7-12 Math Common Core Resources

Fordham’s review does not unequivocally say the standards are higher, either. They may be higher than some state standards but they are certainly lower than the best of them

Nor are the Common Core standards necessarily clearer.

Andrew Porter, dean of the University of Pennsylvania’s Graduate School of Education

conclusion was stark: Those who hope that the Common Core standards represent greater focus for U.S. education will be disappointed by our answers. Only one of our criteria for measuring focus found that the Common Core standards are more focused than current state standards…Some state standards are much more focused and some much less focused than is the Common Core, and this is true for both subjects. We also used international benchmarking to judge the quality of the Common Core standards, and the results are surprising both for mathematics and for [ELA].… High-performing countries’ emphasis on “perform procedures” runs counter to the widespread call in the United States for a greater emphasis on higher-order cognitive demand.

with only somewhat less redundancy in the middle grades

There is much to criticize about them, and there are several sets of standards, including those in California, the District of Columbia, Florida, Indiana, and Washington, that are clearly better.

Where this gap is most obvious, and most important, is in laying the foundation for college readiness in mathematics early, by grade 6 or 7. Judging by state standards, few people see a connection between elementary school mathematics and college math, let alone really understand how the foundation is built.

et Common Core is vastly superior—not just a little bit better, but vastly superior—to the standards in more than 30 states.

the standards don’t rank in terms of quality in the middle 20 percent of state standards, but, instead, fall in the top 20 percent.

Fewer than 15 states are explicit about the need for students to know the single-digit number facts (think multiplication tables) to the point of instant recall. States love to have kids figure out many ways to add, subtract, multiply, and divide, but often leave off the capstone standard of fluency with the standard algorithms (traditional step-by-step procedures for the addition, subtraction, multiplication, and division of whole numbers).

only 15 states mention common denominators. Common Core does a pretty good job with arithmetic, even a very good job with fractions.

do the math standards resemble those recommended by the National Council of Teachers of Mathematics (NCTM)

There will always be people who think that calculators work just fine and there is no need to teach much arithmetic, thus making career decisions for 4th graders that the students should make for themselves in college. Downplaying the development of pencil and paper number sense might work for future shoppers, but doesn’t work for students headed for Science, Technology, Engineering, and Mathematics (STEM) fields.

There will always be the anti-memorization crowd who think that learning the multiplication facts to the point of instant recall is bad for a student, perhaps believing that it means students can no longer understand them. Of course this permanently slows students down, plus it requires students to think about 3rd-grade mathematics when they are trying to solve a college-level problem.

There will always be the standard algorithm deniers

Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared. Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older.

You learn Mathematical Practices just like the name implies; you practice mathematics with content.

At present, it seems that the majority of people in power think the three pages of Mathematical Practices in Common Core, which they sometimes think is the “real” mathematics, are more important than the 75 pages of content standards, which they sometimes refer to as the “rote” mathematics

NCTM followed shortly with its 2006 Curriculum Focal Points, a document that finally focused on what mathematics is all about: mathematics. Since then, NCTM seems to have regressed, as evidenced by its 2009 publication Focus in High School Mathematics, a document that is full of high-minded prose yet contains little rigor or specificity.

The Common Core mathematics standards are grade-by-grade‒specific and hence are more detailed than the NCTM 2000 standards, but they do resemble them in setting their sights lower than our international competitors, by, for example, locking algebra into the high school curriculum.

And they contain inexplicable holes even when compared to the much shorter NCTM Curriculum Focal Points, the major one being the absence of fraction conversion among their multiple representations (simple, decimal, percent). Other puzzling omissions include geometry basics such as derivation of area of general triangles or the concept of pi. One can argue those can be inferred, but the same can be said regarding all those state standards we acknowledge as “bad”—that all those missing pieces “can be inferred.”

How do the Common Core math standards compare to those in use in the world’s highest-performing nations?

the Common Core standards are not on par with those of the highest-performing nations.

Professor R. James Milgram of Stanford, the only professional mathematician on the Common Core Validation Committee, wrote when he declined to sign off on the Common Core standards: This is where the problem with these standards is most marked. While the difference between these standards and those of the top states at the end of eighth grade is perhaps somewhat more than one year, the difference is more like two years when compared to the expectations of the high achieving countries—particularly most of the nations of East Asia.

Professor William McCallum, one of the three main writers of the Common Core mathematics standards, speaking at the annual conference of mathematics societies in 2010, said, While acknowledging the concerns about front-loading demands in early grades, [McCallum] said that the overall standards would not be too high, certainly not in comparison [with] other nations, including East Asia, where math education excels.

Jonathan Goodman, a professor of mathematics at the Courant Institute at New York University,

“The proposed Common Core standard is similar in earlier grades but has significantly lower expectations with respect to algebra and geometry than the published standards of other countries.”

The enrollment requirements of four-year state colleges overwhelmingly consist of at least three years of high school mathematics including algebra 1, algebra 2, and geometry, or beyond. Yet Common Core’s “college readiness” definition omits content typically considered part of algebra 2 (and geometry), such as complex numbers, vectors, trigonometry, polynomial identities, the Binomial Theorem, logarithms, logarithmic and exponential functions, composite and inverse functions, matrices, ellipses and hyperbolae, and a few more.

What should we make, then, of a recent study purporting to “validate” that Common Core standards indeed reflect college readiness?

Look at California’s standards for example. They are great standards and have been unchanged for over a decade, but many in math education hate them. They think they are all about rote mathematics, but I think such people have little understanding of mathematics.

We, in this country, are still not on the same page about what content is most important, even if everyone says they’ll take Common Core. Without a unified, concerted effort to teach real mathematics, there isn’t much chance of catching up.

In other countries, if you say “learn to multiply whole numbers,” no one questions how this should be done; students should learn and understand the standard algorithm. In the U.S., even if you say “learn to multiply whole numbers with the standard algorithm,” some people will declare wiggle room and try to avoid the standard algorithm.

What, then, are your main areas of disagreement?

Ze’ev refers to Andrew Porter’s work to support his argument that Common Core lacks focus.

he says that 39.55 percent of grades 3‒6 coarse-grained topics for the states are on Number Sense and Operations, but Common Core gets 55.47 percent. To me, that says that Common Core focuses on arithmetic in grades where arithmetic should be the focus, and that the states did not focus on arithmetic.

If Common Core is mediocre, then mediocre is being set at a high standard. There are many states that set a very different, and much lower, standard for mediocre.

I would take these interview comments with a grain of salt. Everyone is an expert.

I can tell you that Ze’ev had not taught and I don’t think has spent any amount of time in the classroom. I served on a committee with Ze’ev evaluating questions for the California Standards Test.

Ze’ev is correct. I thought this long ago. It’s too vague and there is too much wiggle room. The wiggling will be in the downward direction. In fact, they don’t have to wiggle very much. Everyday Math will add a few more units and Math Boxes about standard algorithms, and then they will continue to trust the spiral.

BY FAR the majority of the population did not “get” math when it was taught using the methods and approaches these pompous mathematicians propose. Like so many uninformed “experts” they think that if we just teach math the way they learned it every things will be smooth sailing. But we taught math their way for a very, very long time and we failed. And that’s when the world hd very little technology, far less problems to solve, and agriculture and manufacturing ruled the world. But the world has changed fellas. And we now have scientific research that debunks the didactic, direct, one-way approach to learning math. For one thing we’ve learned that the brain doesn’t learn for the long term the way they propose. Their methods work to pass tests in the short run, but do little to instill knowledge retention and application of the mathematics in solving real problems. If their approaches to learning math worked, we wouldn’t have a very large segment of the adult population, including a lot of elementary teachers, saying things like, I never got math, I hate math, math is too hard.

Thankfully, we’re finally moving toward an educational system that honors the mathematical practices on which the CCSS were developed.

Bottom line… We need to ensure that our students are getting a solid foundation at the early grades to ensure that they are able to engross themselves in deeper, more abstract problems in the future. This, I believe will be enhanced by the common core although I would agree that the standards themselves do not fix the issues.

Probability should not be assessed until grade 7, for instance, the document says, and statistical distributions should not be assessed by materials until grade 6.

In a sign that the new math document will be taken seriously, it has the endorsement of several prominent organizations in the education sphere, including the National Governors Association, the Council of Chief State School Officers, the Council of the Great City Schools, the National Association of State Boards of Education, and Achieve, a national nonprofit that managed the process to develop the common standards. Both the NGA and CCSSO spearheaded that undertaking.

In fact, a group of 20 big-city districts, led by the Council of the Great City Schools, served notice to publishers last month that any materials they purchase must reflect the priorities of the publishers' criteria.

Jason Zimba, a co-author of the document and one of the three lead writers of the math standards, said he anticipates some disagreement.

The other two co-authors of the criteria (and lead writers of the math standards) are William McCallum, a math professor at the University of Arizona, and Philip Daro, an education consultant to states and districts. Both McCallum and Daro also are advisers to Student Achievement Partners.

To be clear, this is not the final word from the standards writers. An "updated" version of the publishers' criteria for math, taking into account feedback, is expected out early next year. In addition, a separate document for high school math will be issued around the same time.

Zimba argues that the single most important element to ensuring the common core's success in improving math education is the emphasis on focus—essentially the idea of covering fewer math topics, but in greater depth.

The criteria document acknowledges upfront that it may be hard for math educators and experts to let go of some topics. "During the writing of the standards, the writing team often received feedback along these lines: 'I love the focus of these standards! Now if we could just add one or two more things,' " it says. "But focus compromised is no longer focus at all. ... 'Teaching less, learning more' can seem like hard medicine for an educational system addicted to coverage."

the new criteria are also aimed at helping to shape professional development pegged to the common-core standards.

The goal of the criteria, the authors say, is not to dictate acceptable forms of instructional resources, suggesting that "materials and tools of very different forms" can be deemed acceptable, including digital and online media.

the guidelines are not binding.

"Ultimately, it's still up to people at the local level. We think it's better to have something to react to than to have nothing out there, ... with people guessing on what they're supposed to do."

Researchers also know that it is not related to overall intelligence.

the introduction of brain-imaging research has given us new and important evidence.

has found that when children are put under math stress, they are unable to execute math problems successfully. The stress impedes their working memory—the area of the brain where we hold math facts. Beilock found that stressful math situations cause worries that compete for the working memory, causing it to be blocked. She also found that math anxiety has an impact on those with high, rather than low amounts of working memory—the very students who have the potential to take mathematics to higher levels.

found that levels of math anxiety did not correlate with grade level, reading level, or parental income. For the most capable students, the research confirms, stress impedes the functioning of their working memory and reduces achievement. Research conducted at Stanford revealed that math anxiety changes the structure and workings of the brain.

some of the students with the highest levels of success were those who indicated the greatest anxiety and made comments such as “I feel nervous. I know my facts, but this just scares me.”

It should not come as a surprise that the highest achievers displayed the greatest anxiety; in fact, neuroscience tells us that these students experience the greatest degree of cognitive dysfunction. But this anxiety does not only affect high-achieving students.

The brain research that has emerged recently could be the impetus for shifting the momentum. But the inclusion of the word “fluency” in the common standards may mean that educators will continue to use these tests, and that they will even be included as part of the new common-core assessments.

There are many good teaching strategies for encouraging fluency in math, but the ones that are effective are those that simultaneously develop number sense—the flexible use and understanding of numbers and quantities—without instilling fear and anxiety. Strategies that involve reasoning about numbers and operations, such as the pedagogical approach called “number talks,” are ideal for developing fluency with understanding.

timed tests also convey strong and negative messages about math, suggesting that math ability is measured by working quickly, rather than thinking deeply and carefully—the hallmark of high-level mathematical thinking.

Educators and policymakers share an important goal: to create math classrooms where students are excited to learn the subject, rather than being stressed and worried about their performance under pressure.

Based on the 1995 TIMSS, we identified common standards from the best-performing countries, which we call "A+ standards." We found an overlap of roughly 90 percent between the common math standards and the A+ standards. If the standards of the world's top achievers in 8th grade mathematics are any guide, then the common standards represent high-quality standards.

we find three key characteristics in the curricula of the highest-performing countries: coherence (the logical structure that guides students from basic to more advanced material in a systematic way); focus (the push for mastery of a few key concepts at each grade rather than shallow repetition of the same material); and rigor (the level of difficulty at each grade level). The common core adheres to each of these three principles.

Unfortunately, when one hears that a state's existing standards are better than the common core, it usually means that those standards include more—and more advanced—topics at earlier grades. But this is exactly the problem the common math standards are designed to correct. It is a waste of time to expose children to content they are not prepared for, and it is counterproductive to skim over dozens of disconnected topics every year with no regard for student mastery.

The disappointing reality is that, while improved from a decade ago, most state math standards fall below the common standards in both coherence and focus.

In debating the utility of the common core, it is very important to recognize that standards are not self-executing.

After including both cut points and how far away a state's standards are from the common core (controlling for poverty and socioeconomic status), we found that the two in combination are related to higher mathematics achievement—an even stronger relationship than was the case when only the measure of similarity was included. In the final analysis, however, the key ingredient in the implementation of standards is whether districts, schools, and, most importantly, teachers, deliver the content to students in a way that is consistent with those standards.

As it stands in many classrooms, teachers are forced to pick and choose among the topics as laid out in the textbook, items on state assessments, and the content articulated in state and district standards—expressions of the curriculum that frequently clash with one another. In our recently completed Promoting Rigorous Outcomes in Mathematics and Science Education, or PROM/SE project—a research and development initiative to improve math and science teaching and learning at Michigan State University—we found tremendous variation in the topics covered in mathematics classes within states, within districts, and even within schools. In fact, the content coverage in low-income districts had more in common with the content delivered in low-income districts in other states than with that of the more affluent districts in their own states. Given how haphazardly standards are implemented, it shouldn't be much of a surprise if the relationship between state standards and student achievement is modest. What's remarkable is that the relationship is as strong as it is.

The essential question is not whether the common core can improve mathematics learning in the United States, but whether we, as a nation, have the commitment to ensure that it does.

It remains to be seen whether the right kind of common assessments and supporting instructional materials will be developed.

The study found that the rate of improvement on math tests for high-achievers slowed in those schools that previously placed students into different classes based on ability level.

"When eliminating remedial math classes, schools are likely to put lower-performing students in algebra classes together with high-performing students," says the study

She suggests that what may be happening is that teachers are adjusting instruction to the "middle students" in a classroom, and so the declines in peer ability levels could result in "less-challenging content and slower-paced instruction."

the study wades into some touchy terrain in examining issues of tracking and mixed-ability grouping.

other points to keep in mind. First, she notes that the study is not saying that mixed-ability grouping will inevitably harm high-achievers, if other interventions are supplied.

it was not accompanied by additional supports for struggling students to master algebra, or professional development for teachers around how to effectively teach the subject in mixed-ability classrooms. (However, in 2003, the district instituted a new policy to provide additional algebra support to low-achieving students.)

Also, prior research by Ms. Nomi and several colleagues at the University of Chicago concluded that the algebra-for-all policy wasn't necessarily much help to low-achieving students either. That earlier research found that although more low-achieving students completed 9th grade with credits in Algebra I and English I, failure rates increased, grades declined slightly, test scores did not improve, and students were no more likely to enter college.

two other recent studies, one focused in California and the other in North Carolina's Charlotte-Mecklenburg school district, came to a similar conclusion, finding that placing struggling urban middle schoolers into algebra not only fails to improve their achievement on state math tests, but also reduces the likelihood that they will take and pass higher-level math courses in high school.

current evidence suggests that the policy led to mixed ability grouping and this, in turn, negatively affected high-achieving students," she said.

The study concludes by offering its findings as a cautionary tale about making changes in course mandates without other assistance.

simply mandating a college-prep curriculum for all students is not sufficient to improve the academic outcomes of all students."

covers the development of the two pieces of the test that will yield students' summative scores in mathematics and English/language arts and be used for accountability purposes: a computer-based end-of-year test and a performance-based assessment given toward the end of the year. The scope of work also includes developing midyear formative assessments that are part of PARCC's system but are optional for states.

A preliminary blueprint of PARCC's English/language arts exam shows that the performance-based assessment, spread over two days, would involve a "research simulation" that asks students to read a suite of texts, including an "anchor" text such as a speech by a prominent historical figure. They would have to answer questions that require them to cite evidence from the text for their answers and write an essay. Another aspect of the performance-based test would require students to "engage" with literature (grades 3-5) or conduct literary analysis (grades 6-11) using a combination of shorter and longer texts. The end-of-year exam would employ six literary and informational texts and ask students to respond to machine-scorable questions, including ones that demand comparison and synthesis of the readings. The end-of-year test in English/language arts would yield at least half of a student's points in that topic. One-third to one-half would come from the performance-based test, according to the preliminary blueprint.

ARCC's math test will include three types of questions: "innovative," machine-scorable, computer-based items; items that call for written arguments or justifications; critiques of mathematical reasoning, or proof that students "attended to precision" in math; and items involving real-world scenarios. The performance-based assessment in math will count for 40 percent to 50 percent of a student's points in that subject, and the end-of-course exam will yield 50 percent to 60 percent of the points. The math exams will focus on solving problems in the "major content areas" at each grade level, as well as demonstrating conceptual understanding, fluency and mathematical reasoning, and applying knowledge to real-world problems. At the high school level, PARCC will develop two series of end-of-course math tests: a traditional one—Algebra 1, geometry, and Algebra 2—and one that integrates those topics. Those parallel pathways reflect choices educators can make about how to design math courses from the common standards. The solicitation document answers a question that had been circulating among some educators of young children. PARCC said that its tests will be given by computer to students in grades 6-11, but those in grades 3-5 will answer questions with pencil and paper because of concerns about younger children's keyboarding skills.

PARCC has contracted with the Dana Center at the University of Texas at Austin to build prototype assessment tasks in math, and with the University of Pittsburgh's Institute for Learning to generate such items in literacy. Those items are slated for release this summer.