Common Core and Education Reform: Why Can't Billy Divide?

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axeil: Feb 14, 2006

As I'm sure everyone here knows, education in America isn't the greatest thing. In 2009, a group of 48 states developed the Common Core curriculum aided by encouragement from the Obama Administration. 43 states have now adopted the standard and are implementing it in classrooms. However, despite the inspiring start, it now seems there is nothing but negative news about it, most notably focused on the math instruction. For example, Louis C.K. showed a bunch of problems from his kid's homework which appeared absolute nonsense and got a lot of play in the media.

However, the New York Times did an article about the reasoning behind the reform and it's based in solid ideas and methods. The issue appears to be the teachers are just as bad at math as everyone else in the country is, and when presented with new methods are unable to accurately adopt them, resulting in even worse instruction than using the traditional method.

http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html?referrer&_r=0

The New York Times posted:

When Akihiko Takahashi was a junior in college in 1978, he was like most of the other students at his university in suburban Tokyo. He had a vague sense of wanting to accomplish something but no clue what that something should be. But that spring he met a man who would become his mentor, and this relationship set the course of his entire career.

Takeshi Matsuyama was an elementary-school teacher, but like a small number of instructors in Japan, he taught not just young children but also college students who wanted to become teachers. At the university-affiliated elementary school where Matsuyama taught, he turned his classroom into a kind of laboratory, concocting and trying out new teaching ideas. When Takahashi met him, Matsuyama was in the middle of his boldest experiment yet � revolutionizing the way students learned math by radically changing the way teachers taught it.

Instead of having students memorize and then practice endless lists of equations � which Takahashi remembered from his own days in school � Matsuyama taught his college students to encourage passionate discussions among children so they would come to uncover math�s procedures, properties and proofs for themselves. One day, for example, the young students would derive the formula for finding the area of a rectangle; the next, they would use what they learned to do the same for parallelograms. Taught this new way, math itself seemed transformed. It was not dull misery but challenging, stimulating and even fun.

Takahashi quickly became a convert. He discovered that these ideas came from reformers in the United States, and he dedicated himself to learning to teach like an American. Over the next 12 years, as the Japanese educational system embraced this more vibrant approach to math, Takahashi taught first through sixth grade. Teaching, and thinking about teaching, was practically all he did. A quiet man with calm, smiling eyes, his passion for a new kind of math instruction could take his colleagues by surprise. �He looks very gentle and kind,� Kazuyuki Shirai, a fellow math teacher, told me through a translator. �But when he starts talking about math, everything changes.�

Takahashi was especially enthralled with an American group called the National Council of Teachers of Mathematics, or N.C.T.M., which published manifestoes throughout the 1980s, prescribing radical changes in the teaching of math. Spending late nights at school, Takahashi read every one. Like many professionals in Japan, teachers often said they did their work in the name of their mentor. It was as if Takahashi bore two influences: Matsuyama and the American reformers.

Takahashi, who is 58, became one of his country�s leading math teachers, once attracting 1,000 observers to a public lesson. He participated in a classroom equivalent of �Iron Chef,� the popular Japanese television show. But in 1991, when he got the opportunity to take a new job in America, teaching at a school run by the Japanese Education Ministry for expats in Chicago, he did not hesitate. With his wife, a graphic designer, he left his friends, family, colleagues � everything he knew � and moved to the United States, eager to be at the center of the new math.

As soon as he arrived, he started spending his days off visiting American schools. One of the first math classes he observed gave him such a jolt that he assumed there must have been some kind of mistake. The class looked exactly like his own memories of school. �I thought, Well, that�s only this class,� Takahashi said. But the next class looked like the first, and so did the next and the one after that. The Americans might have invented the world�s best methods for teaching math to children, but it was difficult to find anyone actually using them.

It wasn�t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious �new math,� only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.

The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them. One 1965 Peanuts cartoon depicts the young blond-haired Sally struggling to understand her new-math assignment: �Sets . . . one to one matching . . . equivalent sets . . . sets of one . . . sets of two . . . renaming two. . . .� After persisting for three valiant frames, she throws back her head and bursts into tears: �All I want to know is, how much is two and two?�

Today the frustrating descent from good intentions to tears is playing out once again, as states across the country carry out the latest wave of math reforms: the Common Core. A new set of academic standards developed to replace states� individually designed learning goals, the Common Core math standards are like earlier math reforms, only further refined and more ambitious. Whereas previous movements found teachers haphazardly, through organizations like Takahashi�s beloved N.C.T.M. math-teacher group, the Common Core has a broader reach. A group of governors and education chiefs from 48 states initiated the writing of the standards, for both math and language arts, in 2009. The same year, the Obama administration encouraged the idea, making the adoption of rigorous �common standards� a criterion for receiving a portion of the more than $4 billion in Race to the Top grants. Forty-three states have adopted the standards.

The opportunity to change the way math is taught, as N.C.T.M. declared in its endorsement of the Common Core standards, is �unprecedented.� And yet, once again, the reforms have arrived without any good system for helping teachers learn to teach them. Responding to a recent survey by Education Week, teachers said they had typically spent fewer than four days in Common Core training, and that included training for the language-arts standards as well as the math.

Carefully taught, the assignments can help make math more concrete. Students don�t just memorize their times tables and addition facts but also understand how arithmetic works and how to apply it to real-life situations. But in practice, most teachers are unprepared and children are baffled, leaving parents furious. The comedian Louis C.K. parodied his daughters� homework in an appearance on �The Late Show With David Letterman�: �It�s like, Bill has three goldfish. He buys two more. How many dogs live in London?�

The inadequate implementation can make math reforms seem like the most absurd form of policy change � one that creates a whole new problem to solve. Why try something we�ve failed at a half-dozen times before, only to watch it backfire? Just four years after the standards were first released, this argument has gained traction on both sides of the aisle. Since March, four Republican governors have opposed the standards. In New York, a Republican candidate is trying to establish another ballot line, called Stop Common Core, for the November gubernatorial election. On the left, meanwhile, teachers� unions in Chicago and New York have opposed the reforms.

The fact that countries like Japan have implemented a similar approach with great success offers little consolation when the results here seem so dreadful. Americans might have written the new math, but maybe we simply aren�t suited to it. �By God,� wrote Erick Erickson, editor of the website RedState, in an anti-Common Core attack, is it such �a horrific idea that we might teach math the way math has always been taught.�

The new math of the �60s, the new new math of the �80s and today�s Common Core math all stem from the idea that the traditional way of teaching math simply does not work. As a nation, we suffer from an ailment that John Allen Paulos, a Temple University math professor and an author, calls innumeracy � the mathematical equivalent of not being able to read. On national tests, nearly two-thirds of fourth graders and eighth graders are not proficient in math. More than half of fourth graders taking the 2013 National Assessment of Educational Progress could not accurately read the temperature on a neatly drawn thermometer. (They did not understand that each hash mark represented two degrees rather than one, leading many students to mistake 46 degrees for 43 degrees.) On the same multiple-choice test, three-quarters of fourth graders could not translate a simple word problem about a girl who sold 15 cups of lemonade on Saturday and twice as many on Sunday into the expression �15 + (2�15).� Even in Massachusetts, one of the country�s highest-performing states, math students are more than two years behind their counterparts in Shanghai.

Adulthood does not alleviate our quantitative deficiency. A 2012 study comparing 16-to-65-year-olds in 20 countries found that Americans rank in the bottom five in numeracy. On a scale of 1 to 5, 29 percent of them scored at Level 1 or below, meaning they could do basic arithmetic but not computations requiring two or more steps. One study that examined medical prescriptions gone awry found that 17 percent of errors were caused by math mistakes on the part of doctors or pharmacists. A survey found that three-quarters of doctors inaccurately estimated the rates of death and major complications associated with common medical procedures, even in their own specialty areas.

One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald�s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W�s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald�s. The �4� in ��,� larger than the �3� in �⅓,� led them astray.

But our innumeracy isn�t inevitable. In the 1970s and the 1980s, cognitive scientists studied a population known as the unschooled, people with little or no formal education. Observing workers at a Baltimore dairy factory in the �80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was �equivalent to shifting between different base systems of numbers.� Throughout these mental calculations, errors were �virtually nonexistent.� And yet when these workers were out sick and the dairy�s better-educated office workers filled in for them, productivity declined.

The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans� innumeracy: school.

Most American math classes follow the same pattern, a ritualistic series of steps so ingrained that one researcher termed it a cultural script. Some teachers call the pattern �I, We, You.� After checking homework, teachers announce the day�s topic, demonstrating a new procedure: �Today, I�m going to show you how to divide a three-digit number by a two-digit number� (I). Then they lead the class in trying out a sample problem: �Let�s try out the steps for 242 � 16� (We). Finally they let students work through similar problems on their own, usually by silently making their way through a work sheet: �Keep your eyes on your own paper!� (You).

By focusing only on procedures � �Draw a division house, put �242� on the inside and �16� on the outside, etc.� � and not on what the procedures mean, �I, We, You� turns school math into a sort of arbitrary process wholly divorced from the real world of numbers. Students learn not math but, in the words of one math educator, answer-getting. Instead of trying to convey, say, the essence of what it means to subtract fractions, teachers tell students to draw butterflies and multiply along the diagonal wings, add the antennas and finally reduce and simplify as needed. The answer-getting strategies may serve them well for a class period of practice problems, but after a week, they forget. And students often can�t figure out how to apply the strategy for a particular problem to new problems.

How could you teach math in school that mirrors the way children learn it in the world? That was the challenge Magdalene Lampert set for herself in the 1980s, when she began teaching elementary-school math in Cambridge, Mass. She grew up in Trenton, accompanying her father on his milk deliveries around town, solving the milk-related math problems he encountered. �Like, you know: If Mrs. Jones wants three quarts of this and Mrs. Smith, who lives next door, wants eight quarts, how many cases do you have to put on the truck?� Lampert, who is 67 years old, explained to me.

She knew there must be a way to tap into what students already understood and then build on it. In her classroom, she replaced �I, We, You� with a structure you might call �You, Y�all, We.� Rather than starting each lesson by introducing the main idea to be learned that day, she assigned a single �problem of the day,� designed to let students struggle toward it � first on their own (You), then in peer groups (Y�all) and finally as a whole class (We). The result was a process that replaced answer-getting with what Lampert called sense-making. By pushing students to talk about math, she invited them to share the misunderstandings most American students keep quiet until the test. In the process, she gave them an opportunity to realize, on their own, why their answers were wrong.

Lampert, who until recently was a professor of education at the University of Michigan in Ann Arbor, now works for the Boston Teacher Residency, a program serving Boston public schools, and the New Visions for Public Schools network in New York City, instructing educators on how to train teachers. In her book, �Teaching Problems and the Problems of Teaching,� Lampert tells the story of how one of her fifth-grade classes learned fractions. One day, a student made a �conjecture� that reflected a common misconception among children. The fraction 5 / 6, the student argued, goes on the same place on the number line as 5 / 12. For the rest of the class period, the student listened as a lineup of peers detailed all the reasons the two numbers couldn�t possibly be equivalent, even though they had the same numerator. A few days later, when Lampert gave a quiz on the topic (�Prove that 3 / 12 = 1 / 4 ,� for example), the student could confidently declare why: �Three sections of the 12 go into each fourth.�

Over the years, observers who have studied Lampert�s classroom have found that students learn an unusual amount of math. Rather than forgetting algorithms, they retain and even understand them. One boy who began fifth grade declaring math to be his worst subject ended it able to solve multiplication, long division and fraction problems, not to mention simple multivariable equations. It�s hard to look at Lampert�s results without concluding that with the help of a great teacher, even Americans can become the so-called math people we don�t think we are.

Among math reformers, Lampert�s work gained attention. Her research was cited in the same N.C.T.M. standards documents that Takahashi later pored over. She was featured in Time magazine in 1989 and was retained by the producers of �Sesame Street� to help create the show �Square One Television,� aimed at making math accessible to children. Yet as her ideas took off, she began to see a problem. In Japan, she was influencing teachers she had never met, by way of the N.C.T.M. standards. But where she lived, in America, teachers had few opportunities for learning the methods she developed.

American institutions charged with training teachers in new approaches to math have proved largely unable to do it. At most education schools, the professors with the research budgets and deanships have little interest in the science of teaching. Indeed, when Lampert attended Harvard�s Graduate School of Education in the 1970s, she could find only one listing in the entire course catalog that used the word �teaching� in its title. (Today only 19 out of 231 courses include it.) Methods courses, meanwhile, are usually taught by the lowest ranks of professors � chronically underpaid, overworked and, ultimately, ineffective.

Without the right training, most teachers do not understand math well enough to teach it the way Lampert does. �Remember,� Lampert says, �American teachers are only a subset of Americans.� As graduates of American schools, they are no more likely to display numeracy than the rest of us. �I�m just not a math person,� Lampert says her education students would say with an apologetic shrug.

Consequently, the most powerful influence on teachers is the one most beyond our control. The sociologist Dan Lortie calls the phenomenon the apprenticeship of observation. Teachers learn to teach primarily by recalling their memories of having been taught, an average of 13,000 hours of instruction over a typical childhood. The apprenticeship of observation exacerbates what the education scholar Suzanne Wilson calls education reform�s double bind. The very people who embody the problem � teachers � are also the ones charged with solving it.

Lampert witnessed the effects of the double bind in 1986, a year after California announced its intention to adopt �teaching for understanding,� a style of math instruction similar to Lampert�s. A team of researchers that included Lampert�s husband, David Cohen, traveled to California to see how the teachers were doing as they began to put the reforms into practice. But after studying three dozen classrooms over four years, they found the new teaching simply wasn�t happening. Some of the failure could be explained by active resistance. One teacher deliberately replaced a new textbook�s problem-solving pages with the old worksheets he was accustomed to using.

Much more common, though, were teachers who wanted to change, and were willing to work hard to do it, but didn�t know how. Cohen observed one teacher, for example, who claimed to have incited a �revolution� in her classroom. But on closer inspection, her classroom had changed but not in the way California reformers intended it to. Instead of focusing on mathematical ideas, she inserted new activities into the traditional �I, We You� framework. The supposedly cooperative learning groups she used to replace her rows of desks, for example, seemed in practice less a tool to encourage discussion than a means to dismiss the class for lunch (this group can line up first, now that group, etc.).

And how could she have known to do anything different? Her principal praised her efforts, holding them up as an example for others. Official math-reform training did not help, either. Sometimes trainers offered patently bad information � failing to clarify, for example, that even though teachers were to elicit wrong answers from students, they still needed, eventually, to get to correct ones. Textbooks, too, barely changed, despite publishers� claims to the contrary.

With the Common Core, teachers are once more being asked to unlearn an old approach and learn an entirely new one, essentially on their own. Training is still weak and infrequent, and principals � who are no more skilled at math than their teachers � remain unprepared to offer support. Textbooks, once again, have received only surface adjustments, despite the shiny Common Core labels that decorate their covers. �To have a vendor say their product is Common Core is close to meaningless,� says Phil Daro, an author of the math standards.

Left to their own devices, teachers are once again trying to incorporate new ideas into old scripts, often botching them in the process. One especially nonsensical result stems from the Common Core�s suggestion that students not just find answers but also �illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.� The idea of utilizing arrays of dots makes sense in the hands of a skilled teacher, who can use them to help a student understand how multiplication actually works. For example, a teacher trying to explain multiplication might ask a student to first draw three rows of dots with two dots in each row and then imagine what the picture would look like with three or four or five dots in each row. Guiding the student through the exercise, the teacher could help her see that each march up the times table (3x2, 3x3, 3x4) just means adding another dot per row. But if a teacher doesn�t use the dots to illustrate bigger ideas, they become just another meaningless exercise. Instead of memorizing familiar steps, students now practice even stranger rituals, like drawing dots only to count them or breaking simple addition problems into complicated forms (62+26, for example, must become 60+2+20+6) without understanding why. This can make for even poorer math students. �In the hands of unprepared teachers,� Lampert says, �alternative algorithms are worse than just teaching them standard algorithms.�

No wonder parents and some mathematicians denigrate the reforms as �fuzzy math.� In the warped way untrained teachers interpret them, they are fuzzy.

When Akihiko Takahashi arrived in America, he was surprised to find how rarely teachers discussed their teaching methods. A year after he got to Chicago, he went to a one-day conference of teachers and mathematicians and was perplexed by the fact that the gathering occurred only twice a year. In Japan, meetings between math-education professors and teachers happened as a matter of course, even before the new American ideas arrived. More distressing to Takahashi was that American teachers had almost no opportunities to watch one another teach.

In Japan, teachers had always depended on jugyokenkyu, which translates literally as �lesson study,� a set of practices that Japanese teachers use to hone their craft. A teacher first plans lessons, then teaches in front of an audience of students and other teachers along with at least one university observer. Then the observers talk with the teacher about what has just taken place. Each public lesson poses a hypothesis, a new idea about how to help children learn. And each discussion offers a chance to determine whether it worked. Without jugyokenkyu, it was no wonder the American teachers� work fell short of the model set by their best thinkers. Without jugyokenyku, Takahashi never would have learned to teach at all. Neither, certainly, would the rest of Japan�s teachers.

The best discussions were the most microscopic, minute-by-minute recollections of what had occurred, with commentary. If the students were struggling to represent their subtractions visually, why not help them by, say, arranging tile blocks in groups of 10, a teacher would suggest. Or after a geometry lesson, someone might note the inherent challenge for children in seeing angles as not just corners of a triangle but as quantities � a more difficult stretch than making the same mental step for area. By the end, the teachers had learned not just how to teach the material from that day but also about math and the shape of students� thoughts and how to mold them.

If teachers weren�t able to observe the methods firsthand, they could find textbooks, written by the leading instructors and focusing on the idea of allowing students to work on a single problem each day. Lesson study helped the textbook writers home in on the most productive problems. For example, if you are trying to decide on the best problem to teach children to subtract a one-digit number from a two-digit number using borrowing, or regrouping, you have many choices: 11 minus 2, 18 minus 9, etc. Yet from all these options, five of the six textbook companies in Japan converged on the same exact problem, Toshiakira Fujii, a professor of math education at Tokyo Gakugei University, told me. They determined that 13 minus 9 was the best. Other problems, it turned out, were likely to lead students to discover only one solution method. With 12 minus 3, for instance, the natural approach for most students was to take away 2 and then 1 (the subtraction-subtraction method). Very few would take 3 from 10 and then add back 2 (the subtraction-addition method).

But Japanese teachers knew that students were best served by understanding both methods. They used 13 minus 9 because, faced with that particular problem, students were equally likely to employ subtraction-subtraction (take away 3 to get 10, and then subtract the remaining 6 to get 4) as they were to use subtraction-addition (break 13 into 10 and 3, and then take 9 from 10 and add the remaining 1 and 3 to get 4). A teacher leading the �We� part of the lesson, when students shared their strategies, could do so with full confidence that both methods would emerge.

By 1995, when American researchers videotaped eighth-grade classrooms in the United States and Japan, Japanese schools had overwhelmingly traded the old �I, We, You� script for �You, Y�all, We.� (American schools, meanwhile didn�t look much different than they did before the reforms.) Japanese students had changed too. Participating in class, they spoke more often than Americans and had more to say. In fact, when Takahashi came to Chicago initially, the first thing he noticed was how uncomfortably silent all the classrooms were. One teacher must have said, �Shh!� a hundred times, he said. Later, when he took American visitors on tours of Japanese schools, he had to warn them about the noise from children talking, arguing, shrieking about the best way to solve problems. The research showed that Japanese students initiated the method for solving a problem in 40 percent of the lessons; Americans initiated 9 percent of the time. Similarly, 96 percent of American students� work fell into the category of �practice,� while Japanese students spent only 41 percent of their time practicing. Almost half of Japanese students� time was spent doing work that the researchers termed �invent/think.� (American students spent less than 1 percent of their time on it.) Even the equipment in classrooms reflected the focus on getting students to think. Whereas American teachers all used overhead projectors, allowing them to focus students� attention on the teacher�s rules and equations, rather than their own, in Japan, the preferred device was a blackboard, allowing students to track the evolution of everyone�s ideas.

Japanese schools are far from perfect. Though lesson study is pervasive in elementary and middle school, it is less so in high school, where the emphasis is on cramming for college entrance exams. As is true in the United States, lower-income students in Japan have recently been falling behind their peers, and people there worry about staying competitive on international tests. Yet while the United States regularly hovers in the middle of the pack or below on these tests, Japan scores at the top. And other countries now inching ahead of Japan imitate the jugyokenkyu approach. Some, like China, do this by drawing on their own native jugyokenkyu-style traditions (zuanyan jiaocai, or �studying teaching materials intensively,� Chinese teachers call it). Others, including Singapore, adopt lesson study as a deliberate matter of government policy. Finland, meanwhile, made the shift by carving out time for teachers to spend learning. There, as in Japan, teachers teach for 600 or fewer hours each school year, leaving them ample time to prepare, revise and learn. By contrast, American teachers spend nearly 1,100 hours with little feedback.

It could be tempting to dismiss Japan�s success as a cultural novelty, an unreproducible result of an affluent, homogeneous, and math-positive society. Perhaps the Japanese are simply the �math people� Americans aren�t. Yet when I visited Japan, every teacher I spoke to told me a story that sounded distinctly American. �I used to hate math,� an elementary-school teacher named Shinichiro Kurita said through a translator. �I couldn�t calculate. I was slow. I was always at the bottom of the ladder, wondering why I had to memorize these equations.� Like Takahashi, when he went to college and saw his instructors teaching differently, �it was an enlightenment.�

Learning to teach the new way himself was not easy. �I had so much trouble,� Kurita said. �I had absolutely no idea how to do it.� He listened carefully for what Japanese teachers call children�s twitters � mumbled nuggets of inchoate thoughts that teachers can mold into the fully formed concept they are trying to teach. And he worked hard on bansho, the term Japanese teachers use to describe the art of blackboard writing that helps students visualize the flow of ideas from problem to solution to broader mathematical principles. But for all his efforts, he said, �the children didn�t twitter, and I couldn�t write on the blackboard.� Yet Kurita didn�t give up � and he had resources to help him persevere. He went to study sessions with other teachers, watched as many public lessons as he could and spent time with his old professors. Eventually, as he learned more, his students started to do the same. Today Kurita is the head of the math department at Setagaya Elementary School in Tokyo, the position once held by Takahashi�s mentor, Matsuyama.

Of all the lessons Japan has to offer the United States, the most important might be the belief in patience and the possibility of change. Japan, after all, was able to shift a country full of teachers to a new approach. Telling me his story, Kurita quoted what he described as an old Japanese saying about perseverance: �Sit on a stone for three years to accomplish anything.� Admittedly, a tenacious commitment to improvement seems to be part of the Japanese national heritage, showing up among teachers, autoworkers, sushi chefs and tea-ceremony masters. Yet for his part, Akihiko Takahashi extends his optimism even to a cause that can sometimes seem hopeless � the United States. After the great disappointment of moving here in 1991, he made a decision his colleagues back in Japan thought was strange. He decided to stay and try to help American teachers embrace the innovative ideas that reformers like Magdalene Lampert pioneered.

Today Takahashi lives in Chicago and holds a full-time job in the education department at DePaul University. (He also has a special appointment at his alma mater in Japan, where he and his wife frequently visit.) When it comes to transforming teaching in America, Takahashi sees promise in individual American schools that have decided to embrace lesson study. Some do this deliberately, working with Takahashi to transform the way they teach math. Others have built versions of lesson study without using that name. Sometimes these efforts turn out to be duds. When carefully implemented, though, they show promise. In one experiment in which more than 200 American teachers took part in lesson study, student achievement rose, as did teachers� math knowledge � two rare accomplishments.

Training teachers in a new way of thinking will take time, and American parents will need to be patient. In Japan, the transition did not happen overnight. When Takahashi began teaching in the new style, parents initially complained about the young instructor experimenting on their children. But his early explorations were confined to just a few lessons, giving him a chance to learn what he was doing and to bring the parents along too. He began sending home a monthly newsletter summarizing what the students had done in class and why. By his third year, he was sending out the newsletter every day. If they were going to support their children, and support Takahashi, the parents needed to know the new math as well. And over time, they learned.

To cure our innumeracy, we will have to accept that the traditional approach we take to teaching math � the one that can be mind-numbing, but also comfortingly familiar � does not work. We will have to come to see math not as a list of rules to be memorized but as a way of looking at the world that really makes sense.

The other shift Americans will have to make extends beyond just math. Across all school subjects, teachers receive a pale imitation of the preparation, support and tools they need. And across all subjects, the neglect shows in students� work. In addition to misunderstanding math, American students also, on average, write weakly, read poorly, think unscientifically and grasp history only superficially. Examining nearly 3,000 teachers in six school districts, the Bill & Melinda Gates Foundation recently found that nearly two-thirds scored less than �proficient� in the areas of �intellectual challenge� and �classroom discourse.� Odds-defying individual teachers can be found in every state, but the overall picture is of a profession struggling to make the best of an impossible hand.

Most policies aimed at improving teaching conceive of the job not as a craft that needs to be taught but as a natural-born talent that teachers either decide to muster or don�t possess. Instead of acknowledging that changes like the new math are something teachers must learn over time, we mandate them as �standards� that teachers are expected to simply �adopt.� We shouldn�t be surprised, then, that their students don�t improve.

Here, too, the Japanese experience is telling. The teachers I met in Tokyo had changed not just their ideas about math; they also changed their whole conception of what it means to be a teacher. �The term �teaching� came to mean something totally different to me,� a teacher named Hideto Hirayama told me through a translator. It was more sophisticated, more challenging � and more rewarding. �The moment that a child changes, the moment that he understands something, is amazing, and this transition happens right before your eyes,� he said. �It seems like my heart stops every day.�

I'm a mathematician by training and during the entire article all I could think was "this is exactly how we were taught complicated math." The "You", "Y'all", "We" method was basically how all our classroom instruction worked. A question was presented, we came to our own conclusions and then discussed them amongst our selves. We then finished with discussing as an entire class with the professor explaining any gaps in our understanding. We used this to do everything from differential equations to number theory to real analysis. If it works for extremely complicated and subtle ideas, I don't see why it wouldn't work for more basic concepts like "why is 1/3 + 1/2 = 5/6?".

The bigger issue is, how do we get buy-in from the schools, parents, administrators, etc. The Japanese guy was able to get everyone on board by basically re-teaching the parents math at the same time he was teaching the children. I think that's a laudable goal but I'm not sure how well it would work here when we have lots of parents using schools as a babysitter.

Consider this a thread to discuss education reform, common core, etc.

# ? Jul 25, 2014 15:40

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# ? Apr 28, 2024 18:57

twerking on the railroad: Jun 23, 2007; Get on my level

Just going to post an interview with the lead author of the CC standards for math.

http://mathbabe.org/2014/02/11/interview-with-bill-mccallum-lead-writer-of-math-common-core/

# ? Jul 25, 2014 15:49

axeil: Feb 14, 2006

Skeesix posted:

Just going to post an interview with the lead author of the CC standards for math.

http://mathbabe.org/2014/02/11/interview-with-bill-mccallum-lead-writer-of-math-common-core/

I found this bit very illuminating:

the interview posted:

Q: OK, but it�s undeniable that CC makes testing easier, do you agree?

A: Yes, and isn�t that a good thing? Having common standards also makes good testing easier. I�d also argue that they make it possible to spend less money on testing, and to make testing more centered on what you actually want. It puts more, not less, power into the hands of the consumers of the test. And that�s a good thing.

A word about testing companies. There�s no question that testing companies are trying to grab their share of money for tests. But before they could get paid for 50 different tests based on 50 different standards. What�s better?

There are two new assessment consortia, groups of states which are developing common assessments based on the standards. The consortia will have more power in the marketplace than individual states had.

I believe that people are conflating two separate issues which I�d like to separate. First, do we do a good job of choosing tests? Second, do the CCSS make that worse?

I believe that the CCSS have the power to make things better, although it�s possible that nobody will take advantage of the �commonness� in CCSS. And I�m not saying I�m not worried � the assessment consortia might do a good job but they might fail or get caught up in politics. The campaign for teacher accountability is causing fear and anger. I think you are right to be suspicious of VAM, for example. But that�s not caused by CCSS. Having common standards gives us power if we use it.

I�d also like to make the point that having common standards helps gives power to small players in curriculum publishing. When 50 different states had 50 different standards, the big publishing players with huge sales forces were able to send people to every state and adapt books to different standards. But now we will have smallish companies able to make something work and prove their worth in Tennessee and then sell it in California or wherever.

Moving textbooks away from "everyone gets textbooks made for CA or TX" would do a world of good.

# ? Jul 25, 2014 15:55

Accretionist: Nov 7, 2012; I BELIEVE IN STUPID CONSPIRACY THEORIES

As a regular dude, that bit about 'answer-getting' gave me flashbacks to Jr. High and I really like advocated approach.

It's like the difference between K - 12 and University was one of learning math in the context of math versus learning math in the context of everything else. I'd liken the difference to learning how to read & write without ever seeing so much as a short story. It wasn't until I was using math for chemistry, biology and statistical analysis that math, as a subject, clicked. Coupling math education to more active engagement would've made all the difference in the world.

# ? Jul 25, 2014 16:27

Stanos: Sep 22, 2009; The best 57 in hockey.

It's really sad to see teachers like Takahashi because it's the EXACT type of people I want teaching but the focus on tests and other problems stifle them. I'm completely terrible at advanced math and the subject is incredibly dry to me and sometimes I wonder if it was just from the rote way it was taught compared to my History and English classes that were far more engaging and interesting. I remember reading an article that I can't find now that goes into 'of course math is loving boring, it's taught in the most boring way possible instead of teaching it like an art form' and I really agreed with it.

Of course it's easier to intimidate teachers to teach to the test or flat out replace them with a TFA gimp so what reason do they have to change? And then all the Very Serious People wonder why education is falling behind.

# ? Jul 25, 2014 18:01

computer parts: Nov 18, 2010; PLEASE CLAP

axeil posted:

I found this bit very illuminating:

Moving textbooks away from "everyone gets textbooks made for CA or TX" would do a world of good.

They're already doing that because it's a lot cheaper to make textbooks these days.

And the common core solution is just "instead of everyone getting textbooks made for Texas it's everyone gets textbooks made for Common Core!"

# ? Jul 25, 2014 18:06

Xyven: Jun 4, 2005; Check to induce a ban

Stanos posted:

It's really sad to see teachers like Takahashi because it's the EXACT type of people I want teaching but the focus on tests and other problems stifle them. I'm completely terrible at advanced math and the subject is incredibly dry to me and sometimes I wonder if it was just from the rote way it was taught compared to my History and English classes that were far more engaging and interesting. I remember reading an article that I can't find now that goes into 'of course math is loving boring, it's taught in the most boring way possible instead of teaching it like an art form' and I really agreed with it.

Of course it's easier to intimidate teachers to teach to the test or flat out replace them with a TFA gimp so what reason do they have to change? And then all the Very Serious People wonder why education is falling behind.

You're probably thinking of A Mathematician's Lament. There are some problems with the article, particularly in how it over-romanticises the subject, but the overall point it makes is good.

# ? Jul 25, 2014 18:43

Stanos: Sep 22, 2009; The best 57 in hockey.

Yep that sure was it, couldn't remember the title to look it up on google.

# ? Jul 25, 2014 18:50

twerking on the railroad: Jun 23, 2007; Get on my level

One thing I'm curious about with respect to "falling behind": I heard at one point that America is not really falling behind... When it comes to educating white people. That when you look at the numbers for whites in the US you get something on the order of Norway or Finland but we just do such an execrable job educating minorities that the us looks pretty poor on the numbers. Of course I haven't really looked at the numbers there but I think it was on an NPR program where they were examining how elites in the country want to shake up education and how they're doing so.

# ? Jul 25, 2014 21:07

computer parts: Nov 18, 2010; PLEASE CLAP

Skeesix posted:

One thing I'm curious about with respect to "falling behind": I heard at one point that America is not really falling behind... When it comes to educating white people. That when you look at the numbers for whites in the US you get something on the order of Norway or Finland but we just do such an execrable job educating minorities that the us looks pretty poor on the numbers. Of course I haven't really looked at the numbers there but I think it was on an NPR program where they were examining how elites in the country want to shake up education and how they're doing so.

Yes.

code:

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Shanghai-China            	570 		580 		613 		587
Singapore                 	542 		551 		573 		556
Hong Kong-China           	545 		555 		561 		554
Asian Americans 	550 		546 		549 		548
Korea, Republic of        	536 		538 		554 		542
Japan                     	538 		547 		536 		540
Chinese Taipei            	523 		523 		560 		535
Finland                   	524 		545 		519 		529
Estonia                   	516 		541 		521 		526
Liechtenstein             	516 		525 		535 		525
Massachusetts All Races 	527 		527 		514 		523
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Canada                    	523 		525 		518 		522
Poland                    	518 		526 		518 		521
Netherlands               	511 		522 		523 		519
Switzerland               	509 		515 		531 		518
White Americans 	519 		528 		506 		518
Connecticut All Races 	521 		521 		506 		516
Vietnam                   	508 		528 		511 		516
Ireland                   	523 		522 		501 		516
Germany                   	508 		524 		514 		515
Australia                 	512 		521 		504 		512
Belgium                   	509 		505 		515 		510
New Zealand               	512 		516 		500 		509
Multiracial Americans 	517 		511 		492 		507
United Kingdom            	499 		514 		494 		502
Austria                   	490 		506 		506 		500
Czech Republic            	493 		508 		499 		500
France                    	505 		499 		495 		500
Slovenia                  	481 		514 		501 		499
Denmark                   	496 		498 		500 		498
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Luxembourg                	488 		491 		490 		490
Spain                     	488 		496 		484 		490
Italy                     	490 		494 		485 		490
Portugal                  	488 		489 		487 		488
Hungary                   	488 		494 		477 		487
Iceland                   	483 		478 		493 		484
Lithuania                 	477 		496 		479 		484
Croatia                   	485 		491 		471 		482
Sweden                    	483 		485 		478 		482
Florida All Races 	492 		485 		467 		481
Russian Federation        	475 		486 		482 		481
Israel                    	486 		470 		466 		474
Slovak Republic           	463 		471 		482 		472
Greece                    	477 		467 		453 		466
Hispanic Americans 	478 		462 		455 		465
Turkey                    	475 		463 		448 		462
Serbia, Republic of       	446 		445 		449 		447
Cyprus                    	449 		438 		440 		442
United Arab Emirates      	442 		448 		434 		441
Bulgaria                  	436 		446 		439 		440
Romania                   	438 		439 		445 		440
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Chile                     	441 		445 		423 		436
African Americans 	443 		439 		421 		434
Costa Rica                	441 		429 		407 		426
Mexico                    	424 		415 		413 		417
Kazakhstan                	393 		425 		432 		416
Montenegro, Republic of   	422 		410 		410 		414
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Uruguay                   	411 		416 		409 		412
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Jordan                    	399 		409 		386 		398
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Colombia                  	403 		399 		376 		393
Indonesia                 	396 		382 		375 		384
Qatar                     	388 		384 		376 		383
Peru                      	384 		373 		368 		375

# ? Jul 25, 2014 21:39

tsa: Feb 3, 2014

Yep we really don't have an education problem, we have a failing inner cities problem. One that cannot be solved by looking at education by itself, but rather a multi-pronged approach dealing with the prison system, drug reform, et cetera. Basically a classic intersection problem.

Stanos posted:

It's really sad to see teachers like Takahashi because it's the EXACT type of people I want teaching but the focus on tests and other problems stifle them. I'm completely terrible at advanced math and the subject is incredibly dry to me and sometimes I wonder if it was just from the rote way it was taught compared to my History and English classes that were far more engaging and interesting. I remember reading an article that I can't find now that goes into 'of course math is loving boring, it's taught in the most boring way possible instead of teaching it like an art form' and I really agreed with it.

Of course it's easier to intimidate teachers to teach to the test or flat out replace them with a TFA gimp so what reason do they have to change? And then all the Very Serious People wonder why education is falling behind.

Do you mean real analysis and beyond? Anyway many people are just going to find math boring and mostly everyone will find the most advanced math incredibly difficult. I found math interesting and the subjects you mentioned to be quite boring so it's most likely just a preference thing anyway.

I think I remember reading that article or one like it. The one I read was by the wolfram alpha guy and shocker his suggestion was to use wolfram alpha starting at a young age and stop having kids do anything that could be done with a computer (derivatives integrals, etc. ). I thought his suggestions would be helpful for people that were terrible at math but bad for producing kids that are really good at it.

tsa fucked around with this message at 21:57 on Jul 25, 2014

# ? Jul 25, 2014 21:54

nonrev: Jul 15, 2012

Some sites I read concerning education policy are former US Assistant Education Secretary Diane Ravitch and Mercedes Schneider a PHD career educator in New Orleans who does pretty detailed analysis of the funding behind Common Core.

Valerie Strauss the Washington post education reporter also has some pretty interesting articles. This one talks about why its somewhat pointless to freak out over comparing national test scores.

# ? Jul 25, 2014 22:07

tsa: Feb 3, 2014

Really the most glaring problem is that we are just now implementing statistics curriculum into middle school and high school. It will be interesting to see how students who go through those classes compare to their slightly older peers who did not.

# ? Jul 25, 2014 22:10

computer parts: Nov 18, 2010; PLEASE CLAP

tsa posted:

Really the most glaring problem is that we are just now implementing statistics curriculum into middle school and high school. It will be interesting to see how students who go through those classes compare to their slightly older peers who did not.

There's been an optional dedicated class to it for forever and some applications (eg, basic probability) have been part of the algebra curriculum for a while now.

# ? Jul 25, 2014 22:14

Rocko Bonaparte: Mar 12, 2002; Every day is Friday!

This is a tangential question, but I wondered if there are any materials out now for those of us curious about these methods that might want to give them a good try. I feel like I got a lovely math education that followed me all the way through getting an engineering degree. There were a few spots here and there where the stuff was making sense. Say, the precalculus class I had senior year was taught by the math department head, and that person was pretty drat good at actually teaching math. There were a few good spots in the early part of college, but it was poo poo for most of the rest of it.

I chalked my problems with math to a general lack of discipline since in a 5-step calculation I'll inevitably bungle it completely, but I'm starting to wonder if I would have benefited from another method. Like, I suspect for other folks that alarm bells start going off when the fuckup arrives, not at the end of two pages of hand calculations when the answer doesn't seem to pan out. That was a major problem for me in college. I'd set everything up correctly, get in a few steps, gently caress up, and follow everything to its illogical conclusion. It only came to me a decade later that, you know, that equal sign means I should be able to smash some numbers in at any time and get the same thing on either side of an equation. I felt like an idiot.

Having that stigma over my head made it much harder for any theory to stick, and I feel myself at a loss for fundamentals I feel like I should have. What I understand now is that if you treat math like a technical skill, you're more likely to become proficient in it than if you consider it to be a talent.

# ? Jul 25, 2014 22:37

Bigup DJ: Nov 8, 2012

Xyven posted:

You're probably thinking of A Mathematician's Lament. There are some problems with the article, particularly in how it over-romanticises the subject, but the overall point it makes is good.

I'm really curious how that's a problem. Could you talk more about it?

# ? Jul 26, 2014 04:23

ikanreed: Sep 25, 2009; I honestly I have no idea who cannibal[SIC] is and I do not know why I should know.

syq dude, just syq!

Bigup DJ posted:

I'm really curious how that's a problem. Could you talk more about it?

I feel like everyone who "gets" math goes through this thought process when they're exposed to other students who "don't". You see people with a half-page of formulas scrawled on notes. You can tell that their plan (to get a good grade) is to go back over that list of formulas and commit as much to memory as possible.

People would be baffled that I'd take no notes, take no "cheat sheets" to tests, and score better than them. I was baffled too, until you start seeing those formulas as what they are: a crutch. A replacement for the natural and true form of mathematics wherein you connect one concept to another by a series of relationships.

The "quadratic formula" has no place in a society with computers, but learning or inferring that a quadratic equation produces two roots because it crosses the x axis twice, and that those roots get closer together as the parabola gets "tighter" and further apart as it gets "lower", and seeing the magic of how those two factors come out of the equation. That is an understanding that can't be broken by a flawed human memory. And can be applied without having hard numbers.

And you can see that relationship in all sorts of things that happen if you start looking. But as a set formula, it's nothing. It's meaningless. And has to be memorized by rote, and fails at being especially useful even for that.

# ? Jul 26, 2014 05:32

Cantorsdust: Aug 10, 2008; Infinitely many points, but zero length.

Xyven posted:

You're probably thinking of A Mathematician's Lament. There are some problems with the article, particularly in how it over-romanticises the subject, but the overall point it makes is good.

As someone who did math competitions throughout childhood, graduated with a degree in mathematics, thought for a long time about being a high school math teacher before abandoning the dream for a better paying career, I can really identify with the article. Mathematics done right is absolutely beautiful. Case in point:

Rocko Bonaparte posted:

I chalked my problems with math to a general lack of discipline since in a 5-step calculation I'll inevitably bungle it completely, but I'm starting to wonder if I would have benefited from another method. Like, I suspect for other folks that alarm bells start going off when the fuckup arrives, not at the end of two pages of hand calculations when the answer doesn't seem to pan out. That was a major problem for me in college. I'd set everything up correctly, get in a few steps, gently caress up, and follow everything to its illogical conclusion. It only came to me a decade later that, you know, that equal sign means I should be able to smash some numbers in at any time and get the same thing on either side of an equation. I felt like an idiot.

There's tons of times where I would be in the middle of a derivation and start turning down a wrong path, but I would stop. Once things started to get complicated, they got ugly, for lack of a better word. With practice, you can absolutely pick out the moment where you've messed up instead of "following everything to its illogical conclusion." It's the same sense you get when you hear an off key note or start a faltering sentence. But you can't build that sense from rote math problems/teaching, because they're usually far too short. You have to be left to flounder around for a page or two so that you can start to build that sense of when things aren't working out.

Alternatively, you have to learn to sketch out the major derivation steps in your head that you instinctively sense to be true, and then work out a path from there. You might have encountered the term "lemma" in geometry referring to such bridging steps. Again, without "freeform" derivation/proofs, you'll never build that skill.

The thing is that beyond a point, math absolutely gets fun. This was a problem I just thought of yesterday, for which I don't know the answer:

You know primes. Sometimes primes come in pairs only two apart, like 11 and 13, or 29 and 31. It's been proven that there are an infinite number of such "twin prime" pairs. But what about primes appearing three apart? Or four apart? Some number n apart? Are there an infinite number of "n prime" pairs for any n? If not, which n's?

This is a fun question. The answer isn't immediately obvious to me, although I would suspect it's true for all n based on beauty alone. But that's an example of a fun "adult-level" math question that without a proper comprehensive math education, kids would never be able to experience.

fake edit: a bit of googling shows that I was wrong, the number of twin primes is only conjectured to be infinite, but there's strong suspicion that it is. My specific question is called Polignac's conjecture, and was first asked in 1849. Thus far, the best efforts of mathematicians have shown that it is true for at least one number N for N < 246. It remains unanswered. Cool!

Cantorsdust fucked around with this message at 05:42 on Jul 26, 2014

# ? Jul 26, 2014 05:37

Bigup DJ: Nov 8, 2012

ikanreed posted:

I feel like everyone who "gets" math goes through this thought process when they're exposed to other students who "don't". You see people with a half-page of formulas scrawled on notes. You can tell that their plan (to get a good grade) is to go back over that list of formulas and commit as much to memory as possible.

People would be baffled that I'd take no notes, take no "cheat sheets" to tests, and score better than them. I was baffled too, until you start seeing those formulas as what they are: a crutch. A replacement for the natural and true form of mathematics wherein you connect one concept to another by a series of relationships.

The "quadratic formula" has no place in a society with computers, but learning or inferring that a quadratic equation produces two roots because it crosses the x axis twice, and that those roots get closer together as the parabola gets "tighter" and further apart as it gets "lower", and seeing the magic of how those two factors come out of the equation. That is an understanding that can't be broken by a flawed human memory. And can be applied without having hard numbers.

And you can see that relationship in all sorts of things that happen if you start looking. But as a set formula, it's nothing. It's meaningless. And has to be memorized by rote, and fails at being especially useful even for that.

Oh yeah I agree completely! I'm asking what the problem is with it "over-romanticising" the subject, and more importantly how he came to that conclusion in the first place. He was using it as a pejorative and I'm curious why.

Edit: For the record though I don't agree that it over-romanticises maths, or that there is such a thing as over-romanticisation unless it's papering over something which has the potential to do serious harm to people.

Bigup DJ fucked around with this message at 06:00 on Jul 26, 2014

# ? Jul 26, 2014 05:53

axeil: Feb 14, 2006

Cantorsdust posted:

There's tons of times where I would be in the middle of a derivation and start turning down a wrong path, but I would stop. Once things started to get complicated, they got ugly, for lack of a better word. With practice, you can absolutely pick out the moment where you've messed up instead of "following everything to its illogical conclusion." It's the same sense you get when you hear an off key note or start a faltering sentence. But you can't build that sense from rote math problems/teaching, because they're usually far too short. You have to be left to flounder around for a page or two so that you can start to build that sense of when things aren't working out.

This is why I think I always seemed to learn math better when we were doing a proof by contradiction. The goal was to get an answer that didn't make any sense. My abstract math professor was always amused that my strategy was usually to prove why something couldn't be a, b or c and therefor had to be the only remaining option, d. It all seemed very Sherlock Holmes to him.

In that vein, math is just solving lots of really neat puzzles and mysteries. If we taught math that way I think kids would be way more interested. Everyone likes solving puzzles, and even more people like solving puzzles and knowing they got the correct answer. That seems way more satisfying to me than "you did this algorithm correctly."

Cantorsdust posted:

You know primes. Sometimes primes come in pairs only two apart, like 11 and 13, or 29 and 31. It's been proven that there are an infinite number of such "twin prime" pairs. But what about primes appearing three apart? Or four apart? Some number n apart? Are there an infinite number of "n prime" pairs for any n? If not, which n's?

This is a fun question. The answer isn't immediately obvious to me, although I would suspect it's true for all n based on beauty alone. But that's an example of a fun "adult-level" math question that without a proper comprehensive math education, kids would never be able to experience.

fake edit: a bit of googling shows that I was wrong, the number of twin primes is only conjectured to be infinite, but there's strong suspicion that it is. My specific question is called Polignac's conjecture, and was first asked in 1849. Thus far, the best efforts of mathematicians have shown that it is true for at least one number N for N < 246. It remains unanswered. Cool!

Related to this, there's the story of the math professor writing an unsolved problem on the board in class and a student arriving late and thinking it was part of an assignment. He ends up solving it and using it as his Ph.D thesis. That, to me, is what math is all about.

I don't think we can give grade school kids that sort of stuff, but even in undergrad we did our own research about obscure areas of math (the boolean lattice and weird number theory things) which made it extremely exciting anytime we found something, since we were reasonably sure we were first. If we gave grade school kids that same sense of elation when they have "Aha!" moments it'd make math much more interesting.

axeil fucked around with this message at 06:01 on Jul 26, 2014

# ? Jul 26, 2014 05:57

emfive: Aug 6, 2011; Hey emfive, this is Alec. I am glad you like the mummy eating the bowl of shitty pasta with a can of 'parm.' I made that image for you way back when. I’m glad you enjoy it.

Cantorsdust posted:

You know primes. Sometimes primes come in pairs only two apart, like 11 and 13, or 29 and 31. It's been proven that there are an infinite number of such "twin prime" pairs. But what about primes appearing three apart? Or four apart? Some number n apart? Are there an infinite number of "n prime" pairs for any n? If not, which n's?

I'm pretty sure number theory is something that makes people go crazy. Not everybody, but the entire subject area is a minefield of whoa relationships and interconnections. Even really simple things like φ(n) are just weird.

Maybe it's just that I personally am borderline nuts.

# ? Jul 26, 2014 06:12

axeil: Feb 14, 2006

emfive posted:

I'm pretty sure number theory is something that makes people go crazy. Not everybody, but the entire subject area is a minefield of whoa relationships and interconnections. Even really simple things like φ(n) are just weird.

Maybe it's just that I personally am borderline nuts.

Best in class demo we ever did in any class, not just math, was my number theory class where we broke RSA by hand. We took a 4 (or maybe 8? or 6?) digit semi-prime and walked through how effectively factoring that number would allow you to decode information. That was mind-blowing.

And I suppose it would've been illegal 20 years ago when cryptography was considered a munition.

# ? Jul 26, 2014 06:17

twerking on the railroad: Jun 23, 2007; Get on my level

Cantorsdust posted:

As someone who did math competitions throughout childhood, graduated with a degree in mathematics, thought for a long time about being a high school math teacher before abandoning the dream for a better paying career, I can really identify with the article. Mathematics done right is absolutely beautiful. Case in point:

There's tons of times where I would be in the middle of a derivation and start turning down a wrong path, but I would stop. Once things started to get complicated, they got ugly, for lack of a better word. With practice, you can absolutely pick out the moment where you've messed up instead of "following everything to its illogical conclusion." It's the same sense you get when you hear an off key note or start a faltering sentence. But you can't build that sense from rote math problems/teaching, because they're usually far too short. You have to be left to flounder around for a page or two so that you can start to build that sense of when things aren't working out.

Alternatively, you have to learn to sketch out the major derivation steps in your head that you instinctively sense to be true, and then work out a path from there. You might have encountered the term "lemma" in geometry referring to such bridging steps. Again, without "freeform" derivation/proofs, you'll never build that skill.

The thing is that beyond a point, math absolutely gets fun. This was a problem I just thought of yesterday, for which I don't know the answer:

You know primes. Sometimes primes come in pairs only two apart, like 11 and 13, or 29 and 31. It's been proven that there are an infinite number of such "twin prime" pairs. But what about primes appearing three apart? Or four apart? Some number n apart? Are there an infinite number of "n prime" pairs for any n? If not, which n's?

This is a fun question. The answer isn't immediately obvious to me, although I would suspect it's true for all n based on beauty alone. But that's an example of a fun "adult-level" math question that without a proper comprehensive math education, kids would never be able to experience.

fake edit: a bit of googling shows that I was wrong, the number of twin primes is only conjectured to be infinite, but there's strong suspicion that it is. My specific question is called Polignac's conjecture, and was first asked in 1849. Thus far, the best efforts of mathematicians have shown that it is true for at least one number N for N < 246. It remains unanswered. Cool!

That's kind of neat that you happened upon this one since there was no n it was known for until last year when it was uncovered by a guy who was not a professor but instead an adjunct teacher in New Hampshire.

As far as crypto as....munitions... OP, did you grow up in Soviet Russia or something? RSA has the name because it was discovered by academics and is now the base for the entire online transaction system of the internet.

# ? Jul 26, 2014 14:29

karthun: Nov 16, 2006; I forgot to post my food for USPOL Thanksgiving but that's okay too!

Skeesix posted:

As far as crypto as....munitions... OP, did you grow up in Soviet Russia or something? RSA has the name because it was discovered by academics and is now the base for the entire online transaction system of the internet.

What does Rivest, Shamir and Adleman have to do with the US considering crypto to be a munition?

# ? Jul 26, 2014 14:44

Polygynous: Dec 13, 2006; welp

Yeah, wiki summary of the subject here if you're not old enough to remember Netscape 3. :v:

# ? Jul 26, 2014 14:53

tsa: Feb 3, 2014

computer parts posted:

There's been an optional dedicated class to it for forever and some applications (eg, basic probability) have been part of the algebra curriculum for a while now.

There's a new curriculum in the pipeline. The current state of affairs is woefully inadequate for how important of a subject it is becoming in nearly every branch of science and beyond.

Cantorsdust posted:

The thing is that beyond a point, math absolutely gets fun. This was a problem I just thought of yesterday, for which I don't know the answer:

You know primes. Sometimes primes come in pairs only two apart, like 11 and 13, or 29 and 31. It's been proven that there are an infinite number of such "twin prime" pairs. But what about primes appearing three apart? Or four apart? Some number n apart? Are there an infinite number of "n prime" pairs for any n? If not, which n's?

This is a fun question. The answer isn't immediately obvious to me, although I would suspect it's true for all n based on beauty alone. But that's an example of a fun "adult-level" math question that without a proper comprehensive math education, kids would never be able to experience.

fake edit: a bit of googling shows that I was wrong, the number of twin primes is only conjectured to be infinite, but there's strong suspicion that it is. My specific question is called Polignac's conjecture, and was first asked in 1849. Thus far, the best efforts of mathematicians have shown that it is true for at least one number N for N < 246. It remains unanswered. Cool!

I think that's fun, but I'm a math guy. I'd guess a lot of kids would find that just as boring as any other math thing. Actually I know most would because I try those fun things from time to time. At the end of the day math is a very abstract topic and there's going to be a lot of people who either don't get it or aren't interested in abstract things like that. Much like abstract art isn't a lot of people's cup of tea.

# ? Jul 26, 2014 15:14

PT6A: Jan 5, 2006; Public school teachers are callous dictators who won't lift a finger to stop children from peeing in my plane

axeil posted:

And I suppose it would've been illegal 20 years ago when cryptography was considered a munition.

It's still fairly restricted. If you distribute a program containing certain sorts of cryptography from the US, or from a server based in the US (like, oh, all the servers for Google Play and the Apple App Store), you need to get an export license from the US government. This includes the use of things as trivial and common as HTTPS, it appears.

# ? Jul 26, 2014 15:43

Xyven: Jun 4, 2005; Check to induce a ban

Bigup DJ posted:

Oh yeah I agree completely! I'm asking what the problem is with it "over-romanticising" the subject, and more importantly how he came to that conclusion in the first place. He was using it as a pejorative and I'm curious why.

Edit: For the record though I don't agree that it over-romanticises maths, or that there is such a thing as over-romanticisation unless it's papering over something which has the potential to do serious harm to people.

It's the same reason it's a problem to over-romanticize anything, it obscures the truth and replaces facts with dreams. The fact is, for most people math is just going to be a tool they use, and pretending that it's not is just naive. The article draws comparisons between art and math, and while there is an element of expression in proofs, proving theorems isn't what math is used for by the vast majority of the population. The analogy to art is fallacious because art is not the fundamental backbone to all the science and technology that drives the modern world. You cannot treat math the same as art, because while someone who does not understand or appreciate art is only missing out on some aspects of culture, someone who does not understand math cannot fully function in society.

Another problem with romanticising math is that, frankly, a lot of it is drudgery. It is EXTREMELY important to keep track of all the little details when you are doing a proof or deriving an equation, and a missed term or forgotten minus sign can completely change your result. In many advanced subjects of math, such as analysis, differential equations, probability/measure theory, your work is 99% manipulating equations and keeping track of fiddly little details and 1% finding a neat trick that makes a proof work. Right now our education system basically focuses solely on those little details, but Lockheart's article leaves non-mathematicians with the impression that they should be ignored almost entirely. In reality these details should be a large part of math education, although not to the exclusion of all else.

And again, I do not disagree with the primary point of the article, I just think that he presents an unrealistic picture of a subject that is often dull and tedious.

# ? Jul 26, 2014 16:18

Vermain: Sep 5, 2006

Figured it might be relevant: Jacobin has an article about the failure of the charter school experiment in New Orleans. It's got the same sort of soul-grinding factory method of education mentioned above:

quote:

This curriculum was delivered almost exclusively through direct instruction � what TFA corps members refer to as the �five step lesson plan,� and educator and philosopher Paulo Freire calls �banking education,� wherein students are treated as passive and empty receptacles into which information can be deposited. In nearly every lesson Sondel observed, teachers stood in front of students to introduce new content or an isolated skill, after which students were asked to parrot, practice, and then perform their newly acquired knowledge on worksheets and multiple-choice assessments. There were no student debates, projects, or science experiments.

In a literacy lesson, for example, a teacher started by reviewing the definitions of figurative language. The teacher then projected on the Smartboard sentence after sentence, poem after poem, and, finally, a short story while students raised their hands and waited to be called on to identify idioms, similes, and personification.

After this series of questions and answers, the students sat silently at their desks, read four short passages, and identified figurative language on multiple-choice questions. The students were not asked to read the poem, analyze the story, or discuss the purpose of metaphors. After the lesson, upon being asked if students practice this skill in their independent reading or writing activities, the teacher responded, �You know the problem with that is then they have a difficult time identifying metaphors on the test.�

Perhaps because there was little inherently interesting or relevant to students about the curriculum or the classroom activities, teachers often attempted to control rather than engage students in lessons.

There were, for example, specific expectations about where students should put their hands, which direction they should turn their heads, how they should stand, and how they should sit � practices referred to at one school as SLANT (Sit up, Listen, Ask and Answer Questions, Nod, and Track the Speaker) and at the other as SPARK (Sit up straight, Pay attention, Ask and answer questions, React to show I�m following along, Keep tracking the speaker). Students were kept silent, or what teachers called �level zero,� through most of the day.

Silence seemed to be especially important in the hallways. At the sound of each bell at the middle school, students were expected to line up at �level zero� with their faces forward and hands behind their backs and, when given permission, step into the hallway and onto strips of black duct tape. There they waited for the command of an administrator: �Duke, you can move to your next class! Tulane, you can walk when you show me that you are ready!�

Students then marched until they reached the STOP sign on the floor, where their teacher checked them for hallway position before giving them permission to continue around the corner. Throughout this process, students moved counter-clockwise around the perimeter of the hallway (even if they were going to a classroom one door to the left).

This system of control was administered through intricate systems of reward and punishment. Elementary students received and lost stars for each �behavioral infraction.� In one classroom, a teacher circulated the room with a timer in her hand while students read silently. Every three minutes, after the buzzer, she put a single goldfish on the desk of each student who had remained silent. In another classroom, a teacher silently glared at a student and then typed into his iPhone, which was connected through Class Dojo � an online behavior management system � to his Smartboard. Numbers would increase and decrease on little avatars representing each student.

At the middle school, stars matured into fake money that students could use to buy access to brass band and spoken word performances. When they were not compliant, or did not have enough money to attend the weekly celebration, they were sent to the �behavior intervention room,� where they were expected to copy a piece of text word for word on lined paper. One particular afternoon, the text in question was Martin Luther King, Jr.�s �I Have a Dream� speech.

I think "behaviour intervention room" is one of the most Orwellian things I've read this year. I can't believe this actually exists.

# ? Jul 26, 2014 17:07

PT6A: Jan 5, 2006; Public school teachers are callous dictators who won't lift a finger to stop children from peeing in my plane

Jesus Christ, what are they trying to do: make it the most like a prison that it can possibly be?

# ? Jul 26, 2014 17:11

Vermain: Sep 5, 2006

PT6A posted:

Jesus Christ, what are they trying to do: make it the most like a prison that it can possibly be?

Factories. They're being run as businesses, producing the commodity of workers. That's the simplest way of putting it. Not coincidentally, the most "efficient" way of producing a worker is also the one which most mangles the human spirit.

# ? Jul 26, 2014 17:17

Cantorsdust: Aug 10, 2008; Infinitely many points, but zero length.

Xyven posted:

It's the same reason it's a problem to over-romanticize anything, it obscures the truth and replaces facts with dreams. The fact is, for most people math is just going to be a tool they use, and pretending that it's not is just naive. The article draws comparisons between art and math, and while there is an element of expression in proofs, proving theorems isn't what math is used for by the vast majority of the population. The analogy to art is fallacious because art is not the fundamental backbone to all the science and technology that drives the modern world. You cannot treat math the same as art, because while someone who does not understand or appreciate art is only missing out on some aspects of culture, someone who does not understand math cannot fully function in society.

Another problem with romanticising math is that, frankly, a lot of it is drudgery. It is EXTREMELY important to keep track of all the little details when you are doing a proof or deriving an equation, and a missed term or forgotten minus sign can completely change your result. In many advanced subjects of math, such as analysis, differential equations, probability/measure theory, your work is 99% manipulating equations and keeping track of fiddly little details and 1% finding a neat trick that makes a proof work. Right now our education system basically focuses solely on those little details, but Lockheart's article leaves non-mathematicians with the impression that they should be ignored almost entirely. In reality these details should be a large part of math education, although not to the exclusion of all else.

And again, I do not disagree with the primary point of the article, I just think that he presents an unrealistic picture of a subject that is often dull and tedious.

The thing is, math isn't needed as just a tool to use for the vast majority of people. We have calculators, computers, google, etc to fill the role of that. There's no reason to make basic arithmetic the core focus of a math curriculum. It would be like saying "the vast majority of people use art for making sketches to explain something in a lecture or presentation, so let's remake the art curriculum to focus on mastering the quick sketch."

Also, with math, the two goals of high level concepts and low level arithmetic are not mutually exclusive. The low level arithmetic is exclusively derived from the high level concepts, and starting with the concepts first can help to explain the arithmetic. Adding is just iterated counting. Multiplication is just iterated adding. Exponents are just iterated multiplication. And more exotic operations can be described iterating on that, etc.

A more concrete example: you probably memorized a bunch of equations for distance/speed/acceleration in your high school physics class. Before calculus, these equations might have seemed arbitrary, but once you learned calculus, you saw the connection between them: speed is the derivative of distance, acceleration is the derivative of speed, jerk the derivative of acceleration, etc. And suddenly, you didn't even need to memorize the equations anymore! You now understood why they were what they were, and you could rederive them if necessary.

And loving lol at advanced math being 99% manipulating equations and keeping track of fiddly little details and just 1% being the neat trick that makes the proof. On paper, maybe, but that 1% neat trick is the core of the proof and will take up the majority of the time to come up with. The equations are just writing down the argument you're making in your head. I can't tell you the number of times where I would make some sign error or something, get halfway through the derivation, and realize the sign is opposite what I want. But instead of accepting the arithmetic blindly, my sense of mathematics/aesthetics would tell me to backup and find the mistake somewhere, because what I've derived has to be right, it's just the arithmetic that's wrong. This statement is as absurd as saying that writing a story is 99% using a pen to make words and 1% coming up with characters and a plot. Anyone who's done high level math would tell you otherwise. Lockheart is telling non-mathematicians that those aspects can be ignored almost entirely because most mathematicians ignore those aspects almost entirely. Seriously, university math professors are hilariously forgetful and error-prone sometimes. That's why poo poo like Matlab / Mathematica / Wolfram Alpha exist. Let computers handle the bookkeeping, as they should.

You can argue, "but what about the engineers and scientists who are using math instead of playing with it?" Well, that's what computers are for. And before computers, that's what sine tables and slide rules were for. I cannot think of a single example in all of history where a scientist, engineer, or mathematician became known for their amazing skill in arithmetic. But I can remember Gauss as a schoolboy discovering that the sum of a series of consecutive integers from 1 to n equals n(n+1)/2. There's a reason people automated the process of arithmetic and focused on the high level stuff instead. Arithmetic is perfect for a computer: it requires perfect accuracy but no thought. So why are we making children focus on arithmetic instead of math?

edit: I guess what I'm arguing is that math is fundamentally closer to being an art or a language as it is a science. It needs to be taught like an art or a language.

Cantorsdust fucked around with this message at 17:48 on Jul 26, 2014

# ? Jul 26, 2014 17:42

Nessus: Dec 22, 2003; After a Speaker vote, you may be entitled to a valuable coupon or voucher!

Cantorsdust posted:

Arithmetic is perfect for a computer: it requires perfect accuracy but no thought. So why are we making children focus on arithmetic instead of math?

Because what is desired from these students is accuracy, but no thought, I guess. Of course, that implies there's some kind of a goal here, and I'm pretty sure that at no point was any of this actually designed from the top down to accomplish these goals, they are just side effects of a desire to "do more with less" (by very strictly defining what is more (higher test scores) and what is less (costs).)

# ? Jul 26, 2014 17:53

Bel Shazar: Sep 14, 2012

Cantorsdust posted:

It would be like saying "the vast majority of people use art for making sketches to explain something in a lecture or presentation, so let's remake the art curriculum to focus on mastering the quick sketch."

I, for one, would expect to see good value from courses designed to help children create effective visual representations of their ideas.

# ? Jul 26, 2014 18:02

computer parts: Nov 18, 2010; PLEASE CLAP

Bel Shazar posted:

I, for one, would expect to see good value from courses designed to help children create effective visual representations of their ideas.

They'd be taught on Visio if anything.

# ? Jul 26, 2014 18:06

Bel Shazar: Sep 14, 2012

Sadly true...

# ? Jul 26, 2014 18:07

nigel thornberry: Jul 29, 2013

The teachers in my family are convinced common core is just an elaborate way to close down public schools and gut the teacher's unions so we can have non-union charter schools. Maybe someone who knows more about it can say whether this is at all accurate or not.

# ? Jul 26, 2014 18:33

Xyven: Jun 4, 2005; Check to induce a ban

Cantorsdust posted:

The thing is, math isn't needed as just a tool to use for the vast majority of people. We have calculators, computers, google, etc to fill the role of that. There's no reason to make basic arithmetic the core focus of a math curriculum. It would be like saying "the vast majority of people use art for making sketches to explain something in a lecture or presentation, so let's remake the art curriculum to focus on mastering the quick sketch."

Understanding that 1/4 < 1/3 is using math as a tool, understanding how to estimate a 15% tip is using math as a tool, looking at a result that your excel spreadsheet spits out at work and realizing it is off by several orders of magnitude is using math as a tool. There are things that you need to be able to do without a calculator in order to function. And again, the analogy between math and art is terrible, because while art is neat, it is not the underlying mechanic for all of modern science and technology.

Cantorsdust posted:

And loving lol at advanced math being 99% manipulating equations and keeping track of fiddly little details and just 1% being the neat trick that makes the proof. On paper, maybe, but that 1% neat trick is the core of the proof and will take up the majority of the time to come up with. The equations are just writing down the argument you're making in your head. I can't tell you the number of times where I would make some sign error or something, get halfway through the derivation, and realize the sign is opposite what I want. But instead of accepting the arithmetic blindly, my sense of mathematics/aesthetics would tell me to backup and find the mistake somewhere, because what I've derived has to be right, it's just the arithmetic that's wrong. This statement is as absurd as saying that writing a story is 99% using a pen to make words and 1% coming up with characters and a plot. Anyone who's done high level math would tell you otherwise. Lockheart is telling non-mathematicians that those aspects can be ignored almost entirely because most mathematicians ignore those aspects almost entirely. Seriously, university math professors are hilariously forgetful and error-prone sometimes. That's why poo poo like Matlab / Mathematica / Wolfram Alpha exist. Let computers handle the bookkeeping, as they should.

Writing IS 99% sitting down and putting words on paper, making sure they flow, making sure they all fit. You appeal to your sense of mathematics to recognize when something is wrong without acknowledging where that intuition came from. It comes from a strong understanding of arithmetic and algebraic manipulation that you built up over years of paying close attention to details while doing problems.

Cantorsdust posted:

You can argue, "but what about the engineers and scientists who are using math instead of playing with it?" Well, that's what computers are for. And before computers, that's what sine tables and slide rules were for. I cannot think of a single example in all of history where a scientist, engineer, or mathematician became known for their amazing skill in arithmetic. But I can remember Gauss as a schoolboy discovering that the sum of a series of consecutive integers from 1 to n equals n(n+1)/2. There's a reason people automated the process of arithmetic and focused on the high level stuff instead. Arithmetic is perfect for a computer: it requires perfect accuracy but no thought. So why are we making children focus on arithmetic instead of math?

edit: I guess what I'm arguing is that math is fundamentally closer to being an art or a language as it is a science. It needs to be taught like an art or a language.

If you do not have a strong understanding of what the computer is doing, you will be unable to recognize when it spits out clearly nonsensical results. There are also countless examples of times where sitting down and using a computer to do a simple calculation or double check something is a ludicrous waste of time.

Nobody is remembered for their amazing skill in arithmetic because it is a prerequisite for further accomplishments. Neither are authors remembered for their great spelling abilities. It's just that you need a solid foundation in order to build up more advanced ideas. Understanding arithmetic is like knowing how to spell; it is something you need to know in order to be a functional human being.

I'm sure that math as seen from the eyes of a research mathematician really closely resembles art, but this is the minority view of a fraction of a percentage of the population. Chemists could also describe what they do as close to an art, but it doesn't matter because for most people chemistry is a tool. Math is similar to art in some ways, but that does not mean that math is an art, any more than two sets containing the same object implies that they are both the same set.

Let me reiterate that I am not saying there is no room for creativity and expression in math, or that it should be completely ignored. I am saying that Lockheart presents an unrealistic view of math because he has the viewpoint of a research mathematician.

# ? Jul 26, 2014 18:35

hobbesmaster: Jan 28, 2008

the boston bomber posted:

The teachers in my family are convinced common core is just an elaborate way to close down public schools and gut the teacher's unions so we can have non-union charter schools. Maybe someone who knows more about it can say whether this is at all accurate or not.

That is the goal of the people implementing it, as the nyt article says it isn't the goal of the authors.

# ? Jul 26, 2014 18:35

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# ? Apr 28, 2024 18:57

nigel thornberry: Jul 29, 2013

drat that sucks. I mean I guess the authors are well-intentioned? They are still breaking a working system for some people who hate unions and hate having to pay teachers a fair amount with reasonable security.

# ? Jul 26, 2014 18:41

The Something Awful Forums > Discussion > Debate & Discussion > Common Core and Education Reform: Why Can't Billy Divide?

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