# Resources

## Publications

While improving the quality of early learning experiences is a worthy investment in the future, real world practices have only recently begun to catch up with this idea. MIELL works to increase the connection between research and practice through:

- our partnership with Colorado's early childhood system-building efforts.
- our original research on innovative classroom- and home-based interventions
- our work integrating and translating the best research in the early childhood field around the country and the world

Broadly speaking, MIELL's research agenda is focused on the contribution that adults make in creating stimulating and nurturing environments for young children, thereby establishing a foundation for lifelong learning. Our publications are described below:

## Douglas Clements, PhD

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***Social Policy Report-- PK-3: What Does it Mean for Instruction?**

**Social Policy Report-- PK-3: What Does it Mean for Instruction?**

"PK–3” has become a rallying cry among many developmental scientists and educators. A central component of this movement is alignment between preschool and the early elementary grades. Many districts have made policy changes designed to promote continuity in children’s educational experiences as they progress from preschool through third grade— to provide children with a seamless education that will sustain the gains made in preschool and lead to better developmental and learning outcomes overall. This report proposes a conceptualization of productive continuity in academic instruction, as well as in the social climate and classroom management practices that might affect children’s social-emotional development. It also considers ways in which schools might seek to achieve continuity in parents’ and children’s experiences. Finally, the report proposes specific state and district policies and school practices that are likely to promote continuous and meaningful learning experiences.

https://www.du.edu/marsicoinstitute/media/documents/socialpolicyreportclements.pdf

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***Book Chapter: Developing Young Children's Mathematical Thinking and Understanding **

**Book Chapter: Developing Young Children's Mathematical Thinking and Understanding**

Recent research and developmental work has suggested that learning trajectories can help early childhood educators respect children's developmental processes and constraints, and their potential fix thinking about and understanding mathematical ideas.

Clements, D.H., & Sarama, J. (2015). Developing young children’s mathematical thinking and understanding (pp. 331-344). New York, NY: Routledge.#####
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***Book Chapter: Methods For Developing Scientific Education**

**Book Chapter: Methods For Developing Scientific Education**

Many types of studies contribute to the field of education. But too few, in our opinion, go to the heart of the educational enterprise-developing scientifically based practices, pedagogies, programs, and policies. Whether developed for children or their teachers, these are the main malleable factors that affect the quality of children's educational experiences. In this chapter, we describe why be believe that this type of research-and-development program should take precedence in early childhood education and then describe a framework for such a program.

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***Discussion from a Mathematics Education Perspective**

**Discussion from a Mathematics Education Perspective**

This special issue includes research gems in the domain of preschool mathematics education. Most share several features, such as their perspective on research methodology and their view of mathematics thinking and learning. They address the cognitive architecture and processes, and the developmental levels through which young children pass.

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***Making Early Math Education Work for All Children**

**Making Early Math Education Work for All Children**

Many children don’t learn the count words in their home language or how to count things accurately or the meanings of and words for written number symbols. They must see and hear and try these for months and years as they gradually extend their count word list, increase their counting accuracy, and learn more concepts about number. How can preschools and kindergartens support all of this extensive learning? The Common Core State Standards do not specify methods of teaching. But the NRC report does identify effective teaching-learning practices.

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***Processes in the Development of Mathematics in Kindergarten Children from Title 1 Schools**

**Processes in the Development of Mathematics in Kindergarten Children from Title 1 Schools**

This study examined how well nonverbal IQ (or ﬂuid intelligence), vocabulary, phonological awareness (PA), rapid autonomized naming (RAN), and phonological short-term memory (STM) predicted mathematics outcomes.

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***The Importance of the Early Years**

**The Importance of the Early Years**

Research provides findings—some surprising—about the importance of math for young children. Doug Clements and Julie Sarama explore these and suggest ways to build up children's mathematical concepts and skills.

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***The Building Blocks of Early Mathematics: Learning Trajectories for Young Children**

**The Building Blocks of Early Mathematics: Learning Trajectories for Young Children**

What are the mathematical and educational building blocks of early mathematics? What role should these building blocks play in early education, and why? Webinar for Early Childhood Investigations with 2,100 participants.

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***Preschoolers Getting in Shape**

**Preschoolers Getting in Shape**

Iris and Lauren are drawing triangles. Iris draws this on the chalkboard. Lauren says, "That's not a triangle! It's too skinny!" Iris responds, "I'm telling you, it is a triangle. It's got three straight sides, see? One, two, three! It doesn't matter that I made it skinny." Studies from around the world confirm that children can learn much more about geometry at earlier ages than most people assume. What does it mean to know shapes? What levels of mathematical thinking do preschoolers develop? How can we help them learn more sophisticated ways of thinking?

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*Learning and Teaching Early Math: The Learning Trajectories Approach (2nd Ed.)*

Everyone knows that effective teaching involves "meeting the students where they are" and helping them build on what they know. But that's easier said than done. Which aspects of the mathematics are important, which less so? How do we diagnose what a child knows? How do we build on that knowledge—in what directions, and in what ways? We believe that learning trajectories answer these questions and help teachers become more eﬀective professionals. Just as important, they open up windows to seeing young children and math in new ways, making teaching more joyous, because the mathematical reasoning of children is impressive and delightful. Learning trajectories have three parts: a specific mathematical goal, a path along which children develop to reach that goal, and a set of instructional activities that help children move along that path. So, teachers who understand learning trajectories understand the math, the way children think and learn about math, and how to help children learn it better. Learning trajectories connect research and practice. They connect children to math. They connect teachers to children. They help teachers understand the level of knowledge and thinking of their classes and the individuals in their classes as key in serving the needs of all children.

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*Play, Mathematics, and False*
** Dichotomies**

Some worry that the push for quality education even partially driven by a desire to improve achievement may deprive children of important childhood experiences. Others worry that unstructured play without teacher engagement does little to develop children's minds, particularly for children at high risk of academic failure. Let's stop the cycle of "abuse"—or at least confusion—that stems from false dichotomies in early education. "Play vs. academics" is arguably the main one. Of course children should play. But this does not mean they should not learn, and even play, with mathematics.

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*Building Blocks, Volumes 1 and 2*

Building Blocks builds on young students' math experiences by integrating engaging, hands-on activities and interactive technology into instruction. Building Blocks ranges from designated math activities to circle and story time to help kids relate their informal math knowledge to more formal mathematical concepts.

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*Background Research on Early Mathematics*

The National Governors Association hosted an expert roundtable meeting focused on state policies to strengthen early mathematics education – from early childhood through 3rd grade. A growing body of research shows that early mathematics proficiency is a predictor of later student achievement. This meeting brought together national experts and state policy leaders to discuss how state policy could more explicitly promote high-quality mathematics teaching and learning in both early childhood and early elementary settings.

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*Book Chapter - Lessons Learned in the Implementation of the TRIAD Scale-Up Model: Teaching Early Mathematics with Trajectories and Technologies*

Although the successes of research-based, visionary educational practices have been documented, equally recognized is the failure of these practices to be implemented at a scale that affects more than a trivial portion of children. Further, there may be no more challenging educational and theoretical issue than scaling up educational programs across a large number of diverse populations and contexts in the early childhood system in the United States, avoiding the dilution and pollution that usually plagues such efforts to achieve broad success. In this chapter, we describe a model of scale-up at the school district level and its initial evaluation. Although our intent is that the model should apply to all subject matter domains and grade levels, any evaluation must involve a specific instantiation. Our evaluations have focused on early childhood mathematics. Therefore, we begin with background information on the need for models of scale-up, especially in early childhood education, as well as a consideration of the particular needs in mathematics education. Next, we introduce the theoretical framework, the model we developed, and the research corpus on which they were based. We then summarize the empirical evaluations we have conducted of this model. In the final section, we summarize what we have learned and describe implications and challenges for the field.

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*Book Chapter - Rethinking Early Mathematics: What Is Research-Based Curriculum for Young Children?*

We believe that researchers and practitioners can work together to ameliorate this situation and develop, evaluate, and use valid research-based approaches. To support such collaborative activity, we have developed two major conceptual tools. The first is a set of learning trajectories that describe how children learn major topics in mathematics and how teachers can support that learning. Based upon studies in fields ranging from cognitive and developmental psychology to early childhood and mathematics education, these guide the creation of standards, curricula, and teaching strategies. They also are at the core of the second conceptual tool, a framework for developing curricula and teaching strategies. This framework describes criteria and procedures for creating scientifically-based curricula.

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*Book Chapter - Solving Problems: Mathematics for Young Children*

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*Instructional Practices and Student Math Achievement: Correlations From A Study of Math Curricula*

This brief is directed to researchers and adds to the research base about instructional practices that are related to student achievement. Additional evidence on these relationships can suggest specific hypotheses for the future study of such instructional practices, which, in turn, will provide research evidence that could inform professional development of teachers and the writing of instructional materials. The results of this study revealed a pattern of relationships largely consistent with earlier research, but not in every case. Results that are consistent with previous research include increased student achievement associated with teachers dedicating more time to whole-class instruction, suggesting specific practices in response to students' work (1st grade only), using more representations of mathematical ideas, asking the class if it agrees with a student's answer, directing students to help one another understand mathematics, and differentiating curriculum for students above grade level (2nd grade only). Less consistent results were found in three 2nd-grade results, and include lower achievement associated with teachers' higher frequency of eliciting multiple strategies and solutions; prompting a student to lead the class in a routine; and with students more frequently asking each other questions. These findings suggest that practices associated with higher achievement gains include both student- centered and teacher-directed practices; however, some student-centered practices were associated with lower achievement gains.

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*Longitudinal Evaluation of a Scale-Up Model for Teaching Mathematics with Trajectories and Technologies: Persistence of Effects in the Third Year*

Using a cluster randomized trial design, we evaluated the persistence of effects of a research-based model for scaling up educational interventions. The model was implemented in 42 schools in two city districts serving low-resource communities, randomly assigned to three conditions. In pre-kindergarten, the two experimental interventions were identical, but one included follow-through in the kindergarten and first-grade years, including knowledge of the pre-K intervention and ways to build upon that knowledge using learning trajectories. Students in the experimental group scored significantly higher than control students (g = .51 for those who received follow-through intervention in kindergarten and first grade; g = .28 for non–follow-through), and follow-through students scored significantly higher than non–follow-through students (g = .24).

Clements, D.H., Sarama, J., Wolfe, C.B., & Spitler, M. E. (2013). Longitudinal evaluation of a scale-up model for teaching mathematics with trajectories and
technologies: Persistence of effects in the third year. *American Educational Research Journal, 50*(4), 812 - 850. doi: 10.3102/0002831212469270.

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*Young Children's Understandings of Length Measurement: Evaluating A Learning Trajectory*

This study investigated the development of length measurement ideas in students from prekindergarten through 2nd grade. The main purpose was to evaluate and elaborate the developmental progression, or levels of thinking, of a hypothesized learning trajectory for length measurement to ensure that the sequence of levels of thinking is consistent with observed behaviors of most young children.

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*Math in the Early Years*

This issue of The Progress of Education Reform reveals five surprising findings about the strong relationship between early math instruction and later student achievement. Researchers have found that early knowledge of math not only predicts later success in math, but also predicts later reading achievement even better than do early reading skills. The paper concludes with implications and recommendations for state policy that will support the development of early math competencies and young children.

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*Book Chapter - Learning and Teaching Early and Elementary Mathematics*

Today's early childhood and elementary teachers work with more diverse students than ever before. Further, they have received little instructional support for the subject many are least prepared to teach - mathematics. Additional pressure to teach mathematics has emerged from recent research indicating that early mathematics learning is foundational for later academic success - possibly equal to or even more so than early learning of literacy.

Clements, D.H., & Sarama, J. (2012). Learning and teaching early and elementary mathematics. In J. S. Carlson & J. R. Levine (Eds.), Instructional strategies for improving student learning: Focus on early mathematics and reading (Vol. 3 of Psychological perspectives on contemporary educational issues, pp. 107-162). Charlotte, NC: Information Age Publishing.

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*Book Chapter - Mathematics Learning, Assessment, and Curriculum*

A discussion of what teachers should know about mathematics learning, assessment, and curriculum to develop children's interest and competence in mathematics, based on a select but substantial body of research on young children's learning of mathematics

Clements, D. H., & Sarama, J. (2012). Mathematics learning, assessment, and curriculum. In R. C. Pianta, L. Justice, S. W. Barnett & S. Sheridan (Eds.), Handbook of Early Education (pp. 217-239). New York, NY: Guilford.

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*Early Mathematics Assessment: Validation of the Short Form of a PreKindergarten and Kindergarten Mathematics Measure*

In recent years, there has been increased interest in improving early mathematics curricula and instruction. Subsequently, there has also been a rise in demand for better early mathematics assessments, as most current measures are limited in their content and/or their sensitivity to detect differences in early mathematics development among young children. In this article, using data from two large samples of diverse populations of prekindergarten and kindergarten children, we provide evidence regarding the psychometric validity of a new theory-based early mathematics assessment. The new measure is the short form of a longer, validated measure. Our results suggest the short form assessment is valid for assessing prekindergarten and kindergarten children's numeracy and geometry skills and is sensitive to differences in early mathematics development among young children.

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*Book Chapter - Measurement*

Parmar, R.S., Garrison, R., Clements, D.H., & Sarama, J. (2011). Measurement. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 197-218). Reston, VA: National Council of Teachers of Mathematics

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*Early Childhood Mathematics Intervention*

Preschool and primary grade children have the capacity to learn substantial mathematics, but many children lack opportunities to do so. Too many children not only start behind their more advantaged peers, but also begin a negative trajectory in mathematics. Interventions designed to facilitate their mathematical learning during ages 3 to 5 years have a strong positive effect on these children's lives for many years thereafter.

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*Focus in Grade 2: Teaching with the Curriculum Focal Points*

Focus in Grade 2: Teaching with Curriculum Focal Points describes and illustrates learning paths for the mathematical concepts and skills of each grade 2 Focal Point as presented in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. It includes representational supports for teaching and learning that can facilitate understanding, stimulate productive discussions about mathematical thinking, and provide a foundation for fluency with the core ideas. This book also discusses common student errors and misconceptions, reasons the errors may arise, and teaching methods or visual representations to address the errors.

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*TEAM - Tools for Early Assessment in Mathematics*

Tools for Early Assessment in Math (TEAM) is an assessment screening tool for students in Grades Pre–K–2. This tool can be used to determine where a student is proficient in math skills by providing meaningful diagnostic reports and by prescribing additional activities to accelerate their learning of these skills based on the reported data.

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*Early Childhood Teacher Education: The Case of Geometry*

For early childhood, the domain of geometry and spatial reasoning is an important area of mathematics learning. Unfortunately, geometry and spatial thinking are often ignored or minimized in early education. We build a case for the importance of geometry and spatial thinking, review research on professional development for these teachers, and describe a series of research and development projects based on this body of knowledge. We conclude that research-based models hold the potential to make a significant difference in the learning of young children by catalyzing substantive change in the knowledge and beliefs of their teachers.

Clements, D.H., & Sarama, J. (2011). Early childhood teacher education: The case of geometry. *Journal of Mathematics Teacher Education, 14*(2), 113-148.

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*Mathematics Learned by Young Children in an Intervention Based on Learning Trajectories: A Large Scale Cluster Randomized Trial*

This study employed a cluster randomized trial design to evaluate the effectiveness of a research-based intervention for improving the mathematics education of very young children. This intervention includes the "Building Blocks" mathematics curriculum, which is structured in research-based learning trajectories, and congruous professional development emphasizing teaching for understanding via learning trajectories and technology. A total of 42 schools serving low-resource communities were randomly selected and randomly assigned to 3 treatment groups using a randomized block design involving 1,375 preschoolers in 106 classrooms. Teachers implemented the intervention with adequate fidelity. Pre- to posttest scores revealed that the children in the Building Blocks group learned more mathematics than the children in the control group (effect size, g = 0.72). Specific components of a measure of the quantity and quality of classroom mathematics environments and teaching partially mediated the treatment effect.

Clements, D.H., Sarama, J., Spitler, M.E., Lange, A.A., & Wolfe, C.B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. *Journal for Research in Mathematics Education, 42*(2), 127-166.

## Julie Sarama, PhD

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*Preschoolers Getting In Shape*

Studies from around the world confirm that children can learn much more about geometry at earlier ages than most people assume. What does it mean to know shapes? What levels of mathematical thinking do preschoolers develop? How can we help them learn more sophisticated ways of thinking?

Sarama, J., & Clements, D. H. (2015). Preschoolers getting in shape Exploring math and science in preschool (pp. 35-37). Washington, DC: National Association for the Education of Young Children.

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***Scaling Up Early Mathematics Interventions: Transitioning with Trajectories and Technologies**

**Scaling Up Early Mathematics Interventions: Transitioning with Trajectories and Technologies**

Transitions in the early years have substantial effects on children's success in school. Moreover, lack of consideration of continuity and alignment may mislead both resaerchers and politicians to assume preschool effects 'fade', when it may be that poor transitions to primary school are to blame. We hypothesize that most present educational contexts are unintentionally and perversely aligned against early interventions.

Sarama, J., & Clements, D. H. (2015). Scaling up early mathematics interventions: Transitioning with trajectories and technologies. In B. Perry, A. MacDonald & A. Gervasoni (Eds.), *Mathematics and Transition to School* (pp. 153-169). New York, NY: Springer.

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*Sustainability of a Scale-Up Intervention in Early Mathematics: A Longitudinal Evaluation of Implementation Fidelity*

Fidelity of implementation and the sustainability of effects of a research-based model for scaling up educational interventions. The model was implemented by 64 teachers in 26 schools in 2 distal city districts serving low-resource communities, with schools randomly assigned to condition. Although a logical expectation would be that, after the cessation of external support and professional development provided by the intervention, teachers would show a pattern of decreasing levels of fidelity, these teachers actually demonstrated increasing levels of fidelity, continuing to demonstrate high levels of sustained fidelity in their implementation of the underlying curriculum 2 years past exposure. Different profiles of variables predicted separate aspects of sustainability.

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*Book Chapter - Lessons Learned in the Implementation of the TRIAD Scale-Up Model: Teaching Early Mathematics with Trajectories and Technologies*

Although the successes of research-based, visionary educational practices have been documented, equally recognized is the failure of these practices to be implemented at a scale that affects more than a trivial portion of children. Further, there may be no more challenging educational and theoretical issue than scaling up educational programs across a large number of diverse populations and contexts in the early childhood system in the United States, avoiding the dilution and pollution that usually plagues such efforts to achieve broad success. In this chapter, we describe a model of scale-up at the school district level and its initial evaluation. Although our intent is that the model should apply to all subject matter domains and grade levels, any evaluation must involve a specific instantiation. Our evaluations have focused on early childhood mathematics. Therefore, we begin with background information on the need for models of scale-up, especially in early childhood education, as well as a consideration of the particular needs in mathematics education. Next, we introduce the theoretical framework, the model we developed, and the research corpus on which they were based. We then summarize the empirical evaluations we have conducted of this model. In the final section, we summarize what we have learned and describe implications and challenges for the field.

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*The Impacts of an Early Mathematics Curriculum on Emerging Literacy and Language*

Competence in early mathematics is crucial for later school success. Although research indicates that early mathematics curricula improve children's mathematics skill, such curricula's impacts on oral language and early literacy skills are not known. This project is the first to investigate the effects of an intensive pre-kindergarten mathematics curriculum, Building Blocks, on the oral language and letter recognition of children participating in a large-scale cluster randomized trial project. Results showed no evidence that children who were taught mathematics using the curriculum performed differently than control children who received the typical district mathematics instruction on measures of letter recognition, and on two of the oral language (story retell) subtests, sentence length and inferential reasoning (emotive content). However, children in the Building Blocks group outperformed children in the control group on four oral language subtests: ability to recall key words, use of complex utterances, willingness to reproduce narratives independently, and inferential reasoning (practical content).

Sarama, J., Lange, A., Clements, D.H., & Wolfe, C.B. (2012). The impacts of an early mathematics curriculum on emerging literacy and language. Early Childhood Research Quarterly, 27(3), 489-502.

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*Book Chapter - Mathematics for the Whole Child*

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*Book Chapter- Walking The Same Broad Path (With Side Trips)*

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*Mathematics Knowledge of Low-Income Entering Preschoolers*

For more than a century, researchers have surveyed the specific mathematics skills of children entering school. With increasing numbers of children entering preschool (especially programs designed for children at risk), there is a need for such studies of younger children, especially those from low-resource communities (LRC). We review previous work and report two studies investigating the mathematics knowledge and competencies of children entering preschools in two states in the U.S., using theoretically-based assessments emphasizing psychological developmental progressions. Results suggest that children are acquiring mathematical concepts and skills at younger ages than previous generations. Children from LRC enter preschool with a range of mathematical skills and concepts upon which educators can build, but are not achieving their full potential. These results are intended to assist those responsible for developing standards, writing curricula, and assessing and teaching all children by providing updated information about what children know when they enter school, including the specific levels of achievement along research-based developmental progressions.

Sarama, J., & Clements, D.H. (2011). Mathematics knowledge of low-income entering preschoolers. *Far East Journal of Mathematical Education, 6(*1), 41-63.

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*Evaluation of a Learning Trajectory for Length in the Early Years*

Measurement is a critical component of mathematics education, but research on the learning and teaching of measurement is limited, especially compared to topics such as number and operations. To contribute to the establishment of a research base for instruction in measurement, we evaluated and refined a previously developed learning trajectory in early length measurement, focusing on the developmental progressions that provide cognitive accounts of the development of children's strategic and conceptual knowledge of measure. Findings generally supported the developmental progression, in that children reliably moved through the levels of thinking in that progression. For example, they passed through a level in which they measured length by placing multiple units or attempting to iterate a unit, sometimes leaving gaps between units. However, findings also suggested several refinements to the developmental progression, including the nature and placement of indirect length comparison in the developmental progression and the role of vocabulary, which was an important facilitator of learning for some, but not all, children.

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*Book Chapter - Geometry*

Sarama, J., Clements, D.H., Parmar, R.S., & Garrison, R. (2011). Geometry. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 163-196). Reston, VA: National Council of Teachers of Mathematics.

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*Book Chapter- Technology*

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*Book Chapter- The Mathematical Lives of Young Children*

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**Building Blocks and Cognitive Building Blocks: Playing to Know the World Mathematically*

*Building Blocks and Cognitive Building Blocks: Playing to Know the World Mathematically*

Drs. Sarama and Clements explore how children’s play supports the development of mathematical ideas and skills. We discuss research that suggests how adults can support children’s representation of their play and thus its mathematization. They begin by observing children to see how much and what kinds of mathematics we can actually ﬁnd in the free play of children. Next, we brieﬂy review children’s development of diﬀerent types of play and describe ways adults can support and guide each of these in order to encourage children’s mathematical development.

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*"Concrete" Computer Manipulatives in Mathematics Education*

The use of "concrete manipulatives" in mathematics education is supported by research and often accepted as a sine qua non of "reform" approaches. This article reviews the research on the use of manipulatives and critiques common notions regarding concrete manipulatives. It presents a reformulation of the definition of concrete as used in educational psychology and educational research and provides a rationale of how, based on that reformulation, computer manipulatives may be pedagogically efficacious. The article presents 7 hypothesized, interrelated affordances of manipulatives and briefly reviews evidence for their empirical validity.

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*Teaching Math in the Primary Grades*

Children's thinking follows natural developmental paths in learning math. When teachers understand those paths and offer activities based on children's progress along them, they build developmentally appropriate math environments. The authors explain math learning trajectories and why teaching math using the trajectories approach is effective. A chart gives examples of instructional tasks for the learning trajectory for addition and subtraction.

Sarama, J., & Clements, D.H. (2009). Teaching math in the primary grades: The learning trajectories approach. *Young Children, 64*(2), 63-65.

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*Book Chapter- Learning and teaching geometry with computers in the elementary and middle school*

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*Book Chapter- Linking research and software development*

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*Effects of a Preschool Mathematics Curriculum: Summative Research on the Building Blocks Project*

This study evaluated the efficacy of a preschool mathematics program based on a comprehensive model of developing research-based software and print curricula. Building Blocks, funded by the National Science Foundation, is a curriculum development project focused on creating research-based, technology-enhanced mathematics materials for pre-K through grade 2.