success in mathematics. Of course, many activities overlap these topic areas and could be counted in either if there is a balanced focus on both. The time spent on number and operations and on geometry and measurement can also include connections to data analysis and patterns, as listed in Curriculum Focal Points and discussed in the chapters of Part II.

The kind of learning involved in various number and operation components and in various aspects of geometry and measurement is different, as we describe. Major themes of these variations in the kinds of learning are the need for achieving fluency, the use of patterns, generalizing, and extending. All of these require many repeated experiences with the same numbers and related similar tasks. This is part of what makes learning mathematics require so much time focused on mathematical content.

Mathematics is a participant sport. Children must play it frequently to become good at it. They do need frequent modeling of correct performance, discussion about the concepts involved, and frequent feedback about their performance. Both modeling and feedback can come from other students as well as from adults, and feedback also sometimes comes from the situation. All children must have sustained and frequent times in which they themselves enact the core mathematical content and talk about what they are doing and why they are doing it. In mathematics learning, effort creates ability.

REFERENCES AND BIBLIOGRAPHY

Clements, D.H., and Sarama, J. (2007). Early childhood mathematics learning. In F.K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age.

Clements, D.H., and Sarama, J. (2008). Experimental evaluation of a research-based preschool mathematics curriculum. American Educational Research Journal, 45, 443-494.

Clements, D.H., Sarama, J., and DiBiase, A. (2004). Engaging Young Children in Mathematics: Findings of the 2000 National Conference on Standards for Preschool and Kindergarten Mathematics Education. Mahwah, NJ: Erlbaum.

Fuson, K.C. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R.T. Putnam, and R.A. Hattrup (Eds.), The Analysis of Arithmetic for Mathematics Teaching (pp. 53-187). Hillsdale, NJ: Erlbaum.

Fuson, K.C. (1992b). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 243-275). New York: Macmillan.

Ginsburg, H.P. (1983). The Development of Mathematical Thinking. New York: Academic Press.

National Association for the Education of Young Children and National Council of Teachers of Mathematics. (2002). Early Childhood Mathematics: Promoting Good Beginnings. A joint position statement of the National Association for the Education of Young Children and National Council of Teachers of Mathematics. Available: http://www.naeyc.org/about/positions/pdf/psmath.pdf [accessed August 2008].

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.



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