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.

Common Core Standards

The process is moving right along. In my son's high school honors classes, the teachers are required to cite CCSS book, chapter and verse for everything. Some teachers are annoyed and are directly teaching the kids their opinions. Also, our local paper talked about how the lower schools are finishing their scope and sequence document for CCSS math with help from the Dana Center at the U. of Texas. The Executive Director is Uri Treisman, whose philosophy is summed up by: "To Treisman, high school algebra is the burial ground for the aspirations of many students in part because "almost no one uses the content of these courses in their subsequent university courses." So why are these gatekeeper courses determining who goes to college? Kids who drop out of these math classes are the same kids who drop out of school, says Treisman. At the other end of the spectrum, students who need more challenges aren't finding them in high school math." "almost no one uses the content of these courses in their subsequent university courses." I had to reread that a few times. The Dana Center is working with the Partnership for Assessment of Readiness for College and Career (PARCC) on their CCSS test - which happens to be the test that our state will use. I've been trying to find some sample questions, but have only seen a few scanned examples that I could barely read. Everybody is waiting for test developers to quantify what CCSS means by "fluency", "understand", "reason abstractly", "solve", "represent", and a host of other vague terms. "Fluency" sounds pretty explicit (and it is specifically used sparingly in the standard), but we will have to wait for the sample tests. Then, we will have to wait to see how different states define the low cutoff levels. I find it instructive that some seem only worried about whether the standards will ensure that nobody has to take remedial courses in college. They are trying to find out whether colleges will buy into accepting their test results - and their cutoffs. There's the rub. I also found this commentary on fluency. "Although we do find that students who are fluent in facts have fewer obstacles when engaging in more complex problem-solving, we also find that many students who are fluent in facts still struggle with such problem-solving, and many who are not yet fluent are able to generate highly sophisticated solution strategies to different types of problems." Fluent students have fewer obstacles with more complex problem solving, but they still struggle with problem solving. And many who aren't fluent are able to "generate highly sophisticated solution strategies". Figure that out. They can't let go of their idea that understanding is somehow not connected with fluency. The haves will continue to get help at home and completely ignore CCSS. They will get on the AP calculus track to a STEM career and the rest get to look forward to college without remediation.

adverbials are your friends

Do I need a $165 book on adverbials?

I'm sure I need a second life so I can actually read a $165 book on adverbials.

thinner, part 2

In the wake of my discovery re: weight and cognition, I've decided to harp on my vegans-are-thinner experience. Re-harp, that is.

AND SEE: thinner
AND: the Ed diet

arithmetic and the brain

ABSTRACT

Recent studies in human neuroimaging, primate neurophysiology, and developmental neuropsychology indicate that the human ability for arithmetic has a tangible cerebral substrate. The human intraparietal sulcus is systematically activated in all number tasks and could host a central amodal representation of quantity. Areas of the precentral and inferior prefrontal cortex also activate when subjects engage in mental calculation. A monkey analogue of these parieto-frontal regions has recently been identified, and a neuronal population code for number has been characterized. Finally, pathologies of this system, leading to acalculia in adults or to developmental dyscalculia in children, are beginning to be understood, thus paving the way for brain-oriented intervention studies.

Arithmetic and the brain.
Dehaene S, Molko N, Cohen L, Wilson AJ.
Curr Opin Neurobiol. 2004 Apr;14(2):218-24.

help desk

2 seats open; 5 candidates running; maybe 1600 people will vote. Two candidates are running as a slate, with posters around town saying "Two seats, two votes."

There will be some bullet-voting. No idea how much.

What's the fewest votes a candidate would need to win if the race is close?

good news from the state tests for once

Robert Pondiscio on Pineapplegate 

The kids interviewed in the Times seem to know exactly what's wrong with the pineapple item.

most Berkeley students do not know

Brad De Long:

MOST BERKELEY STUDENTS DO NOT KNOW THAT 2^5=32 AND 2^10≈1000
How can they expect to survive in the modern world without knowing these things?
And why haven't they learned them?

To which I can only say: Brad DeLong has not been paying attention!

I've just started reading the comments. Calculators in grade school aren't faring well thus far.

From the first Comment:

The best students are great, but the quantitative reasoning skills of even the average student at a good university are worse than those of a typical waiter/waitress 40 years ago.
invhand

brain atrophy in teens with Type 2 diabetes

We also measured the brain, the hippocampus... What we found was that among the kids with Type 2  diabetes, their hippocampi are smaller in volume, very significantly. For those of you who work in Alzheimer's disease, the difference between the kids with Type 2 diabetes and the control kids [who were obese but did not have Type 2 diabetes] is about the same as that between a normal elderly and one with mild cognitive impairment. So the volume difference is around 12% So this is not a small, little thing. So what will happen to these children, whether we're actually seeing permanent damage or not, we don't know.

Their frontal lobe regions are also affected, and they have more overall brain atrophy than the control group. And remember, the control group was an obese control group.

Impact of Obesity and Metabolic Disease on Brain Structure and Function 5/5/11
Antonio Convit, M.D.
NYU

I haven't watched the entire lecture, but I gather that the reason he tells us to remember that the control group is obese is that we can also expect to see brain changes in obese teens who have not developed Type 2 diabetes, which would mean that the brains of teenagers with Type 2 diabetes are even more different from the brains of normal-weight adolescents.^*

The lecture - the few minutes I've watched of it - is horrifying.

I had no idea.

^*update (4/5/2012)

Right. Obesity in and of itself, without Type 2 diabetes, is linked to brain atrophy. Sounds like overweight may be as well, at least in people over 70.

They found that obese individuals [over age 70] had, on average, 8 percent less brain tissue than people of normal weight, while overweight people had 4 percent less tissue. According to Thompson, who is also a member of UCLA's Laboratory of Neuro Imaging, this is the first time anyone has established a link between being overweight and having what he describes as "severe brain degeneration."

"That's a big loss of tissue, and it depletes your cognitive reserves, putting you at much greater risk of Alzheimer's and other diseases that attack the brain," he said. "But you can greatly reduce your risk for Alzheimer's if you can eat healthily and keep your weight under control."

btw, remember back when I was bugging everyone about vegan diets making you thin?

Well, they do. I adopted a vegan weight-loss diet in September 2009, lost 11 pounds, and have basically kept it off ever since -- without even being a vegan. I need to get back on track, but still: even part-time veganism makes you thinner than full-time non-veganism.

what's wrong with this picture?

Education Is the Key to a Healthy Economy By GEORGE P. SHULTZ AND ERIC A. HANUSHEK

not your father's bell curve

Sticky wages in 2011:

The tall bar in the middle represents workers who had a wage increase of $0
People to the right of the bar had pay raises
People to the left of the bar had pay cuts
Why Has Wage Growth Stayed Strong?
By Mary Daly, Bart Hobijn, and Brian Lucking

I've been meaning to post this for a while now.

I find this chart fascinating. I have never, not once in my entire lifetime, seen a curve that looked like this -- although they must be out there.

Here's the San Francisco Fed write-up:

Researchers generally point to asymmetries in the distribution of observed wage changes among individual workers as evidence of nominal wage rigidities. Figure 2 plots an example of this type of wage change distribution in 2011. The dashed black line shows a symmetric normal distribution. The blue bars plot the actual distribution of nominal wages.

The figure’s most striking feature is the blue bar that spikes at zero, indicating the large number of workers who report no change in wages over a year. This spike stands out in the distribution of actual wage changes, suggesting that, rather than cutting pay [in line with reduced earnings], employers simply kept wages fixed over the year. This is supported by the large gap to the left of zero between the actual distribution of wage changes and the dashed black line representing the normal distribution. This gap suggests that the spike at zero is made up mostly of workers whose wages otherwise would have been cut.

Once you start thinking about sticky wages, you see them everywhere.

For instance, Ed and I were watching an episode of Friday Night Lights a couple of weekends back, and the story line, which spanned multiple episodes, revolved around budget cuts to the schools. First the cuts were rumored, then the cuts were announced, and then there was rending of clothes and tearing of hair and parents mobbing board meetings to demand that cuts happen to other people's programs, not theirs. (And, yes, this sequence of events does sound familiar).

It was high drama.

Teachers would be FIRED!

Football teams would be MERGED!

Fire and flood, death and despair!

Two years ago, right up to the moment I found out about  downward nominal wage rigidities over the business cycle, ^* this story line would have made perfect sense to me. Like everyone else on the planet (including pick-up farm workers and their employers in India, apparently), I simply took it as a given that people can be cut but wages can't. In hard times, fear and loss (and television drama) follow directly from this belief.

But once you know about the money illusion, a school budget crisis loses most of its oomph as dramatic premise. Watching the pandemonium onscreen, I was unmoved. I kept thinking, "How about a wage freeze? How about a furlough? How about a wage cut?"

"How about everyone sit down and do some arithmetic and, while you're at it, figure out that it's not like the Dillon School District is a family where the sole breadwinner just lost their^** job. You've still got money coming in, you just don't have as much money coming in next year as you did this year. (Or, if Dillon is anything like Westchester County, you don't have as big an increase coming in next year as you did this year, but you've still got an increase.) So everyone's gonna have to make do with less, but nobody's gonna starve, not unless you insist on firing the young teachers so the old teachers don't have to take a cut."

No dice.

The story arc ends with teachers getting fired and football teams getting merged, and nobody says 'boo' about the possibility of 'shared sacrifice' and the like.

I realize that, in real life, employees do take freezes and furloughs to keep everyone on the job. But they don't do it often, as the chart reveals.

^* I have now consumed so much macroeconomics that I can read formulations like downward nominal wage rigidities over the business cycle almost as fast as I can read twinkle, twinkle, little star.
^**It pains me to write "the sole breadwinner just lost their job," as opposed to "the sole breadwinner just lost his job," but I think the time has come.

Direct Instruction in Grammar

From the study guide for the California Teachers of English Learners Teacher Certification Exam:

"Direct instruction in grammar and spelling has had disappointing effects on students' writing.  Teachers have not achieved much success with extensive error correction either.  The most successful teaching of language conventions has been the presentation of well-written materials.  A good reader becomes a good writer as the self editing process develops and good models are available.  A teacher is most likely to be successful if he/she keeps a variety of well-written and easily understood examples of both written and spoken English available to the students."

Study guide was prepared by XAMonline.com and has no affiliation with the California Teacher Credentialing Dept. nor the testing companies that California uses.  The above quoted passage doesn't seem that far out of line from what I've seen presented in ed school classes.

Susan S on Facebook

One problem I had was with my special ed son. He is very "young" even though he's grown. He mostly likes coupons and anime, and kid stuff, so I never worried about him and rarely checked.

Well, I noticed one day that he had a couple of men that he barely knew that were his new "friends". They were friends of friends of friends, or so the story goes. He really didn't know them. Well, they were engaging him (trying to set up a meet) that was disturbing to me. He was uncomfortable, too, so he did unfriend them and refused to answer. But I was surprised that he had allowed them in.

If a photo gets tagged, you can get it untagged, but if it has gone out to a bunch of people, I don't think you can get it back. So, someone can have a picture of you that you have no control over, and then tag it and send it to many.

Another time, and I think this was Yahoo and not FB, I was alerted that a friend of mine had commented at some blog. It even showed me her comment. I asked her if she knew about it and she said no. We both had to change our privacy settings (and the rest of our family's) to stop it. Again, I think that was Yahoo.

Someone out there probably knows more than me, but those were just a few of my experiences.

Facebook rules

Terri W:
Do not post or say anything on any network, email, chat, text, website, WHATEVER unless you are willing for the whole world to see it.

chemprof:
For Facebook, I'd actually use a slightly different rule (borrowed from a friend who was going through a lot and who loved posting on Facebook): "Only good goes on Facebook." When I see my students getting into trouble, that's the rule they break.

Does your student know 10/9 is a fraction?

American elementary and middle school mathematics programs are poor in myriad ways. They lack breadth and depth. They lack reasoning. They lack precision. An object lesson in all of the above is the common use of analogy to teach math, even in grades 5-8.

 Math by analogy is when teachers substitute ideas completely unrelated to math in order to make some concept "easier". Usually, this is because they themselves do not understand the meaning behind what they are teaching, so they cannot explain it accurately. Math by analogy substitutes presumed common context for reasoning. Yet most young students don't share enough common context to build the analogous connection anyway, even if they can abstract away from the literal -- something most children cannot do. And if you are are ELL, it is probably entirely worthless. Plus, the analogy is by definition imprecise, so its correctness will break down with even the slightest scrutiny.

 You see math by analogy in both big and little examples, from the use of it to "explain" greater than and less than to its use in teaching place value. The most common analogy I see used by teachers and their books is that "a fraction is part of a whole".This analogy has devastating results. I routinely (in 100% of classrooms not using Singapore math, in more than 50% of the students) hear:

•  1. "there's no such thing as ten ninths." that's the majority response in classroom after classroom. Why? Because a fraction is PART of a whole. How can a part of a whole be bigger than the whole? What's the whole then?  
• 1b. therefore, they believe no fraction can be bigger than 1.
• 2. "You can't divide 6 things among 7 people." 6 things isn't one whole. It's 6. 
• 3. "three thirds is A Whole." Not one. 
•  3b. Therefore, they don't know 3 divided by 3, written as a fraction, is 1. I often hear of students who ask "is this a division problem or a fraction problem?" 

 Additionally they don't know decimals are fractions. how could 1.2 be a fraction? Twelve tenths isn't a fraction, remember?

 These problems are so severe because these students have teachers who manage not to notice these errors. No problems in their books, no lesson script in the teachers guides illuminates this to the teacher. They only see the most trivial of problems. 10/9 is beyond the pale.

 The correct explanation is that a fraction is a number. What number? A number defined on the number line as follows:

1/3 is the point on the number line when you break the unit length into 3 equal length parts, and take 1 part. the endpoint of that part is 1/3.

4/3 is the point on the number line when you break each unit length into 3 equal length parts, and take 4 parts. the endpoint of those parts is 4/3.

 Yes, teachers will need to build up to this. They should do so.

Terri W is of two minds

re: Fill-in-the-blank has a really bad idea

I'm of two minds whenever I hear of particularly dumb ideas being floated in the school systems.

On the one hand, I'm thinking, "Cool, this gives my kids a leg up over the competition."

Then on the other hand, I realize that the overwhelming vast majority of the next generation of citizens are being educated with these cockamamie ideas and I think: "We are so hosed."

I'm on both sides of that fence with just one (typical) kid!

C. is now an official Math Victim of U.S. schools (including his Catholic high school, sad to say). And, at the same time, he's way out in front in the verbal realm compared to most h.s. seniors in America (also thanks to his Catholic high school, as well as to family background).

Speaking of 'official,' C. has just now reached the point of maturity at which he realizes, without being told by his mother: Holy ****, I don't know any math. 

He has two weeks left in high school. He came home the other day, said he'd had a talk with his math teacher (statistics!), who had convinced him he needed to take math in college. So he was wanting to know whether NYU has remedial math courses.

K-12 kids don't know, while they're K-12 kids, that they're going to be sorry they didn't learn math or writing or science or whatever else they didn't learn. I remember Glen, I think it was, once saying that we have to advocate for our children's future selves: for the selves they're going to be, eventually.

That's exactly right.

The Daily has a really bad idea

I love The Daily. Love, love, love. Read it every day.

That said, today's issue has an op-ed on the high cost of college that proposes, as a solution:

"Why not reduce undergraduate education to three years, while tacking an extra year onto high school?"^*
Josh Barro

Good God Almighty, as my father would have said.

Five years of public high school?

FIVE YEARS of Hunger Games in English and Core-Plus in math?

And here I was thinking everyone should skip senior year and go re-take algebra and English at their local community college instead.

(Is high school cheaper than community college? High school teachers here earn a great deal more than instructors in community colleges, but I don't know whether that's the case elsewhere.)

and see:
the founder, chairman, and CEO of Netflix has a really bad idea
Larry Summers has a really bad idea
David Brooks has a really bad idea
David Brooks has a really bad idea, part 2
All is forgiven.

^* That's the pull. Full passage: Why not adopt the Quebec model and reduce undergraduate education to three years, while tacking an extra year onto high school? 

Core-Plus Students at Michigan State

In summary, our data show a clear decline in the level of Michigan State University mathematics courses taken by Core-Plus graduates. The existence of that decline is statistically significant at any reasonable level. The decline in course level is accompanied by a decline in average grades for all but the very top students, as well as a decline in the percentages of those who eventually passed a technical calculus course. These trends occur in data that include students from a variety of communities. The data from individual high schools show that the timing of these declines corresponds precisely to the implementation of the Core-Plus program.
A Study of Core-Plus Students Attending Michigan State UniversityRichard O. Hill and Thomas H. Parker Thomas Parker

a cohesive paragraph about cohesive paragraphs

I spend a lot of time thinking about cohesion in writing.

student work

pretty good!