2. Previous Studies
• Less focus on Errors kids make
• Classic Give-N study
o Significant association between age and knower-level
• Debate on the concept of kids knowing how to “count” but not really
know the concept of number
o Say the last word in a different tone
o Significance of last word
3. Brief Literature Review
• Wynn (1992)
o Longitudinal study shows that very early on, children know that the
counting words each refer to a distinct, unique numerosity, although they
do not know yet to which numerosity each word refers. Despite this
knowledge, it takes children a long time to learn how the counting system
represents numerosity.
• Sarnecka, B. W., & Carey, S. (2008)
o Compared CP to Subset knowers, this study shows
• Many children answer the question “how many” with the last
word used in counting, despite not understanding how
counting works
• Only children who have mastered the CP principle, or are
short of doing so, understand that adding means moving
forward, whereas subtracting means going backward
• Only CP-knowers understand that adding exactly 1 to a set
means moving forward exactly 1 word in the list, whereas
subset-knowers do not understand the unit of change.
4. Why study errors?
• Errors can tell us what children understand about
counting
o Counting as an important tool for acquiring the concept of
number
More specifically
• What kinds of mistakes?
• More descriptive and quantitative
• We studied errors that correspond with
the 3 counting principles
5. Coding
• Dual Coders (Mary & Harry)
• Reliability Coding (Three coders)
• The original coding was then converted into a binary system
for analysis
Methods
Give-N (NOCO)
• A simple counting game, Give-N, was used for the
previous study, NOCO and filmed.
• Give-N involves asking children to have an X number of
fish go swimming, then asking them to count to check
6. More Background
• Old NOCO videos
• Three Counting Principles– Stable Order, One to One, Cardinal
Principle (=Last word; Gelman and Gallistel, 1978)
Examples
• SO: 137_NOCO_AV (03:12)
• ONEONE:77_NOCO_KV (06:28)
• CP: 134_NOCO_AL (03:00)
7. Description of the Sample
• 100 kids (F=63, M=37) between 34.8 month-
and 52.5-month old from NPS or other
preschools
• Other demographic info was not included,
but a majority speak English as their primary
language
10. Question: What is the developmental
trajectory of the three counting principles?
Hypotheses
(1)If a child is a CP knower, then they should answer Give-
N questions correctly (= not making any counting
errors);
(2)If a child is a subset knower, they will likely demonstrate
some combination of these mistakes.
12. An issue with the previous analysis:
The number of trials was not controlled for, so children
with higher knower levels had completed more trials and
(e.g., 3-knowers had more trials than 1-knowers so it’s more likely for
them to have a higher ‘proportion correct’)
● One solution is to create a normalizing variable →
N + 1 vs. numbers that children know (e.g., N and
N-1; N=knower level)
13. Here is an analysis using the normalizing variable N for
knower level to control for number of trials (subset-knowers only)
14. • SO: correct 88.74% of the time
• ONEONE: correct 63.82% of the time
• CP: correct 44.00% of the time
Result on N + 1
15.
16. Discussion
• Stable Order appears to be learned first
• 4k and CP knowers less accurate on 1-1 than on
the cardinal principle (last word)
→ CP knowers have more to learn?
• Some of the Experimenters pointed or corrected
children when counting
17. More questions to ask
• Subset-knowers grabbed the right number, but counted wrong
(e.g., grabbed two for two, but counted four, said two). What
does this mean? Why don’t children take this contradictory
information in learning about counting?
• What if you provide them with feedback? Especially children
who understand last-word.
