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.

- 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!

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.

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