Everyday Mathematics and the Common Core State Standards for Mathematics
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Since the release of the Common Core State Standards for Mathematics (CCSSM) in June 2010, the authors of Everyday Mathematics (EM) have been working to understand these new standards. In this article, Andy Isaacs, director of EM revisions, outlines how the EM authors went about responding to the CCSSM. He then answers some frequently asked questions about the CCSSM edition of Everyday Mathematics.
Because the Everyday Mathematics authors kept abreast of the development of the CCSSM, we were not caught unaware by its release in June 2010. Fortunately, we were in a good position to respond effectively, because in the spring of 2010 we were finishing more than two years of work updating the third edition of Everyday Mathematics. This “copyright update” focused on improving EM’s tools for assessment and differentiation, expanding the range of technology available for the program, and aligning EM with NCTM’s Pre-K–8 Curriculum Focal Points (2006) and other key documents.
The release of the CCSSM meant, on the one hand, that the copyright update was obsolete for the more than 40 states adopting the new standards. On the other hand, a substantial amount of good work had gone into the creation of the copyright update, much of which would be useful in a CCSSM edition of EM. More importantly, we had an experienced team of authors and editors available to begin work immediately on a CCSSM edition of EM. So, in June of 2010, we put aside completion of the copyright update and turned our full attention to CCSSM. I doubt that any other curriculum project in the nation had a similarly qualified group of authors and editors available to begin work on CCSSM curriculum development on such short notice.
First we carried out detailed studies of how EM aligns with CCSSM. It was immediately obvious that there are huge areas of agreement. CCSSM’s Standards for Mathematical Practice, for example, are very much aligned with what EM has always been about. Its emphasis on developing both skills and understanding also fits well with EM. And almost all of EM’s learning trajectories and content standards are consistent with CCSSM. These large areas of agreement did not surprise us to the extent that both EM and CCSSM are based on the same body of U.S. and international research in mathematics education and the learning sciences.
We also recognized that there are areas where the CCSSM has useful ideas that can improve EM. The CCSSM’s emphasis on unit fractions, for example, is based on compelling research into children’s developing understanding of rational numbers and is something the new CCSS edition of EM incorporates. The CCSSM’s emphasis on strategies for math facts is also helpful and pushed us to make the development of such strategies more explicit at earlier grades. There is much in CCSSM that helped us improve EM.
The CCSSM’s Standards for Mathematical Practice proved to be particularly powerful because they helped us address a problem with EM that we had been working on for more than a year. The Grade-Level Goal (GLG) structure we introduced in the third edition of EM provides a useful framework for tracking instruction, practice, and assessment of factual knowledge, procedural skills, and conceptual understandings. However, a well-known problem with such frameworks is that they tend to under-emphasize higher-level process outcomes such as problem solving, reasoning, and communication (because it is easier to write goals and objectives for lower-level outcomes than for higher-level outcomes). As we worked on the copyright update, we had been wrestling with this problem, trying to find a way to raise the profile of higher-level processes in our GLG framework.
These processes are precisely what CCSSM’s Standards for Mathematical Practice emphasize, so creating a CCSS edition of EM has given us a great opportunity to raise the profile of these higher-level outcomes—which have always been central to EM but can get lost when too many goals focus on specific facts, skills, and concepts. Figuring out how to operationalize the Standards for Mathematical Practice was not easy, both because breaking higher-level standards into smaller, more explicit goals is inherently difficult and because CCSSM’s Standards for Mathematical Practice are not always transparent. But, after nearly two years of work, we have made good progress toward making explicit how EM can be used to meet the Standards for Mathematical Practice. We recently released a set of documents to help EM teachers translate the ideals of CCSSM’s Standards for Mathematical Practice into a daily classroom reality. (These documents are available on the McGraw-Hill Education website at https://www.mheonline.com/emcrosswalk; use password: CCSS2007support (case sensitive).)
We also identified several areas where we differ with the CCSSM writers. One involves our approach to data. The CCSSM largely ignores data until Grade 6, at which time children are introduced to mean absolute deviation, interquartile range, and a variety of other fairly sophisticated topics. Most statistics educators believe such an approach enters the realm of statistics at too high a level, as the American Statistical Association’s 2007 Guidelines for Assessment and Instruction in Statistics Education (GAISE) report makes clear.
Working with data before grade 6 is important for a number of reasons:
- To build a foundation for data work in later grades
- To help children make connections between mathematics and the world around them
- To support the development of number sense
- To support early algebraic thinking by, for example, exploring functional relationships in bivariate data
- To connect mathematics to science, social studies, and other disciplines
We, therefore, made a deliberate decision to maintain work with data in all grades beyond what is required by CCSSM’s Content Standards. But this apparent mismatch between EM and CCSSM is less serious than it might appear since CCSSM’s Standards for Mathematical Practice require extensive work with modeling and real-world applications at every grade. While not all modeling and applications involve data, a great many do. If children are to model the world around them with mathematics, then data must be involved. So it remains important to include work with data in any CCSSM-aligned program for grades K–5.
As we worked on our analysis and attended to the public comments and debate about CCSSM, it also became evident that certain aspects of the CCSSM have sometimes been misinterpreted. For example, some people seem to think that the CCSSM requires that technology, including calculators, be eliminated in the elementary grades. The reason for this misinterpretation of the CCSSM 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 CCSSM writers had no such thing in mind. In fact, one of the CCSSM 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 the CCSSM bans technology from K–6 would be a mistake.
Another common misconception is that the CCSSM requires a particular kind of “focus.” That is, some people are interpreting the CCSSM as requiring that topics be treated once in depth so that mastery is achieved immediately and forever. This interpretation of “focus” does not reflect research into how people learn. One can refer to our bibliography at http://everydaymath.uchicago.edu/about/research-results/distributed_practice_bib.pdf that highlights the research underlying EM’s approach to distributed practice. That research is reflected in the program’s spiral. There is nothing to indicate that this research-based approach conflicts with anything in the CCSSM.
In any case, when the CCSSM specifies topics for a given grade, the intent is clearly that those topics are to be mastered no later than at that grade—but such mastery must be based on preparatory work in earlier grades and must be followed with appropriate practice, applications, and extensions in later grades. That the CCSSM writers were thinking in terms of multi-grade learning trajectories is clear both from the CCSSM document itself and also from talks the CCSSM writers have given since its release and in subsequent documents created about learning trajectories. CCSSM’s standards are thus similar to EM’s Grade-Level Goals, which are also mastery goals for the grade at which they are listed—and for which important preparatory and follow-up work is built into the program.
One way to think about “focus” is related to a question we often receive about “in-depth” instruction. EM certainly does aim at in-depth understanding, namely, in-depth understanding that is long lasting. That is, EM is based on the fact that one does not achieve in-depth understanding that endures by focusing exclusively on a topic until the children seem to have mastered it. When that is done, it is easy to confuse short-term performance with long-term learning.
In-depth understanding is achieved by repeatedly returning to a topic after time has passed—focus over time, if you will—so that learners will have to remember what they learned before. A curriculum built to do that, as EM is, will be more efficient at developing in-depth, long-lasting understanding, but may also appear to be less “focused” at the lesson level because several topics may be treated in a single day. Such a curriculum will also enable students to perform at high levels on standardized tests, which typically include questions on a wide variety of topics, as well as in real life, where problems involving fractions, for example, don’t come at the end of the chapter on fractions.
Our analysis of EM in light of the CCSSM also identified a number of areas in which what the CCSSM requires is simply different from what EM provides. In such cases, we made adjustments to help EM teachers meet the CCSSM’s requirements. Consequently, one of the changes in the CCSS edition of EM is that the schedule for fact mastery has been accelerated, though with provisions for “catching” those children who fail to achieve mastery on the schedule stipulated by the CCSSM.
The number of changes we have made in EM to better align it with the CCSSM is quite large. Our planning documents for these revisions ran to hundreds of pages. Many of the changes we have made are subtle. Most often, we adjusted existing activities in various ways to bring them in full agreement with what the CCSSM requires. Other times we replaced activities with new activities and sometimes we wrote entirely new lessons and projects. It’s always hard to quantify the percentage of new material in a new edition, but a majority of pages in the Teacher’s Lesson Guides were touched in some way: sometimes just with a marginal note pointing out or clarifying a connection to the CCSSM, sometimes with extensive reworking of an activity, sometimes with entirely new material replacing or supplementing what was there before. As a result of all this work, every lesson in EM supports important standards in the CCSSM, and the program as a whole aligns very well with the CCSSM’s requirements both for in-depth treatment of concepts and skills and for the development of mathematical practices.
We feel confident that no other program has made such an extensive and thorough effort to align itself with the CCSSM. And our work in that regard continues as we research new approaches and ideas that will improve EM in the future.
Frequently Asked Questions
What is the difference between the standards in CCSSM and EM’s Grade-Level Goals?
The standards in CCSSM are goalposts, destinations to be reached no later than certain designated points in time. They mark important things to aim at over long periods. Standards are not necessarily articulated in coherent trajectories; they are simply endpoints, though, if standards are well conceived, they should not be at odds with research-based developmental learning trajectories. So, for example, the standards in CCSSM are not explicitly connected in multi-grade learning trajectories.
EM’s Grade-Level Goals, on the other hand, are explicitly connected in learning trajectories that span several grades. EM’s Grade-Level Goals are also often more detailed than the standards in CCSSM. EM’s Grade-Level Goals provide a framework that connects instruction, assessment, and interventions in the program. EM’s Grade-Level Goals, instructional activities, assessments, and interventions delineate pathways or learning progressions that ensure that children will meet the standards laid out in CCSSM.
How should we go about aligning EM to the CCSSM?
The EM authors have developed a variety of resources to help teachers and students address the Standards for Mathematical Content as well as the Standards for Mathematical Practice. The largest of these resources is the new 2012 copyright update (the CCSS Edition) of the 3rd Edition (2007) of Everyday Mathematics that is fully aligned with the CCSSM. This new edition became available in summer 2011.
Along with the CCSS update, another tool, the EM Crosswalk to the CCSSM, was launched as a free online resource. The crosswalk is a document designed for teachers to have access to all additions and changes made to the 3rd Edition of Everyday Mathematics in order to align 100% to the CCSSM. To access the crosswalk go to the following URL and type in the password given, which is case sensitive:
password: CCSS2007support (case sensitive)
The same URL provides access to a document about EM’s alignment to the CCSSM’s Standards for Mathematical Practice. This document translates the CCSSM’s complex descriptions into everyday language that is accessible to teachers and students. Specific questions for the initial units are available at that site. We recently completed sets of guiding questions related to the Standards for Mathematical Practice for virtually every EM lesson in all grades. These updated documents are currently available for free download through the Everyday Mathematics Virtual Learning Community (VLC): http://vlc.cemseprojects.org/. Thousands of EM teachers have already begun to access the many resources available at the VLC. Note that you will have to establish a free account in order to access the VLC resources, but we think you will find the short process well worth it.
My district thinks “we’ll need to unspiral EM” this year and teach only to the CCSSM. The discussion went into how teachers would need to pick and choose which lessons/parts of lessons they will teach. Not only will this be a teacher’s nightmare when it comes to planning and using math journals and home links, but it will turn EM into something else completely! Any advice or help in how to keep EM intact while honoring the guidelines my district has directed pertaining to the national core standards?
Rather than trying to break the curriculum down into individual standards, the EM authors recommend focusing on the Standards for Mathematical Practice and tightening up the portions of EM that are closely aligned with the CCSSM but are often implemented inadequately. Even with the many rich problems in EM, teachers often over-scaffold the instruction, doing much of the thinking for the students. Strengthening the implementation of the Standards for Mathematical Practice, combined with EM’s rigorous and coherent content, will do more to help your instruction align with the CCSSM than tearing the program apart.
What did you take out of EM as you aligned with the CCSSM? How will we have time to teach the new material if nothing has been removed?
For the CCSS edition, we took out some things, moved quite a lot of things, increased or decreased emphasis on certain topics at certain grades, and modified the treatment of many topics. The net addition of content was, in the end, quite modest, on the order of five or ten new lessons out of the more than 600 lessons in the program. We have been working with many teachers who have been able to get through the 2007 edition with time to spare, so we expect teachers will be able to get through the CCSS edition as well.
Research has shown a couple of things that are important to keep in mind. One is that time on task is hugely important. The more time you spend on something, the more children are going to learn. If you want children to learn more mathematics, the first step is to allot more time to it. One hour a day, every day, should be the minimum. Even better would be 75 or 90 minutes a day.
Another important research finding is that distributed instruction and practice over time is more effective than concentrated practice. This finding is well established in the research literature on learning, but is counterintuitive and often unfamiliar. Most people think concentrated study is more effective. As the daughter of one of the EM authors said, when she got an A in Latin, “See, Dad, cramming works.” But cramming doesn’t work if your goal is long-term learning rather than a short-term boost in performance for a test. What you’ve crammed, you quickly forget. You have at best the illusions that you have learned something.
EM is built on the principle of distributed instruction and practice. This is entirely in harmony with the CCSSM, which is based on learning progressions that span multiple grades and which also does not dictate which pedagogical approach is to be used.
Rather than adopting a distributed approach, some teachers get bogged down trying to teach to mastery skills that are just being introduced. This slows down progress through the curriculum so that teachers feel even more time pressure. We’re working with teachers and schools in Chicago and across the nation to better explain the distributed approach and why it’s more effective. We’re also working with teachers and schools that are changing how they organize for instruction. Many teachers have had success “borrowing” time for mathematics from across the school day. During the literacy block, for example, students who are not meeting with the teacher can do Math Boxes or Journal pages or even play math games. Many teachers have also had success using a “centers” approach with EM. All this can help with an overcrowded school day in which finding sufficient time for mathematics can be difficult.
Shouldn’t there be more changes to be 100% aligned as you claim? The teachers are not feeling like they are reaching or will reach mastery with the limited changes and additional material to cover that isn’t at their grade level.
As we did our alignment to the CCSSM during the planning phases of this edition, we found that in many cases EM was well aligned with the standards, but needed some clarification to make this alignment explicit. One example is Standard 6.EE.9, which reads “Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.” In G6 EM, students already did much work using equations, graphs, and tables to represent real-world problems. We were simply not using the terms dependent variable and independent variable. It did not make sense to add a lot of work to “cover” this standard—it was a matter of including appropriate vocabulary to work the students were already doing. We found that much of the coverage that we added was of this type: clarifying and pointing out where we already are well aligned to the standards.
How have EM’s assessments changed to reflect the added CCSSM content to the program?
New “Recognizing Student Achievement” assessments were added in the Teacher’s Lesson Guides, where appropriate, to assess new CCSSM content that was added to the program. Whenever a Grade-Level Goal was changed to better fit CCSSM – when memorization of the multiplication facts was moved from fourth grade to third, for example – unit Progress Checks and assessments in the Assessment Handbook were updated to reflect the new goal. But whenever EM’s assessments gave adequate information about progress towards mastery of the Grade-level Goals, those assessments were not changed. These changes preserve the basic structure of EM’s assessment system, which is designed to track progress towards Grade-Level Goals.
Assessment in the context of the Common Core State Standards is a moving target. The U.S. Department of Education has funded two large consortia of states, PARCC and SBAC, to develop new assessments aligned with CCSSM. Those assessments are currently scheduled for use beginning in the 2014–2015 school year, but details of the consortia’s plans are not yet available. In the meantime, most states continue to use state tests based on pre-CCSSM standards. Some states are using their current assessments this year, modified assessments better aligned to CCSSM in 2013 or 2014, and assessments from PARCC or SBAC in 2014–2015. In this evolving environment, the EM authors are closely following the work of the consortia so that EM’s assessments can be adjusted to better complement instruments from the assessment consortia. The authors are also working with EM’s publisher to create interim solutions such as revised assessments tools and items that will be embedded in the program, test item generation software, and an online assessment and differentiation system.
Grade 6 has a lot of changes but not a lot of new Journal pages. Why?
We tried to make as many changes as possible within the context of existing lessons, without adding extra pages that needed to be completed. In some instances, this was as simple as including appropriate vocabulary in the lesson itself. When we added new lessons or activities, we also added new journal pages. When a lesson or part of a lesson was revised, the journal pages were revised to reflect the content in the lessons, but additional pages were not necessarily added.