kitchen table math, the sequel: Teaching Geometry According to the Common Core Standards

Saturday, May 19, 2012

Teaching Geometry According to the Common Core Standards

Teaching Geometry According to the Common Core Standards,  by H. Wu.

from the preface:

This document is a collection of grade-by-grade mathematical commentaries on the teaching of the geometry standards in the CCSS (Common Core State Standards) from grade 4 to high school. The emphasis is on the progression of the mathematical ideas through the grades. It complements the usual writings and discussions on the CCSS which emphasize the latter's Practice Standards. It is hoped that this document will promote a better understanding of the Practice Standards by giving them mathematical substance rather than adding to the verbal descriptions of what mathematics is about. Seeing mathematics in action is a far better way of coming to grips with these Practice Standards but, unfortunately, in an era of Textbook School Mathematics, one does not get to see mathematics in action too often. Mathematicians should have done much more to reveal the true nature of mathematics, but they didn't, and school mathematics education is the worse for it. Let us hope that, with the advent of the CCSS, more of such e fforts will be forthcoming.

there is a seamless transition from the geometry of grade 8 to high school geometry in the CCSS. The concepts of rotation, reflection, translation, and dilation taught in grade 8--basically on an intuitive level--become the foundation for the mathematical development of the high school geometry course. In the process, students get to see, perhaps for the fi rst time, the mathematical signi ficance of rotation,
reflection, translation, and dilation as well as the precise meaning of congruence and similarity. Thus the latter are no longer seen to be some abstract and shadowy concepts but are, rather, concepts open to tactile investigations.


Because rotation, reflection, translation, and dilation are now used for a serious mathematical purpose, there is a perception that so-called "transformational geometry" (whatever that means) rules the CCSS geometry curriculum. Because "transformational geometry" is perceived to be something quaint and faddish--not to say incomprehensible to school students--many have expressed reservations about the CCSS geometry standards.

The truth is di fferent. For reasons outlined above, the school geometry curriculum has been dysfunctional for so long that it cries out for a reasonable restructuring. The new course charted by the CCSS will be seen to ful ll the minimal requirements of what a reasonable restructuring ought to be, namely, it is minimally intrusive in introducing only one new concept (that of a dilation ), and it helps students to
make more sense of school geometry by making the traditionally opaque concepts of congruence and similarity learnable. One cannot overstate the fact that the CCSS do not pursue "transformational geometry" per se. Transformations are merely a means to an end: they are used in a strictly utilitarian way to streamline the existing school geometry curriculum. One can see from the high school geometry
standards of the CCSS that, once reflections, rotations, reflections, and dilations have contributed to the proofs of the standard triangle congruence and similarity criteria (SAS, SSS, etc.), the development of plane geometry can proceed along traditional lines if one so desires.

Such knowledge about the role of reflections, rotations, etc., in plane geometry is fairly routine to working geometers, but is mostly unknown to teachers and mathematics educators alike because mathematicians have been negligent in sharing their knowledge. A successful implementation of the CCSS therefore requires a massive national e ffort to teach mathematics to inservice and preservice teachers. To the extent that such an eff ort does not seem to be forthcoming as of April 2012, I am posting this document on the web in order to make a reasonably detailed account of this knowledge freely available.


Anonymous said...

Wu also wants to try to encourage school districts not run by Gypsy Supers or people who aspire to become lucrative Gypsy Supers to actually push the content of CCSS.

If he has been listening to what Brad Frindell is pushing or Bill McCallum among others he is saddened to put it mildly.

Math via learning tasks with no fixed answer is a terrible thing. As is the math curriculum for CCSS funded by the Gates Foundation.

And yes I have seen it.

Student of History
a/k/a invisible serfs collar

SteveH said...

"Brad Frindell"

Do you mean Brad Findell?

If I take out they stuff about "Gypsy Supers" (whatever they are), I get:

"Wu also wants to try to encourage school districts to actually push the content of CCSS."

What's your point, that the content is bad or that he should be pushing something else?

"Math via learning tasks with no fixed answer is a terrible thing."

Yes, that's on the terrible thing list at KTM, but that's nothing new.

"And yes I have seen it."

Seen what, exactly?

You need to get to the point. You can't wink and nudge about things like David Coleman now heading the College Board. I find his comments interesting, but articles like the one about him in the NYT are poor. Analyze the scenario of what might happen, not just what people say.

Anonymous said...

A Gypsy Super is an administrator who just keeps moving up in salary and job titles promising to promote ever more aggressive forms of radical education reform.

A district still teaching content is suddenly got a new super taking it to Transitional Outcomes Based Education. He stays one or 2 years and then goes off for more money in a bigger district to push the more aggressive transformational OBE. Values and life roles is all the taxpayers get for all those property taxes.

Here's how it works with specifics. Fulton County Georgia is at Traditional OBE and refused to sign onto Race to the Top. Charlotte is at Transitional. Dallas is at Transitional. Cobb County, with over 100,000 students also refuses to sign onto Race to the Top.

The Race to the Top district agreement bound the school districts to shift to transitional OBE. So Fulton and Cobb suddenly get what I call Gypsy Supers from Charlotte-Meck (one of the deputies and former Chief of Staff). Cobb gets Dallas Super who takes a nice six figure retirement and then takes $250,000 or so new Super job there.

Cobb and Fulton get shoved to Transitional OBE by these new Gypsy Supers. New Cobb Super had tried to get Super job in Las Vegas. He was a finalist. They were going to Transformational.

Montgomery County Deputy goes to Reno to take it to Transformational. Stays less than 2 years. Wins Natl Super award for the new Trans OBE pilot for districts. Never uses that term but works just like William Spady designed it. Now he brings the pilot to Charlotte-Meck so they can be shifted to Transformational.

Dallas school district hires Colorado Springs super. Co Springs is already at Transformational.

More money. Constant moves away from content. Gypsy Supers is a good term to describe the behavior.

And Steve if you keep being snarky to me, why on earth would I voluntarily tell you which well-known colleges and universities have said civic engagement is now to be the centerpiece of all undergrad coursework. And I do mean ALL. Hard science included.

It is Findell. It gets typo'd a lot. He was heavily involved with Georgia's math disaster before moving to Ohio. Now he is insisted that CCSS is not about the content in standards. It's about the applications. In projects and group work. Which will then constitute the only assessment. No tests.

That's why Wu wants to keep focus on content. He wants the districts to follow content language, not where groups like NCTM and PARCC are pushing.

SteveH said...

You are using terms like traditional, transformational, and transitional OBE without definition. OBE has been around for a long time. It's just being adapted to CCSS by the usual suspects.

"Values and life roles is all the taxpayers get for all those property taxes."

This has long been a problem. What specific mechanism of CCSS makes it worse?

"More money. Constant moves away from content."

No. I see slight moves towards not letting schools get away with no content and skills. The problem with CCSS is that it lets schools stop way too soon. They continue to talk about fancy learning ideas to hide low expectations, and they continue to shift the onus of learning onto students.

" ... why on earth would I voluntarily tell you which well-known colleges and universities have said civic engagement is now to be the centerpiece of all undergrad coursework. And I do mean ALL. Hard science included."

Why would you hide this from anyone if it's so important? However, this is just a vague comment backed up with no explanation of how this gets implemented in practice, even if people are spouting such silliness. I don't look at our K-12 mission statement to see what is going on. There is a world of difference between what people say and what actually goes on.

"That's why Wu wants to keep focus on content. He wants the districts to follow content language, not where groups like NCTM and PARCC are pushing."

Wu hopes that CCSS can be a vehicle to focus on content. He might win a few battles, but NCTM, PARCC, and low cutoffs will be the main driving forces. The tests will continue to be fuzzy and K-6 schools will continue to to pump kids along to the same old high school filter.

So, what, specifically, is it about CCSS that is really new? What is it, specifically, about Coleman's leadership of the College Board that will change things? To me, CCSS is just the latest battle between higher and lower expectations. Talk is used to hide the fact that it's still the same old thing.

Anonymous said...

Steve-Good luck with life.

I have tried to alert you and you just want to argue.

SteveH said...

You are alerting us to what you consider to be serious new issues. You have not made your case. Many of the things you talk about have been going on for years.

If you want to talk about the implications of David Coleman's presidency of the College Board, that might be interesting. I would like to know how he is going to match up his beloved AP courses and curricula with the lower CCSS standard. Does he see the SAT going away over the long term, replaced by AP test scores? It would be so much better for my son to focus on AP classes rather than both AP and SAT. So much better.

The question then is whether AP will be watered down to meet the lower CCSS standards. Coleman will ruin the College Board if he does that. They will lose the SAT and many colleges will devalue AP test scores, especially if 50% of the students get 5's. They will move to the ACT. However, his move towards having the AP tests replace the SAT could be a real coup over ACT and IB.

If they reduce the difficulty of AP, they could make the test scores nonlinear (like the SAT), so that very few get 5's. They could keep some tests difficult, like calculus, chemistry, and physics, but make the others easier. BTW, I just found out that AP Music counts as a math course at our high school.

My view is that if the College Board stops being able to separate students at the top end (however poorly it's done), then colleges will just not need them anymore. They will lose a lot of income and be completely co-opted by the K-12 education thought world.

My best guess is that he is planning for the long term elimination of the SAT and replacing it with AP. I'm for that. Maybe.