From the ashes of New Math

[Previously posted at the teaching blog]

One of my big concerns with the education reform debate, particularly as it regards mathematics, is that a great deal of the debate consist of words being thrown around that have a positive emotional connotation, but which are either vague or worse yet mean different things to different participants in the discussion.

As a result, you have a large number of "supporters" of common core who are, in fact, promoting entirely different agendas and probably not realizing it (you might be able to say the same about common core opponents but, by nature, opposition is better able to handle a lack of coherence) . I strongly suspect this is one of the causes behind the many problems we've seen in Eureka math and related programs. The various contributors were working from different and incompatible blueprints.

There's been a great deal of talk about improving mathematics education, raising standards, teaching problem-solving, and being more rigorous. All of this certainly sounds wonderful, but it is also undeniably vague. When you drill down, you learn that different supporters are using the same words in radically different senses .

For David Coleman and most of the non-content specialists, these words mean that all kids graduating high school should be college and career-ready, especially when it comes to the STEM fields which are seen as being essential to future economic growth.

(We should probably stop here and make a distinction between STEM and STEAM – science technology engineering applied mathematics. Coleman and Company are definitely talking about steam)

Professor Wu (and I suspect many of the other mathematicians who have joined into the initiative) is defining rigor much more rigorously. For him, the objective is to teach mathematics in a pure form, an axiomatic system where theorems build upon theorems using rules of formal logic. This is not the kind of math class that most engineers advocate; rather it is the kind of math class that most engineers complain about. (Professor Wu is definitely not a STEAM guy.)

In the following list taken from this essay from Professor Wu, you can get a feel for just how different his philosophy is from David Coleman's. The real tip-off is part 3. The suggestion that every formula or algorithm be logically derived before it can be used has huge implications, particularly as we move into more applied topics. (Who here remembers calculus? Okay, and who here remembers how to prove the fundamental theorem of calculus?)

All of Professor Wu's arguments are familiar to anyone who has studied the history of New Math in the 60s. There is no noticeable daylight between the two approaches.

I don't necessarily mean this as a pejorative. Lots of smart people thought that new math was a good idea in the late 50s and early 60s; I'm sure that quite a few smart people still think so today. I personally think it's a very bad idea but that's a topic for another post. For now though, the more immediate priority is just understand exactly what we're arguing about.

The Fundamental Principles of Mathematics

I believe there are five interrelated, fundamental principles of mathematics.
They are routinely violated in school textbooks and in the math education
literature, so teachers have to be aware of them to teach well.

1.  Every concept is precisely defined, and definitions furnish the basis for logical
deductions. At the moment, the neglect of definitions in school mathematics has reached the point at which many teachers no longer know the difference between a definition and a theorem. The general perception among the hundreds of teachers I have worked with is that a definition is “one more thing to memorize.” Many bread-and-butter concepts of K–12 mathematics are not correctly defined or, if defined, are not put to use as integral parts of reasoning. These include number, rational number (in middle school), decimal (as a fraction in upper elementary school), ordering of fractions, product of fractions, division of fractions, length-area-volume (for different grade levels), slope of a line, half-plane of a line, equation, graph of an equation, inequality between functions, rational exponents of a positive number, polygon, congruence, similarity, parabola, inverse function, and polynomial.

2.  Mathematical statements are precise. At any moment, it is clear what is known and what is not known. There are too many places in school mathematics in which textbooks and other education materials fudge the boundary between what is true and what is not. Often a heuristic argument is conflated with correct logical reasoning. For example, the identity √a√b = √ab for positive numbers a and b is often explained by assigning a few specific values to a and b and then checking for these values with a calculator. Such an approach is a poor substitute for mathematics because it leaves open the possibility that there are other values for a and b for which the identity is not true.

3.  Every assertion can be backed by logical reasoning. Reasoning is the lifeblood of mathematics and the platform that launches problem solving. For example, the rules of place value are logical consequences of the way we choose to count. By choosing to use 10 symbols (i.e., 0 to 9), we are forced to use no more than one position (place) to be able to count to large numbers. Given the too frequent absence of reasoning in school mathematics, how can we ask students to solve problems if teachers have not been prepared to engage students in logical reasoning on a consistent basis?

4.  Mathematics is coherent; it is a tapestry in which all the concepts and skills are logically interwoven to form a single piece. The professional development of math teachers usually emphasizes either procedures (in days of yore) or intuition (in modern times), but not the coherent structure of mathematics. This may be the one aspect of mathematics that most teachers (and, dare I say, also math education professors) find most elusive. For instance, the lack of awareness of the coherence of the number systems in K–12 (whole numbers, integers, fractions, rational numbers, real numbers, and complex numbers) may account for teaching fractions as “different from” whole numbers such that the learning of fractions becomes almost divorced from the learning of whole numbers. Likewise, the resistance that some math educators (and therefore teachers) have to explicitly teaching children the standard algorithms may arise from not knowing the coherent structure that underlies these algorithms: the essence of all four standard algorithms is the reduction of any whole number computation to the computation of single-digit numbers.

5.  Mathematics is goal oriented, and every concept or skill has a purpose. Teachers who recognize the purposefulness of mathematics gain an extra tool to make their lessons more compelling. For example, when students see the technique of completing the square merely as a trick to get the quadratic formula, rather than as the central idea underlying the study of quadratic functions, their understanding of the technique is superficial. Mathematics is a collection of interconnecting chains in which each concept or skill appears as a link in a chain, so that each concept or skill serves the purpose of supporting another one down the line. Students should get to see for themselves that the mathematics curriculum moves forward with a purpose.

At the risk of putting too fine of a point on it, this approach tends to produce extremely formal and dense prose such the following (from a company Professor Wu was involved with):

Dilation: A transformation of the plane with center O and scale factor r(r > 0). If
D(O) = O and if P ≠ O, then the point D(P), to be denoted by Q, is the point on the ray OP so that |OQ| = r|OP|. If the scale factor r ≠ 1, then a dilation in the coordinate plane is a transformation that shrinks or magnifies a figure by multiplying each coordinate of the figure by the scale factor.

Congruence: A finite composition of basic rigid motions—reflections, rotations,
translations—of the plane. Two figures in a plane are congruent if there is a congruence that maps one figure onto the other figure.

Similar: Two figures in the plane are similar if a similarity transformation exists, taking one figure to the other.

Similarity Transformation: A similarity transformation, or similarity, is a composition of a finite number of basic rigid motions or dilations. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1.

Similarity: A similarity is an example of a transformation.

Sentence of the day: Greece edition

This is Joseph.

The recent Eurozone stuff requires a bit more blogging than I am prepared for.  But I think that this comment from Ezra Klein puts in perspective just how wrong it all went:

Syriza's strategy, insofar as there was one, uncovered a method of failing that was much more complete and all-encompassing than anyone had thought possible at the start of the process.

The reason that this is bad news is the the European Union has been sold as a partnership.  In a partnership, it is actually bad for one side to lose very, very badly in negotiations.  Not because the person that won will not be objectively better off.  But because a partnership requires mutual benefit, and so a bad deal undermines the strength of the partnership.

Opposite day at the Common Core debate

{previously posted at the teaching blog]

I recently came across this defense of Common Core by two Berkeley mathematicians, Edward Frenkel and Hung-Hsi Wu. Both are sharp and highly respected and when you hear about serious mathematicians supporting the initiative, there's a good chance these two names will be on the list that follows.

Except they don't support it. They support something they call Common Core, but what they describe is radically different than what the people behind the program are talking about. The disconnect is truly amazing. Wu and Frenkel's description of common core doesn't just disagree with that used by David Coleman and pretty much everyone else involved with the enterprise; it openly contradicts it.

The case that Coleman made to Bill Gates and stuck with since then is that "academic standards varied so wildly between states that high school diplomas had lost all meaning". Furthermore, Coleman argued that having a uniform set of national standards would allow us to use a powerful set of administrative tools. We could create metrics, track progress, set up incentive systems, and generally tackle the problem like management consultants.

Compare that to this excerpt from Wu and Frenkel's essay [emphasis added]:

Before the CCSSM were adopted, we already had a de facto national curriculum in math because the same collection of textbooks was (and still is) widely used across the country. The deficiencies of this de facto national curriculum of "Textbook School Mathematics" are staggering. The CCSSM were developed precisely to eliminate those deficiencies, but for CCSSM to come to life we must have new textbooks written in accordance with CCSSM. So far, this has not happened and, unfortunately, the system is set up in such a way that the private companies writing textbooks have more incentive to preserve the existing status quo maximizing their market share than to get their math right. The big elephant in the room is that as of today, less than a year before the CCSSM are to be fully implemented, we still have no viable textbooks to use for teaching mathematics according to CCSSM!

The situation is further aggravated by the rush to implement CCSSM in student assessment. A case in point is the recent fiasco in New York State, which does not yet have a solid program for teaching CCSSM, but decided to test students according to CCSSM anyway. The result: students failed miserably. One of the teachers wrote to us about her regrets that "the kids were not taught Common Core" and that it was "tragic" how low their scores were. How could it be otherwise? Why are we testing students on material they haven't been taught? Of course, it is much easier and more fun, in lieu of writing good CCSSM textbooks, to make up CCSSM tests and then pat each other on the back and wave a big banner: "We have implemented Common Core -- Mission accomplished." But no one benefits from this. Are we competing to create a Potemkin village, or do we actually care about the welfare of the next generation? What happened in New York State will happen next year across the country if we don't get our act together.

[As a side remark, we note that even in the best of circumstances, it's a big question how to effectively test students in math on a large scale. Developing such tests is an art form still waiting to be perfected, and in any case, it's not clear how accurately students' scores on these tests can reflect students' learning. Unfortunately, our national obsession with the test scores has forced teachers to teach to the test rather than teach the material for learning. While we consider some form of standardized assessment to be necessary (just as driver's license tests are necessary), we deplore this obsession. It is time to put the emphasis back on student learning inside the classroom.]

These misguided practices give a bad name to CCSSM, which is being exploited by the standards' opponents. They misinform the public by equating CCSSM with ill-fated assessments, such as the one in New York State, when in fact the problem is caused mostly by the disconnect between the current Textbook School Mathematics and CCSSM. It is for this reason that having the CCSSM is crucial, because this is what will ensure that students are taught correct mathematics rather than the deficient and obsolete Textbook School Mathematics.

It is possible and necessary to create mathematics textbooks that do better than Textbook School Mathematics. One such effort by commoncore.org holds promise: its Eureka Math series will make online courses in K-12 math available at a modest cost. The series will be completed sometime in 2014. [Full disclosure: one of us is an author of the 8th grade textbook in that series.]

The authors have contradicted both major components of Coleman's argument. They insist that we already have a relatively consistent national system of mathematics standards and furthermore they question the reliability of the metrics which Coleman's entire system is based upon.

How can proponents of common core hold such mutually exclusive use and yet be largely unaware of the contradictions?

I suspect it is some combination of poor communication and wishful thinking on both sides. As spelled out in this essay by Wu, the authors desperately want to see mathematics education returned to some kind of Euclidean ideal. A rigorous axiomatic approach where all lessons start with precise definitions and proceed through a series of logical deductions. They have convinced themselves that the rest of the Common Core establishment is in sympathy with them just as they have convinced themselves that the lessons being produced by Eureka math are rigorous and accurate.

I'm trying to make a point about executive compensation (and perhaps implicitly about anti-trust laws)

So I'm going to post this video from Keith Olbermann (I know he can be divisive, but I think he nails this 

Then refer you to this Slate article (you can draw your own conclusions from there):

In a largely symbolic move, the NFL is giving up its nearly 50-year-old tax-exempt status, league officials announced Tuesday. The move extends to the league itself, which had been listed as a nonprofit trade group under Section 501(c)(6) of the tax code since 1966, and not the 32 teams that make up pro football, which are already taxed.

The vast majority of the NFL’s $9.5 billion revenues go to those teams, as NFL Commissioner Roger Goodell noted in a letter to owners and members of Congress announcing the move that was reported by Bloomberg.

“Every dollar of income generated through television rights fees, licensing agreements, sponsorships, ticket sales, and other means is earned by the 32 clubs and is taxable there,” Goodell wrote. “This will remain the case even when the league office and Management Council file returns as taxable entities, and the change in filing status will make no material difference to our business.”

As several commentators have noted, though, the move means that Goodell will not have to report his salary—he made $44 million in 2012 and $35 million in 2013—which invariably gets brought up every time he screws something up, which is quite often.

Video accompaniment

This post on class attitudes has been getting quite a bit of attention, which got me thinking about this sketch from the College Humor spin-off CH2.

Perhaps "secrets" isn't exactly the right word

At least according to Wikipedia, it seems to involve hiring people to write books, That doesn't seem like it would take up an entire MOOC.

Maybe he can fill in the rest of the time telling about how he created New York's first "great detective hero."

David Brooks -- wrong in the right way

Charles Pierce (writing in the canine persona of Moral Hazard) brings us another dose of godawful from David Brooks

Here's Pierce/Hazard:

"He's writing today about this amazing story of survival told by a woman who escaped the horrific slaughter in Rwanda back in the 1990s. What a saga! Of course, it wasn't enough just to tell a tale of genocide and the indomitable human spirit There had to be something in there that connected to the perilous life of a wealthy member of the American opinion elite, beset as he is by the metaphorical machetes of daily life."

And here's Brooks:

Clemantine is now an amazing young woman. Her superb and artful essay reminded me that while the genocide was horrific, the constant mystery of life is how loved ones get along with one another. We work hard to cram our lives into legible narratives. But we live in the fog of reality. Whether you have survived a trauma or not, the psyche is still a dark forest of scars and tender spots. Each relationship is intricacy piled upon intricacy, fertile ground for misunderstanding and mistreatment.

Take a moment to appreciate the metaphors as they mix. We cram lives into legible narratives despite living in a fog of reality in a dark but fertile forest of scars, tender spots, misunderstanding and mistreatment... or something like that. To be perfectly honest, I zoned out for a moment there.

Brooks is capable of extraordinarily sharp and elegant writing, but just as often his prose is abysmal. Sloppy, grandiose and badly argued. He is forgiven these stylistic offenses for the same reason that he is forgiven his substantive ones: because he's wrong in the right way. He plays to the pretensions and class prejudices of the New York Times (and, to a large extent, of the national press in general) while letting the paper congratulate itself for being open to conservative views.

Andrew Gelman recently asked how many uncorrected mistakes would it take for Brooks to be discredited? The answer is, as long Brooks makes his employers and colleagues feel good about themselves, anything up to and possibly including a bodies-in-the-crawlspace incident will be overlooked.

I do not have time to heap an appropriate amount of scorn on this...

But the New York Times is excited to report (at great length) that now a slightly larger small minority of the super rich have come to agree with most social scientists and 60% of the American public that income inequality is a problem.

This is considered newsworthy because the NYT's attitude toward CEOs and hedge fund managers is disturbingly similar to a 14-year-old fan's attitude toward Justin Bieber. The fascination is just as all-consuming and just as annoying.

One of these days, I want to do a serious thread on the paper's increasingly bizarre combination of class insularity and old-school liberalism, and how it creates fertile ground for silly narratives, distorted coverage, and roughly every third column from David Brooks. For now though, I'm just keeping a tally.

"Welcome to Devonian Park"

As mentioned before, during the school year, I keep an eye out for entertaining STEM videos I can post at the teaching blog. Here are a couple in the queue.





This is a really pretty song.

In the great halls of Ithuvania, all of the banquets are catered by Whole Foods

[In case you've forgotten about Ithuvania...]

I'm working on a couple of Whole Foods related threads for the food blog, and I keep coming across these remarkable John Mackey facts. He isn't just your standard crazy CEO; he actually manages to be an ideological chimera, somehow combining the most annoying traits of the left and of the right.  A flaky new-ager and dyed-in-the-wool Randian (“The union is like having herpes. It doesn’t kill you, but it’s unpleasant and inconvenient, and it stops a lot of people from becoming your lover.” [I was going to make a joke about Walmart and sex here, but it just seems like overkill].  An anti-GMO vegan who calls global warming "perfectly natural."

He's also kind of a jerk.

From Nick Paumgarten's profile in the New Yorker:

Two years ago, Mackey passed through one of the roughest stretches of his life. The Bush Administration, in an uncharacteristic spasm of antitrust vigilance, was fighting Whole Foods’ purchase of a competitor, Wild Oats, contending that the merged company would unfairly corner what the Federal Trade Commission called the “premium natural and organic supermarket” sector. Meanwhile, the Securities and Exchange Commission was investigating Mackey: for nearly eight years, he had been secretly logging onto an Internet message board devoted to Whole Foods stock under the sock puppet, or pseudonym, “rahodeb” (an anagram of Deborah, his wife’s name), praising his own company, disparaging Wild Oats, and throwing in a flattering remark about his hair (“I think he looks cute!”). Mackey, for years a media and stock-market sweetheart, was suddenly recast as a monopolist, a fruitcake, and a sneak. The share price fell, and, even though the government eventually let the deal stand (with a few concessions from Whole Foods) and gave the sock puppetry a pass, many wondered how Mackey managed to hold on to his job.

During this period, Mackey sought succor in spiritual practice. He engaged a friend, a follower of the Czech transpersonal psychologist Stanislav Grof, to guide him through a therapeutic session of holotropic breathing. “I had this very powerful session, very powerful. It lasted about two hours,” Mackey said in an inspirational CD set he released last year called “Passion and Purpose: The Power of Conscious Capitalism.” “I was having a dialogue with what I would define as my deeper self, or my higher self.” He had a pair of epiphanies, one having to do with severed relationships that needed healing. The other was that “if I wanted to continue to do Whole Foods, there couldn’t be any part of my life that was secretive or hidden or that I’d be embarrassed [about] if people found out about it. I had to let go of all of that,” he said. “I’m this public figure now.”

All of which would be easier to forgive if Whole Foods wasn't profiting from and aggressively contributing to the pseudo-science and general bullshit of the foodie culture.

From Michael Schulson

Still, there’s a lot in your average Whole Foods that’s resolutely pseudoscientific. The homeopathy section has plenty of Latin words and mathematical terms, but many of its remedies are so diluted that, statistically speaking, they may not contain a single molecule of the substance they purport to deliver. The book section—yep, Whole Foods sells books—boasts many M.D.’s among its authors, along with titles like The Coconut Oil Miracle and Herbal Medicine, Healing, and Cancer, which was written by a theologian and based on what the author calls the Eclectic Triphasic Medical System.

You can buy chocolate with “a meld of rich goji berries and ashwagandha root to strengthen your immune system,” and bottles of ChlorOxygen chlorophyll concentrate, which “builds better blood.” There’s cereal with the kind of ingredients that are “made in a kitchen—not in a lab,” and tea designed to heal the human heart.

Nearby are eight full shelves of probiotics—live bacteria intended to improve general health. I invited a biologist friend who studies human gut bacteria to come take a look with me. She read the healing claims printed on a handful of bottles and frowned. “This is bullshit,” she said, and went off to buy some vegetables. Later, while purchasing a bag of chickpeas, I browsed among the magazine racks. There was Paleo Living, and, not far away, the latest issue of What Doctors Don’t Tell You. Pseudoscience bubbles over into anti-science. A sample headline: “Stay sharp till the end: the secret cause of Alzheimer’s.” A sample opening sentence: “We like to think that medicine works.”

Schulson's piece includes a link to What Doctors Don’t Tell You, but I decided to leave it out. I clicked on it and, trust me, this is not a rabbit hole you want to go down.

For the math nerds in audience (or as we call them here, "the audience")

Let's get real. If you're reading a blog originally called "Observational Epidemiology," the cool kids' boat has sailed.

I've got a post up on the teaching blog on the case against axiomatic rigor in lower level classes, but the best part is probably this anecdote.

A few years ago, when I was teaching math at a big state university, a colleague told me the following.

She was comparing notes with a professor at a nearby school on how their respective real analysis courses were going. She told him that they had just proved that the square root of two was an irrational number. He laughed and said she was way ahead of him; his class had just proved that the square root of two is a number.

John Lott's tertiary defense

I've already wasted way too much time following this exchange between Andrew Gelman and John Lott and reading up on the Lott saga. In retrospect, the man isn't that interesting and I doubt you can find an issue I care less about than gun rights/gun control. Nonetheless, I did notice something about Lott's defense and, having wasted the time following all of those links, I might as well get a post out of it.

Lott was responding to a comparison Gelman drew between him and Michael LaCour. I'm not going to go into the details here (that's what the link at the top of the page is for). What caught my attention was what popped when I checked out the sites Lott provided as support.

The first thing you notice is the tone [from supporter James M. Purtilo]:

However our close observation of Wikipedia points to the company’s willing participation in efforts to promote biased material into “fact.” The company’s business relationships give it high page rank in many search engines, so searches on many terms, disputed or not, naturally draw consumers to Wikipedia material. (Google in particular, a growing icon in politically left-leaning circles, gives high priority to Wikipedia entries.) When controversial topics are ‘frozen’ by Wikipedia editors, they are apparently done so in a form most beneficial to the left wing view, without disclaimer warning a well-intentioned researcher that he or she may be incorporating disputed or unsupported material. When journalists accept such material, whether innocently or by knowingly giving faint diligence to an obligation to get ‘outside’ authoritative sources, the quality of material presented on Wikipedia becomes inappropriately boosted in the eyes of the public. The net effect is a ‘bootstrapping’ process, in which the quality of material which tends to serve liberal political needs is artificially inflated and distributed.

But the main thing that struck me was that the links Lott gave all seemed to attack tertiary sources like Wikipedia and a brief item the Washington Post. The WP focus is particularly odd since pretty much all that writer does is describe a Timothy Noah column from Slate. Lott provides hundreds of words on the Post but I can't find anything on Noah. I also couldn't find any references in the piece to Lott's best-known critic, Steven Levitt, which is strange since Levitt definitely left him an opening.

I'm not sure what the strategy here is. Lott's idea may be to keep the charges from spreading, or perhaps he's just not a very effective debater.





By the way, in the social sciences, Lott vs. Levitt is basically...

I really don't know who to root for.

Mark Thoma continues to shine

Nobody talks about the assumptions behind free markets better than Mark Thoma, who has a new piece on the subject.  It really should be required reading, because it sets the pattern of regulations that all society's develop in context and makes it hard to have an absolutist view on a lot of subjects.

At the same time, it appropriately acknowledges just how powerful this process can be when the necessary preconditions exist.

Cracked, on the other hand, is very good at talking about math

I tend to avoid Cracked.com. Their articles are thoughtful, informative and enormously entertaining. They can discuss complex, even technical subjects in a clear and engaging way. All of this makes the site a dangerous time sink and I just can't afford to hang ouot there.

From 6 Small Math Errors That Caused Huge Disasters

#4. A Huge Walkway Collapses Due to a (Seemingly) Inconsequential Design Change

When designing their newest hotel to be built in downtown Kansas City, the fine people at Hyatt Regency wanted all the bells and whistles in it. The architectural firm in charge of the building design came up with a series of aerial walkways suspended from the ceiling so that guests could people-watch from a heightened vantage point. All in all, it was a pretty nifty feature. Until it suddenly collapsed and killed more than a hundred people.

The Laughably Simple Flaw:

One long rod was replaced with two short ones.

If there's one principle consistent across all human nature, it's that we will always prefer the path of least resistance (i.e., "if you can get away with a half-assed job, do it"). The original plan was for two walkways that were directly on top of one another to both be supported by one very long rod that would anchor into the ceiling. Like so:

Easier to work with, easier to install, works exactly the same. Right?

That little change killed 114 people, injured 216 more and cost $140 million in lawsuits.

Look at the first image again.

One rod, two nuts. Each nut only has to carry the weight of its own platform. Which is good, because each nut (and the welded beam it's screwed to) is only rated to carry the weight of one platform.

Now look at the second image. See the nut we've labeled "OH SHIT"?

That one single nut now has to carry the weight of BOTH platforms, and all the doomed tourists standing on them. Look obvious? Congratulations, because none of the professionals at either company caught it.

And so, one night during a dance competition, the stressed "OH SHIT" nut cleaved clean through the beam and the walkways collapsed.

During the ensuing lawsuits, it came out that neither the steel company nor the engineering firm in charge of construction had even bothered to do a back-of-the-envelope calculation that would have shown them this glaring flaw.

There may be a quality control problem here -- nearing the end of the EngageNY thread

At the teaching blog, a follow-up to the language of Eureka post using the same lesson:

Eureka Math Tips for Parents -- worst SAT prep question ever

Here's the offending passage this time.

[And before you ask, the answer is no, a rectangle and a triangle can't have corresponding parts.]

I have lots more material -- hell, that same page had a different problem that screwed up by omitting the AAA similarity theorem -- but I think that we've already uncovered an unacceptable number of major errors after only a few spot checks. More examples would just be piling on.