Numerosity in preschool – First steps towards mathematics

1. Numerosity in preschool – First steps towards mathematics Kevin F. Miller University of Michigan

2. Numerosity in preschool • An old view • Cross-cultural comparisons • What they can help us see • Representation of numbers • Language and number • Understanding Arabic numerals • Helping children learn • Parental teaching • Remediation

3. An old view • A basic definition of number • Not very amenable to instruction • Connection to other aspects of mathematics not very clear • Led to a very harmful conclusion • “Children at different stages cannot learn the same content. They cannot learn about number, for example, until they reach the concrete operational stage.”* Jean Piaget (1896 – 1980) *Copeland, R. W. (1984). How children learn mathematics. (4th ed.). New York: Macmillan, p. 12

4. Cross-cultural comparisons • Can help us distinguish • universal features and problems of development • Those that depend on particular features and practices • Source of new ideas for working with children • Provide a kind of mirror

5. Numerosity in preschool • Cross-cultural comparisons • What they can help us see • Representation of numbers • Language and number • Understanding Arabic numerals • Helping children learn • Parental teaching • Remediation

6. Learning to count (General rules) • Special features of the representational system • General rules of counting • Gelman & Gallistel (1978) • One-one principle • One name per item • Stable order principle • Say names in same order • Cardinality principle • Last name gives number in the set Rochel Gelman w     

7. Learning to count (Language-specific features) • Human beings are terrible rote learners • An example • 一二三vs.壹贰叁 • How does this apply to preschoolers? • Need to learn the structure of number names • In their language

8. Language and Learning to Count • Children need to learn a system of number names as they learn to count • Not a trivial task

9. Number names in Chinese & English - Part ICounting to Ten • Both languages share an unpredictable list • No way to induce “five” from “one, two, three, four” • Linguistically, learning to count to ten should be of equal difficulty in both languages

10. Number names in Chinese & English - Part IIFrom Ten to Twenty • Chinese has a clear base-ten structure • similar to Arabic numerals: 11 = “10…1” • English lacks clear evidence of base-ten structure • Names for 11 and 12 not marked as compounds with 10. • Larger teens names follow German system of unit+digits name, unlike larger two-digit number names • compare “fourteen” and “twenty-four”

11. Number names in Chinese & English - Part IIIAbove Twenty • Both languages share a similar structure • similar to Arabic numerals: 37 = “3x10 + 7” • For Chinese, this extends previous system • For English, it represents a new way of naming numbers

12. A longitudinal view

13. Learning difficulties reflect language structure ..and they don’t stop here!

14. The Panda’s snack • Language affects only some aspects of early number knowledge • No language difference for counting-principle errors such as double-counting • Mastering number list and understanding numerosity not the same • Producing sets of n items • No language difference

15. Continuing effects • Learning Arabic numerals involves a mapping from verbal number names • Teens continue to cause problems

16. Conclusions • Early mathematical development is a mix of language-dependent and universal factors • Sensitivity to symbol structure begins very early • Base-ten concepts and “teens” are problematic for speakers of English • Foundation for later mathematics

17. Numerosity & the concept of base ten • Representing numbers as tens and ones • Speakers of languages whose numbers are based on Chinese (Chinese, Korean, Japanese), vs. • Speakers of European languages Irene Miura

18. Miura et al. (1993)

19. Conclusions so far • Some aspects of number knowledge are universal • Others depend heavily on language and experience • The English language presents stumbling blocks for preschool children’s learning of the base-ten structure that underlines Arabic numerals and much of later mathematics

20. Numerosity in preschool • Cross-cultural comparisons • What they can help us see • Representation of numbers • Language and number • Understanding Arabic numerals • Helping children learn • Parental teaching • Remediation

21. What parents are doing What are you doing or going to do to prepare your child for school?

22. Project Rightstart • Most middle-class children enter school with • A basic understanding of counting and cardinality • Master of number system to at least 20 • Clear understanding of relative cardinalities of numbers (i.e., that 7 is greater than 5) • By-product of board games, other activities • But some children don’t Robbie Case (1945 – 2000)

23. Project Rightstart • Focus on relative numerosity • Set of games, number-line activities • Kindergarten program • 20 minutes/day for 3-4 months • Mathematical knowledge commensurate with middle-class peers • Gains persisted through the end of first grade Sharon Griffin Griffin, S., Case, R., & Siegler, R. S. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classrom lessons: Integrating cognitive theory and classroom practice . Cambridge, MA: MIT Press/Bradford Books.

24. Conclusions • English-speaking children face some disadvantages in learning about numbers • Middle-class children have experiences that provide them with a basic understanding of numerosity • Not all children have these experiences • But they can be provided • The hole in the sidewalk…