The Common Core Math Standards

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Are they a step forward or backward?



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SUMMER 2012 / VOL. 12, NO. 3

Education Next talks with Ze’ev Wurman and W. Stephen Wilson

More than 40 states have now signed onto the Common Core standards in English language arts and math, which have been both celebrated as a tremendous advance and criticized as misguided and for bearing the heavy thumbprint of the federal government. Assessing the merits of the Common Core math standards are Ze’ev Wurman and W. Stephen Wilson. Wurman, who was a U.S. Department of Education official under George W. Bush, is coauthor with Sandra Stotsky of “Common Core’s Standards Still Don’t Make the Grade” (Pioneer Institute, 2010). Wilson is a professor of mathematics at Johns Hopkins University, served on the National Governors Association-Council of Chief State School Officers “feedback group” for the Common Core standards, and was mathematics author of Stars by which to Navigate? Scanning National and International Education Standards in 2009: An Interim Report on Common Core, NAEP, TIMSS, and PISA.

Education Next: Are the Common Core math standards “fewer, higher, and clearer” than most state standards today? Can you provide some specific examples where you think the Common Core marks a step forward or backward?

Ze’ev Wurman: Common Core standards may in fact be clearer and more demanding than many, though not all, of the state standards they replaced. The Fordham Institute reviewed them last year and found them so. While I have no reason to doubt the technical quality of that review, there is good cause to note what it does not say.

Ze’ev Wurman

It does not say that Common Core standards are fewer. Indeed, if one compares them to the better state mathematics standards like those of Minnesota or California, they are more numerous. Minnesota’s standards fill 42 pages and California’s 59 pages, while the Common Core takes 73 pages even without the advanced statistics or calculus sections that are included in California’s standards. Counting the standards rather than pages, in grades 1 to 4 California has, on average, a few more standards than Common Core, but in grades 5‒8 the Common Core standards are more numerous than California’s.

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. For example, the 2008 report of the National Mathematics Advisory Panel, Foundations for Success, called for fluency in addition and subtraction of whole numbers by the end of grade 3, and fluency in multiplication and division by the end of grade 5. This is also what California calls for, along with high achievers like Singapore and Korea. (Japan and Hong Kong finish with multiplication and division of whole numbers even earlier, by grade 4.) Yet the Common Core defers fluency in division to grade 6. Fractions are touted as the Common Core’s greatest strength, yet the Common Core pushes teaching division of fractions to grade 6 without ever expecting students to master working with a mix of fractions and decimals. Students in Singapore, Japan, Korea, and Hong Kong achieve fluency in fractions and decimals in grade 5.

Nor are the Common Core standards necessarily clearer. They may be clearer than many state mathematics standards, but they still tend to be wordy and hard to read. Table 1 compares a few grade 4 California standards with their Common Core counterparts.

Andrew Porter, dean of the University of Pennsylvania’s Graduate School of Education, recently evaluated the Common Core standards with his colleagues, and their 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.

Another recent analysis, by University of Southern California professor Morgan Polikoff, found the Common Core mathematics standards similarly repetitive, and hence as unfocused across elementary grades as the state content standards they attempt to replace, with only somewhat less redundancy in the middle grades.

In summary, analyses of the Common Core standards find them to be mediocre and not obviously better than many sets of state standards.

W. Stephen Wilson

W. Stephen Wilson: It turns out that nearly everyone was in favor of Common Core standards in mathematics if, and this is a big if, they got to write them. As it turns out, no one got to write the standards. A committee wrote them. Worse, the committee was hired by the very states whose standards would be replaced, so states got first crack at suggesting “corrections” to the standards. The pressures on the writing committee must have been enormous. The only reasonable expectation was that the result would resemble some sort of middle way between the states’ various standards. What is surprising is that 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.

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

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.

Arithmetic is the foundation. Arithmetic has to be a priority, and it has to be done right. A number of things can and do go wrong with state standards for arithmetic in elementary school.

With the introduction of calculators, many states have downplayed the importance of arithmetic, apparently not realizing its true educational value. Instead, they spend time on statistics and probability, both of which Common Core has tossed out of early elementary school. Another thing that states love is geometric slides, turns, and flips, sometimes presented every year in grades K‒11, perhaps under the mistaken belief that they are really doing mathematics.

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). For example, only seven states expect students to know explicitly the standard algorithm for whole number multiplication. Fractions are even harder to find done well. Standards for fractions are generally so vague that nearly everything is left to the reader. Often states expect students to develop their own strategies or a variety of strategies for dealing with fractions. For example, only 15 states mention common denominators. Common Core does a pretty good job with arithmetic, even a very good job with fractions.

EN: Will the Common Core put an end to what has sometimes been termed the “math wars”? In your view, do the math standards resemble those recommended by the National Council of Teachers of Mathematics (NCTM), and what do you make of that similarity (or lack thereof)?

WSW: The end of the math wars! You must be joking.

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, the first line of defense for those who are anti-standard algorithms being just deny they exist. 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.

There will always be people who believe that you do not understand mathematics if you cannot write a coherent essay about how you solved a problem, thus driving future STEM students away from mathematics at an early age. A fairness doctrine would require English language arts (ELA) students to write essays about the standard algorithms, thus also driving students away from ELA at an early age. The ability to communicate is NOT essential to understanding mathematics.

There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.

There will always be people who think that statistics and probability are more important than arithmetic and algebra, despite the fact that you can’t do statistics and probability without arithmetic and algebra and that you will never see a question about statistics or probability on a college placement exam, thus making statistics and probability irrelevant for college preparation.

There will always be people who think that teaching kids to “think like a mathematician,” whether they have met a mathematician or not, can be done independently of 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. They are wrong. You learn Mathematical Practices just like the name implies; you practice mathematics with content.

There will always be people who think that teaching kids about geometric slides, flips, and turns is just as important as teaching them arithmetic. It isn’t. Ask any college math teacher.

The end of the math wars! You must be joking.

ZW: Math wars erupted as a result of the unfocused and mostly math-less 1989 NCTM standards. NCTM rewrote those terrible standards in 2000, yet much of what mathematicians found objectionable remained in place. Only in 2005, with the publication in Notices of the AMS [American Mathematical Society] of “Reaching for Common Ground in K–12 Mathematics Education,” did the two sides make a serious attempt to bridge the chasm. 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.”

What to make of such obvious deficiencies and omissions? Unfortunately, the main authors of the Common Core mathematics standards had minimal prior experience with writing standards, and it shows. While they may have had a long and distinguished list of advisers, they did not seem to have sufficient experience to select the wheat from the chaff. How, otherwise, can one explain their selecting an experimental approach to geometry, teaching it on the basis of rigid motions, that has not been successfully tried anywhere in the world? Simple prudence and an ounce of experience would tell them either to stick to what is known to work or to recommend a trial phase before foisting it sight-unseen on a nation of 300 million.

EN: How do the Common Core math standards compare to those in use in the world’s highest-performing nations? Crucially, on what do you base that assessment?

ZW: It is not difficult to show that the Common Core standards are not on par with those of the highest-performing nations.

Here is what 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.

And here is what a non-American member of the Validation Committee wrote to the Council of Chief State School Officers when declining to validate the standards:

I cannot in all conscience, endorse statements 2 and 3 [(2) Appropriate in terms of their level of clarity and specificity; (3) Comparable to the expectations of other leading nations] The standards are, in my view, much more detailed, and, as Jim Milgram has pointed out, are in important respects less demanding, than the standards of the leading nations.

We also have it straight from the horse’s mouth, so to speak. 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, found exactly that: “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.”

It is also worth mentioning that the standards, in addition to being “[c]omparable to the expectations of other leading nations,” were also supposed to be “[r]eflective of the core knowledge and skills in ELA and mathematics that students need to be college- and career-ready.” That is, at least, what the other Common Core Validation Committee members certified when they signed off on the standards in 2010.

College readiness is defined by what colleges require as prerequisites from their incoming freshmen. 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? The study, led by David Conley, was published more than a year after Common Core standards were already certified as college-ready by…David Conley as a member of the Common Core Validation Committee. Paraphrasing Shakespeare, he doth attest too much.

In summary, the Common Core mathematics standards fail on clarity and rigor compared to better state standards and to those of high-achieving countries. They do not expect algebra to be taught in grade 8 and instead defer it to high school, reversing the most significant change in mathematics education in America in the last decade, supported by the 2008 recommendations of the National Mathematics Advisory Panel, and contrary to the practice of our international competitors. Moreover, their promise of college readiness rings hollow. Its college-readiness standards are below the admission requirement of most four-year state colleges.

WSW: When you are so far behind, comparing the United States with better-performing countries through the incredibly narrow lens of standards doesn’t make a lot of sense. I think Common Core is in the same ball park, certainly not up there with the best of countries, but Common Core isn’t up there with the best state standards either, and what does that mean? 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.

So, let’s just pretend for a moment that Common Core is just as good as the very best. Who, in education circles, will agree with that enough to put it all in practice? The standard algorithm deniers will teach multiple ways to multiply numbers and mention the standard algorithm one day in passing. Korea will say “no calculators” in K–12, a little extreme perhaps, but some in the U.S. will say “appropriate tools” means calculators in 4th grade. 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.

There is one big hope for our international competitiveness. Other countries see that their best STEM students come to the U.S. for graduate school—more than half of our STEM graduate students are foreign—and to start high-tech companies. Instead of thinking that this is possible because of their strong K–12 mathematics education, they erroneously conclude that they should adopt our version of K–12 mathematics education. We just might catch up with these countries without any effort on our part.

EN: What, then, are your main areas of disagreement?

WSW: Ze’ev refers to Andrew Porter’s work to support his argument that Common Core lacks focus. In the corrected version of Porter’s paper, 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.

My only serious disagreement with Ze’ev is his summary that “analyses of Common Core standards find them to be mediocre and not obviously better than many sets of state standards.” 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.

ZW: Steve sees the benefit of having Common Core standards that are better than those of “more than 30 states,” while I see the disadvantage of confining the whole nation to mediocre standards that are worse than those of highly rated states and high-achieving countries.

Taking this a step further, I believe the Common Core marks the cessation of educational standards improvement in the United States. No state has any reason left to aspire for first-rate standards, as all states will be judged by the same mediocre national benchmark enforced by the federal government. Moreover, there are organizations that have reasons to work for lower and less-demanding standards, specifically teachers unions and professional teacher organizations. While they may not admit it, they have a vested interest in lowering the accountability bar for their members. With Common Core, they have a single target to aim for, rather than 50 distributed ones. So give it some time and, as sunset follows sunrise, we will see even those mediocre standards being made less demanding. This will be done in the name of “critical thinking” and “21st-century” skills, and in faraway Washington D.C., well beyond the reach of parents and most states and employers.




Comment on this article
  • Niki Hayes says:

    Finally, I have an overview of clarity on the Common Core standards’ impact on mathematics education, not only for the present but also for the future. Thank you for this analytical and user-friendly article.

    Having served on the TX state committee (grades 3-5) to rewrite our math standards in 2011, I can attest that pressure was great against anyone who believed in teaching standard algorithms and prohibiting the use of calculators in grades K-5. As I was told by one curriculum director for a school district, “There are no such things as standard algorithms.” She proudly announced to me, “Mexico doesn’t use them!” That caught me off guard, I must admit, since Mexico had never entered my mind as a leader in math education.

    We did get a mandate for standard algorithms and no calculators in K-5 written into the proposed document, however, and I anticipate the State Board of Education will not remove those mandates in its approval process.

    Some key state agency leaders started out pushing Common Core as a great resource for us and I kept reminding them that Texas and Alaska were the two states that never signed onto the program. The state commissioner for the Texas Education Agency had made it clear in speeches and articles as to why Texas had not jumped on the Common Core bandwagon. His agency’s workers clearly had not agreed with him and used Common Core wording in the newly proposed TX standards. Our committee members disliked the verbosity of those standards and kept trying to bring clarity to ours so teachers would truly have “direct instruction” about what was in the new “student learning expectations” (SLE’s).

    Clarity over agreement is a worthy goal, I do believe.

  • [...] Education Next interviewed Ze’ev Wurman and W. Stephen Wilson about the Common Core Math Standards which 40 states have signed on to.  They ask are they a step forward or backward?  They are pretty clear that it is a step backward.  Some money quotes… [...]

  • miriam says:

    Thank you for this excellent discussion. I learned a lot.

    But, I still have one question: With the Common Core, will all students have to know arithemetic–especially ‘number facts’ with automaticity? Without a calculator?

  • tom says:

    Have these two gentlemen ever actually taught in an elementary school math classroom? I find it concerning when an educational “expert” dismisses the importance of articulating one’s ideas to others.

    “The ability to communicate is NOT essential to understanding mathematics.” How is a teacher, who uses formative assessments, supposed to know what a child truly understands or is grappling to understand, if they don’t have a window into their thinking. Whether its oral or written, students need to be able to communicate and defend their thinking.

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

  • Myra says:

    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. As a teacher, I am a patient person but Ze’ev really ruffled my feathers with his talk of “Teachers should do this” and “Teachers should do that”. He also insulted a younger teacher by suggesting that she did not have the experience to be a member of the committee. I was so frustrated with him, that I finally suggested that he “walk in my shoes” for a day to see what teaching is really all about. He never took me up on my offer. I would be willing offer it to him again!

    I agree with Tom about taking the comments with a grain of salt.

  • [...] this is background for a very interesting article I saw on Common Core math standards. Ze’ev Wurman and W. Stephen Wilson provide an excellent [...]

  • Barry Garelick says:

    I have taught in a classroom, and you can find out a lot by looking at how students solve problems in class and on quizzes and exams. I really don’t need a written explanation of what they did to solve a problem and why; the math explains it. If it doesn’t explain it, I ask the student what they did.

    For example, if we are covering fractional division and the types of problems where fractional division occurs, I would want a student to be able to show how to find how many 3/4 ounce servings there are in a 1 -1/2 oz cup of yogurt; i.e., show that you divide 1-1/2 by 3/4, and be able to do that calculation. I don’t need a written explanation of what he felt was difficult about it, and how this compares with other problems he’s seen. I just want to see that he knows how to represent the problem and solve it. That’s the type of communication I want.

  • Joye says:

    Thanks so much for this most helpful analysis. For many years, these two gentlemen have probed and studied extensively the issues they addressed. I consider them two of the top experts in the country, along with R. James Milgram at Stanford. I agree with Steve that we will not be able to catch up with other high performing countires. People will choose to interpret the Common Core standards to suit their agendas, just as they did the NCTM Standards in 1989. I have taught algebra for 14 years, and have seen a steady decline in computation skills even in that time period, and it translates into a steady decline in reasoning skills as well. I’m sorry, folks, but they don’t understand it conceptually, either. It gets more and more difficult to teach algebra as the years pass. Students are not getting dumber. They simply are not held to high enough standards in math in the lower grades and it cripples them for life.

  • Cherrie says:

    I ask students to show their work on math problems. They are to show their work in such a way that a teacher or another students can see and follow from start to finish what they have done. That is communication and a teacher should be able to tell from work shown whether or not the student understands what they are doing. If a student doesn’t understand what they are doing, they won’t be able to successfully progress from one step to the next in order to accurately complete a problem.

  • SteveH says:

    “I believe the Common Core marks the cessation of educational standards improvement in the United States.”

    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.

    Also, the original question is flawed:

    “Are the Common Core math standards “fewer, higher, and clearer” than most state standards today? Can you provide some specific examples where you think the Common Core marks a step forward or backward?”

    First, these are not standards, they are guidelines or suggestions. I found the word “fluency” only 4 or 5 times. The standards don’t calibrate this and they don’t say what should happen if this vague fluency does not happen.

    Second, and more importantly, the goal is not necessarily a continuous path. We don’t want a continuous path out of a deep, dark hole of low expectations that puts the onus on the student using the pedagogy of natural learning. Bright urban kids don’t want a continuous path. The goal is not to float all boats up to a mediocre rising standard. The goal is to provide discontinuous paths so that some can show that it’s possible to fly rather than just keeping your head above water.

    Ze’ev’s final warning is true. The standards will institutionalize mediocrity. When parents complain, schools will just point to the standards. Even now, All of my son’s high school classes are tying their syllabi to specific Common Core Standard sections. This is for honors classes.

  • SteveH says:

    The Common Core Standards workplace analysis is also fundamentally flawed. They didn’t need to do their own self-serving analysis to come up with one pseudo-algebra II goal for all in high school. Standards like SAT/ACT and AP already exist. At the low end your have Accuplacer. Look at vocational schools and see exactly what they require. There is not one goal, there are many paths.

    Individual students are not averages, especially once they get to high school. Since this is the goal by the end of high school, there is a lot of wiggle room along the way. The best students don’t get what they need and those at the low end are tortured with endless “Groundhog Day”-like circling of algebra topics year after year after being ruined by bad K-6 math curricula.

    In K-6, the standards might seem higher, but they don’t fix the fundamental flaw of not ensuing mastery at any one point in time. Schools will continue to trust the spiral and assume that kids will learn when they are ready. They pump kids along and let someone else worry about the big filter. They think that there is plenty of time left to hit the pseudo-algebra II goal.

    Can’t you hear the doors slamming shut? The big filter begins in 7th grade when schools start tracking kids in math. The different tracks might send kids off onto paths that have absolutely no connection to any sort of STEM career. They will struggle to pass the CCSS standards in high school. Even if the middle schools use the same textbooks as the algebra in 8th grade track, kids will struggle because it’s not a problem of speed, it’s a problem of gaps and lack of mastery. Because of the problems in K-6 math, it’s all over for so many kids by 7th grade. They will hate math and struggle to meet the CCSS standards. If high schools work hard, they might be able to smooth the process, but the damage has already been done. Students will not be at a base camp to climb the mountain of college math, they will be at their peak with nowhere to go.

    So, for those lucky enough to find themselves on the top math track in 7th grade, they are the winners of the STEM career sweepstakes. I find it incredible that schools don’t survey parents of those who took algebra in 8th grade. We would tell you that we just didn’t turn off the TV or model an interest in education. We ensured that mastery was achieved in a grade-by-grade fashion. This was not just for silly little things like the standard algorithms. It was for all things. It was so annoying to get notes sent home from school telling us parents to work on math facts.

    The best students (and their parents) don’t care about any stinking CCSS or state standards. They care about SAT I and II, and AP calculus. For many students, the state standards are meaningless. For those students without help at home, the average (?) CCSS standard defines the most they can expect, but it’s worse than that. States love to translate standards into proficiency levels and then set low cut-off points. Then they devise a proficiency index that shows how many kids get over the low cutoff. All of a sudden, the stinking low raw percent correct scores on the state tests are translated into not-bad-looking proficiency percentages.

    Beware Ze’ev’s warning!

    Standards will NOT fix education. Choice can, and it can work for many kids right now! It can be discontinuous and absolute, not relative and continuous. Educators in our state have been fighting to prevent charter schools that set higher expectations. They don’t even want to allow initiative to separate kids. It’s not about money.

  • SteveH says:

    “How is a teacher, who uses formative assessments, supposed to know what a child truly understands or is grappling to understand, if they don’t have a window into their thinking. Whether its oral or written, students need to be able to communicate and defend their thinking.”

    For math, the communication is using the language of math. When students hand in a problem set from a proper textbook (I feel I have to say that), one can tell by reading that language. Students (in a proper math course) are taught how that communication should be done. One should be able to follow, mathematically, a sequence of steps from the initial problem statement to the final answer. For most problems, this requires few words.

  • SteveH says:

    “I believe the Common Core marks the cessation of educational standards improvement in the United States.”

    Actually, it’s worse than that. It allows states to offer no alternative paths. Even the schools in our non-urban district have lobbied to prevent “our” students from going to existing charter schools because the schools are “high performing” based on the high percentage of kids who meet the incredibly low proficiency cutoff. Notice the ownership.

    The best students, supported at home, have no need for state standards. For the rest, especially those who have no choice, the top expectation is a low cutoff. And people wonder why there is a gap.

  • Katharine Beals says:

    “How is a teacher, who uses formative assessments, supposed to know what a child truly understands or is grappling to understand, if they don’t have a window into their thinking. ”

    By giving them truly challenging problems and seeing if they consistenly get the right answers!

  • SteveH says:

    “Will the Common Core put an end to what has sometimes been termed the “math wars”? ”

    “WSW: The end of the math wars! You must be joking.”

    Hey, that’s what I say! Steve’s list is a good one.

    I ran into another one to add to the list – a teacher who “detested” even the very little homework his son was getting from Everyday math. My anecdote is different, in 4th grade, I told my son to stop fooling around and get his Everyday Math homework done. Less than 10 minutes later I saw him playing and told him again to do his homework. He said that he already did it. Even the teacher admitted that the goal is just to get a taste and to keep spiraling along. I call it repeated partial learning. I didn’t allow that to happen. Smart parents don’t trust the spiral.

    This is not just a battle over standards. It’s a philosophical and pedagogical battle. Not only that, the battle is about who gets to decide. Those in favor of top-down, integrated, thematic, real world, mixed ability, student-centered, teacher as the guide-on-the-side, and anti drill and kill are the ones in charge, especially in K-8. You would think that they had the onus to show that there is little opinion involved with their beliefs, but it doesn’t happen, and the CCSS standards do not offer any vehicle to question those beliefs.

    In high school, students and parents often have a choice of levels (college prep, honors, AP), and they often have a choice of traditional, IB, or integrated math sequences. Note that virtually nobody heading for a STEM career will be in the integrated math sequence. Parents (and colleges) expect AP or IB. Parents know this, and schools cannot get away with forcing integrated math down students’s throats. Their battle has been lost in high school, but it’s still raging in K-8.

    This choice does not exist in K-6 because nothing is driving the top standards down into the lower grades. The AP calculus math track in our high school finally (!) forced CMP out of our middle school because it was not preparing kids properly for the AP calculus track in high school. There was a curriculum gap to that path. This drove the problem back to the math tracking split at the beginning of 7th grade. It helped the better students (the ones who had help at home), but there was no mechanism to force the high standards for the top path down into the lower grades to help the rest. That’s where you need to fix the STEM problem and the Common Core Standards do nothing to fix that.

    Students and parents get choice in high school and in college, but they are apparently not “informed” enough to make those decisions about K-8. Apparently, education in K-8 is so complex that it takes trained educators to determine “best practices”, even if they can’t separate opinion from reality or level of expectations.

  • Oak says:

    Tom, he was specifically talking about nonsensical essay writing that students are put through, not being able to communicate if they understand a math topic.

  • jim miller says:

    “Thanks for this wonderfully insightful analysis of the Common Core Standards and their apparent and prospective impact on mathematic education. I’m of the view that the common core standards do, in way, bridge the gap between high performing and low performing states but they also cut down any scope of improvements at the state level. Plus I think with the advent and the popularity of digital resources is anyways helping create a level playground for state players. CK-12 for example, a Calif.-based curriculum provider, lets you align high quality math content to your state standards and also gives you room to maybe make enhancements to the content- all for free, So think Common Standards or not we are at present in a space where flexibility is a must, in every facet of education, from learning to teaching to even policy-making.
    http://www.ck12.org/flexbook/

  • [...] level of quality.)There’s more common sense about K12 math instruction in this fascinating dialogue posted at Education Next than in the math sections of the recent report on STEM education by the President’s Council [...]

  • Thomas Jimison says:

    Very well written analysis of the concerns and issues surrounding the ‘new’ core standards. I wish politicians would or more accurately, could, read this article and take heed. You cannot stress enough that ‘technology’ (that’s eduspeak for calculators) should never be allowed in elementary schools and used only to support instruction in high school. I teach senior math courses in HS as well as developmental (remedial) math courses at a local community college.

  • Ze'ev Wurman says:

    Well, when one has Andy Isaacs, the guy in charge of what could be the best-selling math series in the nation (Everyday Mathematics), claim that his textbooks are “aligned “with the Common Core yet publicly write:

    some people seem to think that CCSS requires that technology, including calculators, be eliminated in the elementary grades. The reason for this misinterpretation of CCSS is clear: None of the words “technology,” “calculator,” or “computer” appear in any of the content standards for grades K-6. But in the second decade of the 21st century, a curriculum designed for the world before 1950 would be a disservice to students – and the CCSS writers could have had no such thing in mind. In fact, one of the CCSS Standards for Mathematical Practice reads: “Use appropriate tools strategically.” In today’s world, appropriate tools for mathematics include technology such as calculators and computers. So to believe that CCSS bans technology from K-6 would be a mistake.

    There is little left to be said.

  • Dan says:

    Steve Wilson, Ze’ev Wurman, James Milgram and their compadres are part of a small percentage of the population who were successful learning mathematics under the same system of traditional math education they propose. They’ve spent their life steeped in numbers and formulas as mathematicians, so their rote approach to learning math worked for them, because working with it every day, their mathematics has moved to long-term memory. But what about the larger populace who aren’t mathematicians? What about the firefighter, salesman or nurse whose knowledge of math has faded because it was only placed in short-term memory long enough as non-relatable facts and procedures in order to pass a math test in high school or college? Ask them how to factor an equation with the foil method and watch them fold. Ask most people to restate and use the quadratic equation and you’ll get blank stares. 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. Most adults learned math in the system that these gentlemen support. Ha. It doesn’t work. 50 plus years of proof have shown us. And we don’t live in the 1950s anymore. Thankfully, we’re finally moving toward an educational system that honors the mathematical practices on which the CCSS were developed. But yet these practices (and no, Steven, the definition of practices here is not like practicing facts, it’s a a method or application of an idea) are what they find as the negative part of the standards? Given time it is the more constructivist approach to learning that will result in a higher achieving populace. Real research and teachers who have truly unlearned didactic teaching have proved it. Gentlemen, your approach has failed way too many people for way too long. It may have worked for you, since math is what you live and breathe, but we’re not worried about over-achiever mathematicians like you. We want to help the students who don’t get math the way you want it taught–and that’s most students.

  • Patrick says:

    Using tools strategically… To me that does not mean that we always use calculators and it doesn’t mean that we never use calculators. Just like a good diet, a little of everything is best practice. Students need time to increase their proficiency towards mastery of basic skills, facts, algorithms, etc. Students also need to have access to calculators for when “doing” the calculation is not the end goal of a problem, but instead, a means of moving toward a greater exploration or problem.

    Everyone who has ever taught grades 6 or higher realizes the deficiencies that students have who never have mastered the basic concepts. Spiraling and exposing kids to different concepts/ideas throughout their math careers leaves students in the dark and lacking a true understanding of mathematics and how numbers work together.

    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.

    Teachers at the early grade levels will need to make this a priority and even further, may need some professional development opportunities to see the connections/reasoning behind the transition.

  • Dave says:

    Dan, some historical analysis would do wonders for your argument. The twentieth century was rife with reform, counter-reform, anti-counter-reform, etc. movements, all of it filtered through various layers of government, school administrators and individual teachers – for you to assert that there has been a single monolithic method responsible for a disinterest and/or lack of ability in maths is a little desperate. I think what Wurman and Wilson were largely suggesting is that everything depends on solid foundations in the early years. Middle and high-school topics cannot be approached with ANY method, constructivist or otherwise, without something substantial to build on. Without clear expectations for the basic stuff, and demand follow-through, how do you expect students to tackle the harder stuff? To use an analogy, how can you write an essay if you don’t know the alphabet?

  • Michael Paul Goldenberg says:

    By the way, I’m not a fan of the Common Core. I think the very idea of national standards is a huge mistake, no matter who writes them. They are a blueprint for killing creativity in districts, schools, and classrooms. Let 10,000 methods bloom, not one. Curriculum needs to suit the students and teachers, not a committee or a Math Czar.

  • Hellen Taba says:

    For an interesting, and maybe scary critique of the common core and why we should not jump into this see the article by Christopher Tienken at Seton Hall University: Common Core State Standards: An Example of Data-less Decision Making. It is a humorous as well.

    http://www.aasa.org/uploadedFiles/Publications/Newsletters/JSP_Winter2011.FINAL.pdf

  • gretchen says:

    Having watched Ze’ev in action in California, I have to point out that he is incorrect in his numbers at the very beginning of the article:
    “Counting the standards rather than pages, in grades 1 to 4 California has, on average, a few more standards than Common Core, but in grades 5‒8 the Common Core standards are more numerous than California’s.”

    Here are the actual numbers of standards not counting the practice standards or the CA math and reasoning standards. First number is CA, second is CCSSM

    K: 14/22
    1: 25/20
    2: 31/26
    3: 38/25
    4: 44/28
    5: 27/25
    6: 37/29
    7: 42/24

    Clearly the CCSSM are less standards for California except in Kindergarten which the CCSSM actually focus on creating a solid foundation in number before doing measurement and geometry.

  • Ze'ev Wurman says:

    California K-7 standards are made of so-called “over-standards,” (sometimes called “standard-sets”) that describe content areas in general terms, and each is made up of the actual specific standards. Only those specific standards should count, the fact that is clearly seen if one looks at the test blueprints, which don’t test those over-standards but only the actual standards.

    In contrast, the Common Core has no such hierarchy. Most its standards are detailed and specific, but a few have additional aspects broken down into additional standards. Gretchen mistakenly chose not to count them. The actual counts of the Common Core content standards are 25, 24, 28, 37, 37, 40, 47 and 46 for grades k-7, respectively. In other words, from grade 4 and up the common core ones are more numerous than California’s.

    Perhaps the lesson here is that it is not enough to know how to count. Knowing what to count is as important.

  • Dave says:

    It is worth noting that the CCSS sometimes pack even more standards in than it first appears. Like Ze’ev noted, there are some standards that are split into separate parts. But some also just pack more detail in to make it seem like there are less standards. An example is 2.NBT.2 which is “Count within 1000; skip-count by 5s, 10s, and 100s.” From a programming point of view, this means three separate accomplishments: counting by 5s, counting by 10s, and counting by 100s. This means teaching each set of skip-counting and then assessing each set of skip-counting as you can’t assume that if they have done one set fluently then they can do them all.

    This is a simple example, but this extra packing is repeated throughout the elementary years. It gives the appearance of conciseness, but the specificity means that each standard has to be carefully read to unpack all the components. However, the CCSS is not alone in this as many previous state standards did the same. Personally, I’d rather have a long list of clear, precise statements than a short list of densely written ones. At least I’d know what I was up against.

  • Derek says:

    Look at the third standard in the figure. a/b =? (n+a)/(n+b) Just a typo in this article I hope!!

  • [...] W. Stephen Wilson: The ability to communicate is not essential to understanding mathematics. [...]

  • Ze'ev Wurman says:

    Derek,

    Thanks for noticing. The original standard, and the original quote, have the multiplication (“times”) symbol similar to lowercase x rather than what seems like a plus sign in the html image above. Thanks!

  • Hardin says:

    Dan’s estimation of Ze’ev et al is depressing.
    The fraction of people who have learned math by some hook or crook is not so negligible.
    Unfortunately, few kids get to learn math from anyone who like math, and who can answer their inquisitive questions.
    An enthusiastic early math teacher might send a lot of kids onto a pathway to STEM.
    These standards set a floor: not what kids might do, but a minimum, which even people (such as Dan claims to be) who do not like math are still expected to accomplish.
    An eager learner of math with the least bit of support (or at least removal of shackles) should have no problem exceeding these standards.

  • Pete says:

    with regards to calculators, if the class use a calculator (such as the about to be released new QAMA calculator http://www.qamacalculator.com) that needs students to input a reasonable estimate before they get given the answer, then we reckon the arithmetic skills of students (and maybe teachers as well) will rocket.

    we agree that these arithmetic skills are important, as is an understanding of number that estimation skills can give.

  • Clint Long says:

    “The ability to communicate is NOT essential to understanding mathematics.”
    It is not essential to understanding mathematics, but it is essential for USING mathematics. There have been many instances in my life, when I tried to use mathematics and my inability to communicate my processes and defend their validity caused failure. I will relate one such experience:
    While I was going to college to be a math teacher, I was working as a “grunt” burying service wire. My tools were a shovel and iron bar and I had to locate other utilities in the ground, so the machine used to bury the wire and conduit didn’t hit them. On one particular job, the service wire had to go from the house to the service pole and the conduit needed to travel up the pole to a service box. The conduit with the wire already in it is spooled and loaded on the back of the machine. Everything went smoothly until at the end we found ourselves at the end of the spool, wondering if the conduit left in the spool would reach the service box on the pole. All of the work we did was on or under the ground, so there was no ladder and we didn’t have anything stiff enough to run up the pole to measure it. So, I used some math and similar triangles that I knew worked and was able to get the distance marked on the ground and measured. I knew the conduit was long enough by a foot, which was well within my error tolerances. My problem was convincing the head of my team that what I did was right. I couldn’t explain what I did well enough, in a way that made sense to him. He didn’t believe my math was correct and went with his “gut”. He didn’t think it was long enough, so he went back 10 ft, cut the conduit and spliced on a longer piece using electrician’s tape. This is a horrible practice since it compromises the conduit over time and can later pinch the cable, but the only other choice in the job lead’s mind, was to redo the whole job. A few weeks later, our boss went with an inspector to the job site. The inspector dug a whole to verify the depth of the wire and it just happened to be the exact spot where the lead had spliced the conduit. In the end, we had to redo the whole job and the lead got in some serious trouble.
    Moral of the story: you need to be able to communicate your thinking, or the math remains magic and your solution is easily disbelieved.
    I’m sure our experience with banks collapsing has some stories of actuaries not being listened to, as well. So, if we expect our student to use the math we teach them, they need to be able to communicate their thinking and be able to defend their solution. Otherwise, why are we teaching it to them?

  • Ze'ev Wurman says:

    Pete,

    The QAMA calculator looks interesting. Long time ago, however, I suggested a much simpler design to achieve a similar purpose: a calculator that has a decimal point on the keyboard for input, but no decimal point on the display. It is up to the user to place the decimal point in the right place of the result.

    The basic idea is very similar — the user needs to know the order of magnitude of the expected result. Like with a slide rule.

  • [...] isn’t alone in his skepticism. A recent Education Next article cites concerns from Professor William McCallum, one of the three authors of Common Core’s math [...]

  • [...] isn’t alone in his skepticism. A recent Education Next article cites concerns from Professor William McCallum, one of the three authors of Common Core’s math [...]

  • Jeff says:

    The problem with “Common Core” and any other national/state standards
    As a high school teacher, the problem as I see it is that there are many different potential outcomes for mathematics education, but all of these are trying to be met in one school, with one curriculum.

    Colleges and knowledge demanding careers want students who can think, organize, and use mathematics proficiently to find applications and run them correctly. Their goal is to have an influx of higher level students who are literate with Bloom’s taxonomy and can write their own problems to solve and predict future issues. The ability to communicate math can be vital, or unimportant. It is vital for those who are in need of other inputs and explain applications, but unimportant for the grunt performers of the research and calculations as they need to focus more on accuracy.

    General society wants to feel enabled to use mathematics to make good decisions for their home or business. For instance if I am throwing a birthday party for 20-30 people how much of each item to order, what price to estimate, does quality matter enough to purchase items of greater price, and will I make these decisions well enough using mathematical logic to make all my guests leave happy and my budget intact? They don’t need logarithms but if they can’t quickly figure out the number of hotdog packages to buy they look foolish and aren’t spending their time wisely to prepare. The ability to communicate math is relatively unimportant as the sole proprietor generally makes the decisions and is ultimately responsible.

    General industry wants workers who can use math on the job in spot cases and won’t make decisions which cost their company lawsuits or issues later. Construction guys need to be able to measure fluently, truck drivers need to understand angles and vectors for maneuvaribility, etc. They want students capable of being trained on the job so they need minds which are not necessarily already full of algorithms, but can learn them when and if they are necessary. The ability to communicate math is important here as they need to use teamwork to make projects happen using many different departments of the company.

    The problem is that we have one school to create all 3 of these types of students. We have one curriculum directed by national/state standards. We have standardized tests, some which measure goal 1 (ACT, SAT, AP), and some which try to measure goals 2 or 3 (IL PSAE day 2). These tests are used by state and federal governments to measure schools and threaten them to improve or face consequences. We have parents who cry foul if their student is pigeon holed for any but the outcome they most desire, despite the fact that they themselves might not have been successful in it, or chosen it for themselves if given a choice in high school. We generalize our standards to not offend people who can’t achieve them, rather than help students realize if they can’t achieve one type of standard, there are still limitless opportunities to put their talents to work in another way. In the end we create a ton of mediocre students with no great talents, just a bunch of general knowledge with no ability to adapt themselves to the future. It must be fixed. I still don’t see how it can be with the current public school system and the methods of directing them in place as they are today.

  • Mourad says:

    Something more comprehensive has to haeppn. Change needs to haeppn. Sanity needs to be restored to education. Mainly, no matter what the LAW states, if it isn’t really possible to accomplish, teachers, administrators, school districts, etc. will all find a way to make it appear so on paper. It’s a matter of survival. If they didn’t do that, there would be total chaos. But there is still a level of chaos. Look at all the lawsuits against schools all because education has lost its sanity. I could tell you stories .

  • Karl Wheatley says:

    I love math, but do the common core standards address why so many American kids and adults hate math? If not, why would they work?

  • Ze'ev Wurman says:

    Karl,

    In my opinion the main reason for kids and adults “hating” math is because they were taught by teachers who were incompetent and more often than not hated math themselves. The only way I see to address this problem is to improve teacher education and teacher certification.

    Common Core will not make much difference in itself either way. But some of the work of assessment consortia such as the Smarter Balanced, which said it will focus on Common Core’s “mathematical practices” (3 pages) rather than on its mathematical content (75 pages) seems to play into the hands of clueless schools of eduction that promote practices and disdain content. So this will just make the matters worse.

  • Rayna says:

    I’m an administrator working with a group of K-5 teachers in a New York City school trying to figure out which math curriculum to purchase in light of the CCSS. The teachers are in a panic. We don’t have a standard curriculum, and they’re worried about the higher stakes state exams. They want to know what they’ll be using next year, and the principal wants us to decide before summer break. I’m exhausted trying to convince everyone that purchasing a curriculum that touts, “Aligned to the Common Core,” won’t fix our problem. We need to look at exactly where our students struggle in math, and focus on best practices to address these weaknesses.

    Our very own math war!

    Most likely, I’ll be asked to choose a curriculum.

    So here’s my question to all the esteemed folks who commented here: Is there a decent, newly-published K-5 math curriculum aligned to the common core that could be supplemented with another program that teaches mastery of basic skills?

    I’d really appreciate some advice.

    Rayna in NYC

  • [...] As a practical matter it’s also worth considering whether spending what will amount to $6 billion more taxpayer dollars (the RESPECT Project plus the STEM Master Teacher Corps) will really result in improved teaching, much less better student performance in math and science—particularly since many of the academics and experts who served on the Common Core national standards committees conclude they “are not on par with those of the highest-performing nations.” [...]

  • [...] problem-solving. I am fully aware of the people who the American maths professor, W. Stephen Wilson described in the following way: There will always be people who think that calculators work just fine and there is no need to [...]

  • [...] Wurman and W. Stephen Wilson reviewed the math standards in this piece on Education Next. Since the standards were created by a committee of representatives of all [...]

  • [...] *Summer 2012             THE COMMON CORE MATH STANDARDS             “I believe the Common Core marks the cessation of educational standards improvement in the United States. No state has any reason left to aspire for first-rate standards, as all states will be judged by the same mediocre national benchmark enforced by the federal government.” >>read more>> [...]

  • [...] Many experts conclude that the math standards are vague and incoherent. A recent Education Next article cites concerns from Professor William McCallum, one of the three authors of Common Core’s math [...]

  • [...] we celebrate when the data geeks come out on top. That’s why it was so jarring when I came across this quote from a Johns Hopkins mathematician no less: You never see a question about statistics or [...]

  • Jesse Bluma says:

    What Common Core does not do:

    -Focus on individual needs of students.
    -Identify individual areas of academic, skill, and social weaknesses and strengths.
    -Acknowledge who has been, is, and will be the study body of America.
    -Save tax payer funds, time, and energy.
    -Graduate more students with the skills and knowledge they need to succeed.
    -Provide an alternate path for students, other than college. (Our educational system has not for a long time, does not, and will not under Common Core provide students with the education and skills necessary to become artisans, craftsmen, tradesmen, or work in factories.)

    http://pointeviven.blogspot.com/2012/12/the-origins-of-common-core-how-business.html

  • [...] expert review of the math standards, both pro and con, can be found here4, including an insider s view on the development of the Common Core. An overarching criticism by [...]

  • Vic Podis says:

    Like Ze’ev, I find that the lack of knwledge of basic facts by manu of our students is alarming. It’s not so much that they have not “memoried” the facts, but that taking excess time to figure out what should be rote computation indicats a lack of understanding ofnumber operations. Without a sense of numbers and how they work, it will not matter how far they get or do not get in math.
    My concern is that also that expecting rigorous math classes of students with significant learning disabilities is counter-productive. If these Students leave school only with a good sense of basic computation, measurement, and sense of numbers, their math education will have served them quite adequately.
    I hope that common core is an improvement for math instruction as all of the educational experimentation has last impact upon the students we are attempting to serve.

  • Garry Isbell says:

    I agree with the article that basic arithmetic is the basis for mathmatical fuctions and should be tought with out the use of calculators in the elementary levels. Calculators should be allowed to be used as an assistive tool at the middle and high school levels. It does create some confusion as to why the article explains that Japan, Singapore, etc. use certain common core standards with expectations by certain grade levels, but states in the U.S. all have an array of different standards they go by. Why wouldn’t all the states adopt the same Common Core Standards so we are all on the same page? This is a major issue in my opinion!

  • Janice McCarthy says:

    As an intervention specialist for students in special education I am concerned that the common core standards will not provide for the alternative pathway or necessary skills for my students to be successful in the workplace. Many of these students will not use the mathematical concepts learned in algebra II or attend college in the future. My hope is that they will be able to understand and use basic math concepts necessary for success in everyday life.

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