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State Early Childhood and Pre-K-3 Policies and Best Practices for Early Mathematics

Douglas H. Clements, Julie Sarama, Arthur J. Baroody, Bethdalie Cruz, and Melissa Mincic

The first recommendation by the National Research Council (1) on early mathematics in 2009 was: “A coordinated national early childhood mathematics initiative should be put in place to improve mathematics teaching and learning for all children” (p. 3). Recent research shows that even teachers working with toddlers need more intensive and effective preparation and professional development in mathematics (2, 3) and that these very young children can be competent and confident mathematics learners. Summarized in this brief are key policy issues regarding young children’s mathematical development, including state policies and examples of state documents and initiatives, and research-based recommendations on these policy issues.

State policies on early childhood mathematics differ in their acknowledgement of the need for high-quality, challenging, and accessible mathematics instruction for young children. According to a 2003 National Association for the Education of Young Children survey of state policies on early childhood mathematics (4), policy makers in most states recognized the considerable need for early childhood mathematics. However, 23 states lacked specific math curriculum requirements or recommendations for preschool or kindergarten. As states develop policy agendas strengthen science, technology, engineering, and mathematics (STEM) education across the educational pipeline, there is a critical opportunity to focus on improving early mathematics education.

Incorporating Early Learning in State STEM Strategies As it is, the United States has been falling behind in level of mathematical proficiency since the 1970s. U.S. students currently rank behind 25 countries in math scores and 12 countries in science. A reason for this achievement gap is a lack of rigorous K-12 math and science standards and qualified instructors (5). The number of technology, engineering, and mathematics (STEM) degrees awarded in the U.S. has decreased from 12.4% in 2000-2001 to 10.7% in 2008-2009. In stark contrast, 50% of all degrees awarded in Japan, China, and Singapore are in STEM fields.

The percentage of adults who cannot compute a 10% tip, the interest paid on a loan, and the miles per gallon on a trip is 58%, 71%, and 78%, respectively (6).

Even more alarming than differences in STEM proficiency between the United States and other countries is the achievement gap between states due to a wide variation of expectations among district (7). To address these gaps, many states have already taken the initiative to address state economic performance by implementing policies and suggesting strategies to accomplish higher STEM preparation and proficiency for students. These strategies include: (a) recruitment and retainment of highly qualified teachers through teacher preparation and professional development, (b) adoption of rigorous math and science standards and improved assessments, (c) improved STEM programs, curricula, and opportunities for students. Yet most of these strategies

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have thus far focused on secondary and post-secondary education. As research continues to show the value of high-quality early mathematics education, state policymakers can act to ensure that state STEM agendas incorporate a meaningful focus on early childhood and early elementary education.

Teacher Preparation and Professional Development High-quality, specific teacher preparation and development is a strong predictor of student mathematics attainment (2, 8-11). Research shows that most teachers of mathematics at any age are poorly prepared (11) and that teachers of even the youngest children can benefit from better preparation in mathematics education. Currently, early math is not often emphasized in teacher preparation programs. As a result, pre-service and in-service teachers alike lack content knowledge, such as understanding of mathematical concepts and procedures (learning trajectories’ “goals”). More importantly, they lack mathematics knowledge for teaching—how a student’s thinking about mathematical content develops (learning trajectories’ “developmental progressions”), and how mathematical content can be taught in a meaningful manner (learning trajectories’ correlated “instructional activities”), including how mathematical knowledge is interconnected and connected to the real world (12). They suffer from negative affect, including math anxiety and a lack of confidence in their own mathematical ability and ability to teach mathematics—beliefs that lead to undervaluing or even avoiding the teaching of mathematics (10). Therefore, professional development for early childhood mathematics needs to address content (mathematical) knowledge, particularly mathematics knowledge for teaching, as well as pedagogical knowledge and affective issues (13).

States should ensure that all early childhood teachers of mathematics have successfully completed at least two courses that emphasize all three components of learning trajectories: knowledge of the subject (particularly a profound knowledge of the math taught in the early and elementary years), knowledge of children’s thinking and learning of the different core topics in mathematics (with in-depth knowledge and skills in at least two of the three periods: infants/toddlers, preschool/prekindergarten, and early primary grades), and knowledge of how to design environment, activities, and interactions to engender that thinking and learning. To develop such knowledge, pre-service teachers need multiple courses dedicated to early mathematics (as stated, at least two) and in-service teachers need 50 to 75 hours of professional development in early mathematics (12, 14). State policymakers may review current licensure, accreditation, and professional development policies and consider opportunities to strengthen support for teachers of early mathematics. Lead teachers should be certified with these requirements.

What we now know is that math instruction is far more effective coming from a specialist who understands both the subject matter and the most effective ways in which young children learn math or a least a teacher who works with a mathematics coach (15). A successful program will be one that ensures that early math instructors have specialized knowledge. One solution may be for a school to designate a teacher in each grade who is responsible for teaching only math to all students (e.g., a teacher in a team who specializes in math and teaches all students of that team). States should consider creating a new certification to become an early math specialists (e.g., see the work in Nebraska, Oklahoma).

Of course, teacher preparation should be connected to practice:

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All teacher preparation program components should be grounded in teaching practice. The evidentiary base strongly supports the idea that pre- service teachers gain substantially from more frequent and higher-quality opportunities to gain hands-on experience, explore teaching methods, practice with curricula, and encounter situations and challenges they are likely to confront after the preparation program has ended. Preparation program coursework requirements likely can be streamlined in some areas (e.g., foundations courses) to make room for more practical courses and experiences (16, p. 9).

A policy implication would be a performance assessment of teachers seeking certification, including mathematical competencies (show two ways to divide), interpreting children’s behaviors (e.g., diagnosing errors), and pedagogy (design and implement a research-based activity that addresses a key goal of the CCSS). Finally, states can set standards for research-based professional development that provides ongoing support to educators in improving their teaching of early mathematics. Professional development standards and programs should ensure that teachers understand and can implement the goals for programs, for teaching, and for assessment.

Several states have documented their approaches to employing teachers with not only the content knowledge needed to teach their students, but also with educational training (via license or certification) required to teach in the relevant subject areas.

• Virginia’s 2011 H.B. 1792/S.B. 1270 dictates that school boards employ licensed instructional personnel with qualifications in specific subject areas. Further, the bill states that schools can use the Standards of Learning Algebra Readiness Initiative funding to employ mathematics teacher specialists to provide intervention services if personnel are licensed by the Board of Education (17).

• With H.B. 1918 in 2001, Oklahoma required its Commission for Teacher Preparation to develop and to administer professional development specific to mathematics for any K-3 teachers licensed at the preschool or elementary school age levels before July 2, 2001. Further, this professional development should be based on scientific research as well as meet state law requirements for professional development (18).

• With 2011’s H.B. 1600, Washington State emphasized the importance of elementary and middle school teachers’ proficiency not only in mathematical content but also in effective mathematical instructional methods in order for students to meet more rigorous high school mathematics state standards. The bill also specifies necessary knowledge and skills for elementary mathematics specialists, including mathematical content and pedagogy, the development of professional educator standards and specialty endorsement for elementary mathematics specialists, and school district-university partnerships fostering new training and professional development opportunities specific to mathematical content (19).

• Florida’s H.B. 7165 passed earlier in 2013 changed the State’s Office of Early Learning governance structure by establishing the Office of Early Learning within the Department of Education’s Office of Independent Education and Parental Choice. The bill is intended to increase accountability and transparency in the administration of early learning programs by accomplishing a number of goals (http://www.flsenate.gov/Committees/BillSummaries/2013/html/480) and mandated the

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Office of Early Learning to examine existing performance standards related to mathematical thinking and to develop a plan to provide appropriate professional development and training courses for pre-K teachers (http://www.myfloridahouse.gov/Sections/Documents/loaddoc.aspx?FileName=_h7165er.docx&DocumentType=Bill&BillNumber=7165&Session=2013) (20).

• Iowa’s Early Learning Standards include mathematics and science together as a focal area of development from birth through age 5 years. They include a definition, rationale for inclusion, benchmarks and benchmark examples, suggestions for caregiver behaviors to further develop the specific mathematical skill for each of three specific mathematical skills: comparison and number, patterns, and shapes and spatial relationships (http://www.dhs.state.ia.us/docs/IELS_2-20-006.pdf).

• Pennsylvania’s Early Learning Standards includes ten mathematics standards and children’s expected skills for each standard from birth through kindergarten organized by age group: infant, young toddler, older toddler, pre-K, and kindergarten. Pennsylvania’s Department of Education also provides resources guiding early childhood educators in choosing pre-K and kindergarten mathematics curricula aligned to State standards and provides a suggested mathematics assessment at pre-K and kindergarten levels (http://www.portal.state.pa.us/portal/server.pt/community/departmental_offices/7235/p/1188258).

• Vermont’s Early Learning Standards lists mathematics as one of its eight general learning domains. Specific learning goals include engaging in play to develop mathematical thinking and problem solving, numbers and operations, geometry and spatial sense, and patterns and measurement. Also included are examples of children’s skills illustrating each domain and suggestions of ways in which adults and early learning environments can support the development of children’s mathematical thinking. Goals are aligned with the Vermont Framework of Standards and Learning Opportunities as well as the Head Start Child Outcomes Framework (http://dcf.vermont.gov/sites/dcf/files/pdf/cdd/care/2006-03-29-VELS_booklet.pdf).

• Washington State’s Early Learning and Development Guidelines include developmentally appropriate behaviors denoting children’s fundamental knowledge and skills as well as adult behaviors that can foster the development of children’s knowledge and skills for several developmental domains, including mathematics, from birth through 8 years of age. The Guidelines are aligned with both Washington State Learning Standards, which incorporate the K-3 Common Core State Standards for mathematics, and the Head Start Framework (http://www.del.wa.gov/publications/development/docs/guidelines.pdf).

The policy recommendations of several national organizations echo these recommendations and support strengthening state policies on preparation and professional development for early childhood and elementary educators:

• NAEYC, a professional organization that promotes excellence in early childhood, has developed its own educational expectations for early childhood professionals. In their 2002 position statement for early childhood mathematics (7), as well as a 2010 professional preparation standards document (21), NAEYC indicated that all early childhood professionals should have broad knowledge of development and learning

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across the birth-through-age-8 range; should be familiar with appropriate curriculum and assessment approaches across that age span; and should have in-depth knowledge and skills in at least two of the three periods: infants/toddlers, preschool/prekindergarten, and early primary grades. In early mathematics, these fundamental prerequisites, along with high-quality, challenging and accessible mathematics education, support a strong foundation for future mathematics learning. As children have the capacity to benefit and to succeed when early childhood professionals have specialized training and education, effective, research-based curriculum and teaching practices ought to be the hallmark of this nation’s early mathematics education (7, 22). Policy makers in several states have documented the importance of employing teachers with the content knowledge needed to teach their students and a license for teaching the content.

• The Council for the Accreditation of Educator Preparation recent interim standards (23) argued that teachers of mathematics needed to have subject matter knowledge, including pedagogical content knowledge (cf. 24). Subject matter is important, of course, but it bears repeating that specific content knowledge, related to the content teachers will be teaching (including that which “comes before and after” the content of the age/grade they teach) is that which will support teaching (10, 25).

• An effective policy would ensure that a teacher with special knowledge of mathematics teaches every child. A joint position statement of the Association of Mathematics Teacher Educators (AMTE), the Association of State Supervisors of Mathematics (ASSM), the National Council of Supervisors of Mathematics (NCSM), and the National Council of Teachers of Mathematics (NCTM) stated the following (15, p. 1s):

The AMTE, ASSM, NCSM, and NCTM recommend the use of Elementary Mathematics Specialists (EMS) in PK–6 environments to enhance the teaching, learning, and assessing of mathematics in order to improve student achievement. We further advocate that every elementary school have access to an EMS. Districts, states/provinces, and higher education should work in collaboration to create: (1) advanced certification for EMS professionals; and (2) rigorous programs to prepare EMS professionals. EMS professionals need a deep and broad knowledge of mathematics content, expertise in using and helping others use effective instructional practices, and the ability to support efforts that help all PK–6 students learn important mathematics. Programs for EMS professionals should include foci on mathematics content knowledge, pedagogical knowledge, and leadership knowledge and skills.

Standards and Assessments States have learning standards that define what children should know and be able to do at each stage of early childhood development and at each grade level in K-12 education. These standards are meant to inform development of curriculum, professional development, and assessments. All states have learning guidelines for preschool-age children and most states have learning and development guidelines for infants and toddlers. Presently, 45 states, the District of Columbia, four territories, and the Department of Defense Education Activity have adopted the Common

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Core State Standards (CCSS) in mathematics and English language arts for K-12 education. States also have K-12 learning standards in subjects such as science and social studies.

A report funded by the Heising-Simon Foundation evaluated the mathematics content of pre-kindergarten learning standards addressed in each of the 50 states (26) and the degree to which states’ standards are aligned with two national documents that define expectations for children’s mathematics learning—the Head Start Child Development and Early Learning Framework (HSCDELF) and the CCSS (27). The report emphasized the large variation among the states, in terminology, organization (topics, grade ranges groupings), and orientation. For example, 32 states use a combination of developmental (child-oriented) and disciplinary (mathematics content-oriented) perspectives, 5 focus on disciplinary perspectives, and 13 focus on developmental perspectives. As stated, research supports a combination of the approaches, but not as a weak “a bit of this, a bit of that” strategy, but rather as a strong integration of understanding the goal (the mathematical content), developmental progressions (children’s thinking and learning) and pedagogy (the instructional activities).

The authors suggest such divided orientations may account for the variation. Also, 34 states had a separate mathematics content area; and 13 included mathematics as a sub-section of a broader domain (e.g., “cognition”). Of those, specific mathematical topics were not described in the standards of 19 states. The authors of the Foundation report concluded, “The fact that [some states have] as few as three indicators for mathematics and about half [have] fewer than 20 means that state variation also permits some neglect of this important area of early learning. We regard mathematics as too important an area of development to leave to choice or chance” (26, p. 23).

Many states have early math standards (some acceptable, but all could be improved) and many are currently revising their early learning standards to better align with the CCSS. In addition, emerging cross-state initiatives could support model standards that could be adopted by multiple states for children birth through preschool. As states pursue revisions, there is an opportunity to improve standards for early mathematics. Research shows that these standards should be built on flexible, developmental, guidelines for young children’s mathematical learning. Guidelines should be based on available research and expert practice, focus on and elaborate the big ideas of mathematics, and represent a range of expectations for child outcomes that are developmentally appropriate” (28). These standards should also reflect research-based learning trajectories (1, p. 4).

In early elementary classrooms, experts have found that CCSS are considered stronger than the mathematics standards that most states previously had in place. For example, one study reported that CCSS standards were clearer and more rigorous than the math standards in 39 states and “too close to call” for 11 other states (29). The authors stated that Massachusetts stood out for rigorous standards and serious implementation.

The CCSS in mathematics was built upon previous standards and research, but has both greater focus and coherence. As the CCSS and other state standards and policy documents are revised in coming years, updates could use the most recent research to fine-tune areas (e.g., increased attention to conceptual development of the “big ideas” of mathematics and the integration and illustration of mathematical practices and positive dispositions).

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Educational assessments serve a variety of purposes. Sometimes “assessments” are equated with “high-stakes” standards. Other times, the term suggests a more diagnostic function, as in the identification of children with special needs. Finally, within the classroom, “assessment” serves to guide instruction and learning and gauge whether instruction has been effective or needs modification. Research and expert opinion (28) suggest that the primary goal of assessing young children should be to understand children’s thinking and knowledge and to inform ongoing teaching efforts. Performance tasks (meaningful activities that requires children to synthesize and apply knowledge and skills make a response or create a product) and informal ongoing observations are useful and informative ways of assessing young children’s mathematical learning and should be integrated as appropriate into the early childhood mathematics curriculum.

The purposes of any assessment should determine the content, the methods of collecting evidence, and the nature of the possible consequences for individual students, teachers, schools or programs. In the past, misuse of tests and other instruments in early childhood have stemmed from confusion of purpose. Instruments designed for one purpose, such as identification, may be inappropriate as instruments to measure the success of a program. Assessment that supports early childhood learning should enhance teachers’ powers of observation and understanding of children’s mathematical thinking and learning. It should draw upon a range of sources of evidence of student learning (28, 30). However, in early childhood, group-administered, multiple-choice tests often are not sufficient when used alone (31, 32). Individual assessment, observations, documentation of children’s talk, interviews, samples of student work, and performance assessments that illuminate children’s thinking are useful complements in assessing children’s strengths and needs (32). Careful assessment is especially important for instruction of children with special needs or disabilities (33-38).

A few states have developed instruments specifically designed to support early mathematics instruction. Mississippi has developed a numeracy screening instrument to be used by districts for K-3 students to determine student fluency in understanding numbers and mathematical operations (39). Florida’s Voluntary Prekindergarten (VPK) Assessment includes progress monitoring measures of literacy (i.e., print knowledge, phonological awareness, and oral language/vocabulary) and mathematics knowledge aligned with the Florida Early Learning and Developmental Standards for four-year-olds. The mathematics measure includes 13 assessment items targeting early numeracy skills across three different areas: counting skills, numerical relations skills, and arithmetic reasoning skills (40). The assessments being developed to measure CCSS goals by Smarter Balance (http://www.smarterbalanced.org) and PARCC (http://www.parcconline.org/) are for grades 3-12, and so mainly outside the scope of this brief. The U.S. Department of Education recently awarded over $15 million in enhanced assessment grants to North Carolina, Maryland and Texas, each of which is to develop or improve their kindergarten entry assessments (http://www.ed.gov/news/press-releases/us-department-education-awards-more-151-million-enhanced-assessment-grants-devel). These may be resources for states in the future.

STEM Programs and Curricula Several governors have developed state STEM councils to inform a comprehensive policy agenda to strengthen state STEM policies. Oregon, for example, has recently established a

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permanent council to research and to promote STEM education in their schools (41). The state’s long-term goal is to increase its workforce and to recruit more businesses to Oregon, both of which can be made possible if STEM proficiencies increase for their school-aged populations. States have an opportunity to invite early education stakeholders to join these councils and ensure that policy recommendations include those that focus on early education.

Washington State has similarly created an initiative towards increasing learning opportunities and improving educational outcomes in STEM (42). Via an education innovation alliance, which will partner with various nonprofit state organizations for advice and strategies for implementation of standards, Washington’s governor will be consistently guided through supporting and promoting STEM education from early learning through postsecondary education.

To ensure a program is truly effective, policymakers and school leaders must prioritize investing in high-quality math curricula and instruction that meet the needs of all students. The What Works Clearinghouse (http://ies.ed.gov/ncee/wwc/) documents several interventions and curricula that states could consider. Such work should also consider “early childhood education partnerships…between family and community programs so that they are equipped to work together in promoting children’s mathematics” (1, p. 4). The NSF Math and Science Partnership (MSP) Program, collaborative efforts among institutes of higher education and school districts, might serve as a model.

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Note: We express our appreciation to the Education Commission of the States (ECS) for access to their State Policy Database.

References

1. National Research Council, Mathematics in early childhood: Learning paths toward excellence and equity. C. T. Cross, T. A. Woods, H. Schweingruber, Eds., (National Academy Press, Washington, DC, 2009).

2. B. Perry, Mathematical thinking of preschool children in rural and regional Australia: Looking to the future. Journal of Australian Research in Early Childhood Education 16, 99 (2010).

3. B. Perry, Mathematical thinking of preschool children in rural and regional Australia: An overview. Journal of Australian Research in Early Childhood Education 16, 1 (2010).

4. NAEYC, “State policies that promote early childhood mathematics” (National Association for the Education of Young Children, Washington, DC, 2003).

5. NGA, “Building a science, technology, engineering, and math education agenda: An update of state actions” (The National Governors Association Washington, D.C. , 2011).

6. G. W. Phillips, “Chance favors the prepared mind: Mathematics and science indicators for coparing states and nations” (American Institutes for Research, Washington, DC, 2007).

7. D. H. Clements, C. Copple, M. Hyson. (National Association for the Education of Young Children/National Council for Teachers of Mathematics (NCTM), Washington, DC, 2002).

8. D. H. Clements, J. Sarama, M. E. Spitler, A. A. Lange, C. B. Wolfe, Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. JRME 42, 127 (2011).

9. H. C. Hill, B. Rowan, D. L. Ball, Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal 42, 371 (2005).

10. J. Sarama, D. H. Clements, Early childhood mathematics education research: Learning trajectories for young children. (Routledge, New York, NY, 2009).

11. W. H. Schmidt, N. Burroughs, L. Cogan, “World class standards for preparing teachers of mathematics” (Michigan State University, Lansing, MI, 2013).

12. D. H. Clements, J. Sarama, Learning and teaching early math: The learning trajectories approach. (Routledge, New York, NY, 2009).

13. D. L. Ball, H. Bass, in Multiple perspectives on the teaching and learning of mathematics, J. Boaler, Ed. (Ablex, Westport, CT, 2000), pp. 83-104.

14. K. S. Yoon, T. Duncan, S. Wen-Yu Lee, B. Scarloss, K. L. Shapley, “Reviewing the evidence on how teacher professional development affects student achievement (Issues & Answers Report, REL 2007–No. 033). . Retrieved from ” (U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance, Regional Educational Laboratory Southwest, Washington, DC:, 2007).

15. AMTE, “The role of elementary mathematics specialists in the teaching and learning of mathematics: A joint position of the Association of Mathematics Teacher Educators

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(AMTE), the Association of State Supervisors of Mathematics (ASSM), the National Council of Supervisors of Mathematics (NCSM), and the National Council of Teachers of Mathematics (NCTM)” (Association of Mathematics Teacher Educators, Washington, DC, 2013).

16. J. A. Grissom, S. Vandas, “Teacher preparation and student schievement: Reviewing the evidence (Report 06-2010)” (Missouri P-20 Education Policy Research Center, Harry S Truman School of Public Affairs, University of Missouri, Columbia, Missouri, 2010).

17. Virginia H.B. 1792 S.B. 1270, “Virginia acts of assembly - Chapter 55” (2011). 18. Oklahoma H.B. 1918, “Professional development, mathematics K-3” (2011). 19. Washington H.B. 1600, “Math specialists” (2011). 20. Florida H.B. 7165, “Early learning coalitions and school readiness program” (2013). 21. NAEYC, “2010 NAEYC standards for initial & advanced early childhood professional

preparation programs” (National Association for the Education of Young Children, Washington, DC, 2010).

22. NAEYC, “NAEYC standards for early childhood professional preparation. A position statement of the National Association for the Education of Young Children” (National Association for the Education of Young Children, Washington, DC, 2009).

23. CAEP, “CAEP interim standards for accreditation of educator preparation” (Council for the Accreditation of Educator Preparation, Washington, DC, 2010).

24. L. S. Shulman, Those who understand: Knowledge growth in teaching. Educational Researcher 15, 4 (1986).

25. L. Ma, Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. (Erlbaum, Mahwah, NJ, 1999).

26. C. Scott-Little, S. L. Kagan, J. L. Reid, E. Castillo, “Early mathematics standards in the United States: Understanding their content” (Heising-Simons Foundation, Los Altos, CA, 2011).

27. C. Scott-Little, S. L. Kagan, J. L. Reid, E. Castillo, “Early mathematics standards in the United States: The quest for alignment” (Heising-Simons Foundation, Los Altos, CA, 2012).

28. D. H. Clements, J. Sarama, A.-M. DiBiase, Engaging young children in mathematics: Standards for early childhood mathematics education. (Erlbaum, Mahwah, NJ, 2004).

29. S. B. Carmichael, W. S. Wilson, K. Porter-Magee, G. Martino, “The state of state standards—and the Common Core—in 2010” (Thomas B. Fordham Institute, Washington, DC, 2010).

30. E. Chittenden, in Dialogue on early childhood science, mathematics, and technology education, G. D. Nelson, Ed. (American Association for the Advancement of Science, Washington, DC, 1999), pp. 106-114.

31. K. C. Fuson, in Engaging young children in mathematics: Standards for early childhood mathematics education, D. H. Clements, J. Sarama, A.-M. DiBiase, Eds. (Erlbaum, Mahwah, NJ, 2004), pp. 105-148.

32. M. M. Lindquist, J. N. Joyner, in Engaging young children in mathematics: Standards for early childhood mathematics education, D. H. Clements, J. Sarama, A.-M. DiBiase, Eds. (Erlbaum, Mahwah, NJ, 2004), pp. 449-455.

33. A. Dowker, What works for children with mathematical difficulties? The effectiveness of intervention schemes. (Nottingham, UK, DCSF Publications, 2009).

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34. L. R. Ketterlin-Geller, D. J. Chard, F. Hank, Making connections in mathematics: Conceptual mathematics intervention for low-performing students. Remedial & Special Education 29, 33 (2008).

35. R. Gersten et al., “Teaching mathematics to students with learning disabilities: A meta-analysis of the intervention research” (RMC Research Corporation, Center on Instruction, Portsmouth, NH, 2008).

36. R. J. Wright, G. Stanger, A. K. Stafford, J. Martland, Teaching number in the classroom with 4-8 year olds. (Paul Chapman/Sage, London, 2006).

37. A. Dowker, What works for children with mathematical difficulties? (Research Report No. 554). (University of Oxford/DfES, Nottingham, UK, 2004).

38. S. Baker, R. Gersten, D.-S. Lee, A synthesis of empirical research on teaching mathematics to low-achieving students. Elementary School Journal 103, 51 (2002).

39. Mississippi H.B. 1058, “K-3 literacy and numeracy screening” (2007). 40. Florida Department of Education, VPK assessment. (2005). 41. Senate of Oregon, “New council will promote STEM education for Oregon students”

(2013). 42. State of Washington, “Comprehensive initiative to increase learning opportunities and

improve educational outcomes in STEM” (2013).