By Leland S. Cogan, Nathan Burroughs, and William H. Schmidt

*Researchers have developed a tool to help teachers implement the Common Core by letting standards, not textbooks, guide their instruction.*

Around the world, mathematics teachers rely on textbooks to support their classroom instruction. In other countries, there is a close connection between available textbooks to support teachers’ instruction and the official standards that specify what students are to learn and what teachers are to teach. This connection is much looser in the U.S. as textbook publishers have been faced with producing curriculum materials to support as many different sets of standards as there are states and territories.

The introduction of the Common Core State Standards in mathematics in 2010 sought to address this challenge. However, articles in popular media such as *Education Week, The Washington Post*, and National Public Radio have noted that teachers are confronted with using current textbooks that were written to address last-century standards and don’t line up well with the Common Core (Robelen, 2013, 2014; Chandler, 2014; Herold, 2014; Kotlarek, 2015; Turner, 2014). This creates a major barrier to the success of the new standards. Even newer textbooks reviewed by EdReports.org provide teachers with a substantial challenge to align instruction with the standards (EdReports.org, 2015).

The Textbook Navigator/Journal, developed by the Center for the Study of Curriculum at Michigan State University, is a web-based tool for aligning mathematics instruction with the Common Core. Using the Navigator, teachers can pick a standard and ask which portions of the textbook cover it, or they can use the Navigator to identify which Common Core standards are embodied in a particular lesson in the textbook. The Navigator lets teachers control their mathematics instruction, liberating them from rigidly following textbooks and allowing them to focus on teaching content their students are expected to learn.

The Navigator is based on the results of careful analyses of 34 textbook series and 185 individual mathematics textbooks (See Table 1). We used two criteria for choosing books. First, as part of a 2010 survey, a random sample of school districts in the 45 states that had adopted the Common Core at that time, district curriculum directors were asked to indicate their current textbook series and when they expected to buy a new one. The responses to these questions were used to draw a sample of books that cumulatively would be used by about half of the students in the 45 states. Curriculum directors said most districts didn’t plan to buy a new series for five years or more; one quarter of them are probably waiting a decade or more to buy new books. As a result, a substantial percentage of teachers will be expected to implement the Common Core with older, pre-Common Core textbooks.

Second, the research team sampled recently published books that were marketed as Common Core-aligned. Although not yet in widespread use, we compared older textbooks to a small sample of the books (10) that seemed to best represent the new generation of textbooks.

Once the books were selected, the research team organized a system of coding lessons in the textbooks to assess their alignment to the Common Core mathematics standards. The methodology used in coding the books is based on that used in the original Third International Mathematics and Science Study in 1995 (TIMSS). In TIMSS, trained coders analyzed more than 100 textbooks from about 40 countries. There was high cross-rater agreement with reliabilities ranging from 75% to 90% depending on the level of specificity of the codes. That same methodology was adapted to do research related to a Longitudinal Study of Adolescent Youth and as a part of the 2005 National Assessment of Educational Progress high school transcript study. As part of its effort to study Common Core implementation, the Center for the Study of Curriculum refined the coding framework from the TIMSS study, which focused on mathematics topics, to one that was defined by the Common Core standards.

However, this was done so the two frameworks could be mapped onto each other. This was done as a part of an Ed Researcher article comparing the Common Core math standards to the international benchmark developed in TIMSS. The framework was changed but the basic methodology remained the same.

The textbook coders were rigorously trained and monitored. The same personnel who oversaw the TIMSS textbook coding also oversaw and evaluated the quality of the Common Core textbook coding, randomly sampling coded lessons to ensure quality and consistency. Coders had strong math backgrounds, including math specialist undergraduates and graduate students and former or current mathematics teachers. Coders coded each lesson in the textbook by identifying the Common Core standards that lesson addressed. Coding reflected the focus (the main purpose) of each lesson. Coders were given a broad mandate to identify any standards (including none) present in a lesson — including those from other grades. This approach was designed to reduce the risk of confirmation bias and to let us determine whether the textbook material was “on grade.”

Before turning to a further description of the Textbook Navigator/Journal for teachers, we present some general findings that describe the consistency of a typical textbook with the Common Core. These general findings suggest why teachers sorely need a tool like the Textbook Navigator/Journal.

**Coherence**

International research shows that coherence — that math topics follow a logical sequence both within any single year and across grades — is an important feature of math instruction. A principal goal of the Common Core is to bring greater coherence to U.S. mathematics teaching. In general, textbooks should cover all standards at each grade level to reflect the coherence called for in the standards.

Unfortunately, our analysis indicates that no textbook series covered 100% of the on-grade standards called for in the Common Core. Averaging over all eight grades and all textbook series analyzed, the estimated percentage of Common Core standards appropriate to that grade that were covered was 72%, ranging across the 34 series from 42% to 98%. In other words, textbooks were missing on average one-quarter of the necessary standards at each grade. First-grade textbooks had the higher average fidelity to the Common Core (82%); 6th grade and 5th grade had the lowest average coverage of on-grade standards (about 60%).

There was substantial variation across textbooks in general. One 6th-grade textbook had less than 20% of the grade-appropriate standards in the books’ lessons, compared with seven books where all the standards were covered (three at 7th grade, and two each at 1st grade and 4th grade). Again, no series was 100% compliant with the Common Core at all grade levels. One encouraging note emerges from the data: The post-2011 textbooks we studied covered, on average, a higher percentage of the grade-specific standards (82%) than did the older books (64%).

The failure to adequately cover all of the necessary material at each grade level can have serious long-term effects. Mathematics learning is a cumulative process, with earlier topics providing a critical foundation for understanding later topics. By neglecting to address all of the standards at the proper time, incomplete textbooks will force teachers to search out supplementary materials and pose obstacles to student academic achievement.

**Focus**

A second key feature of the Common Core supported by international research is the concept of focus: At each grade, instruction should center on a small number of key topics or ideas. This allows a more thorough grounding in the material before students move on to more advanced topics. For textbooks to be focused, they should not only cover all grade-specific standards and also refrain from including too many additional topics. The latter has the effect of diverting limited instructional time away from the on-grade mathematics.

Looking first at the number of off-grade standards covered in the textbooks, we found that on average only about half of the total number of standards covered in the lessons were appropriate for that grade. The average is somewhat higher in newer books (63%) than older books (37%), suggesting that Common Core-aligned textbooks are more focused — with about one-quarter fewer off-grade standards covered.

Simply counting the number of standards addressed could be misleading, however. Some standards may only receive minor coverage in textbook lessons; for example they might be brief review lessons. A more accurate gauge of a textbook’s focus is to count the number of instructional days devoted to on-grade standards. Each textbook includes a guide indicating the number of days that should be used for a particular lesson. As another method of examining focus, we used the number of days specified by the textbooks for coverage of a lesson to weight the relative importance of the standard being covered. To standardize the measure across all textbooks, the relative importance of each standard was calculated against a total of 160 instructional days.

Weighting the standards by the number of days, we estimated that on average the analyzed textbooks allocated between 62% and 74% of their class days to the grade-appropriate standards. This means that at least one quarter of the instructional days focused on off-grade standards. In other words, students are likely to spend between eight and 13 weeks on extraneous material.

For some books, the problem is much worse, with as much as two-thirds of the school years spent covering off-grade standards. Meanwhile other textbooks allocate virtually all of their instructional days to grade-relevant Common Core standards.

There is also a substantial difference between newer, Common Core-based textbooks and those that pre-date the Common Core. Older books allocate between 10 and 15 weeks to off-grade material, compared with six to eight weeks for newer books. Clearly the newer textbooks are more focused, allocating most of the 160 days to covering lessons reflecting the Common Core standards. However, a substantial proportion of time is still dedicated to off-grade topics. The Textbook Navigator/Journal is designed to help teachers make more effective use of their current textbook in implementing the Common Core standards.

**The Textbook Navigator**

The coding of the 34 commonly used textbook series summarized above forms the foundation of the Navigator, which is designed to help teachers implement the Common Core by letting the standards, not textbooks, guide the process.

Metaphorically, the Navigator provides a table of contents for each textbook that connects each Common Core standard to all the textbook lessons that cover that standard as well as to the relevant lessons found at other grade levels within that same textbook series. Teachers can use the Navigator in two ways. First, they can use the Navigator to determine which standards are covered in which lessons. Second, teachers can select a given grade-level Common Core standard and will be presented with all of the lessons in the textbook that focus on that particular standard. Using this method, teachers can decide what order they wish to cover the textbook lessons (and which lessons they can and probably should skip), rather than being tied rigidly to the order defined in the textbook. If there are no lessons of the textbook covering a particular standard, the Navigator will point to several free, on-line sources of curricula materials. These sources are included for the purpose of providing teachers a first step in seeking supplementary materials, without necessarily explicitly endorsing them (although we hope to periodically update those links).

**Summary**

*US K-8 textbooks skip one quarter of required math topics.*

Of 34 textbook series analyzed, not one covered 100% of the Common Core math standards at the appropriate grade. A major goal of the Common Core is to bring coherence to U.S. mathematics teaching; a goal that can only be met if textbooks cover all Common Core math standards at the intended grade level. This is crucial because mathematics learning is a cumulative process; earlier topics provide a critical foundation for understanding later topics.

*Only half of a typical classroom K-8 textbook focuses on grade-appropriate math.*

Another key feature of mathematics learning is focus – i.e., instruction at every grade must center on a small number of key concepts/topics. Focused textbooks should not only cover all of that grade’s standards, but they should also refrain from including additional topics. This is confusing to teachers, often wastes precious instructional time, and dilutes students’ attention.

*Newer K-8 math books’ Common Core math coverage is better but not good enough.*

Although Common Core-related math textbooks post 2011 address more grade-specific standards than the older textbooks even the newer books, nonetheless, devote a lot of valuable class time to the wrong kinds of math content.

*Misalignment of U.S. K-8 math books to Common Core creates unequal learning opportunities.*

Especially in the lower grades, most teachers aren’t trained as specialists in mathematics so they typically closely follow the instructional path provided by the textbook; a practice that is common around the world. The consequence is that students fortunate enough to be in districts with more coherent and focused math textbooks receive much better organized math instruction. Thus the great variation in Common Core math coverage in the lessons of various mathematics textbooks for the same grade contributes substantially to educational inequality. These issues were explored in our recent book, *Inequality for All* (Schmidt & McKnight, 2012).

Whether a teacher has a textbook published before the 2011 introduction of the Common Core math standards or one published post 2011, the Textbook Navigator/Journal can provide assistance in navigating the textbook to provide coherent and focused Common Core math-informed instruction to students.

**REFERENCES**

Chandler, M.A. (2014, February 24). Are math textbooks ready for Common Core? *The Washington Post*. www.washingtonpost.com.

EdReports.org. (2015). Series reports. www.edreports.org/reports/grade-level/compare.html

Herold, B. (2014, February 21). Boasts about textbooks aligned to Common Core a ‘sham,’ say researchers. *Education Week.* http://blogs.edweek.org.

Kotlarek, C. (2015, April 07). There is no ‘Common Core math’ — only good and bad teaching materials. *MLIVE*. www.mlive.com.

Robelen, E. (2013, April 25). All is Common-Core aligned at math education exhibit hall, or is it? *Education Week. *www.ewa.org/blog-ed-beat/common-core-recipe-national-curriculum-survey-says-no

Robelen, E. (2014, November 14). Is Common Core a recipe for national curriculum? Survey says ‘no’. ewa.org/Ed Beat blog. www.ewa.org.

Schmidt, W.H. & McKnight, C.C. (2012). *Inequality for all: The challenge of unequal opportunity in American schools. *New York, NY: Teachers College Press.

Turner, C. (2014, June 03). The Common Core curriculum void. National Public Radio. www.npr.org.

**LELAND S. COGAN** (cogan@msu.edu) and **NATHAN BURROUGHS** are senior researchers at the Center for Study of Curriculum. **WILLIAM H. SCHMIDT** is university a distinguished professor and director of the Center for the Study of Curriculum, all at Michigan State University, East Lansing, Mich.

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