Roy Turrentine, an experienced teacher of mathematics in Tennessee, explains why the Common Core standards are misdirecting the teaching of his subject. The creators of the CCSS did a disservice to the standards and to American education by refusing the test the standards in real classrooms with real teachers and real students. By failing to field test the standards, there was no feedback from the world of reality and no opportunity to correct errors. Instead, the standards were sent forth with instructions that they were encased in concrete. Any business that released products that had never been tested in the real world, that had never been subject to make corrections based on experience, would soon be bankrupt. That is why I strongly recommend that every state and every district create committees of its best teachers to review and revise the standards to remove the bugs. Forget the “copyright.” What nonsense! How dare any private organizations assert the right to create national standards and then to exercise a copyright over them! Let them sue.

Roy Turrentine writes:

*I would like to relate my experience with Common Core. I am a classroom teacher in Tennessee. I have advocated more rigor in education for over thirty years.*

*In Geometry,which is my main focus, Common Core seeks to unite the Cartesian approach and the traditional approach to the topics studied. The unfortunate aspect of this approach is twofold.*

*First, the development of the traditional Euclidian approach to Geometry goes back to Euclid himself. His uniting of these concepts created a body of knowledge that has remained intact for centuries. Common Core essentially rejects topics that may only be approached in a Euclidian fashion. Not that they say this. To read the standards you wouldn’t think so. But all the testing depends on the Cartesian approach.*

*Due to this approach, and due to the nature of the testing, only topics that may be approached in the Cartesian manner are treated. Teachers will surely be teaching less, not more. This brings us to the second point. High stakes testing will restrict teachers to practicing in a very specific way. In our training in Tennessee,the emphasis is more on technique in the classroom than it is on what is to be taught.*

*Those of us who teach in high schools across America have long desired rigor. To go to meetings where people seem to feel that this rigor is their idea is nothing short of insulting to those of us who have been trying to unite the disciplines for decades. Every good teacher knows what the ideal is. We have been trying to do this for all of our careers. Having Bill Gates give me his opinion does no one any good. Having his opinion become national policy will not serve anyone.*

*Roy Turrentine*

“That is why I strongly recommend that every state and every district create committees of its best teachers to review and revise the standards to remove the bugs. Forget the “copyright.” What nonsense! How dare any private organizations assert the right to create national standards and then to exercise a copyright over them! Let them sue.”

This is recommendation follows all that we know about good teaching, except for the unforunate fact that it fies in the face of VAM evaluations. Until this albatross is removed from our necks this common sense advise will probably be ignored.

Teaching Less is exactly the point of the Common Core. The triumph of technique over knowledge, inquiry and imagination is the ideology of these corporate technocrat. They definitely will file suit not because they care about the “copyright” or the content but because it is a terrorist tactic at their legal, well-funded disposal. Bill Gates as National Policy does serve some ones with a very precise and relentless agenda.

CCSS is stereotyped “live-in mother-in-law” education:

Move in and presume to take charge because you “know how it should be done.”

LOL

I HOPE everyone READS THIS. iT MAKES ME SICK.

http://www.bloomberg.com/news/2014-01-27/silicon-valley-backed-students-aim-to-expel-bad-teachers.html

Very nicely explained.

I bristle at the word rigor though which has taken on new meaning as used by the reformers. The Reformy-to-English dictionary captures it best:

“Rigor (noun). Difficulty for its own sake, regardless of any applicable research, science or desired outcome. Rigor can be measured by a value-added assessment of the quantity of tears a child produces while he does his homework.”

“Having his opinion become national policy will not serve anyone.”

Oh, but there are those who it serves.

Euclidean? Cartesian? this is as dense as the Common Core.

It doesn’t matter the terminology – it’s the fact common core limits the teaching of geometry by ignoring it’s basic concepts.

Once again, ignorant people creating standards for subjects for which they only have cursory knowledge.

Review the standards and be ready when the window of opportunity opens. Also prepare guidelines for assessment that allows real learning. The time is now, let’s get at it.

Roy is right on target. These are not content standards. They are content and pedagogical standards–one vision of how to teach math. That has no place in standards. Thank you Roy!

The Thomas B. Fordham Institute just announced a webinar that it is holding to present results from a study that it has conducted on how much time is devoted to standardized testing. The intent is, of course, to address the criticism that standardized tests are taking inordinate amounts of time away from instruction.

Now, the folks at Thomas B. Fordham Institute believe in data-driven education. And here’s how you arrive at your data, Thomas B. Fordham style: You decide upon the results that you want to get. Then you design a study that will give you those results. In other words, they take the same approach to their data that states have long been taking to the standardized testing data that they report to the federal government–manipulate your test questions, your raw-score-to-scaled score conversions, and your cut scores so that you get the results you wanted to get.

The Fordham study has not yet been released, but it’s entirely predictable what this study will show–as predictable as a report in Pravada, back in the day, on the health of the Soviet economy (e.g, pig iron production increased 2,000 percent last Saturday).

How much time does standardized testing take out of the school year? Well, if you REALLY WANTED TO ANSWER THE QUESTION, you would include the time that students spend

1. actually taking the tests

2. doing diagnostic and “benchmark” tests to predict how they are going to do on the high-stakes test

3. doing practice tests to familiarize them with test formats and requirements

4. doing prep for the tests

5. doing activities and exercises in their print and online instructional materials that have been designed to imitate test question formats (embedded test prep)

and the time that teachers spend in data chats, trainings in how to improve their students’ test scores, and trainings on using various test prep products and diagnostic and practice tests, in the proctoring of tests, and in the interpretation of test data.

And let’s not forget the RHEELY important training in how to Erase to the Top when the tests are completed.

Talk to an actual teacher, for a change, instead of to the carefully selected members of the panels that Fordham puts together, and you will hear, “Well, it’s February, so our school year is basically over. We will spend the next few months getting ready for the test.

It’s often said that it’s easy to lie with statistics. Well, no. It’s easy to lie with bad statistics. Thomas B. Fordham, Achieve, Students “First,” and other promoters of the test-kids-until-blood-pours-from-their-orifices approach to education have brought lying with cooked statistics and the packaging of these into glossy reports and videos to a high art form. The propaganda ministries of repressive regimes worldwide have much to learn from them.

And, since investigative journalism is basically dead in the United States, advocacy groups like Thomas B. Fordham can count on their press releases on these “studies” to be parroted by the media sans critical examination.

“The fabulous statistics continued to pour out of the telescreen. As compared with last year there was more food, more clothes, more houses, more furniture, more cooking-pots, more fuel, more ships, more helicopters, more books, more babies — more of everything except disease, crime, and insanity. Year by year and minute by minute, everybody and everything was whizzing rapidly upwards.” –George Orwell, 1984

I would estimate that at least 10% (18 days) of the school year is lost to testing at the 3 to 8 level.

Remember Robert, the more you repeat a lie, the truer it sounds. In other words, repeat the dis information enough times and it becomes the truth to the dis informed. Even to people who should know better.

I guess this could be referred to as brain washing.

Robert Shepherd continues to nail this $$$$$Hungry Deform..

“Education Deform”

“Erase to the Top”

Testing is all that teachers worry about….Been there.. Done that…

Schools also hire coaches to up the scores (some have never taught a low-performing class in their lifetime)

Robert…your talk with an actual teacher in February

“Talk to an actual teacher, for a change, instead of to the carefully selected members of the panels that Fordham puts together, and you will hear, “Well, it’s February, so our school year is basically over. We will spend the next few months getting ready for the test.”

Now …talk to the teacher on the Block Schedule….They start getting ready for the test after one 9 weeks……..It feels as if you are stuffing a turkey until it blows up in your face!!

Why doesn’t Bill Gates come forward and defend his standards rather than hide in the shadows???

If you do not know what you are doing…you can not defend yourself..

Less Bill G, more Kenny G.

Roy is right, but there is another serious problem lurking out there.

We do need to upgrade what math we teach and how we teach it (especially beyond the narrow test base content and practice).

But the CCSS, it’s questionable development and flawed implementation will make it almost impossible for legitimate changes to have credibility.

Btw, instead of “rigor” how about “deeper, richer, more challenging, more meaningful”?

Why do we teach Geometry? Do we want students to know the equation of a circle or that parallel lines cut by a traversal form congruent corresponding angles or do we want to try to teach deductive reasoning?

In my opinion, Geometry is the “tool” we use to teach students to reason deductively. It is this logical reasoning that we try to instill in students not Geometry itself. In the 1980’s there was talk of teaching computer programming in lieu of Geometry since programming develops logical deductive reasoning.

With the inception of NCLB and standardized testing, teaching to the test took precedence over the “proof”. Proofs were difficult to grade, so they were left off standardized test and therefore the emphasis on deductive logic began to dwindle.

Now, with the Common Core, we have all but eliminated “the Elements”, the logic of Euclidean Geometry, the true purpose of Geometry! Why? Because it is easier for a computer to grade short answer questions than it is to grade ones ability to reason!

Tim, that was going to be my point also…that it is MUCH easier to grade cartesian geometry questions and answers than it is euclidean proofs. It is all about ease of test creation and grading, NOT what is best for the student’s education.

Why DO we teach Geometry? Why do we teach Shakespeare? Why do we teach anything? Students always ask “why do I have to learn this?” and the answer should not be “because you will need it if you become a …” or “because it helps you develop problem solving skills”. These things are important, no doubt, but I always try to express to my students that math is beautiful, meaningful, and important in and of itself, not because it serves the purpose of “college and career readiness” (blech). The purpose of education should be to expand our minds, not to narrowly focus on “21st century skills” (double blech).

“Why do we teach anything?”

The most important and most overlooked question in the universe!! (Okay maybe a tad over the top but not by much.)

Student: “Why do we need to know this?”

Teacher: “Why would you prefer to be ignorant of it?”

Why geometry?

Well, anyone who wants to do their own home repairs or construction benefit from the study of geometry.

Plus, it’s helpful when you play pool.

And besides – I often quote my lessons from geometry as my personal philosophy:

Equals plus equals yield equals.

The shortest distance between two points is a straight line.

Allison, I was never a science person, but I always loved math. There IS a beauty in solving a proof or figuring out the answer to a complex equation. Even though I was an English/Elementary Education major in college, I took several math classes, including Calculus101 with the Engineering students (and I was competitive). I found it an enjoyable contrast to my humanities course work.

And in high school, where students need to discover their strengths and weaknesses, their passions, and their particular talents, a good cross section of classes, providing a foundation for further pursuit or just an understanding of the basic subject matter, is essential for the development of a well rounded individual.

Ellen,

Well put. I couldn’t agree more! If only teachers had a say as to what and how we educate our youth!

Learning is a lifetime adventure..

Math is like a puzzle….you learn your basics….the perimeter of the puzzle…you then begin building …and eventually you can see how beautiful it all fits together…

They do not build anymore..

They throw it all one one big pile and use what they call the discovery method ..Students are confused by the clutter and the chaos…..They are not learning…

When I taught a proof in Geometry…I taught the students how to interpret the theorems and postulates…we experimented…we put them in our own words…we drew illustrations…eventually…they got it!!

When I taught the Unit Circle….no one had to memorize anything!!

I taught them how to build the Unit Circle….they thank me to this day..

When a student ask me why they had to learn the math..I compared it to a basketball game…You start off by learning to hold the ball..then dribble the ball…shoot the ball….play neighborhood ball…learn the rules of basketball…play the real game of basketball…eventually the game of basketball all comes together…

Why do we learn the name of the first president of the USA?? Why do we learn anything…

????

Exercising the brain is as important as physical exercise…

All of the subjects are important…art….music….P.E….Science..Math..History..English….

Correction..

When a student “asks”

or I should have said…

When students ask

Neanderthal,

I’m sure most teachers would agree with you on the importance of learning for the sake of learning. It is such a shame that the current state of education is killing the love of learning in so many kids. It’s heartbreaking.

Is Mr. Turrentine unhappy with how Common Core includes Euclidian and Cartesian geometry or unhappy with Common Core significantly including Cartesian Geometry? The post does not seem to be clear on this.

We used to teach it all..

But most importantly…we taught the student How to decipher the math…..We would sometimes only get through a few chapters but had enough time to provide a climate conducive to learning……When the students would reach Algebra 2 after having Alg 1….a very quick review was all that was needed as they had learned…really learned the parts of Alg 1 that was taught…..Alg 2 would build on the Alg 1 and they would learn more….

I taught a summer school class of students who had failed in regular school……

I taught them how to solve word problems that summer…I outlined the word problems…compared…analyzed….illustrated…..I then told them that they could take any word problem in the book and solve it by themselves..One student did just that…..He was so excited he solved almost every problem in the book …not a robotic solution…but his own….thinking….illustrating…using the skills that he had learned….

That was one of the most exciting summers in my career…..

Later…I taught a test…Boring…confining…nothing is taught …nothing is analyzed….and therefore nothing is learned…except CONFUSION!

He is unhappy that the testing emphasizes Cartesian at the expense of Euclidean to the point that the latter does not get taught much at all

The author writes:

>I have advocated more rigor in education for over thirty years.

Yet math education in the US has made no forward progress for at least 40 years, which means we continue to graduate students for whom math classes have been nothing more than 40 minutes a day of pointless and mind-numbing rote exercise.

One message on Twitter this week in the form of a geometry proof captured the essence of what’s missing in US education: aptly posed questions that provoke actual thought.

We reposted the question on this blog:

http://fivetriangles.blogspot.com/2014/01/136-congruence-proof.html

There is nothing in Common Core’s standards that would suggest this type of problem, and although it’s middle school caliber, it far surpasses anything that’s appeared in the New York State Regents curriculum for a half century.

Contrary to the author’s opinion, the content problem (in education in general, and math education specifically) isn’t what are ultimately arbitrary distinctions; the issue is what kinds of thought processes that content should engender in students’ minds.

One could characterize the geometry proof as rigorous, but its more important characteristic is that it is not obvious.

Tests, particularly standardized tests, test the obvious, and that’s why it’s possible to teach to the test and why the achievement gap endures: privileged students learn the rules of the game but students from disadvantaged backgrounds continue to be falsely told that the content is what matters.

Ultimately, it does not matter if students get such a geometry proof perfectly correct; it matters that they (hopefully) glimpse the train of thought that goes into completing the proof and that certain insights are necessary along the way.

Over the course of many years, spending class time doing problems like this, and not obsessing over (via tests, say) whether they get all or only one problem correct, not some, but all students will gain significantly far more intellectual benefit simply by participating in and seeing bared the complex process of problem solving.

Many nations have long ago made such a transition to meaningful education; the US has not.

Perhaps those advocating are being ignored.

Sorry, but your Twitter proof would seem just as pointless to MS students as anything else I’ve seen. Math should be taught as a means to an end, math will forever be pointless to the average kid when it continues to be taught as an end.

Requiring logical thought is doable, teaching kids how to think logically is not.

NYS Teacher wrote:

>teaching kids how to think logically is not [doable]

Sorry, too, but we don’t have such a defeatist attitude.

“Requiring logical thought is doable, teaching kids how to think logically is not”…

Thanks goodness Euclid and others did not think so lowly of their students, when they started teaching them how to think using logic.

Geometry teachers can teach students the rules for solving geometric proofs. You can call it teaching kids to think logically but you would be wrong. You have taught kids the to solve geometric proofs. The logic required (rule following really) is specialized and not transferable to other disciplines or different situations. You can pull logic ou of the human brain, you cannot put it in. Math as an academic discipline has been an abject failure (read Robert Shepherd); witness the legions of former math students who will shamelessly tell you that they “sucked at math”. Most students see no beauty in patterns or right answers. Ask any student what they’re learning in math class and you’ll be lucky if they can tell you. And if they can, ask them what its used for and they will simply shrugg their shoulders and give you a quick, “IDK”. Digging holes and filling them up again – for most kids math makes as much sense any other pointless activity: None.

No defending the Twitter proof?

NYS Teacher wrote:

>Geometry teachers can teach students the rules for solving geometric proofs.

Not quite.

There are no rules for thinking logically, and rules for proofs cannot be taught, either, simply because there are infinitely many varieties. For career math teachers, who have not studied beyond Regents Geometry, this may not be true, though, so we see your point. Regents is abominable in its finiteness and that’s part and parcel of the systemic weakness of American K-12 math education. Common Core, too, is nothing more than an a la carte menu of math skills.

We cited the middle school geometry proof only as one example, but the rest of the http://fivetriangles.blogspot.com blog delves into a range of problems for which rules cannot be taught, yet such kinds of problems—without teachable rules or algorithms—are standard fare in the top-performing mathematics nations.

Here’s a high school caliber geometry proof, slightly more complex (read: really difficult).

https://docs.google.com/file/d/0B6lw97EHbvfHMTF2LXVycW51b3M/edit

Although posed simply, it requires genuine insight to solve, and you can reach for rules or algorithms, but you won’t find them.

>Geometry teachers can teach students the rules for solving geometric proofs.

Not quite.

There are no rules for thinking logically, and “rules” for proofs cannot be taught, either, simply because there are infinitely many varieties. For career math teachers, who have not studied beyond Regents Geometry, this may not be true, though, so we see your point. Regents is abominable in its finiteness and that’s part and parcel of the systemic weakness of American K-12 math education. Common Core, too, is nothing more than an a la carte menu of math skills.

We cited the middle school geometry proof only as one example, but the rest of the http://fivetriangles.blogspot.com blog delves into a range of problems for which rules cannot be taught, yet such kinds of problems—without teachable rules or algorithms—are standard fare in the top-performing mathematics nations.

If you like geometry, here’s a high school caliber geometry proof, slightly more complex (read: really difficult).

https://docs.google.com/file/d/0B6lw97EHbvfHMTF2LXVycW51b3M/edit

Although posed simply, it requires genuine insight to solve, and you can reach for rules or algorithms, but you won’t find them.

NYS Teacher wrote:

>Geometry teachers can teach students the rules for solving geometric proofs.

Not quite.

There are no rules for thinking logically, and “rules” for proofs cannot be taught either, simply because there are infinitely many varieties of problems. For career math teachers, who have not studied beyond Regents Geometry, this may not be true, though, so we see your point. Regents is abominable in its finiteness and that’s part and parcel of the systemic weakness of American K-12 math education. Common Core, too, is nothing more than an a la carte menu of math skills.

We cited the middle school geometry proof only as one example, but the rest of the http://fivetriangles.blogspot.com blog delves into a range of problems for which rules cannot be taught, yet such kinds of problems—without teachable rules or algorithms—are standard fare in the top-performing mathematics nations.

If you like geometry, here’s a high school caliber geometry proof, slightly more complex (read: really difficult).

https://docs.google.com/file/d/0B6lw97EHbvfHMTF2LXVycW51b3M/edit

Good luck finding the teachable rule or the algorithm that applies; there is none.

CCSSI Math is correct in saying that basically we haven’t seen improvement in mathematics outcomes for 40 years. This is one part of the curriculum in which there has been a remarkable consensus for a long, long time. The state standards that the CC$$ replace were all basically variants of the NCTM standards, and the CC$$ is yet another variant, but with more emphasis on conceptual understanding and linkages among concepts at earlier levels.

So, how are we doing with our approach to K-12 mathematics. Well, if one looks not at the exam scores of students who have recently completed their educations but, rather, at the mathematics abilities and interest in mathematics of adults–that is, if we look at long-term outcomes–we find that for the vast majority of Americans, their mathematics instruction might as well have been time spent digging holes in the ground and filling them back up again. Most American adults are effectively innumerate. A recent study showed that 63 percent could not calculate a ten percent tip even though doing the calculation simply requires that the decimal point be moved over on place.

Years ago, the Saturday Night Life character Father Guido Sarducci offered what he called his “five-minute university.” For five dollars, he would teach you what you would remember of various subjects a few years after graduation from college. Economics? Supply and demand. Literature? To be or not to be. And so on. Fortunately, it’s not that way in all subjects. But it appears to be in math.

The long-term outcomes of our math instruction, for most adults, are very like those “five-minute university” outcomes. Tweaking our approach by insisting that 3rd graders understand the concepts of the variable and the function isn’t going to change that.

We need to face the fact that for most students, long term, our approach is an utter failure. The way in which we are teaching math is not producing large numbers of adults who enjoy mathematics, who pursue it recreationally, who employ mathematics regularly in their work and personal lives. It’s a truism in publishing that the way to ensure that a book for a popular market will not sell is to include some equations in it.

When the CC$$ in mathematics were prepared, all that happened was that an attempt was made to rationalize and combine existing state standards and to tweak them a bit. No consideration was given to the undeniable fact that judged in terms of long-term outcomes, all those hours spent in math classes were basically a waste of most people’s time. If you believe, as I do, that mathematics is valuable and beautiful and worth knowing, and if you care, as I do, about the fact that most adults have learned, primarily, from their math instruction that math is something they have no interest in and that they are not good at, then you will want to rethink, radically, the learning progression that we’re following.

Here’s what I suspect–that for most people, the cognitive apparatus necessary for explicit conceptual understanding of mathematics does not start to develop until kids are in their teens and is not fully in place until they are in their mid twenties. So, for most kids, their introduction to mathematics in elementary and middle school is an exercise in learning rules for shoving symbols around without having any understanding of what those symbols and rules mean. So, it’s as though we were having kids spend countless hours copying lists of random numbers, erasing them, and recopying them. For most kids, it seems a pointless undertaking. They are bored. They learn, mostly, that they hate it.

I suspect that if we delayed formal mathematics instruction until kids are older and replaced elementary math instruction with exercises designed to increase fluid intelligence (e.g., pattern recognition and manipulation activities), that we would accomplish more in three years of instruction in late middle school and early high school than we are now accomplishing in 12-to-16 years of math instruction.

Clearly, what we are doing, measured in terms of long-term outcomes, is not working at all for MOST students. Again, most adults are effectively innumerate. But they’ve all been through those K-12 math classes–countless hours of them.

Click to access LockhartsLament.pdf

It’s idiotic to think that we can do, basically, what we have been doing and get dramatically different outcomes.

One more thing: We know that young kids have VASTLY varying mathematical ability. There are some few who are ready for explicit abstract thinking at very early ages. These outliers need to be identified early on and given separate courses of instruction. But most kids do not fall into that category.

It would be possible to design an effective curriculum for building in children the neurological foundations for fluid intelligence that are prerequisite to conceptual understanding of mathematics. Such a program for elementary school children would look vastly different from anything that we are doing, systematically, today.

If you want different outcomes, you had better take different approaches. I have suggested one here.

Kids differ. Bullet lists of standards do not. And those bullet lists of standards preclude radical redesigns of learning progressions.

Your take on teaching math should be required reading for every math teacher and math curriculum developer. The CC doubles down on the issues you’ve outlined by framing the pointlessness with convulted. other-worldly word problems and ridiculous demands to verbally explain solutrions – as if showing one’s work is not enough of an explanation. Watch this generation go from “sucking at math” to reallly sucking at math and hating, more than ever, the time spent digging impossibly complex tunnel systems and then filling them back up. If you stick to your principled appraoch (one that applies to a tiny fraction of the students you teach) and continue to waste the time of the majority with your pointless rule following (tricks with numbers or shapes) you will continue to turn millions of kids into innumerate adults who will shmaelessly tell you how they really sucked at math – and really hated their time being wasted. CC does NOTHING to fix this, it will backfire on a scale that will one day embarass us as educators if it is allowed to go on.

Imagine taking a carpentry class and studying hammers. Learning how they transmit force to a nail, learning the principles of the claw hammer as a 1st class lever, comparing hammer weights and designs, explaining in words how they work. But never using it as the tool it was intended to be. Digging holes and filling them in.

Let me hasten to say that the folks who put together the CC$$ in math did a much better job than did those who prepared the embarrassingly amateurish CC$$ in ELA. But their task was easier–tweaking a consensus, and they didn’t consider this key issue of cognitive preparedness for certain kinds of explicitly articulated abstract thinking.

The math standards themselves are at least more concrete than the subjective, abstract ELA standards. However if you could see the CC$$ aligned math modules/worksheets from EnRageNY you would be appalled at just how bad they are. They make math not only pointless but downright painful for the youngest students in the system. They warp and twist math into unrecognizable forms. We are heading down a road that we will regret building.

NY Teacher: Agree with your take on engageNY. Those who wrote the curriculum seem to think they are geniuses coming up with those math models that are just one strategy to teach math which by the way they recreated what already existed. I have never used a math program that I didn’t have to supplement.

They claim it spirals, it is backed up with concrete to abstract to learning the concepts, and support struggling learners. Well, it spirals like they say, but it doesn’t matter when they upped the rigor. Struggling students will be spiraling backwards to the next grade. My students don’t relate to cutting up boxes into fraction units and seeing the relevancy to the word problem application. I don’t see how this supports special ed and ESL students. Additionally, there is so much vocabulary in math to learn and apply that the lesson pace is too fast let alone the rigor.

I guess you get what you get when it’s free.

jon……I agree

Math problems for grades 6,7,8? More like, brain burners for the top 10% of the population who love math and geometry. But ya, spin it like its easy, that always works.

As Einstein said….”You do not understand it unless you can teach it to a 6 year old”….Breakin” it down does not mean you are just making it easy..you are explaining it so it makes sense to someone whose is not as gifted in one area as yourself….

If I do not understand something..I keep asking questions …keep researching….asking someone with more knowledge than myself…until I can finally understand ….I will never understand to the degree of the “master of the subject”…but I will understand…

I have my own aptitudes that I have mastered but I do not own masters of all areas….so we share our gifts…give our best to the world…learn from each other…build your areas of interest…BUILD…..not throw it out in One Big Pile of Clutter..

There has been miles of progress in math education in this country in the last 40 years but the fruits if that knowledge of what works is not being taught in teacher training programs

Look up The Algebra Project sometime

Another proof that in the search for rigor, we achieve the exact opposite.

In New York State we have Regents courses in the main subject areas with a requisite Regents Exam to be passed. In the past, there were also “school” classes in each subject, as well, which only required a local final exam. About fifteen years ago, in the name of rigor (and because many college bound students were taking the school classes instead of the Regents so they would have a higher GPA), then NYS Educational Commissioner Mills decided that EVERYONE was required to get a Regents Diploma and they needed to pass five prescribed Regents Exams (English 11, American History, Global Studies, one math, and one science) to graduate.

The results – many students, especially in the inner city, could not pass all five exams and thus could not graduate. Mills solution was to encourage them to spend five years in high school, if necessary, instead of four. (My son got caught in this mess, which is why we pulled him out and had him get his GED. He wasn’t the only one to take this route.) The upshot was the current low graduation rate in Buffalo, Rochester, and Syracuse (based on the cohort of four years, not five) of less than 50%. We also see students taking the exam three OR MORE times, even in the Suburban schools.

( Ironically this is Regent’s week. The wind chill is 20 to 30 degrees below zero, but the schools have to remain open so those students can retake the exam. (In Buffalo, just the high schools had classes). The exams are offered in June, August, and January.)

The other ramifications is that FIVE is the magic number. Once those five Regents are passed, students do not take the rest of their classes seriously. So, if they’ve passed Biology (Living Environment) they don’t need to bother with any other science Regents Exam, such as Earth Science. The same in Math – after they pass Algebra, they don’t have to worry about Geometry. Oh, they do need school credit, but not the more rigorous Regents Credit. So many, especially in the city, will blow off those Regents finals and stay home instead. Even the smarter students won’t sit for Trigonometry or Physics Regents finals they don’t need to graduate.

And that’s what we call Rigor?

In addition, the exams are prorated or scaled. So, if you get a little over 50% right on the Biology Regents, you pass. Different exams are rated differently, but still, why bother requiring exams which have to be “fixed” so most students can pass? (And even then, many students fail.) When I took the Regents, the score you earned was the score you received. I was proud to do well on a difficult exam. But in those days, only the brightest students were in Regents courses, and we took every appropriate Regents Exam.

We are ruining education in the name of Rigor.

We are ruining education with Bogus Standards.

We are not educating our children the way they deserve to be educated.

I weep.

So glad to here you call out Richard Mills. He single-handedly destroyed the multi-track (Regents v Local diplomas) system that served the students of NYS so well for so many years. Now we have students with IQs in the 60s and 70s taking Regents tests that might as well be written in hieroglyphics. And they consider this the least restrictive environment. Psychologically crushing the spirit of learning disabled children is MORE restrictive than the idiots at state ed know.

And what about those poor ELL students, in this country for only a year or so? Plus we have the refugees with little to no formal education. Oops!

Mills said that requiring every student to receive a Regents diploma in order to graduate was his crowning glory.

Richard Mills – your idiotic idea has ruined so many lives. I only hope the old adage is true – What comes around, goes around. What I really want to say is too derogatory for this blog.

Update:

It was so cold that the school districts had to close school today, but Buffalo, where high school students take public transportation to school, were open for the Regents exams. Today was Global, a comprehensive exam given after two years of instruction – arguably the hardest of the five required Regents. Three quarters of the kids who signed up for the exam struggled against the bitter cold to sit for the test. Students from the suburbs were so desperate, that they made arrangements to take their exam at one of the city schools.

Tomorrow will also be unbearably cold, but this time other district high schools will now be open for the Regents Exams.

Of course, ALL staff to report.

They even put lives in danger for the sake of a test???

Oh, and the next step is to create Common Core Regents Exams.

You draw your own conclusions.

Bet doing Monte Carlo analysis would debunk the deformers.

Click to access 01-1mcbasics.pdf

The Monte Carlo experience – way beyond basic math skills. They used symbols I’d never seen before. I’m sure I could grasp the concepts if you taught me – slowly and with much review.

Do they teach this as science, statistics, or math?

Back in the summer, I read that the type of Geometry that was taught under Common Core had been experimented with in Russia and was deemed a failure. Is that Cartesian Geometry that was being described as a failure?

Cartesian geometry is simply geometry described algebraically–aka coordinate geometry. The increased emphasis in the CCSS on analytical geometry is in keeping with its general emphasis on relations among mathematical concepts. Roy Turrentine is arguing, I believe, for a traditional Euclidian approach that emphasizes proof.

Years ago, when I was a baby editor at an educational publishing house, my boss asked me to do some work on an “informal geometry” text. This was the first time I had ever heard of such a thing. I said to him, “Informal geometry? Now that’s an oxymoron if I ever heard one.” But I am now an advocate for informal approaches, in the early grades, that build students’ neural circuitry for identifying patterns and relations. See Richard Nisbett’s wonderful Intelligence and How To Get It for a great discussion of the astonishing effects of early practice with activities to build fluid intelligence on general cognitive ability.

As it is now, what we are asking kids to do in elementary and middle school is a bit like asking them to turn a tiny Phillips screw with a butter knife. They haven’t the tools for the job yet. And by the time they have those cognitive tools, they have already learned to HATE mathematics.

In every lesson that we teach, we are teaching either the hatred or the love of the subject being taught. It’s important to remember that.

Most people have a deathly fear of giving a speech in public. Most have a similar fear of anything mathematical. We’re teaching that.

You are correct about your suggestion to use informal approaches for teaching in the early years. Also, I always s appreciate your comments about the CC ELA Standards.

Of course, many of the education deformers dismiss, a priori, any discussion of neural plasticity or interventions to improve general intelligence because they subscribe to genetic determinism. They are members of the oligarchical elite, and they love the notion that they are where they are because of their innate, immutable superiority. And so they favor two tracks–training for the proles based on immutable, invariant, top-down bullet lists, and education for the children of the elite.

But switch in the maternity ward their kids with those of the poor and look at the outcomes. The prince becomes the pauper and vice versa.

http://www.slate.com/articles/health_and_science/science/2012/09/how_children_succeed_book_excerpt_what_the_most_boring_test_in_the_world_tells_us_about_motivation_and_iq_.html

Robert –

It’s like the old song:

It’s what you do with what you got,

And never mind how much you got.

It’s what you do with what you got,

That pays off in the end.

Examples of applied Cartesian geometry would be reading longitude/lattitude maps (2 dimensional) or CGI and GPS technology (3 dimensional). Not that any student would ever get that out of a math class. They dialate and rotate and reflect geometric shapes arouns a 4-quadrant graph have no idea why. Just another hole to fill up after digging.

LOVE IT!!!!!!!!!!!!!!!

great!!!

R. Shepherd – I hope you’re writing a book. I have learned more from reading your posts than all of my teacher training combined.

Your hole digging and filling metaphor regarding math is priceless.

But turning a small Phillips head screw with a butter knife gives attempts at early math instruction way too much credit. More like trying to split a Uranium atom with a hammer and chisel.

NY teacher. You are very kind. I, too, enjoy your comments immensely.

Did Cool Hand Luke inspire your metaphor?

“As punishment for his escape, Luke is forced to repeatedly dig a grave-sized hole in the prison camp yard, fill it back in, then be beaten.”

Source of that metaphor:

“If the Treasury were to fill old bottles with banknotes, bury them at suitable depths in disused coalmines which are then filled up to the surface with town rubbish, and leave it to private enterprise on well-tried principles of laissez-faire to dig the notes up again (the right to do so being obtained, of course, by tendering for leases of the note-bearing territory), there need be no more unemployment and, with the help of the repercussions, the real income of the community, and its capital wealth also, would probably become a good deal greater than it actually is. It would, indeed, be more sensible to build houses and the like; but if there are political and practical difficulties in the way of this, the above would be better than nothing.”

–Keynes, The General Theory of Employment, Interest and Money, Book 3, Chapter 10, Section 6.

Thanks

I hadn’t heard the “copyright” bit before. How much more do you need than that for the whole thing to stink of corruption?

If it was a good faith effort done by our government for the benefit of students, they would not put an impediment to using it like that in place.

All part of the whole centralized command and control mindset, Mike.

Funny that anyone would think that a set of hackneyed, received ideas like those instantiated in the Common [sic] Core [sic] State [sic] Standards [sic] would be copyrightable. Well, not funny. Sad. Absurd. Tragic.

While there is a lot of beautiful stuff in synthetic geometry there’s no question that Cartesian geometry is vastly more powerful.

“Cartesian geometry is vastly more powerful” – how so?

Just curious. Example will help.

I thought the same NY Teacher….

ROY……..VERY WELL SAID

Your last paragraph….Absolutely… so… so… so correct!!!!

Those of us who teach in high schools across America have long desired rigor. To go to meetings where people seem to feel that this rigor is their idea is nothing short of insulting to those of us who have been trying to unite the disciplines for decades. Every good teacher knows what the ideal is. We have been trying to do this for all of our careers. Having Bill Gates give me his opinion does no one any good. Having his opinion become national policy will not serve anyone.

The “failed” approach in Russia was trying out transformations and Cartesian geometry approaches in 1960s to mid 1970s.

The following quotes of Shagudin are taken from:

“On the Teaching of Geometry in Russia

Alexander Karp

Teachers College, Columbia University,

New York, USA

Alexey Werner

Herzen State Pedagogical University of Russia,

St. Petersburg, Russia”

=================================================

Addressing the students, Sharygin writes:

“Far from all students feel a great love for mathematics. Some are not

too good at carrying out arithmetic operations, have a poor grasp

of percentages, and in general have reached the conclusion that they

have no mathematical abilities. I have good news for them: geometry

is not exactly mathematics. At least, it’s not the mathematics with

which you have had to deal up to now. Geometry is a subject for

those who like to daydream, draw, and look at pictures, those who

know how to observe, notice, and draw conclusions. (Sharygin, 1997,

pp. 3–4)”

This is what Shargyin writes about the failed approaches calling them anti-geometry –

in his posthumously published article “Do Twenty-First Century Schools Need

Geometry?” Shargyin (2004) identified three basic types of courses

that taught anti-geometry (false geometry and pseudogeometry). The

first type is built on a formal–logical (axiomatic) foundation; the second

type is the practical–applied course with a narrowly pragmatic profile;

and about the third type he wrote: “And yet I am convinced that the

[Cartesian] coordinate method (along with trigonometry) constitutes one of the

most effective means for ruining geometry, and even for destroying

geometry” (p. 75).”

====================================================

Transformations and Cartesian geometry are a major, time consuming part of grade 8 math in NYS – under NCLB and still under CCSS. Ask any 8th grader in NY about their transformations. Easy but pointless as taught.

NY teacher,

It’s off topic here, but in a few posts ago on how NYSUT and Ianuzzi called for a vote of no confidence, you stated succinctly the following:

“Cuomo’s ward is run by steely, unyielding NYSED Commissioner John King Jr. who employs subtle humiliation, condescension, and a mind-numbing daily routine to suppress teachers and outraged parents. Rendo the anti-hero finds that they are more fearful of King and his corporate puppeteers than they are focused on becoming functional adults who know what’s best for their children and students.”

I am truly not sure why you refer to me as an “anti hero” nor do I understand at all why you think I have said or suggested that teachers and parents are more fearful of John King and cannot focus upon the responsibilities of adulthood by protecting their children and students.

If you have followed my posts in this blog for the last year, I must have stated more than 8 times that it will be the PARENTS and VOTERS of NY state who will fight the injustices of this reform movement. Teachers are, of course, part of that population, as many of them send their own offspring to public schools.

Why you have characterized me as saying otherwise or me personally as an “anti-hero” has gone completely over my head.

Was it something else I said or stated?

In the sincere interest of collegial and professional dialogue, I appeal to your better side and account for why you perceive this and have declared it on the blog.

It’s not just a matter of my not agreeing with you, but it’s more, at this point, a matter of me not even understanding your motivations, and, having a very open mind, I’d like to understand your POV more closely and clearly.

If you care to, please reply right here, but if you choose not to, I am also happy to continue the dialogue offline at artwork88@aol.com.

Thank you for your comments, and I look forward to your usual wonderful posts.

Sincerley,

Robert Rendo

Sorry RR. You misread my intention because it was just a poor attempt on my part at equating John King with Nurse Ratched in One Flew Over the Cuckoo’s Nest. It started as a response to the original question, “Who does John King sound like?”. His cold-hearted, relentless, condescending, and mind numbing response to teacher and parental outrage “Now is not the time for delay, the CC is going nowhere” after 16 town hall meetings brought the image of Nurse Ratched to mind. In my haste to carry out the comparison, I guess I miscast you as Randle McMurphy the anti-hero (Nicholson) but the snippet I cut and paste out of Wikipedia simply didn’t read the way I thought it would. Maybe my word choice, anti-hero, was a poor one as well. Yet I do find many teachers who are far too compliant (maybe not fearful) in regards to the insanity being rained down on us and our students by King and the BOR. You have my sincere apology if I offended or misrepresented you. I probably should have written a retraction but I thought at the time it was buried deep enough in the thread that it would go unread.

NY teacher,

There is no need to even think about apologizing. It is facinating that the use of language in different contexts has the potential to render different meanings. It’s not your fault. It’s not my fault, per se.

It’s just an inherent trait of language . . . Who ever is in perfect control of it, save for maybe John Cheever, Jospeh Conrad, and Shakespeare . . . . ?

Thank you for the clarification. No need at all for retractions. But I am grateful to you for clarifying. Please keep posting, as I will keep reading . . . . . I enjoy your posts and fell empowered by them.

Best regards,

Robert

Your posts have done likewise. NY teachers share a unique perspective after the bloodbath inflicted on our 8 to 14 year olds.

The fact the we have been politically orphaned (despite Iannuzzi’s change of heart and Randi’s recanting) has made for an especially frustrating year. This blog has been great for sharing information and venting. Actions speak louder than words and my patience is wearing thin as day after day more unproven, untested, time wasting policies are forced upon us. And as I sit at staff and department meetings I feel as if we are falling deeper and deeper into the rabbit hole. Nearly ten million page views and what do we really have to show for it?

Yes, many posts are encouraging and empowering and I really believe that I have become a better teacher because of the discussions and ideas that have been shared. But when I go back to my building I get the sense that King, and Tisch, and Cuomo, and Coleman, and Duncan, and Gates will prevail, that “Now is not the time for delay, that the CC and APPR ( the true root of the problem) is going nowhere.”

NY teacher,

Thank you again for responding.

I understand your perceptionsm, feelings, and frustrations.

Please understand that this fight is a process, and it’s not going to be won so easily by any far stretch of the imagination.

Please don’t feel defated ever and PLEASE do not give up. Parents are catching on, and they are sensing the maltreatment of children and educators. They have the most important vested interest, and it’s being threatened with all of this narrowing of the curriculum, starvation of funding, test obsession, and predatory treatment of teachers and principals.

The fight AIN’T over.

Anyonw who knows me personally knows I am not pollyanna. In fact, I have a reputation for being a carmudgeon, but even I now sense a shift in the wind, a change in the rhetoric. . . .

You can also look forward to the far too long overdue rhetoric of “fiscal or income inequality” parallel the fight against the corporate takeover of education policy.

Please never give up your “piss and vinegar”, your critical thinking, and your passion for what’s right.

We stand defeated if we think we are . . . . . We stand victorious if we decide to be . . . .

Robert and NY Teacher –

I feel the anger of the parents in the Western New York area, especially in the wealthier suburbs. They are not happy with last year’s assessment results. They are not happy with Arne’s comments. Some of them are beyond angry.

Watch for a significant opt out movement this spring.

I am a Pollyanna – and I don’t intend on falling from the tree branches.

Wonderful, Preeti! Thank you for this information.

I suspect that a lot of early play with coordinate systems and other means of visualizing and doing variations on formal patterns and relations would develop skills that would make later mathematics learning possible. On the relative value of teaching traditional Euclidian v. Analytic Geometry, I remain agnostic. I suspect that the learning of both would be served by delaying formal mathematics instruction and replacing it, in the early grades, with an enormous amount of play to develop fluid intelligence.

Robert, I took both Geometry and Analytical Geometry in high school and they were definitely different types of math. The analytic geometry was helpful in Calculus 101 – the whole 1/2 year course was covered in a one and a half hour class. I could barely keep up, even though I knew what the professor was talking about. I think analytical geometry builds upon Algebra, Advanced Algebra, and Trigonometry more than Geometry (unless they are talking about a watered down version).

Shep

Just caught your Father Guido Sarducci reference and the “5 minute university” bit. As Freud once said, there’s no such thing as a joke.

One of my favorite routines of all time. I think of it often as a teacher.

If you don’t use it you lose it. You left out his geometry lesson: “A squared-a, plus-a, B s-squared-a equall C-squared-a”

Genius Doodler…What a video…..:-)

Thanks Robert…

Enjoyed reading Robert Shepherd’s eloquent comments and…… NYTeachers comments are always Spot on…

what role does the heart play in education?

Mr. Roy Turrentine’s comments are deeply felt, and I sympathize with his anguish completely. But I must say in the same breath that he has been betrayed, first by the educational materials around him on CC geometry and perhaps also by his misreading of the CC standards. The geometry in the CC standards does not at all seek to “unite the Cartesian approach and the traditional approach”. I am here guessing that what Mr.Turrentine meant by the “Cartesian approach” is the approach to reflections, rotations, and translations via coordinates that seems to pervade the literature on “transformation geometry” (I am a geometer by profession and I have no idea what this term is supposed to mean).

I can say, categorically, that the CC geometry uses certain assumptions on reflections, rotations, and translations as the starting point for middle and high school geometry, and these concepts are supposed to be developed WITHOUT any reference to coordinates. Coordinates are introduced only after a sufficient number of theorems have been proved. What CC envisions is nothing other than a more natural approach to Euclidean geometry, one that may be more in line with Euclid’s original intentions than the usual treatment in (mostly unspeakably defective) school textbooks. This is not the place for me to enter into the technical reasons behind my statement above, but perhaps Mr. Turrentine would be so good as to take the trouble to read through the following two documents and let us know whether a treatment of geometry that is FAITHFUL to the true intent of CC meets with his approval on the matter of rigor. (The prefaces of these documents will amplify somewhat the above cryptic statement.) Such a communication from Mr. Turrentine will help clarify some of the current misunderstanding about the CC standards.

Teaching Geometry According to the Common Core Standards (For teachers of grades 4-12 and educators) http://math.berkeley.edu/~wu/Progressions_Geometry_2013.pdf

Teaching Geometry in Grade 8 and High School According to the Common Core Standards (For teachers of grades 8-12 and educators) http://math.berkeley.edu/~wu/CCSS-Geometry.pdf

H. Wu

H WU

Have you ever even stepped into a High School Classroom??????

You must be kidding….

I think I will try-out for the NFL today….Do you have a paper on how I could reach that goal?

Dear H.Wu, it is nice that you are paying attention to this blog. I read your documents, and I want to say, that this sequence your suggest is not age-appropriate, and this is its main flaw.

You seem to assume that children learn in certain way, but the science say they don’t. In fact nearly everything they learn in early age gets erased and rewritten many a time during their childhood. So it is not like you teach fractions on the number line when they are really young and that makes them better at mathematics than if they learn them first by cutting a pie. At every age mathematics(and other subjects) has to be learned in a different way. The brain gets rewired as it is growing, not like a computer program that once you define the objects at the beginning of the program they will stay this way. The approach you define for high school is appropriate for college, really.

I understand that while teaching in Berkeley you encounter students coming from the different parts of the globe and they are really advanced too. But I can assure you that all of them from Russia, India and other countries, followed traditional, age proven, and sound school program, and this is why they succeeded. In USA math education was challenged many a times, for example when New Math was introduced. I know that time you stood against the approach of the New Math as your website contains those references. So I hope that you will change your current approach now, as well and if you are interested in science of how young children learn, you could listen to the voice of educators and learn from the experiences of other countries as well. I am curious about the geometry approach that you suggest. Many people are actually wondering where exactly it is coming from. What is the author that inspired your approach?

In Russia there was Kolmogorov who did a similar approach. His books were translated to many languages, did you happen to study by his textbook? Kolmogorov wrote school textbooks in algebra, calculus and geometry. His school textbooks in Algebra and Calculus were a great success and are still used in Russia and around the world. He also wrote absolutely fabulous textbooks for college in various math areas. However his school textbooks in Geometry were a failure, that he himself admitted, being an honest scientist, he agreed that the approach was not age appropriate for school children.

There was a sad 10 years period in Russian school education when Kolmogorov’s geometry textbook replaced Kiselev’s age proven geometry textbook. After everyone admitted the failure, the Kolmogorov’s textbook was taken out f school and the new textbooks were designed going back to introducing geometry the way Kiselev did. Kiselev wrote in the teachers manual for his geometry textbook that transformations don’t belong in the beginning of the geometry course, and his words came true.

I really hope that you change you approach in the spirit of the distinguished teacher about whom you wrote this book

http://www.amazon.com/Contemporary-Geometry-Memorial-University-Mathematics/dp/1468479520/ref=sr_1_9?ie=UTF8&qid=1391534633&sr=8-9&keywords=Wu+geometry

As J.-Q. Zhong stood against the Cultural Revolution in China, I hope that you will not support Common Core in USA, which is being compared by many to Cultural Revolution in USA.

You are really talented mathematician, and as a scientist, you have to see the facts and you also have to listen to the people’s voice, which is already very loud saying that Common Core approach is wrong.

You used word FAITHFUL in your post. You know that in mathematics there is no place for FAITH. It is only TRUE of it is FALSE. Someone can be FAITHFUL to a boss that pays him or her a salary, but one cannot be faithful in mathematics. So you cannot serve two masters. You can only serve one master. It is only TRUE or it is FALSE like if you drive on the road in Berkeley, there are only two turns – LEFT turn and RIGHT turn. Of course there is U-TURN, but it is not really a turn – it just changes what you can call RIGHT turn or LEFT turn. Sometimes it is very important to take a U-TURN in one’s actions. Berkeley,CA is a very special place that played a very important role for the Democracy in USA in the past.

Money doesn’t bring happiness, it is only people’s true appreciation that brings one happiness and most importantly, honor. So I hope that you take a U-TURN, and I am sure then someone (or may be many) will write a book about you, praising your actions. There is nothing wrong in taking a U-turn. Diane Ravich did take U-turn too. This is what an honest scientist and citizen must do.

With best regards

Preeti

Click to access Karp%20A.,%20Vogeli%20B.%20(eds.)%20Russian%20mathematics%20education..%20Programs%20and%20practices%20(WS,%202011)(ISBN%209814322709)(O)(514s)_MSch_.pdf

Read chapter 3 from this document about geometry.

It really clears all the issues discussed on this page as Russian math education went through a deform for a short period and then reformed back and is doing wonderful now for the last 20+ years. Something to learn from. This describes in details about a dozen of various successful approaches in modern Russian high school geometry textbooks.

Here is introduction:

=================================================

Perhaps the most striking difference between the teaching of mathematics

in Russia and standard mathematics education in the West is

that the former includes a separate course in geometry taught over a

five-year period. It has been over 50 years since it was declared in the

West that “Euclid must go” (cited in Fehr, 1973). Even aside from

this, the “Western” course in geometry was often — and continues

to be — conceived of as occupying only one year and certainly not as

constituting a constant accompaniment for students from sixth grade

on, throughout all of their middle and high school years.

In Russia, Euclid and Euclidean geometry did not go anywhere.

Plane geometry is taught in grades 7–9 (6–8)1 for 2–3 hours per week;

three-dimensional geometry is taught in grades 10 and 11 (9 and 10),

usually for 2 hours per week. The course in plane geometry is thus

intended to occupy over 200 hours of classes, and the course in

three-dimensional geometry approximately 140 hours. In addition,

the mathematics classes in Russian elementary schools and the lower

grades of the so-called “basic schools” (grades 5 and 6) include section

on visual geometry; in other words, students are exposed to what might

be characterized as the informal study of geometry.

The aims and objectives of such a program in geometry have

by no means always been envisioned in the same way, and their

implementation has also varied, so it would be a mistake to suppose

that the history of teaching geometry in Russia is the history of a kind

of stagnation. On the contrary, the teaching of geometry has been and

remains the subject of passionate debate. The authors of this chapter

cannot consider themselves neutral with respect to these debates. For

example, one of them (A. Werner) had occasion to collaborate over

many years with the outstanding Russian geometer A. D. Alexandrov,

initially as a participant in his research seminar, and subsequently as

the coauthor of his textbooks for schools. It should therefore be stated

from the outset that Alexandrov’s views on geometry in general and on

school geometry in particular are particularly close to him. However,

we will attempt to represent other views and approaches that have

existed over the past 50 years in Russian schools as well. Since our

account will necessarily be limited by the size of this chapter, many

mathematical and methodological details will be skipped. On the whole

we will focus mainly on the analysis of textbooks and programs, which

classroom practices in fact follow in many respects, although it is

impossible to describe all the actual and possible varieties of classroom

practices here.

The contents of the course “Geometry” in the most recent programs

at the time of this writing (Standards, 2009) consist of the following

sections (the number of hours recommended by the program for the

study of each section is indicated in parentheses):

Grades 5 and 6: Visual geometry (45 hours). Students are given

a visual sense of basic two-dimensional figures, their construction,

and various ways in which they may be positioned with respect to

one another, as well as measurements of lengths, angles, and areas.

The concept of the congruence of figures and certain transformations

of the plane (symmetries) are discussed. Students are also familiarized

with three-dimensional figures, their representations, crosssections,

and unfoldings, as well as with formulas for determining their

volumes.

Grades 7–9 are devoted to the systematic study of plane geometry,

which includes the following sections:

• Straight lines and angles (20 hours);

• Triangles (65 hours);

• Quadrilaterals (20 hours);

• Polygons (10 hours);

• The circle and the disk (20 hours);

• Geometric transformations (10 hours);

• Compass and straight-edge constructions (5 hours);

• Measuring geometric magnitudes (25 hours);

• Coordinates (10 hours);

• Vectors (10 hours);

• Extra time — 20 hours.

In grades 10 and 11, geometry is studied at the basic and advanced

levels. Second-generation standards for the upper grades are still being

developed, while according to Standards (2004a), at the basic level,

students in grades 10 and 11 were required to study the following

topics in three-dimensional geometry:

• Straight lines and planes in space;

• Polyhedra;

• Objects and surfaces of rotation;

• The volumes of objects and the areas of their surfaces;

• Coordinates and vectors.

The content of each section is quite rich. For each topic, the

programs indicate the basic skill set that the students must acquire.

For example, in the section on “Triangles,” the students must learn to:

• Identify on a geometric drawing, formulate definitions of, and

draw the following: right, acute, obtuse, isosceles, and equilateral

triangles; the altitude, the median, the bisector, and the midpoint

connector of a triangle;

• Formulate the definition of congruent triangles; formulate and

prove theorems on sufficient conditions for triangles to be congruent;

• Explain and illustrate the triangle inequality;

• Formulate and prove theorems on the properties and indications

of isosceles triangles, the relations between the sides and angles of

a triangle, the sum of the angles of a triangle, the exterior angles of

a triangle, and the midpoint connector of a triangle;

• Formulate the definition of similar triangles;

• Formulate and prove theorems on sufficient conditions for triangles

to be similar, and Thales’ theorem;

• Formulate definitions of and illustrate the concepts of the sine,

cosine, tangent, and cotangent of the acute angle of a right triangle;

derive formulas expressing trigonometric functions as ratios of the

lengths of the sides of a right triangle; formulate and prove the

Pythagorean theorem;

• Formulate the definitions of the sine, cosine, tangent, and cotangent

of angles from 0◦ to 180◦; derive formulas expressing the

functions of angles from 0◦ to 180◦ through the functions of

acute angles; formulate and explain the basic trigonometric identity;

given a trigonometric function of an angle, find a specified

trigonometric function of that angle; formulate and prove the law

of sines and the law of cosines;

• Formulate and prove theorems on the points of intersection

of perpendicular bisectors, bisectors, medians, altitudes, or their

extensions;

• Investigate the properties of a triangle using computer programs;

• Solve problems involving proofs, computations, and geometric

constructions by using the properties of triangles and the relations

between them as well as the methods for constructing proofs that

have been studied (Standards, 2009, pp. 36–37).2

It should be noted that although algebra and geometry are taught

as two separate subjects, the course in algebra addresses some topics

(concepts) that pertain to the course in geometry as well. One example

is the section of the algebra course that covers “Cartesian Coordinates

in the Plane”; another is the section on “Logic and Sets” (10 hours)

in the second-generation Standards (Standards, 2009, p. 16), which

belongs to both the course in algebra and the course in geometry.

Comparing the recently published second-generation Standards

for basic schools (cited above) with previously published Standards

(Standards, 2004b) or even earlier programs, we find few differences.

The contents of the course, in terms of the list of concepts and

propositions covered, have remained stable. Naturally, 30 years ago

there was no investigation of the properties of a triangle with the

help of a computer program, mentioned above, nor was such a

problem even posed at the time (nor is it often encountered today in

actual classrooms, by all appearances); but problems involving proofs,

computations, and constructions that require knowledge of the many

theorems studied in the course are assigned and solved today largely as they were years ago.

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I agree. I see the same thing in my NY 4th grade classroom and testing. The emphasis is on the method not the skill.. In math, especially, there are often numerous ways to reach a conclusion, or answer. According to the CCLS and the tests being created we are expected to teach by way of method, never mind ow the student understands the concept. It’s bizarre, backwards, and morally and ethically wrong. I cannot teach this way and have refused to do so. As adults it doesn’t matter how we arrive at a multiplication answer, just that it’s correct. Using an array, taking apart method, or just plain old multiplication, it shouldn’t matter, as long as the student demonstrated comprehension of the content.

Ditto

NYS Teacher – The amount of original research in synthetic geometry over say the last 200 years has been negligible compared with the huge amount of results obtained in Cartesian analytic geometry.

It is possible to develop a beautiful synthetic theory of conic sections and quadrics but I don’t believe anyone has been able to make this work very well beyond that conpared with the enormous amount of research in Cartesian analytic geometry.

It is intereseting to note that Greek mathematicians do not seem to had any idea of the notion of degree of a curve. Their synthetic approach blinded them to this. They did study some higher degree curves which arise as intersections of tori with planes(Bernoulli’s lemniscate can be obtained as such an intersection). But for example they totally missed elliptic curves because they do not arise naturally from a synthetic approach but their significance is obvious once analytic geometry had been developed. Among the curves studied by Greek geometers some are transcendental but there is no evidence that Greek geometers even had any notion of the distinction between algebraic and transcendental curves.

How can your obvious advanced level of undestanding be used to help students learn geometry. Your suggestions in terms of standards and/or curriculum. Plain speak please.

I agree NYS Teacher…

Listening to this mathematical gobbledy-goop is the same as listening to a Dr speak only in medical terms or a Plant Taxonomists using only the scientific names for dandelions and four-leaf clovers…

I respect the Master of Knowledge but to each his own…Problem is..not everyone understands nor do they want or need to understand Bernoulii’s lemniscate…

No disrespect Jim…but it seems to me you are showing off…..Keep it in your Math Meetings…

One of my children has a PHD in Math and the other in Chemistry…I listen to their discussions only if it involves Sports..

INDIVIDUALITY in Public Schools has been kicked to the curb!!

N100

Jim is a prime example, or so it seems, of why intelligent, well educated adults who have never worked with children should STAY OUT OF THE EDUCATION ARENA.

Neanderthal100

People like Jim can’t begin to understand the developmental, emotional, experiential, and motivational range of student we encounter.

When developing geometry in a synthetic manner an approach based on axioms expressing intuitive properties of rotations, reflections and translations as mentioned by Hung-Hsi Wu is probably the best way to go to get a clean non-cumbersome approach that also appeals to our pre-formal geometric intuitions.

The problem with starting with an analytic approach is that while it is very powerful it is not very well-connected to our primordial geometric sense.

I assume “transformational geometry” is more or less an Erlangen program approach.

Robert D. Shepherd – Regarding your proposed maternity ward experiment – Adopted children more closely resemble their biological parents in intelligence, personality and temperment thatn they do their adoptive parents.

Also before you spout some of your drivel on “epigenetics” – Epigenetic mechanisms control the differential expression of chromosomal nuceotides in different body tissues. So although your kidney cells and brain cells have the same chromosomes they do different things expressing different parts of the chromosomal information. All this is very interesting but it has nothing to do with your confused ideas of Lysenkoism. But the way errors in the epigenetic process are not generally important in evolution since the epigenetic switches are reset every generation so epigentic errors are constantly corrected. This is in contrast with copying errors in the replication of chromosomal DNA which will persist until or unless removed by natural selection.

One interesting thing about epigenetics is that epigenetic mechanisms in insects appear to work in a very different way than in most other multicellular organisms. Methylation of the DNA does not seem to be involved although this is the principal epigenetic switch in most other multi-cellular organisms.

Jim, read this. It mentions ongoing research into half a dozen or so epigenetic changes with potentially large effects on general cognitive ability. And this is just the tip of the iceberg.

Click to access Lester%20et%20al%20Annals%20NYAS%202011.pdf

Jim, more for your reading list:

http://www.technologyreview.com/news/411880/a-comeback-for-lamarckian-evolution/

The need to evolve epigenetic controls to allow tissue differentiation may explain why the appearance of multicellular life was so delayed in the evolutionary process.

I am right there with you, Roy. I have been teaching Geometry on and off for nearly 30 years. In my opinion, what Common Core is doing to Geometry is watering it down and making it easy so that more kids can get through it. The rigor is dying – it used to be considered a college-prep course. In at least one major CCSS curriculum, what used to be the entire year of Geometry is stuffed into one unit in Math 1 (integrated). And it is taking nearly all of it to the cartesian plane. Not good for kids and not good in preparing them for future math courses.

When will we ever learn. We can raise single standards and we can lower them and still not solve the problem. Standards must be individual so those who reach the high levels can do so and those who don’t can succeed at a different level. This can only be done locally and only done taking kids from where they are.

The high levels you are talking about must be made available, not forcing the other kids out. The lower levels must be available not forcing those kids out. And who reaches those high levels begins with an exploratory workshop that brings in every child who is interested in the professions that use geometry and allows them to determine what their passion is. By the way, what professions use geometry and why don’t we ever talk in those terms. Everthing isn’t always about a school class.

And algebra should never be a pre requisite because it uses a different type of thinking.

Cap Lee, I completely agree with you on your every point. I do not understand why they require algebra as a pre requisite for geometry, it does not make any sense. Pre-Algebra is quite enough. In fact they have been adding a lot of applications of algebra to geometry lately, and it is killing geometry, it is just becoming another algebra course. I believe algebra applications should belong in algebra, algebra II, pre-calculus, calculus. Geometry should be for geometry only and it can be taught at the same time, if desired, as another math course. And more levels the better, the more choices in math sequences the better.

Until the US Dept. of Education is shut down and until the US withdraws from the UN (who is behind this drive for one size fits all global education) we will NEVER fix education. They do not want to fix education they want to control our children and the future citizens of the world. To mold them into obedient slaves. Dumbed downed, drugged up people are much easier to control. The idea that either of the two suggestions above will ever happen is as impossible as the idea of fixing education with money. Both R and D have been pushing this global education, One World Order for years Bill Clinton is more to blame than Obama. He along with Marc Tucker and others made it formal with the School to Work Act and Goals 2000. Jimmy Carter did it in 1979 by creating the US Dept. of Ed. The creation of the ESEA was again another step toward this progressive, evil agenda. There is only one solution. We need to implode the system from within. The only way to do this is to GET YOUR KIDS OUT OF THE SYSTEM. Many parents are coming to this realization and they are home schooling their kids. In TN we have flexibility to have home school co-ops and anyone can teach your kids….grandma, grandpa, retired neighborhood teacher. We as Americans MUST ban together and help each other. We must help the mother that wants to homeschool but is single, poor and has no choice. If we do not save our children and make this our top priority we will lose our country not just our children. If you really want to you will, if you don’t you will make excuses.

Yawn — yet another reform program for math. I’ve been a math teacher for 17 years and I lost count of how many “new” programs I’ve seen come and go. The kids can barely write their own name and now we’re going to teach them to be metacognitive in the 7th grade. I think it’s all a plot to ensure high enrollments at private schools.