**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

*Mathematics Curriculum, Teacher Professionalism, and Supporting Policies in Korea and the United States: Summary of a Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/21753.

3

MATHEMATICS CURRICULUM IN THE UNITED STATES

The presentations described in this chapter begin with an overview of the education system in the United States and constraints on curriculum reform. In particular, local control of the curriculum requires that multiple influences be aligned if educational reform is to occur. This overview is followed by comments and discussion from all participants. Next are two reflections on the Korean and U.S. presentations. The first is a U.S. perspective from Hyman Bass, a past president of the International Commission on Mathematical Instruction. It describes how the current reform (the Common Core State Standards [CCSS, or Common Core]) occurred in a manner consistent with the principle of local control and gives an overview of the new standards. The second is a European perspective from Gabriele Kaiser, the convener of the 2016 International Congress on Mathematical Education (ICME). This notes the similarity of curricular traditions in continental Europe and Korea and how they differ from those in the United States. These reflections are followed by presentations on the implementation of the current reform and comments from the workshop participants.

Overview of Influences on Curriculum

Janine Remillard of the University of Pennsylvania and Brenda Gardunia of Frank Church High School, Boise, Idaho, gave an overview of the education system in the United States and identified what they considered key issues. To illustrate how the U.S. system operates, they discussed how schools are organized and how the curriculum is structured and described two efforts at curriculum reform.

**Janine Remillard** began with an outline of the presentation, noting that there are some significant differences in the way in which the Korean and U.S. educational systems are structured. The presentation was intended to give a sense of the structure of the U.S. system and in particular how decision making happens, where the authority lies, and why making changes in the United States education system is very challenging.

Remillard discussed four key features of the system: (1) an emphasis on local and regional control of the curriculum; (2) the limited role of the federal government; (3) the distributed, rather than centralized authority; and (4) the strong market influence of commercial publishers.

The United States comprises 50 states and several territories. Each state has a department of education and each state consists of one or, often, many school districts. These vary in size and are determined by region. In each state, the state department of education sets the standards that determine the goals for students and the assessments that are used to gauge whether students have met those goals. The local school districts can select and adopt textbooks to use and provide other instructional guidance. So, much decision making occurs locally.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

*Mathematics Curriculum, Teacher Professionalism, and Supporting Policies in Korea and the United States: Summary of a Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/21753.

The United States does not have a ministry of education. The Department of Education (DoED) is part of the federal government, but it has very limited control over educational decision making. The DoED tends to influence educational decisions in two ways. One is through funding. Less than 10 percent of funding at the state level comes from the federal government, but DoED can set conditions for receipt of funding. Second, DoED can influence policy by commissioning reports. These can raise concerns and identify issues that need to be addressed, but it is the responsibility of the states to take action.

Three major publishers dominate this market. These are Pearson, Houghton Mifflin Harcourt, and McGraw Hill, who, together, are responsible for around 90 percent of textbook sales in the market.^{1} They produce a wide range of textbooks that local school districts can adopt and that influence the curriculum.

Also, there are a number of what Remillard called “nongovernmental organizations” with a stake in education: academic organizations; professional organizations, such as the National Council of Teachers of Mathematics (NCTM); and political organizations, such as the National Governors Association; and other sorts of organizations. These organizations tend to exert their influence by providing expert commentary, by undertaking research and presenting research findings, and sometimes by offering solutions to states and school districts.

Another influence is funding agencies. Some are federal, such as the National Science Foundation (NSF), and some are not. By providing funding opportunities, they can influence what happens in states and school districts by providing resources for curriculum development, as well as for professional development and other activities that happen at the local level. These agencies often work together with some of the nongovernmental organizations.

How Schools and Curriculum Are Organized

**Brenda Gardunia** gave an overview of how schools are organized and how the curriculum is organized. It is difficult to determine what is typical for the United States, so Gardunia described what she considered to be typical in a majority of the schools.

The United States has elementary, middle, and high schools. Typically, elementary schools start with kindergarten and progress through grades 1, 2, 3, 4, and end at grade 5 or 6, when children are 10 or 11 years old. In elementary schools, students stay in one classroom with one teacher who is certified to teach all the subjects that elementary students learn. At some schools, a specialist comes in to teach subjects such as art or physical education.

Middle schools typically include grades 6 through 8. Students are ages 12 to 14. They generally go to different classrooms and different teachers for each subject. Each teacher stays in one classroom. These teachers may have an elementary certification in all subjects or a secondary certification, usually for one subject area.

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^{1} Stephen Noonoo. How “Big Three” Publishers Are Approaching iPad Textbooks. *The Journal*, 2012.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

*Mathematics Curriculum, Teacher Professionalism, and Supporting Policies in Korea and the United States: Summary of a Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/21753.

Students attend high school from grades 9 through 12. Grade 9 is sometimes in the middle school building, and sometimes in the high school building. High school teachers are certified in their subject area. As in middle school, students move from classroom to classroom for different subjects. In high school, students accumulate the credits that they need toward their graduation requirements. And students move into different ability-based mathematics course sequences, which Gardunia referred to as basic, standard, and advanced.

To describe how the curriculum is typically organized, Gardunia began with the K–8 curriculum. It comprises five different content strands:

- Number and Operations
- Data and Probability
- Geometry
- Measurement
- Algebraic thinking

The curriculum is organized in spirals. Each strand is addressed at each grade as students go through elementary school. Ideally, each time students are exposed to a strand, they experience it in more depth and at a more sophisticated level. This organizational approach also includes process strands, such as problem solving, reasoning, and communication.

In high school, most schools offer Algebra I in grade 9, Geometry in grade 10, and Algebra II in grade 11.^{2} Some schools offer an integrated sequence for grades 9 to 11. In both cases, mathematics is often optional in grade 12, and students have a choice of different courses such as Precalculus, Calculus, Statistics, or Discrete Mathematics.

Many schools permit advanced students to enroll in Algebra in grade 8, receiving high school credit for this course. This is followed by Geometry in grade 9, giving students the opportunity to take Calculus before graduating from high school.

Authority for determining graduation requirements varies. A state, district, or sometimes even a school may decide graduation requirements. Some require only two years of mathematics. Most require three years of mathematics, but an increasing number of schools require four years, said Gardunia.

Two Reforms

The next part of the presentation gave two examples of curriculum reform in the United States, illustrating the four key features mentioned at its beginning.

Remillard described the reform that occurred more than 20 years ago with the NCTM’s *Curriculum and Evaluation Standards for School Mathematics.* These standards were developed

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^{2} The National Assessment of Educational Progress states that 47 percent of grade 8 students reported taking “advanced math.” See http://www.brookings.edu/research/reports/2013/03/18-eighth-grade-math-loveless.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

in response to a report that was produced in 1983 called *A Nation at Risk*, which was one of the federally commissioned reports mentioned earlier. It was intended to raise concern about the quality of education in the United States.

The NCTM responded to that report by putting forth a set of standards that called for many changes. In very brief summary, some of the key features were as follows: increased attention to conceptual understanding, problem solving, and reasoning; and decreased attention to the teaching of rote procedures.

The NCTM is an organization that is outside the government, a professional organization without the authority to set policy. However, in state departments of education, there were mathematics curriculum specialists who were members of the NCTM. When the NCTM produced its 1989 standards, state departments of education gradually embraced or slightly adapted those standards and put them in place in their states.

NSF was sympathetic to the calls for change by the NCTM standards and funded a number of projects to develop new curriculum materials that would help teachers instruct in alignment with the NCTM standards.

In those projects, curriculum developers spent several years producing materials. They also established relationships with commercial publishers who marketed the materials to local school districts. That is how support for and examples of what the NCTM standards might look like in a classroom found their way to teachers in local school districts. It is very different from a ministry of education saying: “This is the textbook.”

NSF also funded professional development projects within states and local districts. These projects were intended to help teachers learn about the new standards and teach mathematics differently.

Gardunia described a second example of reform—the Common Core State Standards for Mathematics, which are different in that they are common standards across the United States. (In 2012, 45 states and three territories had formally adopted the CCSS.) The CCSS was a state-led effort coordinated by the National Governors Association (NGA) and the Council of Chief State School Officers, developed in collaboration with teachers, school administrators, and experts, with the intention of providing a clear and consistent framework to prepare students for college and the workforce.

Remillard commented that even though in the 50 states, the NCTM standards did influence what happened in each state’s department of education, there was, over time, significant variation in the standards set by each state. This was a great concern of many in the United States. The Common Core Standards Initiative was a different effort marking the first time that there was a move toward a coherent set of standards for all of, or the majority of, states.

She noted the source of the initiative: a political organization (the NGA). Like the NCTM, this organization does not have policy-making authority. However, it is made up of state governors,

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

so its members have relationships with decision-making bodies. They commissioned a set of standards that they would encourage their states to adopt.

The DoED was also interested in having shared standards. It cannot set standards, but it tied “Race to the Top,” an important funding opportunity, to the adoption of shared standards. Many states wanted to apply for the Race to the Top grants, so they quickly adopted the CCSS.

Commercial publishers and noncommercial curriculum developers who want to develop instructional materials that will be purchased by local districts are finding that they need to make changes in their existing materials. Many of them are developing new materials. So the existence of the CCSS is influencing the development of textbooks and changes in textbooks. Local districts are starting to look to commercial markets to find resources to support their teachers. This is how commercial publishers are influencing the curriculum.

In this case, instead of NSF, a private foundation, the Gates Foundation, has played a very strong role in facilitating the CCSS implementation. It has provided some support for resource development, and earlier support for states to apply for Race to the Top funding.

Remillard summarized: In the United States, we are trying to create a common set of standards, but to do that, all the pieces that influence the system need to be aligned and come into play. Because the system comprises so many distributed pieces, there are many opportunities for slippage—for the pieces to be misaligned, or misinterpreted.

At the end of the presentation, Gardunia described the development of new assessments for the CCSS. In her view, because the United States values individualism and freedom of choice, instead of having one assessment, there are two groups who are in the process of writing assessments. The states are allowed to choose the assessments they will use. These were to be in place for the 2014–2015 school year.

It is still a work in progress, Gardunia concluded, and we are not sure how it will turn out.

Reflections on Korean and U.S. Presentations

After the Korean and U.S. sessions on mathematics curriculum, Hyman Bass of the University of Michigan and Gabriele Kaiser of the University of Hamburg presented their reflections.

**Hyman Bass** said he detected a certain amount of similarity between the United States and Korea for the development of standards in that both referred to a reduction in content. He did not think of it as much as reduction, but as more focus, and greater emphasis on processes or practices. Bass believes that a great deal of content can be covered, and developed in more depth and with more flexibility, if practices are well taught. Covering many small topics with a lot of breadth is not necessarily a way of acquiring more mathematical knowledge.

He offered a personal understanding about the situation in the United States and the Common Core. The fundamental contrast between the two countries is that Korea has a ministry of

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

education staffed by mathematics educators, mathematicians, and others who are relatively stable in their positions. They do not change with each election, and they are professionals who devote themselves to educational problems.

As noted in the U.S. presentation, the United States has a tradition of local control in education that dates to its founding. When the economy was highly regional, that local control made a certain amount of sense. The mandate behind educational improvement in the United States, as in Korea, is mainly economic competitiveness. Both economies are now, at the very least, national, rather than regional. The United States is extremely mobile; students and families move across different regions. So to address educational improvement and reform in localized, uncoordinated ways is totally dysfunctional. It would be like building a national railway system with different gauge tracks in each state so that each train has to stop at the border, unload its cargo, and put it on a different kind of train. It makes no sense. This kind of dysfunctionality is witnessed in the educational system.

Why has the Common Core not usurped local control? The answer is that it is sponsored by the NGA and by the Council of Chief State School Officers. Each of the chief state school officers is like the minister of education for his or her state.

Why would the governors concede this authority and let the standards of their states be developed together with those of other states? If they make their own standards, they have to invest to create those standards, then they have to create assessments for the standards in each of their states—a very expensive proposition. The states already had to collaborate to develop regional assessments because it was too expensive to develop individual state assessments. So they were beginning to suffer the costs of this kind of fragmentation and eventually became convinced that they had to create something that was national in scope.

Another consequence of fragmentation was there were two forces in American education that were national rather than regional: Textbook publishing and commercial testing. Both were driven not by educational policy, but by profit making. To be viable, textbook publishers had to be able to sell their books to each state. Textbooks had to treat the union of all the state curricula, and consequently were very thick books that children had to carry to school or use only at school. One achievement of the Common Core will be to reduce the size and cost of those books. Another will be to make possible a coherent system of teacher education.

In closing, Bass made one last point. There have been many reforms in education in the United States. All have more or less failed, in his opinion. The primary reason for this failure was that in each case the reforms put new demands on teachers. It was assumed that if new standards and new textbooks were written, they could be given to teachers and teachers could be told, “Go do this.” If an airline adopts a new aircraft, their pilots are taken offline for six months to a year to learn how to fly it. They are not given the aircraft and told, “Go fly this.” In the United States, teachers are given mandates that involve new content, new teaching methods, new standards to meet, and new assessments, but little professional development or preparation.

**Gabriele Kaiser** gave her reflection from a European perspective, drawing on her knowledge of U.S. and Korean curricula. She noted that there is a long tradition of international comparative

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

studies. For Europe, the English researcher, Sir Michael Sadler, was very important. He visited French, German, and Prussian schools, and wrote a very famous paper in 1900: “How Far Can We Learn Anything of Practical Value from the Study of Foreign Systems of Education?” He proposed to send future teachers at the end of their studies to German and French schools to study “methods of teaching and systems of education.”

On the other hand, he said, “It is a great mistake to think, or imply, that one kind of education suits every nation alike.” We have to be aware that we can learn from each other, but only to a certain extent.

Kaiser noted that there are significant distinctions between Korea and the United States in curricula, school practice, and teacher education. In considering Korean education, she relied heavily on a special issue of *ZDM* on “The Balance between Foundation and Creativity—Features of Korean Mathematics Education.”

How should continental European traditions be classified with respect to these two countries? (Note that “continental Europe” does not include Great Britain, because it has a very strong and different tradition.) Because content knowledge plays a strong role in continental Europe, Kaiser considers it to have a strong commonality with Korea. In continental Europe, subject-related approaches in mathematics education have priority. These approaches are known as subject-related didactics of mathematics and are associated with German researchers of the 1970s such as Heinz Griesel and Hans-Joachim Vollrath. Especially in Eastern European countries, with their tradition of strong links to the German school system, there was a strong relation to mathematics as a subject and a closely related method of teaching.

In Korea, content knowledge has a similar importance. Where does this come from? What are the famous Korean researchers, what kind of explanations do they give us about where the origin of strong dominance of content knowledge comes from? One common explanation is, of course, Confucian Heritage Culture, which sees a teacher as an expert, as a scholar-teacher.^{3} This kind of education is tied closely to content.

Relationships between U.S. or Western approaches to those in continental Europe are not entirely clear. Progressive education with child-centered approaches dominates in Scandinavian countries, and especially in North and South America, and countries shaped by American influences, for example, the Philippines and Singapore (Nebres, 2006). Individualism is important in this kind of teaching and the “acting subject” is especially prominent. Content is more in the background, although not totally in the background. Anglo-American countries have a tradition of pragmatism (Kaiser, 1999). When it comes to mathematics teaching, argumentation and proof is less important than in European traditions. Mathematical structure is less important in syllabi and daily teaching.

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^{3} In later discussion, Kaiser noted that an article in *ZDM* (Wong, Wong, and Wong, 2012) examines the question of how different kinds of philosophical traditions such as Confucianism, Buddhism, and Daoism might influence mathematics education. This article concludes “it is over-simplistic to draw causal relationships between schools of thought and social phenomena.”

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

Didactics is a common core of continental European tradition, and there is a broader notion of “didactical knowledge” (pedagogical content knowledge) in continental European countries than in Anglo-Saxon and English-speaking countries. Compared to English-speaking countries, this broader, theory-guided knowledge has a very close connection to the subject of mathematics.

Generally speaking, didactics is a constitutive element of continental European educational approaches (Pepin, 1999). In his ICME talk, Bernard Hodgson mentioned three features of didactics described by Winsløw (2007):

1. Its epistemological character, that is, the central focus on specific knowledge in a different form, and the conditions for enabling pupils to acquire it.

2. Its relation to the background science (in this case, mathematics as a science).

3. Its actual and potential roles in relation to teaching and teachers (in particular, didactics as a design science and as a part of teacher education and the professional knowledge base).

These kinds of aspects are very apparent and important when it comes to didactics and its traditions.

These educational orientations influence curricula. In European curricula, the very strong normative orientation influences their structure: There is a long preamble with normatively based goals of mathematics education. There is a strong subject orientation. Forty years ago, there were three mathematical pillars in lower secondary education: number (deliberately called “arithmetic”), algebra, and geometry. Nowadays, these are enriched by probability, which is not called “data,” but rather “probability and data.” In the lower secondary level, algebra dominates. In the upper secondary level, calculus is compulsory in Germany. There is an intensive focus on argumentation and proof.

Continental European curricular traditions and Korean curricula have a remarkable relationship, in contrast to American traditions, amongst others. There are similar pillars in the new Korean curriculum for the lower secondary level (Lew et al., 2012) and continental European curricula (high relevance of functions and structural reflections) despite earlier treatment of other topics in Korea. There is a strong focus on mathematics content in traditional and innovative classrooms (e.g., Pang, 2012). There are intensive reflections on the learning of proofs and their gradual development (Kim and Ju, 2012). There is a strong focus on content knowledge in teacher education curricula (Kwon and Ju, 2012).

Kaiser concluded that these reflect commonalities in didactics of mathematics in Europe and Korea.

Implementation of the CCSS

Mary Kay Stein of the University of Pittsburgh and Stacie Kaichi-Imamura of the Hawaii State Department of Education addressed three questions related to implementation of the CCSS.

- How do the CCSS differ from previous standards?
- What issues are surfacing as the community begins to implement the CCSS?

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

- How have textbooks been developed for the curriculum?

**Mary Kay Stein** began by noting that there are two broad kinds of standards in the United States. As noted earlier, the NCTM started the standards movement for mathematics and other subjects with the 1989 release of the *Curriculum and Evaluation Standards for School Mathematics*. This was followed by other NCTM documents: *Principles and Standards for School Mathematics* (2000); *Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence* (2006); and *Focus in High School Mathematics: Reasoning and Sense Making* (2009). As mentioned earlier, the NCTM cannot mandate adoption of its standards. Nevertheless, Stein remarked, “I think everyone would agree that the NCTM standards have had very far-reaching influences on curriculum frameworks, professional development, classroom practices, and standards in most of the 50 states.”

The state standards came into being in the 1990s, and then were enshrined into law with the No Child Left Behind (NCLB) Act of 2001. In that act, each state was required to establish standards for what students should know and be able to do, and to create assessments for measuring their attainment of those standards. Unlike the NCTM standards, state standards have the force of law. As the NCLB law stands now, all students must have been proficient in mathematics by the spring of 2014, or sanctions would have been applied. This testing and accountability dimension of NCLB is the object of controversy in the United States.

Differences from Previous Standards

Stein described four ways in which the CCSS differ from previous standards.

First, compared to NCTM and state standards, the CCSS represent fewer, clearer, and higher standards.^{4}*Fewer* means less standards per grade level. This was in reaction to the state standards; many of them had become quite bloated as a result of the NCLB law. A commonly heard description of the U.S. curriculum was “a mile wide and an inch deep.”^{5}*Clearer* refers to the simpler and more precise language in which the CCSS are expressed. This is supposed to make it easier for teachers to communicate with each other, and with students, principals, and parents. *Higher* refers to more demanding content. Together these three features should, theoretically, set up teachers to spend *more* time helping students to understand *more* deeply a smaller set of goals for each grade level. Stein commented that she thought that her U.S. colleagues would agree that this is a huge change for U.S. education.

A second feature is that the CCSS aim to develop understanding over time. They do this by defining a set of K–12 pathways that students must be on to meet standards for college and career readiness by the end of high school. The pathways, in turn, are rooted in a set of learning progressions.^{6} Learning progressions are defined as changes in the levels of student thinking as they move toward the goal of instruction in interaction with an ordered set of instructional tasks

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^{4} Gene Wilhoit’s statement. Executive director of the Council of Chief State School Officers. July 1, 2009.

^{5} This description is associated with a study of curricula released in 2000 by the Trends in International Mathematics and Science Study, and thus predates the NCLB.

^{6} The CCSS state: “The development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.”

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

that are intended to encourage the students’ thinking to grow from level to level toward the goal. The teacher’s job is to keep students moving along these pathways. The idea is that the teacher will be helped by the two new assessments that are being developed. The hope is that these assessments will give not only a summative evaluation at the end of each year but also formative evaluation along the way, providing reports to teachers about whether their students are on the pathway or falling behind.^{7}

A third distinguishing feature is the positioning of the standards for mathematical practice (SMPs). At the national presentation at ICME, Mike Shaughnessy, the past president of the NCTM, illustrated ways in which the SMPs grew out of previous NCTM standards. While agreeing that there is some continuity in the substance of the SMPs, Stein argued that there were also some discontinuities. She noted the relative isolation of the SMPs from the content. Compared to past standards, the practice standards are not integrated with the content standards, are less well specified, much less specific than the content standards, and the same across grade levels.

This limited guidance from the CCSS regarding how to integrate the standards for practice with the content standards is a concern. In its absence, guidance will be left to textbook publishers, Stein said. A related concern is that assessments will not give as much weight to practice standards as they do to the content standards. In that case, teachers will get the message that the practice standards are secondary to the content standards.

The fourth feature is the commonness of the Common Core. A related concern is political forces, which continue to challenge the Common Core on the basis of the U.S. Constitution. Another concern is the equality of opportunity that is supposed to emerge from these common standards. It will not materialize on its own. Instead, it will require the commitment of state departments of education, resources, and the professional development of teachers.

Emerging Issues

Stein briefly mentioned two issues related to implementation of the CCSS: (1) instructional programs that can support the transition to the new standards and (2) assessments of the new standards.

She reminded the audience that it was the summer of 2012, and there would be two full school years before the assessments would be administered in 2014–2015. How would schools handle this transition period, especially at the instructional program level? Ideally, along with the CCSS would be appropriate K–12 instructional programs.

Ideally, this coherent instructional system would spring into place full blown at the moment that it was needed. Instead, schools and districts are scrambling to not only build and enact instructional systems but also to assure that students get help if they are not ready for the grade-level knowledge and skills that will be taught.

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^{7} For formative features of the two assessments, see www.parcconline.org/non-summative-assessments and http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/02/Smarter-Balanced-Teachers.pdf.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

The two assessment consortia have been laboring for the most part behind closed doors, yet much depends on these assessments. Past experience suggests that teachers will pay the most attention to the assessments once they are released, and the standards will recede into the background. Teachers will teach to the assessments, and the curriculum will narrow to *what is tested*.

The claim is that teaching to the test—with the Common Core test—will be a good thing, because it supposedly will represent higher, more demanding kind of content and more performance-based activity. Past experience suggests that without additional professional development, teachers will imitate the form of assessment items, but not their substance. A second concern is, Will the assessments yield student data that is useful for guiding teachers?

Other concerns were identified from a practitioner’s viewpoint, that of **Stacie Kaichi-Imamura**, who is the mathematics resource teacher in Hawaii’s Department of Education. Hawaii has been doing in-service work related to the CCSS for the past year and a half.

In the United States, elementary teachers are generalists without the content knowledge demanded by the CCSS. Historically, Hawaiian elementary teachers have relied heavily on textbooks. After the CCSS were released, teachers still used their old textbooks, which were not aligned. This poses a great problem for teachers that do not have the necessary mathematical content knowledge.

Professional development is another challenge. There are more than 13,000 teachers in the state of Hawaii, on several islands. Approximately 9,000 teach mathematics. Many Hawaiian teachers are not tech savvy and not used to having professional development material delivered by video, webinar, webcasting, or other online tools.

The CCSS are very different from the current state standards. Previously, Hawaiian standards changed only slightly from version to version. Although the NCTM process standards should have been a part of classroom teaching, teachers sometimes did not know what the process standards were because they appeared at the very end of the Hawaiian document. The new CCSS give the standards for mathematical practice in the front of the document.

The CCSS are very specific about content at each grade. Because of this, some teachers need help in realizing that their favorite topic or unit may not fall within the standards for their grade.

Textbooks

**Stacie Kaichi-Imamura** described the situation with textbooks. The CCSS were released in June 2010. Publishers have not yet created aligned textbooks. Many textbook companies do key word searches and match or add new material to their textbooks, without removing unnecessary material. And they stamp it “aligned to the Common Core.”

Before the CCSS, textbook companies created textbooks for the larger markets such as Texas, Florida, California, and New York. Publishers also wrote for coverage. Because standards varied

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

from state to state, smaller states, such as Hawaii, had to use these textbooks and make them work. These states had to add or delete material, and tweak lessons, depending on their standards.

From Kaichi-Imamura’s perspective, the tables have been now been turned due to the commonness of the CCSS. Because Hawaii is one of 45 states adopting the CCSS, it is now in a position to ask publishers for what it wants. In particular, it is looking for alignment with the CCSS mathematical practices and problem solving. Hawaii was reviewing instructional materials at the time of the workshop and anticipated the following timetable:

August 2012–January 2013. Initial review by outside vendor. Recommend all material that meets all requirements.

February–March 2013. Second review by stakeholder groups. Will narrow previous list to six per grade band.

April–May 2013. Final review by selected committees of mathematics teachers statewide, to recommend one to two textbooks per grade band.

In the meantime, in response to the lack of CCSS resources, states have been sharing information. Hawaii has been using the following resources:

- Illustrative Mathematics, http://www.IllustrativeMathematics.org.
- Progressions, http://ime.math.arizona.edu/progressions, which illustrate the progressions in the CCSS.
- Open Educational Resources, http://www.oercommons.org.

Comments

The presentations evoked a variety of comments on the CCSS and the challenges of implementing them. Some of those comments are given below.

Promising aspects of the CCSS

- The CCSS have provided opportunities to have more focused conversations about the process of teaching and learning in the classroom.
- In the United States, children are expected to understand the concepts behind the math, but there might be teachers who do not understand them. It is a challenge, but it is promising to see teachers growing with students.
- CCSS provides learning progressions. It is important to provide this information to teachers—why we teach this topic, why this concept is important—looking at it from the viewpoint of students’ learning trajectories.
- A challenge is how to embed the eight mathematical practices, that is, the SMPs, which are new for teachers who learned math with another approach.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

Professional development for the CCSS

- A productive direction for professional development is to document vivid images of instruction in regular content where these practices are integrated.
- In the United States, there are both bottom-up and top-down approaches for teacher development. With CCSS, districts and states have received funding to support this initiative through the Race to the Top grants. That money comes from the federal government to the states, then from the states into the school systems. That is how federal money has been directly affecting the top-down approach. Teachers identify their needs, and then they get together to help build the professional development opportunities and figure out what has to happen. That is at a district level, sort of top and bottom.

Challenges for implementation of the CCSS

- The challenge for the United States in developing textbooks and curriculum is that there is no equivalent of the Korea Institute for Curriculum and Evaluation (KICE). KICE is always monitoring and developing curricula. The United States does not have any similar organization playing this important role.
- The 45 states that have adopted the CCSS have the standards in common, but how the states and local districts support teachers is
*not*necessarily going to be in common. That will be determined at the state level to some extent; to a very large extent it will be determined by the local school districts. - There are about 15,000 education systems in the United States, all of which operate very differently, and all of which have to integrate with different layers of different systems.
- The U.S. education system is highly susceptible to political pressure, and because of that, often not very research-based. Professional development is one of the most difficult pieces of the U.S. system because it is least regulated. Even though there is a lot of professional development activity based on the CCSS occurring all over the country, its quality is highly variable.
- It is known that professional development is most effective when you prepare teachers for the curriculum that they will be teaching. That was impossible to do at scale in the United States because there were so many curricula. There are almost 2,000 different sites in the United States where teachers are prepared. A new Common Core Standards for teacher education and development is needed.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

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**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.

**Suggested Citation:**"3 MATHEMATICS CURRICULUM IN THE UNITED STATES." National Academies of Sciences, Engineering, and Medicine. 2015.