Phil Daro, Bill McCallum, Jason Zimba

You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards.

In the Common Core State Standards, individual statements of what students are expected to understand and be able to do are embedded within domain headings and cluster headings designed to convey the structure of the subject. “The Standards” refers to all elements of the design—the wording of domain headings, cluster headings, and individual statements; the text of the grade level introductions and high school category descriptions; the placement of the standards for mathematical practice at each grade level.

The pieces are designed to fit together, and the standards document fits them together, presenting a coherent whole where the connections within grades and the flows of ideas across grades are as visible as the story depicted on the urn.The analogy with the urn only goes so far; the Standards are a policy document, after all, not a work of art. In common with the urn, however, the Standards were crafted to reward study on multiple levels: from close inspection of details, to a coherent grasp of the whole. Specific phrases in specific standards are worth study and can carry important meaning; yet this meaning is also importantly shaped by the cluster heading in which the standard is found. At higher levels, domain headings give structure to the subject matter of the discipline, and the practices’ yearly refrain communicates the varieties of expertise which study of the discipline develops in an educated person.

Fragmenting the Standards into individual standards, or individual bits of standards, erases all these relationships and produces a sum of parts that is decidedly less than the whole. Arranging the Standards into new categories also breaks their structure. It constitutes a remixing of the Standards. There is meaning in the cluster headings and domain names that is not contained in the numbered statements beneath them. Remove or reword those headings and you have changed the meaning of the Standards; you now have different Standards; you have not adopted the Common Core.

Sometimes a remix is as good as or better than the original. Maybe there are 50 remixes, adapted to the preferences of each individual state (although we doubt there are 50 good ones). Be that as it may, a remix of a work is not the same as the original work, and with 50 remixes we would not have common standards; we would have the same situation we had before the Common Core.

Why is paying attention to the structure important? Here is why: The single most important flaw in United States mathematics instruction is that the curriculum is “a mile wide and an inch deep.” This finding comes from research comparing the U.S. curriculum to high performing countries, surveys of college faculty and teachers, the National Math Panel, the Early Childhood Learning Report, and all the testimony the CCSS writers heard. The standards are meant to be a blueprint for math instruction that is more focussed and coherent. The focus and coherence in this blueprint is largely in the way the standards progress from each other, coordinate with each other and most importantly cluster together into coherent bodies of knowledge. Crosswalks and alignments and pacing plans and such cannot be allowed to throw away the focus and coherence and regress to the mile-wide curriculum.

Another consequence of fragmenting the Standards is that it obscures the progressions in the standards. The standards were not so much assembled out of topics as woven out of progressions. Maintaining these progressions in the implementation of the standards will be important for helping all students learn mathematics at a higher level. Standards are a bit like the growth chart in a doctors office: they provide a reference point, but no child follows the chart exactly. By the same token, standards provide a chart against which to measure growth in childrens’ knowledge. Just as the growth chart moves ever upward, so standards are written as though students learned 100% of prior standards. In fact, all classrooms exhibit a wide variety of prior learning each day. For example, the properties of operations, learned first for simple whole numbers, then in later grades extended to fractions, play a central role in understanding operations with negative numbers, expressions with letters and later still the study of polynomials. As the application of the properties is extended over the grades, an understanding of how the properties of operations work together should deepen and develop into one of the most fundamental insights into algebra. The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.

This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events.

I love the urn analogy. Here in the high school trenches I’m hearing less from colleagues about students not needing to know how the light switch works (just flip the switch) because between Marzano, ACT, National Boards, and now CCSS, the answer is resounding, “Oh yes they do!” My curiousity is compelling me to ask exactly what are we leaving out that was a mile wide? Can you suggest a resource to learn about that?

I guess I could be considered in the high school trenches as well. As a former HS teacher now working in K-12 I am seeing the changes in the grade level standards, but I am a bit lost in the high school standards. I, too, would like to hear more about the high school standards and what is being considered left out?

The mile-wide inch-deep problem looks different in high school. In earlier grades its a matter of having too many topics. In high school its a matter of having too many separately memorized techniques, with no overall understanding of the structure to tie them altogether. So narrowing and deepening the curriculum is not so much a matter of eliminating topics, as seeing the structure that ties them together. For example, if students see that the distance formula and the trig identity sin^2(t) +cos^2(t) = 1 are both manifestations of the Pythagorean theorem, they have an understanding that helps them reconstruct these formulas rather than memorize them. The CCSS high school standards in the Algebra and Function categories are arranged under headings like “Seeing Structure in Expressions” and “Building Functions” in order to help teachers and curriculum developers see this coherence.

Thank you for the response! Your reply was just the thing I needed in our first attempt to address the CCSS Shifts to every single hs math teacher in our district. While I do work with some super fabulous professors, your blog has come in quite handy to support the work I do with math teachers. Looking forward to the high school progressions!!!

Excellent post! I have found the Progressions Documents in your Tools sections to be extremely useful in my work with teachers. I’ll be attending the CCSS PD Workshop in Tucson starting tomorrow and look forward to leaving on Sunday with new tools in my toolkit!

As always and once again, you have eloquently stated the advantage and benefits of the U.S. moving to the CCSS model. Very well put, and thank you for all of your time and efforts on behalf of the mathematics education community in the United States.

Thank you! Thank you!! Well said. I have tried to explain this many times, but with little success. Your analogy is great and will help me tremendoulsy. Thank you for all your work.

Having teachers look at one or two individual standards is the route that many states are taking (including mine). No, they don’t even consider the whole cluster, let alone the domain or progressions. It’s so disheartening. Thank you for explaining how counter-productive this approach is for helping teachers understand the “whole urn.”

Good Morning All,

As has been said, our gratitude goes out to Bill and the writing team and our hope is that the CCSS WILL make a difference. I too have been devouring the Progressions documents. Here is a question. Has anyone or does anyone intend to make a Progression document in chart form by standards? I know it may take away from the coherence of the clusters, but think it would help teachers examine where individual students and groups of students are coming from, and where they are heading. Thank you for your consideration.

Nancy, I don’t know of such a chart, but I have used the Illustrative Mathematics project display of the standards to navigate to explore them with teachers. If you click on a progression you get all the grades in that progression, and can expand and collapse the cluster headings. It’s useful when you want to compare two similar clusters in consecutive grade levels, because you can collapse all the others and see the two side by side. I’ve done this on my iPad projected on a screen, where you can zoom in and out.

Nancy, are you thinking of a graphic like this (translation shown here on pages 80 and 81) from Japanese teachers’ manuals?

I, too, am looking for a sequencing document such as the one you described. Have you had any luck locating one? If I can’t find one that makes sense to me, I’ll make by own, but I’d love to see what others have come up with…

Me too!! Through the standards and the progressions documents I’m starting to see how the standards are sequenced, but it would be so helpful to have a graphic that laid out those connections.

I have tried what Bill recommends: Nancy, I don’t know of such a chart, but I have used the Illustrative Mathematics project display of the standards to navigate to explore them with teachers. If you click on a progression you get all the grades in that progression, and can expand and collapse the cluster headings. It’s useful when you want to compare two similar clusters in consecutive grade levels, because you can collapse all the others and see the two side by side. I’ve done this on my iPad projected on a screen, where you can zoom in and out.

This is good. In the future, perhaps a logical progression by mathematical standard will be written. I am patiently waiting.

Thanks, Bill, for another informative post on the common core and your take on the standards, the mis-treatment of them via fragmentation happens in lower grades also. We all become better teachers when we know the progression of standards as you’d mentioned, the charge is to get teachers on board teaching with better strategies and less focus on standardized state tests.

Bill, Can’t get funding to attend your workshop from Vermont. Will some or all of the information be available afterwards?? Please.

Constance C. Cannon,

Math Consultant

Constance,

I live in Montpelier and am also a math consultant in Vermont. I’m at the Tucson airport as I write this having attended the CCSS workshop, which was wonderful. I’m sure Bill will get back to you as per his plans to share the work that the group accomplished over this weekend. However, I’d be happy to share my experience with you and perhaps we can collaborate to get the word out about the Common Core in Vermont. My email is elaine.watson0729@gmail.com

Elaine

Constance,

We are sorry to have missed your expertise this weekend as we created professional development units together in Tucson. Many great tools and PD modules came out of the workshop. We are now in the period of revision by the authors and a bit of formatting before the modules are ready to share. Once they are ready we will be holding a much larger workshop to disseminate the information to even more regions, states, and districts nationwide. We will have information on the blog when the registration for the larger workshop at the end of April becomes available. Registration should be open soon!

And wonderful to meet you Elaine. I hope some collaboration can occur in VT!

Ellen

Thanks so much. I look forward to your modules.Connie

Good Morning,

The teachers in our public school system are asking me about three technical math concepts that they no longer see in the CCSS-M. If anyone has information, please let us know. In MA, our math framework had a term “Number Sentence” (equation) in the lower grades. I notice that equations are called equations at first mention. We also had “Stem-and-Leaf” plots in the Data Strand. Finally, an 8th grade teacher asked me about “Line of Best Fit.” I looked up all three terms in THE BIG BOOK (an index of CCSS-M terms) and they were not there. I am all ears. Thank you.

Dear Nancy, you are correct that number sentences are just called equations in CCSS-M, and that stem-and-leaf plots are not mentioned. As for lines of best fit, if you read the Grade 8 Statistics and Probability standards you will see that students informally fit lines to scatter plots and informally assess the closeness of the fit. More formal approaches using regression and correlation coefficients are in the High School standards on Statistics and Probability in the S-ID domain. What is the Big Book? I haven’t heard of it before.

Your reply is very helpful, thank you. Common Core State Standards for Mathematics (Massachusetts) Big Book was created by Center for Hands-On Learning. It lists almost everything included in the grade level standards and where they are found. It is a comprehensive index of mathematical language and where each vocabulary term/ concept is located. http://www.standards-toolbox.org/index.php

Best,

~Nancy

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While I see the concept of not unpacking the “big idea” standards, what might this mean for standards-based report cards?

Well, a separate reporting for the cluster headings would be a good idea, I guess.

In what ways do the ideas you present here reflect research on activating students as self-assessors and, ultimately, including students in the instructional decision-making process? At this point, your argument seems squarely at odds with the research of Wiliam, Black, Stiggins, Chappuis, and Clarke.

Can you please cite the specific education research consulted in making your argument? If no specific education research was consulted, can you please explain why?

Can you please elaborate on how this essay is at odds with the research you cite?

There are few practices in education that get the kinds of results associated with formatively assessing students and including students in the process of deciding how they and their teachers might close the gap between where students find themselves at any point in time in relation to where they are supposed to “end up” along the path towards mastering standards.

The researchers and practitioners I referenced assert that unless it is crystal clear to a student where they need to “end up” the benefits of formatively assessing students and including students in the process of instructional decision-making cannot be realized.

If a student is presented with any kind of standard in any subject, but the student doesn’t understand what the standard means, then teachers and students need to “unpack” the standard so that it does make sense to students. Once students and teachers clearly understand the way(s) a standard describes mastery of knowledge, reasoning, or performances, then both may engage in a diagnosis of the strengths and weaknesses a student’s work reflects in relation to the standard(s) attempting to be mastered. This diagnosis informs what gaps may exist between current understanding and mastery, and informs both teachers and students how they might close those gaps.

Your essay seems to treat all “unpacking” processes as a hammer smashing an urn; however, the word “unpacking” also refers to a process of analyzing standards and expressing them in a way that makes sense to students. This may, for certain students in certain grades, mean that a single standard may need to be expressed in its component parts; however, and this is what I believe you and yours are asserting, if expressing a complicated standard to a student leads either student or teacher to permanently shift perspective to a micro-level, and forget the macro-level from which the standard came, then we have problems.

In essence, just as mathematicians and students can “step back and shift perspective” and see “complicated objects as composed of several objects” in making sense of mathematics, so too can educators do the same with standards. Educators and students can unpack standards for the purposes of formatively assessing and student self-assessment and, as long as they “maintain an oversight of the process” and are aware of how “intermediate results” relate to the overall task at hand “unpacking” does not necessarily lead to a loss in the overall structure of the standards.

Thanks Seth, this helps me understand your perspective and understand a possible miscommunication in our original essay. We are certainly not opposed to detailed analysis of the standards, which is what I think you mean by “unpacking”. Using the urn analogy, I would say it is very useful to look closely at the details of the depiction and try to figure out what the story is (this is, indeed, the subject of Keats’ poem “Ode to a Grecian Urn”, which inspired the analogy). So if that’s what you mean by unpacking I’m all for it, although I would probably call it something different: more like designing an itinerary through closely observed details, with the design of the itinerary supported by research about what itineraries work best for kids.

It’s a complex process to design a curriculum to support the standards, and I don’t think kids should be “presented with the standards.” Rather, they should be presented with a curriculum which supports learning the knowledge described by the standards. On page of 8 of the documented at corestandards.org, it says

“These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.”

Seth:

I can appreciate your point. Anecdotally, I have noticed that my students who understand why we are learning a particular topic and how it fits into a larger picture, seem to do better at general problem solving and using the said topic knowledge to solve a variety of problems. However, inorder to communicate to students the larger picture and the “why are we learning this stuff” questions, teachers must have clear, understandable, standards. So far, I’m not finding this with these “new” Common Core Standards.

“Unpacking,” while maybe a little vague in terms of the process, makes sense to me. But “designing an itinerary through closely observed details, with the design of the itinerary supported by research about what itineraries work best for kids” as put by Mr McCallum, is confusing to me.

I hope that whatever new standards emerge that they are clear, concise, and complete. I hope that the result will not be that we end up teaching less.

John

Bill,

Thank you so much the article, “The Structure is the Standards”. It says so eloquently what I have been trying to explain to teachers for the past year. There are so many layers to, and connections between these new math standards.

The question we are teaching our students to ask themselves this year is: “What do I already know that can help me figure out this new problem?” Those connections that can be made to previous learning are what make the new standards so powerful. We are discovering that students are arriving at deeper levels of mathematical understanding, even in the early grades, as our teachers become more and more intentional about setting up learning experiences that allow students to make understanding for themselves. It has become clearer than ever the importance of teaching for understanding, rather than just teaching isolated skills.

I do have a question for you. Is fourth grade the first time that students specifically have to make conversions within the same measurement system? I have read everything that I can find and do not see anything at the third grade level that asks students to make conversions. A question came up about this recently and I just want to make sure that I have the correct answer.

Thank you!

Bill and others,

Kathy here from Vermont. I was just asked to define rigor, focus and coherence as they are used in the Common Core mathematics. I have some thoughts but I would appreciate your ideas before I get too deep into this work.

Kathy,

Elaine here from Vermont. A good resource to frame the ideas of rigor, focus, and coherence is the Hunt Institute videos featuring Bill McCallum and Jason Zimba.

http://www.youtube.com/watch?v=2rje1NOgHWs&feature=related

The ones I would recommend are:

“The Importance of Coherence in Mathematics”

“The Importance of Focus in Mathematics”

“Helping Teachers with Focus and Coherence”

“The Mathematical Standards and the Shifts They Require”

In the last one, the idea of rigor is, in my interpretation, defined as a cross section of focus and coherence. Rigor is accomplished through giving students conceptual understanding, procedural skill and fluency, and problem solving skills. Rigor is also accomplished by implementing the content standards that have a high cognitive demand for reasoning, sense making, and justification.The practice standards provide students with a habits of mind that help them to make sense of the content standards. Focus on fewer things gives kids more time to master the important understandings that are stepping stones in the stream of coherence. Mastery of one topic (given enough focus for students to learn) informs the next topic in the stream or progressions (helping students to make connections rather than memorize isolated facts.)

I look forward to working with you in our small state of Vermont to help teachers understand how to implement the CCSS-M!

Thaank you Elaineand now that you mention them , I know exactly what you are talking about. I think they might be on the DOE wiki..

i have actuallywatched them all before. i am working on a using them as an organizing thread for some work we need to do next week.

Talk to you tomorrow.

Kathy

Sorry…I didn’t mean to imbed the video in the reply! I just copied the URL. Oops!

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Another view of standards-based instruction: in the Bangor Daily News. The instructional organization described sounds a lot to me like what used to be called IPI (Individually Prescribed Instruction).

Bill:

I got this article sent to me by our department chair via our district math person. I can’t believe how poorly written it is. It is as if you didn’t even proof your paper. The run-on sentences never ended. There were even missing words. “Standards are a policy document, after all, not a work art.” While your message might be apt, it is lost in the lack of communication ability.

When I hear that the new standards replace “a mile wide and an inch deep” with less wide but more depth, I only hear less. The standards we currently use were an attempt to make our classes more rigorous. When they emerged some 20 years ago, I remember hearing the complaints from primary grade teachers about how much core subjects they would have to cover. They seemed upset because they wouldn’t be doing those fun and creative projects they had spent a lot of time developing and loved to do.

It feels like the pendulum is now swinging back in the other direction, less rigor, more art.

— John

Thanks for pointing out the typo. I’m happy to correct any others that you spotted.

Dr. McCallum,

Help me to understand how much latitude instructors should take with the sequence of the CCSS standards. I’m in NC and we are “rolling out” these standards with the so-called Essential Standards in other subject areas. I am not a mathematician or even a teacher of math. However, as an instructional leader (principal), I am endeavoring to understand as much as I can about the standards. (And math was my favorite subject in school)

For years I’ve have believed that we attempted to cover too many topics within our math curricula. When I read William Schmidt’s paper, A Coherent Curriculum I was relieved to find out that people a lot more learned than I saw what I considered to be part of the problem.

Some of the progressions are obvious, but some are not.

Fortunately, NC has developed some “unpacking” documents which are helpful in preparation for implementation. Help me with understanding what should and should not be done with these standards where the importance of sequence may not be so obvious.

Ron, a few thoughts. First, the sequencing of the standards is not intended to dictate their sequence in the curriculum. There’s a paragraph on page 5 that makes this point:

The progressions documents offer more guidance about sequencing, but even they don’t get at how domains might be intertwined. For example, in elementary school it would make sense to treat the Operations and Algebra Thinking and the Number and Operations in Base Ten domains in parallel, catching the many connections between them, rather than in sequence.

So, I don’t have a simple answer to your question. Ultimately sequencing is a matter of curriculum design, which take time if done well. All I can say is that the progressions documents are designed to help in this endeavor.

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Hello Dr. McCallum, Thank you for your article. I am a high school math teacher in Wisconsin trying to figure out how to teach the standards in my district, and I feel sometimes it’s a struggle to even understand the standards. I am hoping these progressions documents will help. One question I have about the standards is why are some of them, like F.IF.7 are so long and mention multiple topics, while others, like S.ID.5, are more narrow and specific. Are some meant to be ongoing while others are only taught at select points in the curriculum?

I also am wondering how teachers are supposed to implement the standards with curriculum that, as you say, “which supports learning the knowledge described by the standards” when no such curriculum seems to exist. Right now the options are to search through our current curriculum to figure out what matches the standards or to create it completely on our own. Both are a lot of work and imperfect as they rely on our correct interpretation of the standards.

Sorry that can’t frame all this into a specific question, but any suggestions or resources you can give me would be greatly appreciated.

Thank you, Alice

Your analogy might have more meaning if instead of examining urns you examined content and its distribution relative to standards when it comes to structure.

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