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Curricular Coherence in the Age of Open Educational Resources

By Matt Larson, NCTM President August 22, 2016

From its very founding, NCTM has actively promoted the use of high-quality curricular materials to support effective mathematics teaching and student learning. A critical feature of high-quality curricular materials is that they are coherent. Coherence, with respect to mathematics curriculum, generally means that connections are clear and receive emphasis from one year to the next, from one concept to another, and from one representation to another. High-quality materials are coherent pedagogically, logically, and conceptually. More than 15 years ago, NCTM enunciated the Curriculum Principle in Principles and Standards for School Mathematics (NCTM, 2000): “A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.” Fourteen years later, NCTM reinforced the importance of curricular coherence in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014): “An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world.” NCTM is certainly not alone in advocating curricular coherence. The authors of the Common Core State Standards for Mathematics identified coherence as one of their guiding principles and organized the content standards into clusters and domains that weave content together from grade to grade or topic to topic to make conceptual connections and coherence more obvious to teachers and curriculum developers alike. The increasing availability of online instructional materials—some of which are of high quality and some of which are not, and many of which can be downloaded at no cost—has added a new dimension to the curricular landscape for mathematics teachers and school districts. Some of the most engaging conversations about mathematics teaching today are taking place within online communities where teachers share instructional resources and ideas that they have either created themselves or found on their own online. A recent survey by the RAND Corporation found that the vast majority of math teachers, at both the elementary and secondary levels, reported they used materials that they developed or selected themselves to implement the Common Core State Standards for mathematics. There is no question that this practice is widespread. The dilemma is that while districts, schools, and teachers have greater access than ever to tools and resources for selecting and developing instructional materials, the skill required to develop a high-quality curriculum is both complex and often underappreciated. The widespread availability of online tasks therefore makes having and working with a coherent curriculum at the school and district level even more important because it is the curriculum that establishes the learning goals in a coherent progression and helps teachers see and understand the multiple pathways that students might take through the progression. NCTM itself has published online materials that provide examples of curricular resources that encourage teachers to integrate high-quality mathematical tasks and problems into their mathematics instruction. These materials stand as examples for teachers and schools in cases where the core materials may lack highly engaging, high-cognitive demand tasks or lessons. NCTM’s recent publication of exemplar Activities with Rigor and Coherence (ARCs) is an example of one such online resource. Each ARC is a series of lessons that addresses a mathematical topic and demonstrates the vision of instruction that Principles to Actions describes in detail. ARCs integrate a wide array of NCTM resources and include

community features that offer opportunities for social interaction, feedback, and ratings. Ideally, teachers who select online instructional resources and engage in online community discussions would not be working in isolation but within well-developed professional learning communities in their schools. This sustained colleague-to-colleague communication would increase the likelihood of the selection of high-quality tasks that fit within mathematical learning trajectories and support the school and district’s curricular goals for students. Whether such collaborative task selection is feasible or not, the selection of online materials should be done in such a way that the instructional materials used in classrooms are situated within an overall coherent curriculum. That lessens the chance that students’ learning experiences devolve into a mere “collection of activities” rather than a coherent, well- designed curriculum. Stated very simply, the danger in online curricular selection is the undercutting of curricular coherence by the introduction of disjointed tasks that are of questionable quality, do not fit within the mathematical learning progression, and are incoherent. Perhaps the greatest danger is the potential for vast inconsistencies in instruction and highly variable learning experiences for students that in turn can lead to differences in student learning outcomes. Without question, curricular coherence is highlighted and enhanced when teachers work collaboratively and regularly with colleagues at the school level to plan instruction, implement the task, anticipate student work, respond to student learning needs, and provide consistency in curricular aims and instruction for students—no matter what teacher students might be assigned. Easy access to online tasks and communities makes the need to work collaboratively with colleagues in local professional learning communities more critical than ever before in the interest of safeguarding consistency in student learning experiences and outcomes.

Acknowledgement I would like to thank NCTM’s Emerging Issues Committee for its thoughtful work on a framing paper that was the basis for this President’s Message.

References National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: Author.

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Curriculum An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world.

What is meant by curriculum? In many cases, educators and community members use the terms curriculum and textbooks interchangeably, just as many often collapse the distinction between standards and curriculum. Standards are statements of what students are expected to learn. Standards are the ends. A curriculum is the program used to help students meet the standards, including instructional materials, activities, tasks, units, lessons, and assessments. The curriculum is the means.

Standards should be designed with intended learning progressions (or trajectories) across the pre-K–12 spectrum and beyond. The design of a curriculum implies a “sequence of thoughts, ways of reasoning, and strategies that a student employs when learning a topic” (Battista 2011). For example, in CCSSM, mathematical ideas build developmentally year by year on what came before, with students making connections to prior topics while laying a foundation for future learning (Daro, Mosher, and Corcoran 2011). Consequently, curricula based on CCSSM should be designed so that students and teachers can make mathematical connections across content topics that capitalize on CCSSM’s underlying structure, so that, for example, students can appreciate the use of a geometric model when exploring a number pattern or the use of ratios when analyzing a probability problem. The broad view of learning progressions in any set of college- and career-ready standards must guide both the work of schools and districts in developing curricular frameworks and other instructional resources and the efforts of developers of textbooks and other instructional materials.

Mathematics curricula can be characterized from both a horizontal and a vertical perspec- tive. From a horizontal perspective, teachers need an in-depth understanding of the math- ematics and materials that they use to teach a particular course or grade level. From this perspective, fourth-grade teachers need a deep understanding of all the content to be ad- dressed that year, the concepts and skills that need to be taught, how the topics connect with one another, how the mathematics content is sequenced, how much time might be needed for each topic, what tools (such as textbooks, materials, and technology) are available to support the content, and how to assess student understanding of the fourth-grade content standards.

From a vertical perspective, fourth-grade teachers need to understand what the students have learned in the past, how this year’s curriculum builds on students’ prior knowledge and experiences, and how the mathematics content that is studied this year will lay the foundation for topics that students will explore in fifth grade and beyond. A vertical understanding of the curriculum helps teachers engage in dialogue with colleagues who teach in grade levels below or above their own grade level (or with colleagues who teach the previous or the next

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course in a high school sequence) so that they can examine strengths and weaknesses of the overall program to prioritize the needs of students.

Obstacles Content included in textbooks influences what is taught and emphasized by teachers in the classroom (Schmidt, Houang, and Cohen 2002; Tarr et al. 2006). Some textbooks are effectively organized to focus on big mathematical ideas, such as those outlined in CCSSM and state or provincial standards, and to emphasize connections among topics. Unfortu- nately, others are less effectively organized, and schools often place too much emphasis on adhering to the content and sequence of such materials. Moreover, some teachers’ lack of deep understanding of the content that they are expected to teach may inhibit their ability to teach meaningful, effective, and connected lesson sequences, regardless of the materials that they have available.

Grade-level mathematics content standards are too often treated as checklists of topics. When they are regarded in this light, mathematics content becomes nothing more than a set of isolated skills, often without a mathematical or real-world context and disconnected from related topics. A typical traditional high school mathematics course sequence that spends a year on algebra, a year on geometry, and another year on algebra frequently focuses on covering a list of topics rather than on presenting a coherent program that “uncovers” those topics, establishing connections among them throughout the three years. Similar effects can be seen in other grades when the school year is organized into disjointed units addressing different domains of mathematics.

Even with the best curriculum model, lesson planning in some classrooms is conducted on a day-to-day basis, blindly sequenced by sections in a textbook, with little attention to the broader curriculum, contextual applications of the mathematics, or progressions of the topics and how they fit together. Furthermore, curriculum maps and pacing guides often dictate the topic, and sometimes even the page number of a book, to be addressed on each day of the school year, without regard for differences among students and classes. Teachers using pacing guides tend to feel rushed and, as a result, they often omit rich and challenging problem-solving tasks that are essential for developing deeper mathematical understanding (David and Greene 2007).

The table on the next page compares some unproductive and productive beliefs that influence the implementation of an effective curriculum. It is important to note that these beliefs should not be viewed as good or bad, but rather as productive when they support effective teaching and learning or unproductive when they limit student access to important mathematics content and practices.

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Beliefs about the mathematics curriculum

Unproductive beliefs Productive beliefs

The content and sequence of topics in a textbook always define the curriculum. Everything included in the textbook is important and should always be ad- dressed, and what is not in the book is not important.

Standards should drive decisions about which topics to address and which to omit in the curriculum. How a textbook is used depends on its quality—i.e., the degree to which it provides coherent, balanced instruction in content aligned with standards and provides lessons that consistently support implementation of the Mathematics Teaching Practices.

Knowing the mathematics curriculum for a particular grade level or course is sufficient to effectively teach the content to students.

Mathematics teachers need to have a clear understanding of the curriculum within and across grade levels—in other words, student learning progressions—to effectively teach a particular grade level or course in the sequence.

Implementation of a pacing guide ensures that teachers address all the required topics and guarantees continuity so that all students are studying the same topics on the same days.

Curriculum maps and pacing guides at- tempt to ensure coverage of content but do not guarantee that students learn the mathematics. Adequate time to provide for meaningful learning, differentiation, and interventions must be provided for students to develop deep understanding of the content.

Mathematics is a static, unchanging field. Mathematics is a dynamic field that is ever changing. Emphases in the curriculum are evolving, and it is important to embrace and adapt to appropriate changes.

The availability of open-source mathe- matics curricula means that every teacher should design his or her own curriculum and textbook.

Open-source curricula are resources to be examined collaboratively and used to support the established learning progressions of a coherent and effective mathematics program.

Overcoming the obstacles A mathematics curriculum is more than a collection of activities; instead it is a coherent sequencing of core mathematical ideas that are well articulated within and across grades and courses. Such curricula pose problems that promote conceptual understanding, problem solv- ing, and reasoning and are drawn from contexts in everyday life and other subjects.

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Designing standards and curriculum

In light of the sheer quantity of mathematics that could be addressed in any grade or course, it is important to make careful choices about what specific mathematics to include. Those designing curriculum standards and related documents need to carefully consider whether topics remain in the curriculum because of tradition, or, more important, whether they are necessary in promoting students’ readiness for college, careers, and life. Some topics may warrant increased attention, given their prevalence in students’ future use of mathematics in postsecondary study or the workplace. For example, as NCTM argued in Focus in High School Mathematics (2009), statistics is increasingly recognized as essential for students’ success in dealing with the requirements of citizenship, employment, and continuing educa- tion (Franklin et al. 2007; College Board 2006; American Diploma Project 2004). Likewise, discrete mathematics, algorithmic thinking, and mathematical modeling may warrant addi- tional attention, given their importance in computer science and related fields. Mathematical curricula also need to reflect changing emphases within the field of mathematics. As the report Mathematical Sciences in 2025 (National Research Council 2013a, p. 2) states,

Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more. This work involves the integration of mathematics, statistics, and computation in the broadest sense and the interplay of these areas with areas of potential application.

Finally, curriculum design needs to take into consideration the amount of new content to be introduced in a particular grade or course so that sufficient time will be available to teach concepts and procedures, using the Mathematics Teaching Practices. That is, sufficient time is needed to—

• engage students in tasks that promote problem solving and reasoning to make sense of new mathematical ideas;

• engage students in meaningful mathematical discussions; and

• build fluency with procedures on a foundation of conceptual understanding.

One of the positive features of CCSSM is its focus and coherence in grades K–8 and the delay of expected fluency in standard computational algorithms. These features provide instructional time for students to build conceptual understanding and proficiency in the mathematical practices and to develop fluency in standard computational algorithms that is based on their understanding of properties, operations, and the base-ten number system.

Implementation of curriculum

Teachers who are well prepared in their knowledge of mathematics, students’ thinking, and the school’s curriculum are positioned to appreciate how mathematical thinking develops

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over time and are equipped to help students connect topics to strengthen understanding (Ball, Thames, and Phelps 2008). Also, when teachers recognize the importance of developing students’ proficiency with the mathematical practices, they can more effectively select and implement appropriate tasks that emphasize mathematical thinking throughout the pre-K–12 years. Instructional materials and tasks selected by schools have a significant influence on what students learn and how they learn it (Stein, Remillard, and Smith 2007). Consequently, teachers need high-quality professional development to maximize the effectiveness of these materials, since even the best textbooks and resources can be misinterpreted or misused.

Given the central role of textbooks as a resource and their potential for supporting instruc- tion, textbook selection should not be taken lightly. This process should consider not only whether textbooks “cover” standards but also whether their development of content reflects learning progressions focused on conceptual understanding and emphasizes the mathemat- ical practices (Bush et al. 2011; NGA Center and CCSSO 2013). As discussed earlier, the Mathematics Teaching Practices promote students’ conceptual understanding and proficiency in the mathematical practices. Thus, another important selection criterion is the extent to which a textbook’s lessons consistently support these teaching practices.

Appropriate use of textbooks—whether to teach from them lesson-by-lesson almost exclu- sively or whether to treat them as one resource among many—depends on the quality of the textbook, as defined above. If a textbook develops mathematical topics in a coherent manner, based on learning progressions, and features lessons that consistently support the Mathematics Teaching Practices, then teaching primarily from that textbook makes sense, and significant omissions or deviations can decrease, rather than enhance, the quality of instruction (Banilower et al. 2006). Conversely, if a textbook does not provide such support, then the only option is to treat it as one of many resources and supplement it as needed.

Some schools develop pacing guides to ensure that instruction addresses all the required standards in the school year and spends an appropriate amount of time on each topic. Al- though these resources can help teachers with long- and short-term planning, the needs of individual classes and students should have priority over rigid curricular schedules. Collab- oration among teachers throughout the school year can result in appropriate adjustments and adaptations of pacing guides to address student strengths and weaknesses.

Structuring units—and lessons within the units—around broad mathematical themes or ap- proaches, rather than lists of specific skills, creates coherence that provides students with the foundational knowledge for more robust and meaningful learning of mathematics. In partic- ular, attention to the mathematical practices provides students with important mathematical tools that they need to navigate mathematical situations and contexts. In planning lessons, teachers should also consider the intended standards and the developmental needs of the stu- dents. Consequently, careful consideration should be given to appropriate ways to sequence a series of lessons. Daily lesson plans should take into account the broader perspective of what

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students learned in the past and where they are headed in the future, as well as the contexts that can be used to motivate students and help them understand why particular topics are important.

High school mathematics

Efforts to achieve curricular coherence in mathematics at the high school level are partic- ularly challenging, given the typical sequence of courses and topics, in which the study of geometry is often isolated as a separate course and statistics is grafted onto courses in stand- alone units rather than naturally connected to related topics (e.g., using a visual line of best fit to lead into a formal study of linear functions). Some schools have successfully reconfigured their programs as integrated sequences of courses that address algebra, geometry, statistics, probability, and discrete mathematics topics across all grade levels, allowing students to revisit these topics at increasingly sophisticated levels and make connections among them. Such reconfiguration requires developing mathematical reasoning and helping students see how, for example, a probability problem can be solved by use of a geometric model, or how geometric transformations of shapes can be performed through the use of matrices in algebra.

All high schools should reevaluate their mathematics programs to determine whether the cur- rent sequence of courses is preparing students for the demands of a workplace that will re- quire more than the mastery of isolated mathematics skills. Such a reevaluation might require that teachers know “how and why mathematical models are derived,” how to “create their own models,” and how to “think about the relationship between the models and the mathe- matics that is integrated” (Keck and Lott 2003, p. 131). Efforts to build coherence across the high school curriculum are of paramount importance.

Connecting and revising the curriculum

The mathematics curriculum should not only be coherent but also make connections from the mathematics curriculum to other disciplines. For example, A Framework for K–12 Science Education (National Research Council 2012) and the subsequently released Next Genera- tion Science Standards (National Research Council 2013b) have significant importance for mathematics. The scientific and engineering practices have a great deal in common with the mathematical practices outlined in CCSSM, and indeed, “Using Mathematics and Computa- tional Thinking” is listed as one of the science practices. Furthermore, mathematical con- cepts underlie much of science—for example, “Scale, proportion, and quantity” is listed as one of seven crosscutting concepts.

Finally, all curriculum-related documents (national, state or provincial, and local) need to be periodically revisited to ensure that they reflect changing priorities related to the mathematics that students need to learn, as well as new research into effective learning progressions. Al- though a level of stability in such documents is necessary to allow progress toward the goals

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that they establish, specific mechanisms should also be put in place to track changes that are needed, so that the documents can be regularly updated.

Illustration Effective attention to curriculum involves periodic monitoring, with course revisions as needed. Consider, for example, a high school mathematics department that engages in professional development during the summer to revise a unit on congruence for the coming year. This is one of the topics that the department believes that students have not learned at the intended level in the previous two years. The teachers recognize that CCSSM includes standards for using transformations to help students make sense of congruence and that some of the approaches in the adopted textbook series do not adequately address these standards. Moreover, they know that some approaches to the topic that the book includes are not neces- sary to address the standards. As a result of reviewing student performance, the teachers in the department agree that they can omit two of the sections of a chapter in the book.

At one meeting, the teachers note that when the students are in middle school, they study the idea that the translation, reflection, or rotation of a figure produces a congruent shape. They also notice that an earlier chapter in the high school book involves the exploration of parabolas and how the locations of the curves, as well as their shapes, are related to their equations. They decide that rather than studying quadratic equations and parabolas early in the year and then separating this topic from congruence, they can link the content of the two chapters to make both topics more meaningful for their students.

As a group, the teachers agree to position transformations as the foundation of the unit. In the students’ examination of parabolas, they will embed some review of transformations. Then, building on this theme, they will have students investigate congruence through the lens of transformational geometry. Although all this content is in the standards, they are able to reorder and restructure the material in the textbook and ancillary materials to meet the needs of the students more effectively.

When they teach the sequence of lessons that they have prepared as a team, the teachers will continually ask students to switch the lenses that they use—from looking at a situation alge- braically to exploring how it connects with the geometry that they have been studying. Once they have an outline for accomplishing the goal and have made a tentative schedule for each lesson, they recognize that the next step is to identify appropriate tasks that will build the students’ conceptual understanding and mathematical reasoning. They investigate tasks that are offered in the textbook as well as tasks from other curricular resources, such as websites, and they map out a restructured unit that will help students make connections and achieve at a higher level. In the coming school year, they will gather data on student success and revise the plans as needed for the future.

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Moving to action Making the Curriculum Principle a reality will require all stakeholders to focus on helping students achieve challenging standards by implementing a coherent curriculum. Teachers need to enter into dialogue with colleagues to become more familiar with the mathem

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