numerosity in preschool – first steps towards mathematics kevin f. miller university of michigan
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Numerosity in preschool – First steps towards mathematics
Kevin F. Miller
University of Michigan
Numerosity in preschool
An old view
Cross-cultural comparisonsWhat they can help us see
Representation of numbers
Language and number
Understanding Arabic numerals
Helping children learnParental teaching
Remediation
An old viewA basic definition of numberNot very amenable to instructionConnection to other aspects of mathematics not very clearLed 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 comparisonsWhat they can help us see
Representation of numbers
Language and number
Understanding Arabic numerals
Helping children learnParental teaching
Remediation
Learning to count (General rules)Special features of the representational system
General rules of countingGelman & 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
Numeral 1 2 3 4 5 6 7 8 9 10 Chinese (written)
一 二 三 四 五 六 七 八 九 十
Chinese (spoken) yi er san si wu liu qi ba jiu shi
English one two three four five six seven eight nine 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
Number names in Chinese & English - Part ICounting to Ten
Numeral 11 12 13 14 15 16 17 18 19 20 Chinese (written)
十一 十二 十三 十四 十五 十六 十七 十八 十九 二十
Chinese (written)
shi yi shi er shi san shi si shi wu shi liu shi qi shi ba shi jiu er shi
English eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen 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”
Number names in Chinese & English - Part II From Ten to Twenty
Language Rule Example Chinese (written)
三十七
Chinese (written)
Decade unit (two,three,four,five,six,seven,eight,nine) + ten (shi) + unit
san shi qi
English Decade names (twen,thir,for,fif,six,seven,eight,nine) + ty + unit thirty-seven
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 IIIAbove Twenty
A longitudinal view
1 2 3 4 5 6 7 8 9 1011120
10
20
30
40
50
60
70
80
90
100
110
Med
ian
Ab
stra
ct C
oun
tin
g
Month
China
US
1 2 3 4 5 6 7 8 9 1011120
10
20
30
40
50
60
70
80
90
100
110
Month
China
US
1 2 3 4 5 6 7 8 9 10 11120
10
20
30
40
50
60
70
80
90
100
110
Month
China
US
2-year-olds 3-year-olds 4-year-olds
Learning difficulties reflect language structure
..and they don’t stop here!
The Panda’s snack Producing Sets of 4
0
20
40
60
80
100
3 4 5
Age
% C
orre
ct
US
China
Producing Sets of 12
0
20
40
60
80
100
3 4 5
Age
% C
orre
ct US
China
• 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
Teens numbers
0
20
40
60
80
100
K 1 2
Grade
% C
orre
ct
US
China
Larger 2-digit numbers
0
20
40
60
80
100
K 1 2
Grade
% C
orre
ctUS
China
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 universalOthers depend heavily on language and experienceThe 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 comparisonsWhat they can help us see
Representation of numbers
Language and number
Understanding Arabic numerals
Helping children learnParental teaching
Remediation
What parents are doing
0
10
20
30
40
50
60
70
80
90
Champaign Beijing
Per
cent
age
of P
aren
ts
MathReading
What are you doing or going to do to prepare your child for school?
Project RightstartMost 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 RightstartFocus on relative numerosity
Set of games, number-line activities
Kindergarten program20 minutes/day for 3-4 months
Mathematical knowledge commensurate with middle-class peersGains 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…