Julie Sarama

Engaging Young Children in Mathematics : Standards for Early Childhood Mathematics Education

Contents: Preface. Part I: Major Themes and Recommendations. D.H. Clements, Major Themes and Recommendations. Part II: Elaboration of Major Themes and Recommendations. Section 1: Standards in Early Childhood Education. S. Bredekamp, Standards for Preschool and Kindergarten Mathematics Education. C.E. Copple, Mathematics Curriculum in the Early Childhood Context. Section 2: Math Standards and Guidelines. K-H Seo, H.P. Ginsburg, What Is Developmentally Appropriate in Early Childhood Mathematics Education? Lessons From New Research. K.C. Fuson, Pre-K to Grade 2 Goals and Standards: Achieving 21st-Century Mastery for All. A.J. Baroody, The Role of Psychological Research in the Development of Early Childhood Mathematics Standards. A.J. Baroody, The Developmental Bases for Early Childhood Number and Operations Standards. L.P. Steffe, PSSM From a Constructivist Perspective. C. Sophian, A Prospective Developmental Perspective on Early Mathematics Instruction. D.H. Clements, Geometric and Spatial Thinking in Early Childhood Education. D.H. Clements, M. Stephan, Measurement in Pre-K to Grade 2 Mathematics. Section 3: Curriculum, Learning, Teaching, and Assessment. K. Richardson, Making Sense. S. Griffin, Number Worlds: A Research-Based Mathematics Program for Young Children. A. Klein, P. Starkey, Fostering Preschool Childrens Mathematical Knowledge: Findings From the Berkeley Math Readiness Project. J. Sarama, Technology in Early Childhood Mathematics: Building Blocks as an Innovative Technology-Based Curriculum. B. Casey, Mathematics Problem-Solving Adventures: A Language-Arts-Based Supplementary Series for Early Childhood That Focuses on Spatial Sense. Section 4: Professional Development. R. Feiler, Early Childhood Mathematics Instruction: Seeing the Opportunities Among the Challenges. J.V. Copley, The Early Childhood Collaborative: A Professional Development Model to Communicate and Implement the Standards. J. Sarama, A-M. DiBiase, The Professional Development Challenge in Preschool Mathematics. Section 5: Toward the Future: Implementation and Policy. M.M. Lindquist, J.M. Joyner, Mathematics Guidelines for Preschool. Appendices: Reactions. D. Borkovitz, Afterword. J. Ware, Engaging Young Children in Mathematics. Agenda--Main Conference. Agenda--Follow-up Conference. State Standards. Biographical Sketches.This chapter is based on papers presented at the symposium “Linking Research and the New Early Childhood Mathematics Standards” at the Research Presession of the annual meeting of the National Council of Teachers of Mathematics, Chicago, April 2000; the Conference on Standards for Preschool and Kindergarten Mathematics Education (sponsored by the National Science Foundation and Exxon Mobil Foundation), Arlington, VA, May 2000; and a follow-up conference in Dallas, TX, October, 2000.

Learning Trajectories in Mathematics Education

Many successful recent approaches to developing innovative mathematics curricula and to conducting research on learning and teaching mathematics have used the construct of “learning trajectories” as a foundation. However, the developers and authors have interpreted and applied this idea in different ways, leading to a need for discussions of these variations and a search for clarifications and shared meanings. Further, the construct of learning trajectories is less than a decade old, but palpably has many roots in previous theories of learning, teaching, and curriculum. The purpose of this special issue is to present several research perspectives on learning trajectories with the intention of encouraging the broader community to reflect on, better define, adopt, adapt, or challenge the concept. This brief article introduces learning trajectories from our perspective. The other articles provide elaboration, examples, and discussion of the construct. They purposefully are intended to be illustrative, exploratory, and provocative with regard to the learning trajectories construct; they are not a set of verification studies.

Experimental Evaluation of the Effects of a Research-Based Preschool Mathematics Curriculum

A randomized-trials design was used to evaluate the effectiveness of a preschool mathematics program based on a comprehensive model of research-based curricula development. Thirty-six preschool classrooms were assigned to experimental (Building Blocks), comparison (a different preschool mathematics curriculum), or control conditions. Children were individually pre- and posttested, participating in 26 weeks of instruction in between. Observational measures indicated that the curricula were implemented with fidelity, and the experimental condition had significant positive effects on classrooms’ mathematics environment and teaching. The experimental group score increased significantly more than the comparison group score (effect size = 0.47) and the control group score (effect size = 1.07). Early interventions can increase the quality of the mathematics environment and help preschoolers develop a foundation of mathematics knowledge.

Young Children's Concepts of Shape.

We investigated criteria preschool children use to distinguish members of a class of shapes from other figures. We conducted individual clinical interviews of 97 children ages 3 to 6, emphasizing identification and descriptions of shapes and reasons for these identifications. We found that young children initially form schemas on the basis of feature analysis of visual forms. While these schemas are developing, children continue to rely primarily on visual matching to distinguish shapes. They are, however, also capable of recognizing components and simple properties of familiar shapes. Thus, evidence supports previous claims (Clements & Battista, 1992b) that a prerecognitive level exists before van Hiele Level 1 (“visual level”) and that Level 1 should be reconceptualized as syncretic (i.e., a synthesis of verbal declarative and imagistic knowledge, each interacting with the other) instead of visual (Clements, 1992).

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.

Development of a measure of early mathematics achievement using the Rasch model: the Research‐Based Early Maths Assessment

There are only a few instruments to assess mathematics knowledge and skills in children as young as three to four years of age, and these instruments are limited in scope of content. We describe the development of a theoretically based, empirically tested instrument designed to measure the mathematical knowledge and skills of children from three to seven years of age, emphasising its submission to the Rasch model. After using the data to refine the instrument, they fit the model well, with high reliability. These data also provided empirical support for the developmental progressions for most topics. We conclude with a description of the research’s contribution to theory and empirical research regarding young children’s development of specific mathematical competencies.

Effects of a Pre-Kindergarten Mathematics Intervention: A Randomized Experiment.

Abstract Research indicates that a socioeconomic status-related gap in mathematical knowledge appears early and widens during early childhood. Young children from economically disadvantaged families receive less support for mathematical development both at home and in preschool. Consequently, children from different socioeconomic backgrounds enter elementary school at different levels of readiness to learn a standards-based mathematics curriculum. One approach to closing this gap is the development and implementation of effective mathematics curricula for public preschool programs enrolling economically disadvantaged children. A randomized controlled trial was conducted in 40 Head Start and state preschool classrooms, with 278 children, to determine whether a pre-kindergarten mathematics intervention was effective. Intervention teachers received training that enabled them to implement with fidelity, and a large majority of parents regularly used math activities teachers sent home. Intervention and control groups did not differ on math assessments at pretest; however, gain scores of intervention children were significantly greater than those of control children at posttest. Thus, the intervention reduced the gap in childrens early mathematical knowledge.

Young Children's Composition of Geometric Figures: A Learning Trajectory

The purpose of this research is to chart the mathematical actions-on-objects young children use to compose geometric shapes. The ultimate goal is the creation of a hypothetical learning trajectory based on previous research, as well as instrumentation to assess levels of learning along the developmental progression underlying the trajectory. We tested both the developmental progression and the instrument through a series of studies, including formative studies (including action research by 8 teachers) and a summative study involving 72 children ages 3 to 7 years. Results provide strong support for the validity of the developmental progressions levels and suggest that children move through these levels of thinking in developing the ability to compose 2-dimensional figures. From lack of competence in composing geometric shapes, they gain abilities to combine shapes-initially through trial and error and gradually by attributes-into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes).

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).

Network of Influences in an Implementation of a Mathematics Curriculum Innovation

There is widespread interest in reform in U.S. mathematics education, engendered by influences from comparative educational research (National Center for Education Statistics, 1996; Stigler, Lee and Stevenson, 1990) to position documents from prestigious organizations (National Council of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991; National Research Council, 1989). One route to such reform is the development and implementation of innovative curricula. We studied the adoption of one such innovation, emphasizing teachers’ social construction of knowledge and beliefs. The innovation was designed by two of us (Clements and Meredith, 1994; Clements and Sarama, 1995) as part of a research project funded by the National Science Foundation (NSF). 1 In this context, we developed a software environment with correlated curriculum materials. During the field testing phase of development, we worked with a school that was attempting to alter its mathematics program according to recent reform recommendations. Given the apparent convergence of goals of the school staff and the designers, we believed the adoption process began with a good chance of success. We planned to assess the effect of the innovation on the knowledge and beliefs of students and teachers. During the project, however, concerned and confused about what we observed, we altered the direction of the research study to examine, and attempt to positively affect, the implementation of the innovation.