Counting Error Study

Mary Toomey and Harry Yahao JiangApril 2016

Previous Studies

• Less focus on Errors kids make

• Classic Give-N studyo Significant association between age and knower-level

• Debate on the concept of kids knowing how to “count” but not really know the concept of numbero Say the last word in a different toneo Significance of last word

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.

Why study errors?

• Errors can tell us what children understand about countingo 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

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

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)

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

Results

• Non-Knower: 8• 1-Knower: 11

• 2-Knower: 36

• 3-Knower: 16

• 4&5-Knower: 9

• CP-Knower: 19

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.

Results

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)

Here is an analysis using the normalizing variable N for knower level to control for number of trials (subset-

knowers only)

• SO: correct 88.74% of the time

• ONEONE: correct 63.82% of the time

• CP: correct 44.00% of the time

Result on N + 1So far, our results are consistent with what previous studies have shown.

But the most interesting thing in this N vs. N + 1 analysis is N + 1, that is, what do children fail on N + 1? No studies have looked at this. Once a child fails on a number, you go back down, but you don’t ask why. Here, we’re looking at precisely this question.

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

Says something about the potential order of the different counting principles – one interesting finding

Do you think this says anything about what CP-knowers understand about counting (given that most assume CP-knowers understand the counting principles)

Does it tell us about what subset-knowers know about counting?

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.

What else can we get out of this data?

Questions?