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Hiné Mizushima for Quanta Magazine

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Chapter 1: PLAN

Meet the New Math, Unlike the Old Math

The latest effort to overhaul math and science education offers a fundamental rethinking of the basic structure of knowledge. But will it be given time to work?

If we could snap our fingers and change the way math and science are taught in U.S. schools, most of us would. The shortcomings of the current approach are clear. Subjects that are vibrant in the minds of experts become lifeless by the time they’re handed down to students. It’s not uncommon to hear kids in Algebra 2 ask, “When are we ever going to use this?” and for the teacher to reply, “Math teaches you how to think,” which is true — if only it were taught that way.

To say that this is now changing is to invite an eye roll. For a number of entrenched reasons, from the way teachers are trained to the difficulty of agreeing on what counts in each discipline, instruction in science and math is remarkably resistant to change.

That said, we’re riding the next big wave in K-12 science and math education in the United States. The main events are a pair of highly visible but often misunderstood documents — the Common Core math standards and the Next Generation Science Standards (NGSS) — that, if implemented successfully, will boldly remake the way math and science are taught. Both efforts seek to recast instruction in the fundamental ideas and perspectives that animate the two fields.

“What we did in reorganizing the content of school mathematics was long overdue,” said Phil Daro, one of three lead authors of the Common Core math standards.

The changes go beyond the contentious new methods of teaching arithmetic that have grabbed headlines and threatened to blunt the momentum of Common Core math. Both documents developed out of decades of academic research on how children learn, and they reflect similar priorities. They exhibit an elegant rethinking of the basic structure of knowledge, along with new assertions of what’s important for students to be able to do by the time they finish high school.

“Overall, there’s a movement towards more complex cognitive mathematics, there’s a movement towards the student being invited to act like a mathematician instead of passively taking in math and science,” said David Baker, a professor of sociology and education at Pennsylvania State University. “These are big trends and they’re quite revolutionary.”

Pedagogical revolutions are chancy endeavors, however. The Common Core math standards were released in 2010 and NGSS in 2013. Now, years on, even enthusiastic early adopters of the Common Core like the state of New York are retreating from the standards. While the ultimate impact of both the Common Core and NGSS is still uncertain, it’s clear these standards go beyond simply swapping one set of textbooks for another — to really take hold, they’ll require a fundamental rethinking of everything from assessments to classroom materials to the basic relationship between teachers and students.

The Old New Math

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NGSS and the Common Core are a significant departure from the way science and math have been taught, but they didn’t come out of nowhere. In fact, they’re consistent with a trend that’s been slow-boiling for a half-century.

In a 2010 paper, Baker and colleagues analyzed 141 elementary school math textbooks published between 1900 and 2000. They found that what kids were learning changed considerably during that period. Until the 1960s, basic arithmetic accounted for 85 percent of math instruction. By the end of the century that proportion had dropped to 64 percent, with the balance of instruction devoted to more complex topics like advanced arithmetic and geometry.

“When you step back historically and sociologically, it’s clear education has really ratcheted up along these cognitive dimensions,” Baker said. “The idea that education is like men’s ties and just goes through this cycle of wide and thin is not true.”

Pedagogy has shifted as well. During the same period in which students began to learn more complex mathematics, leaders in science and math education launched complementary pushes to teach students to think more like real scientists and mathematicians. These efforts included the “New Math” of the 1960s and similar plans that decade to teach science as an “enquiry into enquiry,” as one leading expert of the time put it. Later manifestations of the impulse away from rote instruction include curricular standards created by the National Council of Teachers of Mathematics in the 1980s and the enthusiasm for “inquiry-based” science in the 1990s.

All of these initiatives had the right idea, but their implementation was off, say developers of NGSS and Common Core math. “Inquiry” is a habit of mind among scientists, but in the 1990s it was taught as its own curricular topic: Last week we learned about DNA, this week we’re going to learn about inquiry.

“Inquiry became almost an empty word, where it didn’t really matter what the inquiry was about,” said Heidi Schweingruber, director of the Board on Science Education at the National Academies of Sciences, Engineering, and Medicine, which provided guidance for the development of NGSS.

The same problem happened in math. For the last 50 years, reformers have wanted to teach kids to reason mathematically, to think nimbly about topics like quadratic equations that otherwise come off flat. Instead, in programs that employed the New Math, students often ended up playing logic games.

“The push toward conceptual understanding and understanding rich mathematical ideas sometimes ended in practice with students just engaged in activities and messing around,” said Robert Floden, dean of the College of Education at Michigan State University.

It’s not surprising that ambitious changes like these would be hard to implement. After all, teaching kids to adopt a scientific mindset is a subtler and more complex task than having them memorize the parts of a cell. For one thing, it requires teachers who inhabit that mindset themselves, and they’re harder to find. For another, it takes a more patient perspective than the prevailing one in public education, which expects teachers to post a learning objective on the board before each class and end every unit with a multiple-choice test.

Less Is More

How does one adjust the course of a curriculum that’s been gathering inertia for decades? The developers of NGSS and Common Core math started by reducing the mass of content that had accumulated over the years, often in haphazard fashion.

“Mainly, the U.S. mathematics curriculum prior to the Common Core was a geological accretion of additions, mostly, and [some] compressions over 50 years,” Daro said. “There was a lot of mathematical junk food and traveling down rabbit holes and up cul-de-sacs.”

Schweingruber made a similar point. “The U.S. has a mile-wide, inch-deep curriculum with tons and tons of things and ideas for kids to learn, but not an opportunity to go in depth,” she said.

As the authors got down to work on Common Core in 2009 and on NGSS a year later, some of their first discussions were about what to leave in and what to take out. “It required some argument on the part of folks in the framework about what that baseline really would look like,” Schweingruber said.

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The final documents omitted a number of familiar topics. The NGSS writers eliminated instruction in the rote formula for stoichiometry calculations (the process for quantifying elements at different stages of a chemical reaction) from the high school chemistry curriculum. Daro and his collaborators on Common Core math, William McCallum of the University of Arizona and Jason Zimba of Student Achievement Partners, decided the technique of “simplifying” answers didn’t add much to mathematical understanding, so they took it out.

By removing content, the creators of Common Core math and NGSS hoped to expose core disciplinary ideas. A good example of this is how the Common Core teaches proportionality. Before, proportionality occupied about 10 percent of math instruction in grades six and seven. The main outcome of all that instructional time was that given two equivalent fractions, students could cross-multiply in order to find a missing term.

“What they’re learning is: The way you find the fourth number is by setting up this gadget called a proportion,” Daro said. “That’s not really learning anything about proportionality, that’s learning how to get answers to problems in this chapter.”

Common Core math doesn’t mention cross-multiplying, and it cuts out the special case of finding a missing fourth term. Instead, it focuses on the idea of a ratio, which begins modestly in sixth grade and develops all the way through calculus. Students begin by looking at a table of equivalent ratios — also presented as a double number line — and progress to the understanding that the slope of a line is a ratio.

“[The Common Core writers] said, look, let’s figure out what’s important about fractions and choose a path through them, which leads to ratio and proportion, which leads to linear functions, which leads to aspects of algebra,” said Alan Schoenfeld, a professor of education and mathematics at the University of California, Berkeley.

The understanding of slope as a ratio feeds into an even more fundamental emphasis in Common Core math: the analysis of functions. By thinking about the slope of a line as a ratio, students get in the habit of analyzing the parts of a linear function so they can see how changes in elements of the function affect the relationship between inputs and outputs.

Daro sees this shift from solving equations to analyzing functions as one of the biggest conceptual changes in the Common Core.

“The important line of progress is the line that begins with the theory of equations, a 19th-century central focus, to calculus and analysis, which is 20th-century [mathematics],” he said. “It’s a move from spending almost all your time solving equations towards analyzing functions.”

Stumbling Blocks

The change from solving equations to analyzing functions seems benign, but that has not stopped the Common Core from becoming a charged political issue. Currently 42 states plus the District of Columbia use the standards, with adoption motivated in part by financial incentives provided by the Obama administration’s Race to the Top initiative — a top-down tactic that has helped fuel blowback. There have been plenty of other complications, too, from parents complaining that they don’t know how to help their first-graders with their math homework, to concerns that the assessments that accompany the Common Core are too hard. As a result, even stalwart adopters are questioning whether the standards work. In December 2015, Governor Andrew Cuomo of New York announced that his state would undertake a “total reboot” of the Common Core math standards in the coming years.

The designers of NGSS, which came out three years after the Common Core without any kind of federal mandate, say they learned from the contentious rollout of the earlier standards. So far, 17 states plus the District of Columbia have adopted NGSS and 11 more states have implemented standards that are similar to varying degrees.

“The Common Core got people to sign on and implement standards before the standards were there, and I think that backfired,” Schweingruber said. “I feel like the intent of the standards is to improve what happens to kids in classrooms, and if that happens even before a state formally adopts, that’s fine with me.”

Still, NGSS has had its controversies. The document includes standards related to climate change and evolution, which has motivated opposition in conservative states. And, politics aside, the standards necessitate sweeping changes to the way science is taught.

Like Common Core math with its long-running development of core concepts, NGSS reframes science in terms of a small number of basic ideas that inform the scientific perspective. These include “structure and function,” “patterns,” “cause and effect,” “stability and change,” and “systems and systems models.”

“Even at a young age you’re going to have a workable knowledge of energy so you can apply it,” said Joseph Krajcik, a professor of science education at Michigan State and the lead author of the NGSS physical science standards. “At a third-grade level you might know that as something is moving, it has energy, and the faster it’s moving, the more it can do something. It’s a nascent idea of what energy is, and it builds across time.”

Steven Gross

Jason Zimba explaining the Common Core’s standard addition algorithm to teachers in Denver this past May.

This slow-building approach is at odds with some aspects of public education. It’s not uncommon for districts to require that each class period address a discrete objective, and teachers are expected to measure whether students learned it at the end of the period. The authors of Common Core math and NGSS don’t see their disciplines fitting into that structure.

“One insight we got is that there’s almost no mathematics worth learning that breaks into lesson-size pieces,” Daro said. “You have a three- or four-week sequence and treat it with coherence. It’s about systems and structures, not small facts and small methods. It’s about how it all works together.”

Schweingruber agrees. “Some of these ideas in science are hard to get quickly,” she said. “It took humans hundreds of years, so why would kids figure them out quickly?”

The same mismatch between the standards and the way public education is set up occurs in another major area: assessments. Because standardized tests often drive instruction, it’s hard to expect teachers to teach differently unless students are tested differently.

“Teachers are starting to make changes in their classrooms,” Schweingruber said, “but if they’re still looking out toward a large-scale test their kids will have to take that is completely contrary to what they’re doing in the classroom, that can be problematic.”

There is progress in that direction. Two recent initiatives, the Partnership for Assessment of Readiness for College and Careers and the Smarter Balanced Assessment Consortium, are developing standardized tests that incorporate a greater variety of question types, like constructed response questions in which students are asked to explain their reasoning, and technology-enhanced questions in which, for example, students manipulate a line on a graph to make it match a given algebraic function.

“You’re seeing a deeper push for conceptual understanding and the ability to apply mathematics, and assessments are on their way to becoming equipped to actually assess that,” said Robert Kaplinsky, a math teaching specialist and consultant in Southern California.

The New Science

On the first and third Thursdays of every month, science teachers from around the country gather for #NGSSchat, a Twitter conversation about how to implement the new science. Topics for discussion have included how to incorporate reading and writing into science instruction and how to use technological tools alongside the standards. The July chats focused on “storylining,” which is emerging as a popular technique for bringing the standards to life in the classroom.

In a storyline, a teacher begins by introducing students to a phenomenon that prompts questions that students will investigate over the course of about two months. The question needs to be related to science, but accessible enough to grab students right away, and broad enough that it can’t be answered by a Google search. One storyline asks students to explain the biology behind the death of the Georgia high school football player Zyrees Oliver in 2014 after he drank too much fluid during practice. Another storyline asks simply: How does a seed grow into a tree?

“The storyline needs to be complex enough that it’s not going to just be a one-day or several-day event,” said Tricia Shelton, a high school science teacher in Kentucky and co-organizer of the NGSS chats who has been active in the implementation of NGSS. “It’s a necessity that it forces students to make those connections between many pieces of science in a coherent way.”

With storyline science, there are correct explanations, but there’s no right answer. A teacher’s job becomes less about handing down facts and more about establishing a classroom environment in which students can gather evidence and formulate arguments, with nudges along the way. This is a significant change from the way teachers have traditionally understood their role in the classroom. During the July 7 chat, some participants doubted their ability to make the shift. “[Teachers] are woefully unprepared [for] engaging in inquiry driven lessons. Local [teachers’] collaboration essential,” one contributor tweeted.

This concern is not lost on the NGSS developers.

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“For some elementary teachers it will be like I’m doing science in a real way for the first time ever,” Schweingruber said. “For high school teachers I think one of the biggest shifts will be the emphasis on kids carrying out investigations and making decisions. That’s a real shift in your role as a teacher.”

Shelton thinks the instructional changes entailed by NGSS are too big to internalize in isolated chunks of professional development.

“Face-to-face learning is super essential, but you can’t get enough in one or two days,” she said. “You need some kind of sustained system to try things out in your own classroom and then a support network that you can go back to. Without that support I think it’s hard to make that big shift.”

Along with professional networks, teachers also need curricular materials that fit the NGSS approach — textbooks, assessments and lab equipment that are well-suited to the basic method of gathering evidence and building arguments. One classroom technique that has gained currency is the building and analysis of models — functions that tune an input with some number of parameters and produce an output that describes phenomena in the world. It’s sophisticated work more often performed by professional researchers than 10th-graders.

“The first time I constructed a model was in graduate school,” Krajcik said. “It’s very challenging to say to a kid: How would you explain how all the parts work together? That’s tough.”

Constructing models may be complicated, but it’s also a perfect way for students to learn how to bring together multiple forms of evidence in the service of a larger scientific argument. The Concord Consortium, an educational research organization based in Massachusetts, is currently working with Krajcik’s group at Michigan State to create a tool called SageModeler that, in its simplest form, lets students drag and drop icons to create conceptual models to explain real-world events.

“The SageModeler tool allows [students] to construct a representation of some phenomenon and test it out,” said Dan Damelin, co-creator of SageModeler. “They can see what are the results of my setting up this model of how I think things work.”

The first unit for the software, which will be pilot-tested in the spring, follows the storyline-style question: “Why Do Fishermen Need Forests?” It allows middle school students to investigate the causes and consequences of ocean acidification.

Prior to building an ocean acidification model, students will read about topics like deforestation, receive some direct instruction about the distinction between acids and bases, and carry out experiments that will give them a tangible sense of the factors involved. These could include exhaling into a jar of water containing a pH indicator (and observing that, as the water absorbs carbon dioxide, its pH declines) or conducting experiments to understand the role of photosynthesis in carbon sequestration.

Once the students have a feel for the factors contributing to ocean acidification, they’ll start to construct their models by pulling images from a clip art database to represent the variables they want to include: a car to represent carbon dioxide emissions, trees to represent carbon-dioxide-absorbing plants, shellfish to represent shellfish health, a fishing boat to represent the fishing economy. After students have defined relationships between the variables, they’ll run the model, graph the resulting data, and then refine their work to better approximate real-world data — in this case, data from the marine research center Station Aloha in Hawaii that can be dragged into SageModeler for a side-by-side comparison.

Teaching in this fashion can be exciting, but it will take sustained commitment for these techniques to ripple through the 100,000 or so public schools in the United States. In order for the new science and math standards to succeed, the entire education ecosystem will need to pull in that direction, from writers of standards to textbook publishers to professors in education schools to curriculum leaders running professional development sessions, to teachers swapping lesson ideas online. Just as the core concepts in math and science require repeated encounters over many years to be fully absorbed, a new practice of math and science teaching will need time to become established.

“I hope we give it the time,” Schweingruber said. “One problem in education reform is, people have unrealistic expectations about how quickly you change it. If you know it’s a huge ship, you have to give it some time before you decide it’s not working.”

Correction: This article was revised on October 5, 2016, to reflect that while Heidi Schweingruber co-directed a study that led to the framework that the Next Generation Science Standards are based on, she was not co-director of NGSS itself.

This article was reprinted on Wired.com.

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  • As a parent of a 6th grader who has been taught common core math since 1st grade, I would like to say that it took many years to come to appreciate this "new" math. Now that he is in 6th grade, the things he is learning are astonishing me – I'm quite certain he has more mathematical knowledge already than I graduated high school with. (Though the article is right, it seems his 1st grade homework was the hardest to help him with – it was such a shift from how we were taught the basics as kids!)

  • <blockquote>“When are we ever going to use this?” </blockquote>

    In short, in from a little to a lot in nearly all of your life.

    Why? Now pure/applied math, science, engineering, and technology are very important for nearly everyone. The importance has risen quickly and enormously since year 1900 and is rising quickly now.

    But pure/applied math, science, engineering, and technology are all heavily dependent on math including Algebra 2.

    Here are some examples of where material in Algebra 2 get used:

    (1) In Algebra 2 you will learn more about manipulations of algebraic expressions. Well, you will need to do a lot of that in trigonometry, analytic geometry, calculus, linear algebra, differential equations, and more, and you will need all of those subjects for a huge fraction of science, engineering, and technology.

    (2) You will learn about the binomial theorem and the binomial coefficients, and those play big roles in probability, statistics, and computer science. For this last, look at the nice, advanced work with the binomial coefficients in D. Knuth's <i>The Art of Computer Programming.</i> There Knuth used the binomial coefficients to analyze how fast computer algorithms run.

    (3) You will learn about complex numbers, and they are crucial in the fundamental theorem of algebra about roots of polynomials, differential equations, AC circuit theory in electrical, electric power, electronic engineering, signal processing for radar, sonar, and much more, linear algebra, and quantum mechanics.

    For example, for wired Internet access, why do we use optical fibers or coaxial cables instead of just the old copper wire used for voice telephone? The answer is some basic work in Maxwell's equations in electricity and magnetism and in terms of complex numbers.

    In summary, if your career is going to be close to or in pure/applied math, science, engineering, and technology, definitely you will need to know Algebra 2 plus much more in math. If you are unsure about your career, and nearly no one in K-12 can be sure, knowing Algebra 2 will let you have your options more open. No matter what you do, Algebra 2 will help you be a better informed citizen. E.g., as a parent, will may have to consider this question for your children.

  • Encouraging indeed that this is being taken so seriously. Notwithstanding the benefits of math—not just for future scientists, inventors, engineers of every kind, software developers, physicists, doctors, astronomers, statisticians, etc etc etc, but for anyone who lives above mud-hut level—there's the question of fragile democracy, so easily thrown away by voters who don't think rationally and the political lice who adore exploiting those same people. Better math and science education breeds rationality and respect for evidence—that old fashioned concept called Truth. It's good for us, our societies and our species.

    That said, no initiative will work without decent resources. In the UK some counties are even considering a four-day school week because the government won't fund education properly. I think the US has some serious resource problems too. Well educated kids are a better long-term guarantor of a nation's survival than aircraft carriers.

    Furthermore, any tightly prescriptive curriculum becomes damaging in the long run, because it takes away good teachers' single greatest power—the power to explain as they see fit. I'm guessing that Quanta's scientifically literate readership have all had the experience of trying to learn something tricky and, maybe after despairing, found that a good lecturer would say, "Well, look at it this way instead …" and for the light to go on. It's why you can get stuck in one textbook and be rescued by another, because the same thing is explained in a different way. It's a teacher's golden skill.

    By all means find great ways to teach math. But let good teachers, paid well, with the time to explain to small class sizes, implement your policy.

  • Crazy idea: At every class there are some students who are ahead on English but behind in Math etc. How about allowing students to be at different grades for different classes.
    For example imagine a student is at grade 3 for English, at grade 2 for Science, Grade 1 for Math. That student would finish English first and would need study only Science and Math later. Afterwards would finish Science and would need to study only Math afterwards. This would allow student to focus on hardest classes in the end w/o worrying about other classes.

  • @Frank

    Not a bad idea, but it would have to be handled very carefully. For one, having a class with many different ages of children makes managing the classroom harder, especially at younger ages where even a difference of one year results in large behavioral changes. You also have to worry about older kids being embarrassed about being assigned to a classroom with mostly younger kids–especially if they still struggle at that level. Keeping kids of similar ages in the same classroom would require many more teachers, which a lot of schools can't afford.

    I don't think the sequencing works out at the end of a student's school career. Would your hypothetical student be taking Math 6 and 7 simultaneously after finishing English and Science?

    The separating of students by skill does happen in high schools (standard, college-prep, honors, Advanced Placement, etc.), so it's not so crazy an idea.

  • If you ever have kids, then yes you will need this.

    As an engineer with a Masters in Physics, and I'm old enough to have suffered under the original "new math" I find the "new new math" puzzling. Like "why"?

    Recently the oldest (16) is in Pre-Calc. Where they seem to be spending a lot of time on algebraic manipulation of rates. Something I can do in seconds with a quick differential. Takes hours constructing rates problems. I don't speak or think that language. Example: A tub with 2 nozzles, with both nozzles on fills in 55 minutes. If the larger nozzle is off, the smaller nozzle can fill the tube in 120 minutes. What time would be needed for just the larger nozzle to fill the tub.

    Having said that, it's amazing to watch them on their TI graphing calculators plot solutions out and zoom in on the intercepts. Having worked at the company that made the Lunar Lander Flight Computer (before my time but it was in the display case in the lobby) the march of tech is amazing.

  • I wonder how this will work overall. When I first saw common core ideas getting expressed, I thought they were a little silly. Then someone showed an example that many people poo pooed. The example I see on the board in the picture above is similar.

    The example made me say "Oh look someone thinks they can make people learn faster by teaching them relativity first!" Relativity is awesome. 0 is awesome. It might be easier, but I am not sure that it benefits us overall. People with skills in these areas sit down to help the kids do their homework and discover that strange methods are being used. Instead of leveraging what we have, we clip it. There are continuous generations of children who are taught by teachers who have been taught to teach math and not by people who like math.

    The people who like math have taken other jobs.

  • using terms like 'new' vs 'old' doesn't seem to get to the substance of the question – which I think is how can you get say, understanding of science, to emerge in a human mind, regardless of age – i took undergraduate science research as an elective in college because i didn't want to wait until i got to grad school to see if it worked for me. i was also aware that simply regurgitating 'science' out of textbooks really missed the mark of how to think about science, how to identify an interesting problem and then work on it to learn something new (and could i do that). even realizing that, the research was challenging in many ways. one take away that might be useful for nurturing true understanding from one mind to another might be to consider the 'finding forrester' approach: take a microcosm of work, give to a budding learner and have them start by reproducing the work, allowing for the instinctual jump to happen in their mind, when they take it as their own and find new discoveries and ways to do it

  • I have severe doubts as to the effectiveness of the Common Core pedagogy, based upon my own learning experiences as a child. I was not and have never been a "memorizer;" in fact, when I was Physics student at MIT, my antipathy to memorization was a benefit, because I always understood the physics down to an operative intuitive level, rather than depending upon memory of solved problems. Many valedictorians were "memorizers," and generally did not make it past their second year at MIT. That being said, however, it is important to understand when memorization IS essential to understanding. Memorization of childhood multiplication tables is essential, for example, because solving real problems in real life involves knowing (by estimation) when one has arrived at the WRONG answer. Such estimations very, very often require nothing more than elementary multiplication or division. My experience with most high school graduates, today, is that they have absolutely no ability to estimate real life results involving even trivial arithmetic. And that is my problem with Common Core, and its excessive allocation of student time to highly-suspect "reasoning processes," at the expense of "memorized by exercise" facility with basic mathematical and algebraic operations.

    The Common Core pedagogy might help the child that is somewhat gifted arrive at a high level more quickly, but it will be little help, I suspect, to many more children that are struggling. The idea that all children are programmed internally to approach mathematics in the same way is a foolish one. Some will be much better served, with much less angst, by simple memorization of multiplication tables and facility at very basic algebraic operations, and may well go on – as their cognition develops – to integrate that comfortable and unfearful basis into more advanced mathematics.

    There seems to be this idea that cognitive development in children proceeds along a simple, well-understood trajectory. That is an adult conceit, and does disservice to the remarkable adaptivity of a child's developing mind, and its tendency to ingrate information and extend its learning perimeter in disconnected leaps, with delayed consolidation.

  • While state adoption of NGSS is low, as suggested by the map, I think there is hope that it will pick up steam. Here in Pitt County Schools in North Carolina, they are going to begin implementing NGSS at the district level because science curriculum coordinator has been pushing hard for them, and he seems to have others on the staff very excited about moving instruction in the district in a new direction. I would hope that similar things might or may already be happening in areas where the state has not yet taken the initiative.

  • I'm extremely puzzled by the constant desire to fiddling with "pedagogy". First of all, most of educational 'research' are of questionable quality. Who really knows what is the cause and effect? Secondly, the root cause of America's education problem cannot be solved by 'methodology' change. Plenty of kids from the higher social-economical class do just fine. Lastly, the idea of teaching "thinking" is well and good, but without mastering the basics it is all for nothing. It is like teaching tennis by teaching how to think like a tennis player. Yes it is more fun to the students compared with hours of 'rote' practice. Ultimately they may perform well in a multiple-choice exam (which is designed according to the pedagogy. Herein lies the dilemma. The people who design the methods should not be the same people who design the tests) but they cannot play tennis at all.

  • I'm troubled that any single minded approach to math and science to the exclusion of all others is a mistake. In addition, not every child is equipped intellectually to this approach. I'm particularly concerned that the value of an algorithm or useful heuristic seems to be a casualty in this shift in philosophy. Neuroscience seems to be telling us we think in narratives, and reason by connecting and verifying those narratives. If so, this approach to education has some gaping questions to be answered.

  • I find it hard to reconcile the idea of "New Math" while we are still teaching Euclidean Geometry. It's time to abandon Euclid, fix the sequence of math courses, and add Discrete Math as a high school course.

  • The reason these new pedagogical methods are met with such resistance is that there is an implicit assumption that they are correct without the presentation of any proof of real world results. Can these theories not be tested on sample populations to prove their effectiveness and worth before they are adopted statewide? How about using the scientific method? The theories are intuitively attractive to me but there must be a better way to prove them out without disrupting the lives of millions of students, teachers and parents.

  • I really dislike the common core math. I have seen students cry and parents despair. And I love math. It was taught wrong in your first example both ways. If you want to teach someone, you have to start where they are, not where the book says they should be.

  • This is complete rubbish. How did any innovation ever happen…in mathematics, science, the arts or any other field…if the prior intellectual framework was such an inhibiting factor. The educational structure must not only express the truth of the discipline but also correspond to the stage of development of the mind to which it is being taught. Only a small percentage of the world thinks like a scientist or mathematician or artist or whatever. And an even rarified group thinks like all of them…as you are asking a fourth grader to do. These are the geniuses among us. To use that as the standard is an absurdity beyond belief. This way of thinking has been applied at the school my daughter attends. It started 2 years ago. Now, I am receiving notes from the teachers stating the difficulties the 4th graders are having with basic arithmetic. Maybe she will be a scientist. Maybe she will be a mathematician. Maybe an artist. I don't know. I would be happy with any of these. If she does pursue one of these disciplines it will be proof that her manner of thought was so inclined. That said, I know that she will need to pay for things at the grocery store and work out sale deductions at the department store. That is certain of everyone. And her elevated scientific thinking as taught by unscuentific educators has made even that problematic. Thanks!

  • I love Maths and Science – just because I LOVE them. Does one have to always have a reason for liking something or doing something – I think, one can just LOVE something for its own sake!

  • Unfortunately this sounds like exactly what some of the critics seam to fear. A bunch of intellectuals using kids to experiment with their interesting new ideas. Of course, if the experiment fails, those kids don't get another chance. The experimenters can try something else on a different set of kids.

    "If you know it’s a huge ship, you have to give it some time before you decide it’s not working.”"

    Actually no. It means you can only make very small, incremental changes in the ships course. It also means you have to be very certain of your course before you start out because if you are too far off you will never get to your destination.

    Of course light speedboats are more interesting to drive.

  • Cross multiplying always works; no matter how big the problem gets. Try using the new math with large problems. Those old, dead, Greeks knew what they were doing.

    I used the tangent function to build a hexagon roof on the pump-house. I never thought I would use trig, from high school, on the farm, 30 years later….. lol….

  • As a former high school Mathematics and Physics teacher I have a problem with the example given. It is assumed that one way or the other is the way to teach proportionality whereas it is a combination of the two methods that would truly be best. The "new" method teaches the concept whereas the "old" method gives a quick and dirty way of actually finding a new ratio. The "new" method falls down if the denominator is not a multiple of 7 but the "old" method allows for any denominator. I have been a teacher for over 34 years and have seen a myriad of curricula come and go but have always had the same question: How did the NASA engineers get us to the moon with old, outdated math and science teaching in their backgrounds? Modern education theory seems to be based on the assumption that we have always been doing it wrong but we have accomplished some pretty awesome things in the meantime.

  • It would be interesting to see what the common core looked like if it was teacher generated consolidation of the actual teaching practices around the country. The emphasis would have been developing a "common" core instead of an opportunity for a few scholars to put their newest theories into practice.

  • I could not agree more with Ross Williams comment. "Experts" and top down have always been suspect of the actual practitioners. What would the standards look like if you had to have actual classroom practice from K-12 informing the document both in substance and examples?

  • As a parent of 4 kids, I was annoyed by their teachers' active disapproval of flash cards for memorizing basic math facts (thru multiplication). There was no expectation that accuracy or timed tests were at all useful. In response, I bought an Apple IIc and the floppy program MATH BLASTER (with programmable lists). Every weekday after school my kids played Math Blaster, working their way thru progressively programmed sets of problems. Their practice time was limited to 15 minutes per day. They did not consider this an imposition, or intellectually degrading. For their entire public school lives they remained at the top of their class(es) in Math. This MATH BLASTER experience in memorization and instant recall was carried on only thru the Multiplication Facts, but their early success, and comfort with math led them to achieve almost perfect scores on their SATs. Studying Math is like studying Ballet, or piano: the beginner focuses on the basic skills and becomes "competent", and only after that can we expect him/her to develop a sophisticated ability to analyze and verbally explain diverse approaches to solving a particular problem.

  • Thanks for the very informative article. I am an elementary teacher and feel much more confident when teaching Language Arts and Social Studies rather than math and science. This article helped my put the pieces together a bit better to see the whole picture of what we as educators are trying to accomplish in the fields of science and math. When reading about the inquiry piece of science I was reminded of the Project Based Learning I do in Social Studies lessons. It gave me some ideas of some directions to proceed in during Science.

  • Math learning requires two skills:
    1. memorization
    2. pattern matching

    And nothing else. Thinking is not involved unless you are working on something new, some new discovery.

    The old math emphasized memorization. The new math emphasized pattern matching. This new new math re-emphasizes the pattern matching.

  • Modeling Instruction is aligned with the Next Generation Science Standards. Modeling Instruction is used by high school and middle school science teachers nationwide. More than 60 Modeling Workshops are held each summer, and most are two or three weeks long (80 to 90 contact hours).
    Modeling Workshops thoroughly address most aspects of science teaching, including integration of teaching methods with course content. Workshops incorporate up-to-date results of physics and science education research, best curriculum materials, use of technology, and experience in collaborative learning and guidance. Workshops focus on all 8 science practices and cross-cutting concepts of the NRC Framework for K-12 Science Education (2012).
    Participants are introduced to Modeling Instruction as a systematic approach to design of curriculum and instruction. The name Modeling Instruction expresses an emphasis on making and using conceptual models of phenomena in science as central to learning science. Math instruction is integrated seamlessly in each course by an emphasis on mathematical modeling.
    In each workshop, content for an entire semester course is reorganized around models to increase its structural coherence. Participants are supplied with a complete set of course materials and work through activities alternately in roles of student or teacher. Teachers use computers as scientific tools to collect, organize, analyze, visualize, and model real data.
    Modeling Workshops are listed at http://modelinginstruction.org/ under the “teachers” tab. The American Modeling Teachers Association (AMTA) oversees them: the AMTA is a grassroots professional organization of, by, and for teachers who use Modeling Instruction.

  • The key is to let a child pursue his/her natural curiosity. When my son was in first grade, he brought home a sack of "number rods". This was a visual way to experience addition — the red and blue rods, placed end to end, were as long as the yellow rod. To help him associate the rod lengths with numbers, I took a large sheet of paper, drew a line on it, and marked off the nonnegative integers on the line to match the rod lengths. I mentioned to my son that every place on that line had a name. Some had easy names, like 1 or 2, while others had really fancy names. Then I stood back and let him play for a while. Although it was helpful to associate numbers with the rods, what I really wanted to do was see where my son's curiosity led him.
    After a few minutes of experimenting with rods and numbers on the line, my son took a (conveniently placed) pencil, put a dot roughly halfway between 0 and 1 on the line, and asked me, "What do you call this?". "I'm glad you asked that question," I replied, as I grabbed a cookie and broke it in half. Several cookies and a large pile of crumbs later, my son had an intuitive grasp of what fractions were, so I stood back and let him play around for a while.
    A few minutes later, I saw this sly look cross my son's face. He took the pencil, placed a dot to the left of zero (which I just happened to place in the middle of the line I had drawn), and asked, "What do you call this?". I'm sure he thought he had stumped me. "I'm glad you asked that question," I replied, as I added negative integers to the line. We talked about "adding backwards" by reversing the "direction" of the number rods. After a few minutes, my son had an intuitive grasp of negative numbers, so I stood back and let him play around for a while.
    Several minutes later, my son got That Look on his face again. He grabbed the pencil, put a dot OFF THE LINE, turned to me with a big grin on his face, and said, "Okay, Dad, what do you call THIS??". "I'm glad you asked that question," I replied as I added a Y axis to the sheet of paper. We spent the next ten minutes or so talking about 2D coordinate systems and how we could use the "number rod direction" idea from earlier to add two directed line segments (vectors, although I didn't refer to them as such) together to get a third one. Although I would have been astounded had my son imagined a point above the paper, I would have replied with "I'm glad you asked that question," grabbed a ruler, stood it on its end, and added a Z axis.
    If children are allowed to follow their natural curiosity, (1) they will come up with some pretty sophisticated ideas, and (2) they will more readily grasp the underlying concepts because the ideas were "theirs" in the first place.

    My granddaughter just started first grade. I can't wait until she brings home a sack of number rods.

  • After 1st grade and after much stress associated with the ghastly Common Core, we placed our son in a private, non Federally funded, non Common Core private school. He is now thriving (and has been ADVANCED one grade in math) and we do not have to concern ourselves with the massive data collection that feeds the beast that is Common Core. As parents, you must recognize CC for what it truly is, child abuse.

  • This article champions a style of teaching that requires high level students with IQ above 100 , have above average resources, and are in classrooms that are uniform with others of equal attributes and behavior. This is simply not possible either legally or pragmatically. The great error of 'big thinkers' in education is the ironic bias that forgets that half of children have IQ under 100 and that many children have learning issues or behavior issues . Using a 'discovery based' teaching method fails to consider that public school is not graduate school in population or goals. The failures of science and math in schools today is not the failure to create tomorrow's PHDs, the failure is to create a population that can make change from a dollar or who know the Earth circles the sun. Public school education has to be based on teaching the basics in a simple and straightforward manner that creates a population with basic abilities. Before people roar that I underestimate students, I challenge you to go to any public high school, and ask seniors in the hallway at random to make change , figure a fraction as one needs to do in carpentry or ask them the most basic questions of science. That will ground you in education reality.

  • Where is the advance testing of these innovations taking place? Why must the entire student population of the nation become a laboratory for trying out new concepts in teaching? Is there not some way of trying out these methods on real students before the whole experiment goes nationwide? Would a new software system or manufacturing technology be rolled out before thorough testing? How many more generations of students must be guinea pigs for systemic failures in education? The testing of these things on small representative samples of real children should be an area of intense focus and development before any more educational innovations become widely adopted.

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