Early Math Publications
Divided into early math development and early math interventions, Marsico Institute researchers publish articles about children's development and learning of early math. We also publish about interventions, pursuing knowledge about what works for whom and in what context.
Early Math Development

2021
LENGTH MEASUREMENT IN THE EARLY YEARS: TEACHING AND LEARNING WITH LEARNING TRAJECTORIES
Measurement is a critical component of mathematics education, but research on the learning and teaching of measurement is limited. We previously introduced, refined, and validated a developmental progression – the cognitive core of a learning trajectory – for length measurement in the early years. A complete learning trajectory includes instructional activities and pedagogical strategies, correlated with each level of the developmental progression. This study evaluated a portion of our learning trajectory, focusing on the instructional component. We found that the instruction was successful in promoting a progression from one level to the next for 40% of the children, with others developing positive new behaviors (but not sufficient to progress to a new level). We identified specific characteristics of the interventions’ tasks and teacher–child interactions that engendered learning. The use of standard or multiple nonstandard units may play different roles at different points in the learning trajectory.
 Julie Sarama, Douglas H. Clements, Jeffrey E. Barrett, Craig J. Cullen, Aaron Hudyma & Yuly Vanegas (2021) Length measurement in the early years: teaching and learning with learning trajectories, Mathematical Thinking and Learning, DOI: 10.1080/10986065.2020.1858245

2020
YOUNG CHILDREN’S ACTIONS ON LENGTH MEASUREMENT TASKS: STRATEGIES AND COGNITIVE ATTRIBUTES
Researchers often develop instruments using correctness scores (and a variety of theories and techniques, such as Item Response Theory) for validation and scoring. Less frequently, observations of children’s strategies are incorporated into the design, development, and application of assessments. We conducted individual interviews of 833 prekindergartens to Grade 3 children using 70 length measurement assessment items, recording both correctness and strategy use. We performed qualitative analyses of children’s strategies to validate or refine our hypotheses of how items align with strategies, and the concepts and procedures underlying them. We then used these analyses to refine: (a) our definitions of those strategies, concepts, and procedures; (b) the items in the assessment instrument; and (c) the alignment of both. The results will form the foundation of a ComputerAdaptive Test for length measurement, which we intend to validate in future research.
 Douglas H. Clements, Holland Banse, Julie Sarama, Curtis Tatsuoka, Candace Joswick, Aaron Hudyma, Douglas W. Van Dine & Kikumi K. Tatsuoka (2020) Young children’s actions on length measurement tasks: strategies and cognitive attributes, Mathematical Thinking and Learning, DOI: 10.1080/10986065.2020.1843231
EXAMINING AND DEVELOPING FOURTH GRADE CHILDREN’S AREA ESTIMATION PERFORMANCE
We present our findings with respect to our two research questions: 1) How well do children at the ICS and ARCS levels of the LT for area measurement estimate areas of rectangles and what is the nature of their estimates? And 2) To what extent can ICS and ARCS level children’s area estimation performance be quantitatively improved through targeted instruction emphasizing spatial structuring with numeric feedback? In the following sections we first describe fourthgrade children’s area estimates of rectangles including the direction of errors. We found a clear tendency to make underestimates rather than overestimates. Next, we describe the changes in children’s estimates of areas of rectangles. Children showed marked improvement after as little as two sessions.
 Eames, C. L., Barrett, J. E., Cullen, C. J., Rutherford, G., Klanderman, D., Clements, D. H., Sarama, Julie, and Van Dine, D. W. (2020). Examining and developing fourth grade children’s area estimation performance. School Science and Mathematics, 120(2), 6778. doi: 10.1111/ssm.12386
EFFECTS ON MATHEMATICS AND EXECUTIVE FUNCTION OF A MATHEMATICS AND PLAY INTERVENTION VERSUS MATHEMATICS ALONE
Early education is replete with debates about “academic” versus “play” approaches. We evaluated 2 interventions, the Building Blocks (BB) mathematics curriculum and the BB synthesized with scaffolding of play to promote executive function (BBSEF), compared to a businessasusual (BAU) control using a 3armed cluster randomized trial with more than 1,000 children in 84 preschool classrooms across three districts (multiracial or multiethnic, low income, 27% English Language Learner). Impact estimates for BBSEF were mixed in sign, small in magnitude, and insignificant. Most impact estimates for BB were positive, but only a few were statistically significant, with more in the kindergarten year (delayed effects), including both mathematics achievement and executive function (EF) competencies. Gains in both mathematics and EF can be mutually supportive and thus resist the fadeout effect.
 Clements, D. H., Sarama, J., Layzer, C., Unlu, F., & Fesler, L. (2020). Effects on mathematics and executive function of a mathematics and play intervention versus mathematics alone. Journal for Research in Mathematics Education, 51(3), 301333. doi:10.5951/jresemtheduc20190069

2019
STRATEGY DIVERSITY IN EARLY MATHEMATICS CLASSROOMS
Strategic processes are a form of procedural knowledge in which a child knows how to enact a given strategy that improves their capability in problem solving or learning. The solution strategies children use are critical components of their learning, especially in mathematics. Children vary substantially in their knowledge and use of different strategies, and much research has focused on intraindividual strategy variability. However, we do not know if classrooms that evince a broader variety of strategies across children are related to higher mathematics achievement. We investigated the diversity of arithmetical strategies within classrooms and examined the relations between strategy diversity and mathematical achievement as children moved from preschool to kindergarten and first grade. These analyses were applied to data from a largescale experiment involving 1,305 children from 42 schools and 106 classrooms. We created and applied a new method of measuring classroom strategy diversity and related this measure to children’s concurrent and subsequent math achievement. We found that early strategy diversity was strongly related to achievement, but in subsequently, less diversity was so related. We compared these results to the predictions of three theoretical categories and found that our results mainly supported one.
 Clements, Douglas & Dumas, Denis & Dong, Yixiao & Banse, Holland & Sarama, Julie & DayHess, Crystal. (2019). Strategy Diversity in Early Mathematics Classrooms. Contemporary Educational Psychology. 60. 101834. DOI: 10.1016/j.cedpsych.2019.101834.
EVALUATING THE EFFICACY OF A LEARNING TRAJECTORY FOR EARLY SHAPE COMPOSITION
Although basing instruction on learning trajectories (LTs) is often recommended, there is little direct evidence regarding the premise of a LT approach—that instruction should be presented (only) one LT level beyond a child’s present level. We evaluated this hypothesis in the domain of early shape composition. One group of preschoolers, who were at least two levels below the target instructional LT level, received instruction based on an empirically validated LT. The counterfactual (skiplevels) group received an equal amount of instruction focused only on the target level. At posttest, children in the LT condition exhibited significantly greater learning than children in the skiplevels condition, mainly on neartransfer items; no childlevel variables were significant moderators. Implications for theory and practice are discussed.
 Clements, D. H., Sarama, J., Baroody, A. J., Joswick, C., & Wolfe, C. B. (2019). Evaluating the efficacy of a learning trajectory for early shape composition. American Educational Research Journal, 56(6), 25092530. doi: 10.3102/0002831219842788
DOUBLE IMPACT: MATHEMATICS AND EXECUTIVE FUNCTION
This article reviews the research implications of the correlation between the development of executive function and the development of early math. Executive function is a combination of inhibitory control, cognitive flexibility, and working memory. These important skills are considered the “air traffic control” of the mind, thus supporting and are supported children’s experiences with early math.
 Joswick, C., Clements, D. H., Sarama, J., Banse, H., & DayHess, C. A. (2019). Double impact: Mathematics and executive function. Teaching Children Mathematics, 25(7), 416426. Clements, D. H., Sarama, J., Swaminathan, S., Weber, D., & TrawickSmith, J. (2018). Teaching and learning Geometry: Early foundations. Quadrante, 27(2), 731.

2018
WHAT IS THE LONGRUN IMPACT OF LEARNING MATHEMATICS DURING PRESCHOOL?
The current study estimated the causal links between preschool mathematics learning and late elementary school mathematics achievement using variation in treatment assignment to an early mathematics intervention as an instrument for preschool mathematics change. Estimates indicate (n = 410) that a standard deviation of interventionproduced change at age 4 is associated with a 0.24SD gain in achievement in late elementary school. This impact is approximately half the size of the association produced by correlational models relating later achievement to preschool math change, and is approximately 35% smaller than the effect reported by highly controlled ordinary least squares (OLS) regression models (Claessens et al., 2009; Watts et al., 2014) using national data sets. Implications for developmental theory and practice are discussed.
 Watts, T. W., Duncan, G. J., Clements, D. H., & Sarama, J. (2018). What is the longrun impact of learning mathematics during preschool? Child Development, 89(2), 539–555. doi:10.1111/cdev.12713

2017
DIFFERENTIAL EFFECTS OF THE CLASSROOM ON AFRICAN AMERICAN AND NONAFRICAN AMERICAN'S MATHEMATICS ACHIEVEMENT
Schenke, K., Watts, T. W., Nguyen, T., Sarama, J., & Clements, D. H. (2017). Differential effects of the classroom on African American and nonAfrican American's mathematics achievement.. Journal of Educational Psychology, 109(6), 794811.
DOES EARLY MATHEMATICS INTERVENTION CHANGE THE PROCESSES UNDERLYING CHILDREN'S LEARNING?
Watts, T. W., Clements, D. H., Sarama, J., Wolfe, C. B., Spitler, M. E., & Bailey, D. H. (2017). Does early mathematics intervention change the processes underlying children’s learning? Journal of Research on Educational Effectiveness, 10(1), 96115. doi: 10.1080/19345747.2016.1204640

2016
LEARNING EXECUTIVE FUNCTION AND EARLY MATHEMATICS: DIRECTIONS OF CAUSAL RELATIONS
Clements, D. H., Sarama, J., & Germeroth, C. (2016). Learning executive function and early mathematics: Directions of causal relations. Early Childhood Research Quarterly, 36, 79–90. doi: 10.1016/j.ecresq.2015.12.009.
WHICH PRESCHOOL MATHEMATICS COMPETENCIES ARE MOST PREDICTIVE OF FIFTH GRADE ACHIEVEMENT?
Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., Wolfe, C., & Spitler, M. E. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement?Early Childhood Research Quarterly, 36, 550560. doi: http://dx.doi.org/10.1016/j.ecresq.2016.02.003

2015
PRESCHOOLERS GETTING IN SHAPE
Sarama, J., & Clements, D. H. (2015). Preschoolers getting in shape Exploring math and science in preschool (pp. 3537). Washington, DC: National Association for the Education of Young Children.
PROCESSES IN THE DEVELOPMENT OF MATHEMATICS IN KINDERGARTEN CHILDREN FROM TITLE 1 SCHOOLS
Foster, M. E., Anthony, J. L., Clements, D. H., & Sarama, J. (2015). Processes in the development of mathematics in kindergarten children from Title 1 schools. Journal of Experimental Child Psychology, 140, 56–73.
BOOK CHAPTER: DEVELOPING YOUNG CHILDREN'S MATHEMATICAL THINKING AND UNDERSTANDING
Clements, D.H., & Sarama, J. (2015). Developing young children’s mathematical thinking and understanding. (pp. 331344). New York, NY: Routledge.
DISCUSSION FROM A MATHEMATICS EDUCATION PERSPECTIVE
Clements, D. H., & Sarama, J. (2015). Discussion from a mathematics education perspective. Mathematical Thinking and Learning, 17(23), 244252. doi: 10.1080/10986065.2015.1016826

2014
THE IMPORTANCE OF THE EARLY YEARS
Clements, D.H., & Sarama, J. (2014). The importance of the early years. In R.E. Slavin (Ed.), Science, Technology, & Mathematics (STEM) (pp. 59). Thousand Oaks, CA: Corwin.
LEARNING AND TEACHING EARLY MATH: THE LEARNING TRAJECTORIES APPROACH (2ND ED.)
Clements, D.H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.). New York, NY: Taylor & Francis.

2013
BACKGROUND RESEARCH ON EARLY MATHEMATICS
Clements, D.H., & Sarama, J., & Baroody, A. J. (2013). Background research on early mathematics. Washington, DC: National Govenors Association. Retrieved from http://www.nga.org/cms/home/ngacenterforbestpractices/meetingwebcastmaterials/pageedumeetingswebcasts/col2content/maincontentlist/strengtheningearlymathematics.html
BOOK CHAPTER  SOLVING PROBLEMS: MATHEMATICS FOR YOUNG CHILDREN
Clements, D.H., & Sarama, J. (2013). Solving problems: Mathematics for young children. In D. R. Reutzel (Ed.), Handbook of research based practice in early education (pp. 348363). New York, NY: The Guilford Press.
BOOK CHAPTER  RETHINKING EARLY MATHEMATICS: WHAT IS RESEARCHBASED CURRICULUM FOR YOUNG CHILDREN?
Clements, D.H., & Sarama, J. (2013).Rethinking early mathematics: What is researchbased curriculum for young children? In L. D. English & J. T. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 121147). Dordrecht, Germany: Springer.
MATH IN THE EARLY YEARS
Clements, D. H., & Sarama, J. (2013). Math in the early years [ECS Research Brief: The progress of educational reform]. Denver CO: Education Commission of the States.
YOUNG CHILDREN'S UNDERSTANDINGS OF LENGTH MEASUREMENT: EVALUATING A LEARNING TRAJECTORY
Szilágyi, J., Clements, D. H., & Sarama, J. (2013). Young children's understandings of length measurement: Evaluating a learning trajectory. ZDMThe International Journal on Mathematics Education, 44(3), 581620.

2012
BOOK CHAPTER  MATHEMATICS FOR THE WHOLE CHILD
Sarama, J., & Clements, D.H. (2012). Mathematics for the whole child. In S. Suggate & E. Reese (Eds.), Contemporary debates in childhood education and development (pp. 7180). New York, NY: Routledge.

2011
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. 163196). Reston, VA: National Council of Teachers of Mathematics.
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. 197218). Reston, VA: National Council of Teachers of Mathematics
MATHEMATICS KNOWLEDGE OF LOWINCOME ENTERING PRESCHOOLERS
Sarama, J., & Clements, D.H. (2011). Mathematics knowledge of lowincome entering preschoolers. Far East Journal of Mathematical Education, 6(1), 4163.
EVALUATION OF A LEARNING TRAJECTORY FOR LENGTH IN THE EARLY YEARS
Sarama, J., Clements, D.H., Barrett, J.E., Van Dine, D.W., & McDonel, J.S. (2011). Evaluation of a learning trajectory for length in the early years. ZDMThe International Journal on Mathematics Education, 43, 667680. doi: 10.1007/s1185801103265.
Early Math Interventions

2021
IS INTERVENTION FADEOUT A SCALING ARTEFACT?
To determine whether scaling decisions might account for fadeout of impacts in early education interventions, we reanalyze data from a wellknown early mathematics RCT intervention that showed substantial fadeout in the two years after the intervention ended. We examine how various orderpreserving transformations of the scale affect the relative mathematics achievement of the control and experimental groups by age. Although fadeout was robust to most transformations, we were able to eliminate or even reverse fadeout by emphasizing differences in scores near typical levels of firstgraders while treating differences elsewhere as unimportant. Such a transformation lowers treatment effects at preschool age and raises them in first grade, relative to the original scale. The findings suggest substantial implications for interpreting the effects of educational interventions.
 Sirui Wan, Timothy N. Bond, Kevin Lang, Douglas H. Clements, Julie Sarama, Drew H. Bailey, Is intervention fadeout a scaling artefact?, Economics of Education Review, Volume 82, 2021, 102090, ISSN 02727757. doi.org/10.1016/j.econedurev.2021.102090

2020
MATHEMATICS AND EXECUTIVE FUNCTION COMPETENCIES IN THE CONTEXT OF INTERVENTIONS: A QUANTILE REGRESSION ANALYSIS
Research on young children’s development of executive function (EF) and early mathematics has established relationships between the two, but studies have not investigated whether these relations differ for children with different outcomes in mathematics and EF, especially in the context of interventions. To examine the homogeneity of those relations and the intervention effects, we conducted quantile regression analyses on data from a large study of two prekindergarten interventions: the Building Blocks math curriculum alone (BB), or BB with scaffolding of play to promote executive function (BBSEF). Results revealed that EF competencies have a larger positive relationship to mathematics for children with low math competence compared to children with medium or high competence. The significant predictive relationship of mathematical competencies on EF did not vary for children with different levels of EF abilities at prekindergarten and varied only slightly at kindergarten. Also, interventions had similar immediate effects on math and EF for children with various abilities. The BB intervention had a larger positive delayed effect on math and EF competencies for children with low scores relative to children with high scores. The delayed effect of the BBSEF intervention was similar for children with different levels of EF and math competencies.
 Dong, Yixiao & Clements, Douglas & Sarama, Julie & Dumas, Denis & Banse, Holland & DayHess, Crystal. (2020). Mathematics and Executive Function Competencies in the Context of Interventions: A Quantile Regression Analysis. The Journal of Experimental Education. DOI: 10.1080/00220973.2020.1777070.

2019
PRESCHOOL MATHEMATICS INTERVENTION CAN SIGNIFICANTLY IMPROVE STUDENT LEARNING TRAJECTORIES THROUGH ELEMENTARY SCHOOL
Perhaps more than at any other time in history, the development of mathematical skill is critical for the longterm success of students. Unfortunately, on average, U.S. students lag behind their peers in other developed countries on mathematics outcomes, and within the United States, an entrenched mathematics achievement gap exists between students from more highly resourced and socially dominant groups, and minority students. To begin to remedy this situation, educational researchers have created instructional interventions designed to support the mathematical learning of young students, some of which have demonstrated efficacy at improving student mathematical skills in preschool, as compared with a businessasusual control group. However, the degree to which these effects last or fade out in elementary school has been the subject of substantial research and debate, and differences in scholarly viewpoints have prevented researchers from making clear and consistent policy recommendations to educational decision makers and stakeholders. In this article, we use a relatively novel statistical framework, Dynamic Measurement Modeling, that takes both intra and interindividual student differences across time into account, to demonstrate that while students who receive a shortterm intervention in preschool may not differ from a control group in terms of their longterm mathematics outcomes at the end of elementary school, they do exhibit significantly steeper growth curves as they approach their eventual skill level. In addition, this significant improvement of learning rate in elementary school benefited minority (i.e., Black or Latinx) students most, highlighting the critical societal need for researchbased mathematics curricula in preschool.
 Dumas, D., McNeish, D., Sarama, J., & Clements, D. (2019). Preschool mathematics intervention can significantly improve student learning trajectories through elementary school. AERA Open, 5(4), 15. doi:10.1177/2332858419879446

2018
EFFECTS OF THREE INTERVENTIONS ON CHILDREN'S SPATIAL STRUCTURING AND COORDINATION OF AREA UNITS
We examine the effects of 3 interventions designed to support Grades 25 children's growth in measuring rectangular regions in different ways. We employed the microgenetic method to observe and describe conceptual transitions and investigate how they may have been prompted by the interventions. We compared the interventions with respect to children's learning and then examined patterns in observable behaviors before and after transitions to more sophisticated levels of thinking according to a learning trajectory for area measurement Our findings indicate that creating a complete record of the structure of the 2dimensional arrayby drawing organized rows and columns of equalsized unit squaresbest supported children in conceptualizing how units were built, organized, and coordinated, leading to improved performance.
 Cullen, A. L., Eames, C. L., Cullen, C. J., Barrett, J. E., Sarama, J., Clements, D. H., & Van Dine, D. W. (2018). Effects of three interventions on children's spatial structuring and coordination of area units. Journal for Research in Mathematics Education, 49(5), 533574.
RISKY BUSINESS: CORRELATION AND CAUSATION IN LONGITUDINAL STUDIES OF SKILL DEVELOPMENT
Developmental theories often posit that changes in children's early psychological characteristics will affect much later psychological, social, and economic outcomes. However, tests of these theories frequently yield results that are consistent with plausible alternative theories that posit a much smaller causal role for earlier levels of these psychological characteristics. Our paper explores this issue with empirical tests of skill building theories, which predict that early boosts to simpler skills (e.g., numeracy or literacy) or behaviors (e.g, antisocial behavior or executive functions) support the longterm development of more sophisticated skills or behaviors. Substantial longitudinal associations between academic or socioemotional skills measured early and then later in childhood or adolescence are often taken as support of these skillbuilding processes. Using the example of skillbuilding in mathematics, we argue that longitudinal correlations, even if adjusted for an extensive set of baseline covariates, constitute an insufficiently risky test of skillbuilding theories. We first show that experimental manipulation of early math skills generates much smaller effects on later math achievement than the nonexperimental literature has suggested. We then conduct falsification tests that show puzzlingly high crossdomain associations between early math and later literacy achievement. Finally, we show that a skillbuilding model positing a combination of unmeasured stable factors and skillbuilding processes is able to reproduce the pattern of experimental impacts on children's mathematics achievement. Implications for developmental theories, methods, and practice are discussed.
 Bailey, D. H., Duncan, G. J., Watts, T. W., Clements, D. H., & Sarama, J. (2018). Risky business: Correlation and causation in longitudinal studies of skill development. American Psychologist, 73(1), 8194.
DO HIGH QUALITY KINDERGARTEN AND FIRST GRADE CLASSROOMS MITIGATE PRESCHOOL FADEOUT?
Prior research shows that shortterm effects from preschool may disappear, but little research has considered which environmental conditions might sustain academic advantages from preschool into elementary school. Using secondary data from two preschool experiments, we investigate whether features of elementary schools, particularly advanced content and highquality instruction in kindergarten and first grade, as well as professional supports to coordinate curricular instruction, reduce fadeout. Across both studies, our measures of instruction did not moderate fadeout. However, results indicated that targeted teacher professional supports substantially mitigated fadeout between kindergarten and first grade but that this was not mediated through classroom quality. Future research should investigate the specific mechanisms through which aligned preschoolelementary school curricular approaches can sustain the benefits of preschool programs for lowincome children.
 Jenkins, J. M., Watts, T. W., Magnuson, K., Gershoff, E. T., Clements, D. H., Sarama, J., & Duncan, G. J. (2018). Do HighQuality Kindergarten and FirstGrade Classrooms Mitigate Preschool Fadeout? Journal of Research on Educational Effectiveness; J Res Educ Eff, 11(3), 339–374. https://doi.org/10.1080/19345747.2018.1441347
HISPANIC DUAL LANGUAGE LEARNING KINDERGARTEN STUDENTS' RESPONSE TO A NUMERACY INTERVENTION: A RANDOMIZED CONTROL TRIAL
This study evaluated the impact of the Spanish version of the Building Blocks software program and vocabulary on kindergarten mathematics outcomes. Participants included 270 Hispanic dual language learners from lowincome communities. Relative to children in the computer assisted instruction (CAI) literacy control group, those in the Building Blocks CAI group evidenced higher posttest scores for Spanish mathematics, but not for English mathematics, after controlling for pretest numeracy. There were also main effects of English vocabulary and Spanish vocabulary predicting posttest mathematics scores in English and Spanish, after controlling for covariates. These findings support the use of the Building Blocks software as a supplemental method for improving the mathematics competencies of Hispanic dual language learners from lowincome backgrounds.
 Foster, M. E., Anthony, J. L., Clements, D. H., Sarama, J., & Williams, J. J. (2018). Hispanic dual language learning kindergarten students response to a numeracy intervention: A randomized control trial. Early Childhood Research Quarterly, 43, 83–95. doi:10.1016/j.ecresq.2018.01.009
EVALUATION OF THREE INTERVENTIONS TEACHING AREA MEASUREMENT AS SPATIAL STRUCTURING TO YOUNG CHILDREN
We evaluated the effects of three instructional interventions designed to support young children’s understanding of area measurement as a structuring process. Replicating microgenetic procedures we used in previous research with older children to ascertain whether we can build these competencies earlier, we also extended the previous focus on correctness to include analyses of children’s use of procedural and conceptual knowledge and examined individual differences in strategy shifts before and after transitions, enabling a more detailed examination of the hypothesized necessity of development through each level of a learning trajectory. The two experimental interventions focused on a dynamic conception of area measurement while also emphasizing unit concepts, such as unit identification, iteration, and composition. The findings confirm and extend earlier results that seeing a complete record of the structure of the 2D array— in the form of a drawing of organized rows and columns—supported children’s spatial structuring and performance.
 Clements, D. H., Sarama, J., Barrett, J. E., Van Dine, D. W., Cullen, C. J., Hudyma, A., Dolgin, R, Cullen, A. L. & Eames, C. L. (2018). Evaluation of three interventions teaching area measurement as spatial structuring to young children. The Journal of Mathematical Behavior, 50, 2341. doi:10.1016/j.jmathb.2017.12.004

2017
INTERVENTIONS IN EARLY MATHEMATICS: AVOIDING POLLUTION AND DILUTION
Sarama, J., & Clements, D. H. (2017). Interventions in early mathematics: Avoiding pollution and dilution. Advances in Child Development and Behavior, 53, 95126.
THE DREME NETWORK: RESEARCH AND INTERVENTIONS IN EARLY CHILDHOOD MATHEMATICS
DayHess, C. A., & Clements, D. H. (2017). The DREME network: Research and interventions in early childhood mathematics. Advances in Child Development and Behavior, 53, 142.

2014
SUSTAINABILITY OF A SCALEUP INTERVENTION IN EARLY MATHEMATICS: A LONGITUDINAL EVALUATION OF IMPLEMENTATION FIDELITY
Sarama, Julie, Douglas H. Clements, Christopher B. Wolfe, and Mary Elaine Spitler. "Sustainability of a ScaleUp Intervention in Early Mathematics: A Longitudinal Evaluation of Implementation Fidelity." Early Education and Development 26.3 (2014): 42749.

2013
LONGITUDINAL EVALUATION OF A SCALEUP MODEL FOR TEACHING MATHEMATICS WITH TRAJECTORIES AND TECHNOLOGIES: PERSISTENCE OF EFFECTS IN THE THIRD YEAR
Clements, D.H., Sarama, J., Wolfe, C.B., & Spitler, M. E. (2013). Longitudinal evaluation of a scaleup 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.
BOOK CHAPTER  LESSONS LEARNED IN THE IMPLEMENTATION OF THE TRIAD SCALEUP MODEL: TEACHING EARLY MATHEMATICS WITH TRAJECTORIES AND TECHNOLOGIES
Sarama, J., & Clements, D.H. (2013).Lessons learned in the implementation of the TRIAD scaleup model: Teaching early mathematics with trajectories and technologies. In T. G. Halle, A. J. Metz & I. MartinezBeck (Eds.), Applying implementation science in early childhood programs and systems (pp. 173191). Baltimore, MD: Brookes.
BUILDING BLOCKS, VOLUMES 1 AND 2
Clements, D.H., & Sarama, J. (2013). Building Blocks, Volumes 1 and 2. Columbus, OH: McGrawHill (2 volumes, Teacher's Edition, Teacher's Resource Guide, Assessment Guide).

2012
THE IMPACTS OF AN EARLY MATHEMATICS CURRICULUM ON EMERGING LITERACY AND LANGUAGE
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), 489502.

2011
EARLY CHILDHOOD MATHEMATICS INTERVENTION
Clements, D.H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333(6045), 968970.
MATHEMATICS LEARNED BY YOUNG CHILDREN IN AN INTERVENTION BASED ON LEARNING TRAJECTORIES: A LARGE SCALE CLUSTER RANDOMIZED TRIAL
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 largescale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127166.

2008
EFFECTS OF A PRESCHOOL MATHEMATICS CURRICULUM: SUMMATIVE RESEARCH ON THE BUILDING BLOCKS PROJECT
Clements, D. H., and J. Sarama. "Experimental Evaluation of the Effects of a ResearchBased Preschool Mathematics Curriculum." American Educational Research Journal 45.2 (2008): 44394.