# New York State: looming geometry teacher shortage

Most schools in New York State haven’t offered “Geometry” in decades.

Proof was in a long slow decline when Sequential Mathematics Course 1, 2, and 3 were introduced in the late 1970s. Parallel lines and vertical angles were in Course 1. Triangle congruence was in Course 2. Circles were in Course 3.

And separate as these were, proof withered. As Course II (where some proof remained) was begin phased out in the late 1990s, the proof section of the NYS Regents exam was predictable: choose one 10-point proof from two out of three possible choices, from: 1) logic, 2) Euclidean, or 3) coordinate geometry.

Math B (2000-10) was a rambling exam. All over the place. Sometimes there was proof, sometimes not. A few schools decided not to bother with proof – it was never more than 6 points anyhow – when it showed up.

#### Integrated Geometry

Next year New York State introduces an Integrated Geometry Regents. Algebra starts this year; it likely will be quite easy to pass – it is a graduation requirement. Geometry is not. The people who worked on the geometry committee were old school, tough. The course will be geometry only, a throwback. The State attempted to delay starting these exams, (1, 2, 3) not because of algebra problems, but because of concerns about geometry.

Who will teach this? Few people educated in New York State in recent years got a strong high school foundation in geometry. Colleges don’t, as a rule, teach the Euclidean geometry we associate with high school – at CUNY it is banned as a high school course. Pull teachers out of retirement? Recruit from out of state? Mandate crash-course ‘institutes’ to cover missing topics?

They (who is they?) will go the crash-course route. They will violate a cardinal rule of teaching math: it is not ok for the instructor to know just a little more than the students. They created this mess. And they have no other way out.

But not quite yet. Next year most schools won’t realize they are in trouble. Plenty of NYers think they learned enough in Course 2 to teach geometry (with no extension, no reinforcement in college). We will see results in June 2009, and then administrators will start scrambling.

I found it interesting that the only Geometry course I was required to take was “modern” Geometry – which admittedly was interesting. However, I was concerned by the number of future teachers who didn’t have a solid foundation in Euclidean Geometry (this weakness was apparent when we took History of Math). Sounds like it will be an interesting year in NY next year.

The old teachers are not valued, in fact, they are being pushed to retire. No one cares whether the teachers know the stuff or not. The board of regents will just lower the passing score so everyone can pass.

In regular schools in NYC, yes. But not in the suburbs, upstate, Long Island.

And yes, algebra will have a very low passing score, but no, geometry will not. Your Posmantier was on the committee, among others. There’s nothing but geometry on the exam. I am guessing today that this exam will be harder than Algebra 2/Trig.

I am wondering what “modern” geometry is.

We teach Foundations of geometry (that is mostly taken by preservice teachers) and a practicum that goes with it that is supposed to connect the material they are learning in foundations with the practice of teaching geometry in high school. Right now those two run concurrently which is in my opinion a huge mistake as the students don’t know enough geometry to actually have any reasonable discussion about teaching the topics. Further the actual foundations course is in my opinion not even a glorified version of a high school geometry class, but in fact a shortened condensed version that covers everything from Euclidean geometry, transformations, coordinate geometry, hyperbolic geometry, spherical geometry and trig. Needless to say I am changing it as soon as I teach it, which is next year :)

modern = post-Saccheri? or post-Lobachevski?

is this a course for teachers?

My “modern” geometry course seemed to be whatever interested the professor. It was a, “Comparative study of modern postulates, invariants, and implications of Euclidean, projective, and non-Euclidean geometries.”. We spent two class periods on Euclid. The rest was a smattering of topics. No, it wasn’t a course for teachers. It was the required geometry course for math majors.

Jackie,

I wish projective geometry were taught in high school somewhere. Then again, I could think of a number of topics I would have liked (and I wish the visiting asst prof I took linear algebra/Calc 2/ODE from in college 1st year had been more competent).

The more serious concern I have about all this with geometry is the teaching of proofs, though I suspect I have different concerns from Jonathan. Maybe my experience was unusual, but I recall proofs at all levels being presented as finished products, all cleaned up, with none of the mess of real math creativity. It’s sort of like telling people to produces cakes and pies and (for a Regents diploma) Baked Alaska when all they’ve had is the finished product. I wish I had had more math teachers like Julia Child, complete with blowtorch… and I’m one of the ones who took well to math in my cohort (graduated high school in 1983).

OK, so I’m teaching the geometery class that all of the future teachers (math secondary) have to take, and this is what I’m doing (very different from the last person to teach it). Tell me what you think:

We do almost exclusively Euclidean geometry; we start with axioms and we prove theorems.

I only prove 1-2 of the very hardest theorems, all the rest of the proofs are homework to figure out, and they are presented on the board, and we discuss whether the proof is clear and correct or not, etc..

We don’t get to all of Euclidean geometry–we didn’t make it to the concurrency theorems this time for instance. We go at whatever pace the students in the class are able to muster. My goal is for students to know how to prove theorems by themselves, to have some intuition for what a valid proof is like, and to do it in the context of the geometry they are going to teach.

Does this sound good? Bad? Indifferent? I’d love feedback.

It’s not clear to me what the problem is. Don’t math teachers at the high school level have experience working with axiomatic systems? Haven’t they taken courses in topology, analysis, and mathematical logic, abstract algebra, number theory? (I have no idea, it just seems like this is the kind of thing that a math major would do) Is the problem that they don’t know what constitutes rigor and can’t generalize that to Euclidean geometry? Surely, it’s not the simply the subject matter of Euclidean geometry because historically the whole point of having students study this in particular is that it’s a whole lot easier than any other axiomatic system.

Sherman my junior!

Barely.

I fall back on how I was taught for topic after topic… but not proof. Proof for me was painful, lousy. So I reinvent. (following the lead of people I have found who do a nice job).

Proof for my students is “that which convinces.” And we have different levels of requirement for different courses, different places, different times.

The most basic sort of proof is counter-example, and we encounter it many times before we get to Euclidean proof.

I like the neat finished proofs. Wow. We develop math, moving from one topic, building (often through proof) to the next. I demo’ed the standard “square root of two is irrational” two weeks ago. Sweet. Some kids recreated it (applying to square root of three), some did not, but I think it is important to at least expose kids to these polished proofs.

But I also like mess. I like multiple approaches. Search for “puzzles” here. (Ghost the Bunny is coming up this week with freshmen. We are finishing Terance Tao’s blue-eyed puzzle in my senior elective tomorrow). They are often classroom exercises (dressed up for blogging, but still essentially classroom stuff) allowing for multiple approaches. But more importantly, allowing mess. Allowing not knowing where to start. Allowing experimentation. Allowing dead-ends. I love asking “who can share an approach that looked good, but didn’t work?” I love asking “why do you think you are done?”

All part of math. And either part would be poorer without the other.

Myrtle, often not. And in New York the high school course has largely de-emphasized axiomatic systems. As a result teachers from NY may have had only a weak course in high school, and not realize how much they missed.

Further, a math major is not required in this state to teach math. I assume that this is true in most of the United States.

I don’t know that Euclidean geometry is the easiest axiomatic system. I am going to think about that one.

lsquared,

I don’t know. It sounds ok. What do your students already know?

For me, I had worse-than-modern-New-York geometry, and knew nothing. I took 3 undergrad courses (after I started teaching!) to help me: transformational geometry, “Euclidean and non-Euclidean Geometry” (really a history course, with lots of proof-writing activity, and with geometry as reconsidered by several more recent mathematicians), and non-Euclidean geometry.

As a starting teacher I would have benefited from your course. But it really depends on who your students are, doesn’t it?

My school must be uniuue in that we teach a full term of proofs in Math B (our first term). We go through all the postulates and theorems, not much different than I learned the stuff a zillion years ago.

If Posamentier made up the exam, it will be a doozy. He is a great guy but he has not been in a NYC HS classroom for years. And, geometry is (or at least was) his favorite subject.

It’s not just Al, but he was one on a small committee. And his voice his heard.

The actual exam is being written by a vendor. The guidelines are in the links in this post. We don’t know how they are handling proof, but it probably won’t be full Course II style, and no Math 10 required proof either.

Your school is not unique, but it certainly seems to be in the minority.

Honestly, though, is it watered down in some sections, or consistently strong?

We teach proofs the way they were taught in the old days. For repeaters, we do water them down a little, but the kids still do them.

Geometry is fascinating!

Look at this geometry problem:

http://www.gogeometry.com/problem/p079_triangle_similarity_altitude_circle.htm

I have two complaints about Geometry:

(1) Giving kids a year off of algebra is a disgrace.

(2) Euclidean Geometry lacks entertainment value. I’d rather do the basic point/line/plane stuff high-level and quickly move into things like affine-linear transformations in Euclidean space and the symmetries thereunder. It’s relevant: conservation laws in nature are manifestations of symmetry, it’s pretty: wallpaper groups in Geometer’s Sketchpad, and it’s axiomatic: group theory!

Leave the nine-point circle to Honors electives, I say.

I too have issues with taking a year “off” of algebra. One ends up spending so much time reviewing Alg I in Alg II. We’ve been knocking around the idea of teaching geometry first. Is anyone doing this?

Geometry first? I can think of a number of objections: radicals (pythagorus, 45-45-90, 30-60-90), graphing stuff (coordinate proofs, slope), lots of little bits of equation solving, quadratics (off the top of my head, in solving certain circle questions)…

You may like to see some ancient geometry:

http://sarsen56.wordpress.com/solve-this/

Geometry training in July at Stuyvesant HS indicated that the course would require mostly proofs and understanding transformations using analytical geometry. Yes, most teachers are poorly trained in geometry. One would think the Regents would be somewhat rigorous but we continue to be amazed how little is required to pass most Math Regents exams.

They had no special insights into the course. Their guess is probably based on carefully studying the state documents… but it’s still a guess.

At Lehman College we have funding from the NSF for a new

Mathematics Teacher Transformation Institute, MTTI, where teachers will be prepared to teach the Geometry Regents and learn to serve as leaders at their schools assisting their fellow teachers with content, curriculum development and pedagogy. See our webpage

http://comet lehman.cuny.edu/mtti

We will soon be accepting applications from teams of teachers.

Here is the url again:

http://comet.lehman.cuny.edu/mtti

Applications will soon be available online. Note that this program is initially for public school teachers in the Bronx. We may expand in the future but the current funding from the NSF is for the Bronx only.

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I’ve loaded your blog in 3 completely different browsers and I must say this blog loads a lot quicker then most.

Can you suggest a good internet hosting provider at a honest price?

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