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

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

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

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

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

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

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

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

 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 Number names in Chinese & English - Part I Counting to Ten

 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” Number names in Chinese & English - Part II From Ten to 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 Number names in Chinese & English - Part III Above Twenty

A longitudinal view

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

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

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

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

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

Miura et al. (1993)

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

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

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

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)

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.

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…