What is the Learning and Teaching with Learning Trajectories Tool ([LT]²)?

• [LT]² is an open-access web-based tool for early childhood educators to learn about how children think and learn about mathematics and how to teach mathematics to young children (birth to age 8).

• [LT]² runs on all technological platforms, addresses new ages—birth to age 8 years—and includes new alignments with standards and assessments, as well as our new software for children. [LT]²enables teachers to help children find the mathematics in—and develop the mathematics from—their everyday activities, including art, stories, puzzles, and games. Go to LearningTrajectories.org.

• [LT]² allows teachers, caregivers, parents, researchers, and instructional coaches and professional development providers to seethe learning trajectories for math, as they view short video clips of classroom instruction and children working on math problems in a way that clearly reveals their thinking.

• [LT]² was funded by the Heising-Simons Foundation and the Bill and Melinda Gates Foundation and developed from decades of research conducted by Drs. Julie Sarama and Douglas H. Clements, Distinguished University Professors and Kennedy Endowed Chairs in the Morgridge College of Education at the University of Denver. Creators of the math curriculum, Building Blocks, they also designed professional learning tools to help teachers implement effective and appropriate mathematics education for young children—the “parents” of [LT]². This approach is successful, as shown by large-scale studies and the What Works Clearinghouse, and praised on the cover of The New York Timesand in the Wall Street Journal. If you want to read about two teachers’ use of [LT]², see https://bit.ly/2oQ1Yq4and https://bit.ly/2JnblK5.

A large project that is just being completed is "Scaling Up TRIAD: Teaching Early Mathematics for Understanding with Trajectories and Technologies," the third of a sequence of rigorous evaluations of a model of scaling up successful interventions, in this specific case, to increase math achievement in young children, especially those at risk, by means of a high-quality implementation of the "Building Blocks" math curriculum, with all aspects of the curriculum--content, pedagogy, technology, and assessments--based on a common core of learning trajectories.

The reason this is important is that although the successes of some research-based educational practices have been documented, so too has the inability U.S. schools to successful adopt and scale up these practices. A particularly challenging educational and theoretical issue is scaling up educational programs across the large number of diverse populations and contexts in the early childhood system in the U.S., while avoiding the dilution and pollution that usually plagues such efforts to achieve broad success. With previous funding, Sarama Clements created a research-based model to meet this challenge in the area of mathematics, with the intent to generalize the model to other subject matter areas and other age groups.  The field also needs transferable, practical examples of scale up; empirical evidence of the effectiveness of these examples; and focused research on critical variables--all leading to refined, generalizable theories and models of scale up. Results of the present study indicated high levels of fidelity of implementation resulting in consistently higher scores in the intervention classes on the observation instrument and statistically significant and substantially greater gains in children's mathematics, again with substantial effect sizes in preschool and continuing into kindergarten and 1st grade, significantly more so in the "Follow Through" condition in which Kindergarten and 1st grade teachers also received professional development. 

The National Science Foundation provided us funding to develop and evaluate an innovative curriculum for early children education, preschool to grade 2. The Building Blocks curriculum incorporates both old and new technologies, from blocks and puzzles to multimedia computer programs. Preliminary evaluations show the program's approach of finding the mathematics in, and developing mathematics from, children's every day activity, allows children to learn and do more mathematics than previously assumed. Building Blocks address an urgent need for early childhood mathematics curriculum. Building Blocks' approach is finding the mathematics in, and developing mathematics from, children's activity. The curriculum and software design has an explicit theoretical and empirical foundation, beyond its genesis in someone’s intuitive grasp of children's learning. Materials are truly research-based.

This project is studying the learning and teaching of measurement in early and elementary education. Conducted in collaboration with Jeffrey Barrett form Illinois State University, we are producing research-based developmental progressions in measurement across a seven-year span. These developmental progressions will build on and elaborate existing research-based learning trajectories (Sarama & Clements, 2009).

I directed, with Sarama, an historic, national Conference on Standards for Preschool and Kindergarten Mathematics Education (co-funded by NSF and ExxonMobil Foundation), which resulted in a book, Clements, D. H., Sarama, J., and DiBiase, A.-M. (Eds.). (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. In addition, based on the conference, I proposed and chaired a joint committee that produced a joint National Association for the Education of Young children and National Council of Teachers of Mathematics (NCTM) position statement on early childhood mathematics, Early childhood mathematics: Promoting good beginnings. A joint position statement of the National Association for the Education of Young Children (NAEYC) and the National Council for Teachers of Mathematics (NCTM).

EMERGE: Synthesizing Effective Approaches to Early Childhood Education: Building Blocks and Self-Regulation Julie Sarama and I, along with colleagues Elena Brodrova (McREL) and Carolyn Layzer (Abt Assoc.) were funded by IES to conduct an efficacy trial of a synthesis of a content-focused, or "academic" curricula with more global goals and approaches to early education, by creating and evaluating the integration of two empirically-tested interventions, identifying mediators and moderators of effects. The four-year study is entitled Increasing the efficacy of an early mathematics curriculum with scaffolding designed to promote self-regulation. A cluster randomized trials design is being employed to evaluate the synthesized approach. Early results indicate that some teachers found it difficult to implement the self-regulation strategies. Overall, there were no significant effects on any measures of self-regulation.

What is the Learning and Teaching with Learning Trajectories Tool ([LT]²)?

• [LT]²is an open-access web-based tool for early childhood educators to learn about how children think and learn about mathematics and how to teach mathematics to young children (birth to age 8).

• [LT]²runs on all technological platforms, addresses new ages—birth to age 8 years—and includes new alignments with standards and assessments, as well as our new software for children. [LT]²enables teachers to help children find the mathematics in—and develop the mathematics from—their everyday activities, including art, stories, puzzles, and games. Go to LearningTrajectories.org.

• [LT]²allows teachers, caregivers, parents, researchers, and instructional coaches and professional development providers to seethe learning trajectories for math, as they view short video clips of classroom instruction and children working on math problems in a way that clearly reveals their thinking.

• [LT]²was funded by the Heising-Simons Foundation and the Bill and Melinda Gates Foundation and developed from decades of research conducted by Drs. Julie Sarama and Douglas H. Clements, Distinguished University Professors and Kennedy Endowed Chairs in the Morgridge College of Education at the University of Denver. Creators of the math curriculum, Building Blocks, they also designed professional learning tools to help teachers implement effective and appropriate mathematics education for young children—the “parents” of [LT]². This approach is successful, as shown by large-scale studies and the What Works Clearinghouse, and praised on the cover of The New York Timesand in the Wall Street Journal. If you want to read about two teachers’ use of [LT]², see https://bit.ly/2oQ1Yq4and https://bit.ly/2JnblK5.

A large project that is just being completed is "Scaling Up TRIAD: Teaching Early Mathematics for Understanding with Trajectories and Technologies," the third of a sequence of rigorous evaluations of a model of scaling up successful interventions, in this specific case, to increase math achievement in young children, especially those at risk, by means of a high-quality implementation of the "Building Blocks" math curriculum, with all aspects of the curriculum--content, pedagogy, technology, and assessments--based on a common core of learning trajectories.

The reason this is important is that although the successes of some research-based educational practices have been documented, so too has the inability U.S. schools to successful adopt and scale up these practices. A particularly challenging educational and theoretical issue is scaling up educational programs across the large number of diverse populations and contexts in the early childhood system in the U.S., while avoiding the dilution and pollution that usually plagues such efforts to achieve broad success. With previous funding, Sarama Clements created a research-based model to meet this challenge in the area of mathematics, with the intent to generalize the model to other subject matter areas and other age groups.  The field also needs transferable, practical examples of scale up; empirical evidence of the effectiveness of these examples; and focused research on critical variables--all leading to refined, generalizable theories and models of scale up. Results of the present study indicated high levels of fidelity of implementation resulting in consistently higher scores in the intervention classes on the observation instrument and statistically significant and substantially greater gains in children's mathematics, again with substantial effect sizes in preschool and continuing into kindergarten and 1st grade, significantly more so in the "Follow Through" condition in which Kindergarten and 1st grade teachers also received professional development. 

The National Science Foundation provided us funding to develop and evaluate an innovative curriculum for early children education, preschool to grade 2. The Building Blocks curriculum incorporates both old and new technologies, from blocks and puzzles to multimedia computer programs. Preliminary evaluations show the program's approach of finding the mathematics in, and developing mathematics from, children's every day activity, allows children to learn and do more mathematics than previously assumed. Building Blocks address an urgent need for early childhood mathematics curriculum. Building Blocks' approach is finding the mathematics in, and developing mathematics from, children's activity. The curriculum and software design has an explicit theoretical and empirical foundation, beyond its genesis in someone’s intuitive grasp of children's learning. Materials are truly research-based.

This project is studying the learning and teaching of measurement in early and elementary education. Conducted in collaboration with Jeffrey Barrett form Illinois State University, we are producing research-based developmental progressions in measurement across a seven-year span. These developmental progressions will build on and elaborate existing research-based learning trajectories (Sarama & Clements, 2009).

I directed, with Sarama, an historic, national Conference on Standards for Preschool and Kindergarten Mathematics Education (co-funded by NSF and ExxonMobil Foundation), which resulted in a book, Clements, D. H., Sarama, J., and DiBiase, A.-M. (Eds.). (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. In addition, based on the conference, I proposed and chaired a joint committee that produced a joint National Association for the Education of Young children and National Council of Teachers of Mathematics (NCTM) position statement on early childhood mathematics, Early childhood mathematics: Promoting good beginnings. A joint position statement of the National Association for the Education of Young Children (NAEYC) and the National Council for Teachers of Mathematics (NCTM).

EMERGE: Synthesizing Effective Approaches to Early Childhood Education: Building Blocks and Self-Regulation Julie Sarama and I, along with colleagues Elena Brodrova (McREL) and Carolyn Layzer (Abt Assoc.) were funded by IES to conduct an efficacy trial of a synthesis of a content-focused, or "academic" curricula with more global goals and approaches to early education, by creating and evaluating the integration of two empirically-tested interventions, identifying mediators and moderators of effects. The four-year study is entitled Increasing the efficacy of an early mathematics curriculum with scaffolding designed to promote self-regulation. A cluster randomized trials design is being employed to evaluate the synthesized approach. Early results indicate that some teachers found it difficult to implement the self-regulation strategies. Overall, there were no significant effects on any measures of self-regulation.

ULTIMATE (Understanding Learning Trajectories in Math: Advancing Teacher Education) is a research project funded by the National Science Foundation (NSF, under the title, "Learning Trajectories as a Complete Early Mathematics Intervention: Achieving Efficacies at Economies of Scale").

ULTIMATE will support teachers in deepening their understanding of how children learn mathematics and how to incorporate this understanding into their classrooms to help children develop math ideas and skills—joyfully. Over two decades, Drs. Clements and Sarama have built a professional development tool, called Learning and Teaching with Learning Trajectories, or [LT]² (see a full description in the leftmost column). The DU team will work with teachers, blending high quality in-person professional development with teachers' use of the [LT]² tool. The team will investigate the positive impacts both in supporting teachers and on students' learning of mathematics.

EPIC is a large-scale research project funded by the Institute of Education Sciences (IES, USA Dept. of Education), assessing the impacts of a National Science Foundation-funded curriculum, Connect4Learning (C4L–see more on C4L on this page).

C4L is an interdisciplinary early childhood prekindergarten curriculum, funded by the National Science Foundation, which aims to synthesize research-based approaches in four domains of learning: literacy, mathematics, science, and social-emotional development. C4L uses an interdisciplinary approach to address growing concerns that the majority of preschool instructional time is devoted to literacy at the expense of other content areas, particularly math and science. The aim of the EPIC project is to evaluate the efficacy of the C4L curriculum when it is implemented as intended.

STEMIIEE (STEM Innovation for Inclusion in Early Education) is a new project in the University of Denver’s Morgridge College of Education, in collaboration with the University of North Carolina at Chapel Hill, has been awarded a new grant focused on advancing STEM (Science, Technology, Engineering, and Mathematics) in early childhood, including how to increase access and participation for young children with disabilities. Julie Sarama and Doug Clements, co-directors of the University’s Marsico Institute, will be co-principal investigators on the grant aimed at launching the Center for the Advancement of Early STEM Education (CAESE).  The new award is funded by the Office of Special Education Programs (OSEP), Department of Education, under the title, "Special Education Educational Technology Media, and Materials for Individuals with Disabilities."

Their goal is to ensure that all young children, especially those with disabilities, can engage in STEM and benefit from STEM learning.

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The project will evaluate the usefulness of learning trajectories for improving student achievement in early mathematics.  Learning trajectories have received increasing attention from policy makers, educators, curriculum developers, and researchers and are generally deemed as a useful tool for guiding instructional planning and assessment. For example, they formed the core of the Common Core. However, no research has directly evaluated the contributions of LTs to teaching and learning. The project includes a series of eight randomized clinical trials testing different aspects of LTs. These experiments will determine whether LTs are better than other approaches in supporting young children’s learning. In sum, we are investigating how to bestuse learning trajectories to support young children’s learning.

This is a follow-up to the above "Evaluating the Efficacy of Learning Trajectories in Early Mathematics" project, the brain child of Traci Kutaka, Project Director. Julie Sarama and are are co-PIs along with Dr. Kutaka. Children’s ability to answer assessment questions correctly paints an incomplete portrait of what they know and can do; yet, it remains a common basis for program evaluation. We should study the development of competences by observing the ways children approach and make sense of problems, and how they learn to develop, apply, and generalize problem-solving strategies over time, from one type of problem to the next. This project has two goals. First, the research team will determine how kindergarten children’s problem- solving strategies vary within and across twelve arithmetic problem structures and how they evolve over the course of an intervention. Second, they will construct two novel indicators of instructional efficacy, which jointly account for patterns of strategy use over time, application of strategies to increasingly sophisticated arithmetic problem types, and correctness of response: (1) a Sophistication indicator and (2) a Breadth indicator.

Scalable Professional Development in Early Mathematics: The Learning and Teaching with Learning Trajectories (LT2) Tool

In the course of decades of research-and-development work funded by the NSF and IES, we have developed a powerful tool for professional development of teachers of early mathematics, our “Building Blocks Learning Trajectories” (BBLT ) web application. This software tool presents early childhood mathematics learning trajectories, connecting the three critical components of early childhood mathematics: the mathematical content, how children think about and learn that content, and teaching strategies. It provides scalable access to our research-based learning trajectories via descriptions, videos, commentaries, and interactive experiences.

The Heising-Simons Foundation is funding us to to bolster the usefulness and capacity of this tool, implementing it on different platforms and reaching greater numbers of teachers, caregivers, and parents.

Another new project has been funded by the National Science Foundation (NSF) is a $2 ½ million grant that will fund efforts by Sarama and Clements, along with colleagues (and mother-and-son team) Curtis Tatsuoka and Kikumi Tatsuoka, to create and test a new early mathematics assessment. This assessment will use innovative statistical and computer technology to give teachers more useful and detailed information about children's knowledge of mathematics in less time than existing assessments.  Fast but fully informative assessments help teachers really know their students, and support their use of the powerful teaching strategy of 'formative assessment' or individualizing learning.

Julie Sarama (PI) and I have been funded by the NSF to combine our work on the Building Blocks math curriculum with that of colleagues in other fields.  The Connect4Learning interdisciplinary curriculum will connect four basic domains of learning. In addition to mathematics, the grant includes experts in science (Kimberly Brenneman, Rutgers University), literacy/language (Nell Duke, Michigan State University) and social-emotional development (M. L. Hemmeter, Vanderbilt University). Early childhood is full of debates about subject matter, with arguments arising about new emphases on mathematics taking too much time away from literacy. Science is rarely mentioned. Further, there is little research on whether an emphasis in one area necessarily means less emphasis in others, or whether they can be combined each to the benefit of others. The researchers believe the latter, and believe Connect4Learning curriculum will encourage all children to develop their full potential in all four areas--a potential that is greater than often realized.