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Read the Instructions in the Week 4 Module Summary.
For your INITIAL POST:
- In the title, include information to identify the standard (Grade 8 Math).
- Attach the resources that you identified.
- Justify your selection. Why did you consider the resources high quality or low quality? Why would you use the high-quality resource with your students? Why would you not use the low-quality resource?
You are required to respond to AT LEAST TWO other students. In your response:
- Do you agree with the student’s analysis of the resources? Why or why not?
- Compare these resources with the ones that you found. How are they similar? Different?
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
http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/
http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/
http://www.nctm.org/PtA/
http://www.rand.org/pubs/research_reports/RR1529.html
http://www.rand.org/pubs/research_reports/RR1529.html
http://www.nctm.org/ARCs/
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.
http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/
http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/
http://www.nctm.org/PtA/
http://www.nctm.org/PtA/
PrinciPles to Actions
70 • • •
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
: Curriculum
• • • 71
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.
PrinciPles to Actions
72 • • •
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.
Essential Elements: Curriculum
• • • 73
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
PrinciPles to Actions
74 • • •
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
Essential Elements: Curriculum
• • • 75
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
PrinciPles to Actions
76 • • •
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.
Essential Elements: Curriculum
• • • 77
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 mathematical
expectations of the standards that are guiding their teaching, including discussions of how
these ideas are developed in both horizontal and vertical components of the curriculum. They
need to evaluate the extent to which curricular materials and resources align with and sup-
port meaningful student learning of the content and practices in the standards.
Meanwhile, school administrators can support the implementation of standards by promoting
meaningful professional development that assists teachers in making the most effective use
of curricular materials. Administrators should recognize that pacing guides, textbooks, and
other instructional materials can guide the planning process but should never take the place
of the teacher in determining how to meet the needs of the students in a particular class most
effectively. Finally, curriculum planners at all levels should sequence content to maximize
coherence and connections across unit topics and across grade levels and courses. To accom-
plish this most effectively at the high school level, educators should consider an integrated
approach as a way to help students understand mathematics as a discipline rather than as an
isolated set of courses.
- TOC
- Writing Team
- Preface
- Acknowledgments
- Progress and Challenge
- Effective Teaching and Learning
- Taking Action
- References
Essential Elements
