautiful equilateral triangles are, man.
  • Square roots are a big deal. I need to scaffold up to that if I want them to get special right triangles or Pythagorean theorem.
  • I have an idea that I’m afraid to try. I want to read my cynical, grizzled 10th grade students The Runaway Bunny to illustrate conditional statements. Nearly the whole book is…Bunny: “If you will [x], then I will [y].” Mom Bunny: “If you will [y], then I will [z].” It is a very sweet book, and logical!
Are there any other Geometry teachers out there? What does your school/district/state recommend for the scope and sequence of your course?

NOTES

(1) I may be exaggerating here.

(2) KIPP answer: NEVER-ENDING SCHOOL YEAR. ABOLISH SLEEP. YOU GET THE CRATE IF YOU SO MUCH AS BLINK.

(3) It’s a bit coarse, but “Some Old Hippie Caught Another Old Hippie Tripping On Acid” is definitely memorable.

9 Responses

  1. Gary Rubinstein

    A course based on The Elements would have to focus on books I and III. Things get pretty hairy around book II.5. The running theme would be constructions. Book I even leads up to an interesting one where given a triangle an angle and a line segment, to make a parallelogram which has the angle and the line segment and has the same area as the given triangle. They build up to more complicated constructions, like the regular pentagon, which takes about a half hour to accomplish. When they prove triangles congruent, it is for a purpose, which is to show that the construction works.
    I’ve done some videos of good things from The Elements
    http://www.youtube.com/watch?v=ULWtYXuuBXM
    http://www.youtube.com/watch?v=hRRU9dLqkTw
    http://www.youtube.com/watch?v=T9Csok98COg

    Another approach to Geometry is to use a lot of The Geometer’s Sketchpad and have a lot of experimentation and conjecture making, which would then lead to motivation for more formal proofs.

    The 1985 NCTM yearbook has a great article about a full informal geometry course which would explore some big themes like ‘invariance’ which can be done at various levels of depth.

    http://www.amazon.com/Secondary-School-Mathematics-Curriculum-Yearbook/dp/0873532171/ref=wl_it_dp_o_npd?ie=UTF8&coliid=I2VN2575E7L2YJ&colid=J4H90A5G0G1Y

    • mches

      Gary, thank you for this!

  2. I think you’ve got the wrong idea. To teach geometry as anything other than Algebra Applications with geometric facts is doing your students a huge disservice.

    For example, constructions and even proofs are a massive timewaster. I don’t bother with the first, and usually skip proofs as well.

    As for coordinate geometry, I review it briefly and then bring it up again and again–when teaching transformations, medians, altitudes, and so on.

    Also, I assume you are teaching similar triangles with triangles. It makes much more sense to go from triangles to medians/altitudes/triangle midpoints to similar triangles, then a very brief overview of right triangles (I mean, come on, the kids know the Pythagorean theorem), then return to special right triangles AS similar triangles, which they are, and then trig as the universe of similar triangles.

    If you check my site you can see a lot of this under geometry. I haven’t written up the last yet, it’s on my list.

    • mches

      I will have to take that under advisement then! What is your rationale for pushing aside proofs and constructions?

  3. They will never use either again. I’m
    assuming you are teaching primarily low to mid ability kids whose understanding of algebra is shaky. If you’re teaching an honors class where every kid knows how to factor a quadratic and solve a system, then go to town.

    A few links, if you’re interested:
    Evidence of incoming and outgoing alg 1 ability in geometry classes: http://educationrealist.wordpress.com/2013/06/23/algebra-1-growth-in-geometry-and-algebra-ii-spring-2013/

    And then here’s most of my stuff on teaching geometry:
    http://educationrealist.wordpress.com/?s=Teaching+geometry

    • mches

      Cards on the table, I am not really compelled by the argument that they’ll never use it again, but I think we may have different ideologies flowing through our teaching mojo. I’m sympathetic to the sentiment, though. Algebra is more useful as a math qua math, but I feel like the deductive reasoning and logical statements element of geometry has plenty of applications in less explicitly mathematical settings (law/philosophy, computer science). Constructions — in addition to being a crucial part of the history of the course — are fun. I think there’s something to be said for having fun. Again, bias revealed.

      Reading over my own post, I think I was a little harsh on poor Algebra in my glibness. I definitely need to incorporate algebraic connections as much as I can since most of them will be taking Algebra 2 next year.

      I am looking through a lot of your posts (the Dr. Wu one caught my eye) and I’m thinking a lot more about congruence and similarity as two huge ideas that I want to be sure and get right.

      As an aside: “Me, I often don’t know what I’m teaching the next morning.” I know I’m TFA, but I identify with this a lot. I’m a big mental planner. In the shower, on the car ride home and the car ride over. There’s also something to be said about being able to think on your toes.

  4. “I am not really compelled by the argument that they’ll never use it again,”

    In math, this is a decision you have to make time and time again. Remember, for many low ability kids, you are talking about giving them skills to pass a test to get them out of remediation (or minimize it). There are only so many hours in the year.

    I don’t believe in utility above everything, but when it comes to teaching low ability kids proofs (which they won’t get at all) as opposed to giving them algebra and coordinate geometry work in conjunction with geometric facts, that’s a no brainer for me.

    Congruence, too, they never use again. Similarity they use constantly.

    And a big Yes to your last paragraph.

    • mches

      I will admit that my shift in thinking about this curriculum is driven in part because the only test my kids “need” for graduation now is Algebra 1. Geometry, for better or worse, is gravy in the grand scheme of things. I’m sure they will rue the day they gave someone as capricious as I am such freedom.

      My students were by no means masters of proofs yet, but even if it’s just understanding more general ideas like triangle congruence postulates or transitive property of equality, they could start to see what justification or statements would be necessary to make an incomplete proof complete.

      • I generally introduce proofs with algebra. That is, I go through the equality properties and then demonstrate that every time they solve for x they are actually committing a proof.

        When I teach proofs simply (which I do with my freshmen geometry course, keeping in mind that the bulk of them are also terrible at algebra), I do three step proofs: you are given two facts, you have to add the third. The third is generally going to use the reflexive property, vertical angle, or alternate interior angles. And each year I am stunned at how much they struggle with this very simple process.

        ” the only test my kids “need” for graduation now is Algebra 1.”

        I’m not talking about high school graduation, but college placement.

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