Algebra in Preschool: Emerging Understanding of Patterns in Four-Year-Olds

1. ABSTRACT Reliability &Validity Algebra in Preschool: Emerging Understanding of Patterns in Four-Year-Olds Preschoolers spontaneously engage in a form of early algebraic thinking—patterning. We assessed four-year-old children’s pattern knowledge on three occasions (N = 66). Children could duplicate and extend patterns, and some showed a deeper understanding of patterns by being able to abstract patterns (i.e., create the same kind of pattern using new materials). A small proportion of the children had explicit knowledge of pattern units. Error analyses indicated that some pattern knowledge was apparent before children were successful on items. Overall, findings indicate that young children are developing an understanding of patterns, a key component of algebraic knowledge, before school entry. Good internal consistency: Time 1 α=.82 Time 2 α=.84 Good stability: Time 1 – Time 2: r(63) = .74 Time 1 – Time 3: r(63) = .58 Strong Content Validity: Items rated as important (rating of 3) to essential (rating of 5), with a mean rating of 4.5. Strong Construct Validity Items were at the expected level of difficulty and children were evenly distributed across levels – see Wright Map. Bethany Rittle-Johnson, Emily R. Fyfe, Laura E. McLean & Katherine L. McEldoon CONSTRUCT MAP WRIGHT MAP – FALL DISCUSSION • 4-year-olds can go beyond simple pattern tasks • Many move beyond duplicating and extending patterns • Can abstract pattern and recreate with new materials (although not doing this in school!) • Young children have more than number knowledge • Children are paying attention to structure in the world • Repeating patterns may support algebraic reasoning • We have a good measure for assessing this knowledge • Construct map and assessment captures shifts in knowledge over year of preschool. • Construct modeling approach is powerful BACKGROUND Based on Clements and Sarama (2009) • Modern mathematics has been defined as the science of patterns (Steen, 1998). • Young children spontaneously engage in patterning activities (Ginsburg, Lin, Ness & Seo, 2003). • Knowledge of patterns is a central algebraic topic in consensus documents on mathematics education (e.g., NCTM, 2000). • However, recent National Mathematics Advisory Panel (2008) recommended greatly reducing emphasis on patterning in curriculum. • Note: Used a new cross-classified IRT model to handle sample size around 50 (Cho & Rabe-Hesketh, 2011; Hofman & De Boeck, 2011). METHOD ERRORS IN FALL • Participants: 66 4-year-olds (4.0 to 5.3 years at 1st assessment) • Design: Assessed in Fall and Spring of school year • Assessment: 10 items, each targeted at 1 of 4 levels of construct map (dropped 1 item). • Sample Patterns: • Sample Tasks: • Duplicating – Level 1 Extending – Level 2 • Abstracting – Level 3 Pattern Unit – Level 4 Some pattern knowledge apparent before successful on items REFERENCES • Cho, S.-J., & Rabe-Hesketh, S. (2011). Alternating imputation posterior estimation of models with crossed random effects. Computational Statistics and Data Analysis, 55, 12-25. • Clements, D. H., & Sarama, J. (2009). Other content domains. Learning and teaching early math: The learning trajectories approach (pp. 189-202). New York: Routledge. • Ginsburg, H. P., Lin, C.-l., Ness, D., & Seo, K.-H. (2003). Young american and chinese children's everyday mathematical activity. Mathematical Thinking and Learning, 5(4), 235-258. • Hofman, A., & De Boeck, P. (2011). Distributional assumptions and sample size using crossed random effect models for binary data: A recovery study based on the lmer function. • National Mathematics Advisory Panel (2008). Foundations of Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. • NCTM (2000). Principles and standards for school mathematics. Reston, Va: NCTM. • Steen, L. A. (1988). The science of patterns. Science, 240(4852), 611-616. • Wilson, M. (2005). Constructing measures: An item response modeling approach. Mahwah, NJ: Lawrence Erlbaum Associates. GOALS • Describe 4-year-olds’ knowledge of repeating patterns. • Understanding the pattern unit (i.e., the sequence that repeats over and over). • Use a construct modeling approach (Wilson, 2005) to develop reliable and valid measure. • Develop and test a construct map – a representation of the continuum of knowledge that people are thought to progress through. Improvements FALL TO SPRING Large improvements in proportion correct on Level 1, 2 & 3 items