The Method Of Common Bases Common Core Algebra 2 Homeworks

If we could snap our fingers and change the way math and science are taught in US 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

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,” saidRobert 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 US 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 US 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.”

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.

“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.”

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

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Original story reprinted with permission from Quanta Magazine, an editorially independent division of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences

Math can’t catch a break. These days, people on both ends of the political spectrum are lining up to deride the Common Core standards, a set of guidelines for K-12 education in reading and mathematics. The Common Core standards outline what a student should know and be able to do at the end of each grade. States don’t have to adopt the standards, although many did in an effort to receive funds from President Obama’s Race to the Top initiative.

Conservatives oppose the guidelines because they generally dislike any suggestion that the federal government might have a role to play in public education at the state and local level; these standards, then, are perceived as a threat to local control.

Liberals, mostly via teachers’ unions, decry the use of the standards and the associated assessments to evaluate classroom instructors.

And parents of all persuasions are panicked by their sudden inability to help their children with their homework. Even comedian Louis CK got in on the discussion (via Twitter; he has since deactivated his account).

My kids used to love math. Now it makes them cry. Thanks standardized testing and common core! — Louis CK (@louisck) April 28 2014

In the middle are millions of American schoolchildren who are often confused and frustrated by these “new” ways of teaching mathematics.

Thing is, we’ve been down this path before.

The old New Math

When the Soviets launched Sputnik in 1957, the United States went into panic mode. Our schools needed to emphasize math and science so that we wouldn’t fall behind the Soviet Union and its allegedly superior scientists. In 1958, President Eisenhower signed the National Defense Education Act, which poured money into the American education system at all levels.

One result of this was the so-called New Math, which focused more on conceptual understanding of mathematics over rote memorization of arithmetic. Set theory took a central role, forcing students to think of numbers as sets of objects rather than abstract symbols to be manipulated. This is actually how numbers are constructed logically in an advanced undergraduate mathematics course on real analysis, but it may not necessarily be the best way to communicate ideas like addition to schoolchildren. Arithmetic using number bases other than 10 also entered the scene. This was famously spoofed by Tom Lehrer in his song “New Math.”

I attended elementary school in the 1970s, so I missed New Math’s implementation, and it was largely gone by the time I got started. But the way Lehrer tries to explain how subtraction “used to be done” made no sense to me at first (I did figure it out after a minute). In fact, the New Math method he ridicules is how children of my generation – and many of the Common Core-protesting parents of today – learned to do it, even if some of us don’t really understand what the whole borrowing thing is conceptually. Clearly some of the New Math ideas took root, and math education is better for it. For example, given the ubiquity of computers in modern life, it’s useful for today’s students to learn to do binary arithmetic – adding and subtracting numbers in base 2 just as a computer does.

The New Math fell into disfavor mostly because of complaints from parents and teachers. Parents were unhappy because they couldn’t understand their children’s homework. Teachers objected because they were often unprepared to instruct their students in the new methods. In short, it was the implementation of these new concepts that led to the failure, more than the curriculum itself.

Those who ignore history…

In 1983, President Reagan’s National Commission on Excellence in Education released its report, A Nation at Risk, which asserted that American schools were “failing” and suggested various measures to right the ship. Since then, American schoolchildren and their teachers have been bombarded with various reform initiatives, privatization efforts have been launched and charter schools established.

Whether or not the nation’s public schools are actually failing is a matter of serious debate; indeed, many of the claims made in A Nation at Risk were debunked by statisticians at Sandia National Laboratories a few years after the report’s release. But the general notion that our public schools are “bad” persists, especially among politicians and business groups.

Enter Common Core. Launched in 2009 by a consortium of states, the idea sounds reasonable enough – public school learning objectives should be more uniform nationally. That is, what students learn in math or reading at each grade level should not vary state by state. That way, colleges and employers will know what high school graduates have been taught, and it will be easier to compare students from across the country.

The guidelines are just that. There is no set curriculum attached to them; they are merely a list of concepts that students should be expected to master at each grade level. For example, here are the standards in Grade 3 for Number and Operations in Base Ten:

  • Use place value understanding and properties of operations to perform multi-digit arithmetic.

  • CCSS.Math.Content.3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

  • CCSS.Math.Content.3.NBT.A.2 Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

  • CCSS.Math.Content.3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (eg, 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

There is a footnote that “a range of algorithms may be used” to help students complete these tasks. In other words, teachers can explain various methods to actually accomplish the mathematical task at hand. There is nothing controversial about these topics, and indeed it’s not controversial that they’re things that students should be able to do at that age.

However, some of the new methods being taught for doing arithmetic have caused confusion for parents, causing them to take to social media in frustration. Take the 32 - 12 problem, for example:

Once again, it’s the implementation that’s causing the problem. Most parents (people age 30-45, mostly), remembering the math books of our youth filled with pages of exercises like this, immediately jump to the “Old Fashion” (sic) algorithm shown. The stuff at the bottom looks like gibberish, and given many adults’ tendency toward math phobia/anxiety, they immediately throw up their hands and claim this is nonsense.

Except that it isn’t. In fact, we all do arithmetic like this in our heads all the time. Say you are buying a scone at a bakery for breakfast and the total price is US$2.60. You hand the cashier a $10 bill. How much change do you get? Now, you do not perform the standard algorithm in your head. You first note that you’d need another 40 cents to get to the next dollar, making $3, and then you’d need $7 to get up to $10, so your change is $7.40. That’s all that’s going on at the bottom of the page in the picture above. Your children can’t explain this to you because they don’t know that you weren’t taught this explicitly, and your child’s teacher can’t send home a primer for you either.

Better intuition about math, better problem-solving

As an instructor of college-level mathematics, I view this focus on conceptual understanding and multiple strategies for solving problems as a welcome change. Doing things this way can help build intuition about the size of answers and help with estimation. College students can compute answers to homework problems to 10 decimal places, but ask them to ballpark something without a calculator and I get blank stares. Ditto for conceptual understanding – for instance, students can evaluate integrals with relative ease, but building one as a limit of Riemann sums to solve an actual problem is often beyond their reach.

This is frustrating because I know that my colleagues and I focus on these notions when we introduce these topics, but they fade quickly from students’ knowledge base as they shift their attention to solving problems for exams. And, to be fair, since the K-12 math curriculum is chopped up into discrete chunks of individual topics for ease of standardized testing assessment, it’s often difficult for students to develop the problem-solving abilities they need for success in higher-level math, science and engineering work. Emphasizing more conceptual understanding at an early age will hopefully lead to better problem-solving skills later. At least that’s the rationale behind the standards.

Alas, Common Core is in danger of being abandoned. Some states have already dropped the standards (Indiana and South Carolina, for example), looking to replace them with something else. But these actions are largely a result of mistaken conflations: that the standards represent a federal imposition of curriculum on local schools, that the standardized tests used to evaluate students are the Common Core rather than a separate initiative.

As the 2016 presidential campaign heats up, support for the Common Core has become a political liability, possibly killing it before it really has a chance. That would be a shame. The standards themselves are fine, and before we throw the baby out with the bathwater, perhaps we should consider efforts to implement them properly. To give the Common Core a fair shot, we need appropriate professional development for teachers and a more phased introduction of new standardized testing attached to the standards.

But, if we do ultimately give in to panic and misinformation, let’s hope any replacement provides proper coherence and rigor. Above all, our children should develop solid mathematical skills that will help them see the beauty and utility of this wonderful subject.

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