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Children Logits Items xxx 3 SmallestTower_AAB
| 2.5 Memory_ABB |
xxxxxx 2 | 1.5 |
xxxxxxxxxxxxxxxxxxxxx 1 |
xxxxxxxxx 0.5 | 0 AbstractShape_AABB; AbstractColor_ABB; AbstractColor_AAB; | AbstractColor_AABB; Extend_AABB
-‐0.5 |
xxx -‐1 |
-‐1.5 Extend_ABB |
xxxxxxx -‐2 |
-‐2.5 |
xxxxxxx -‐3 |
-‐3.5 Duplicate_AABB
Preschoolers’ Knowledge of Repeating Patterns Over Time Michael R. Miller, Abbey M. Loehr, Emily R. Fyfe,
Bethany Rittle-Johnson, Laura E. McLean, & Katherine L. McEldoon Vanderbilt University
Patterns & Math • Patterning is a spontaneous,
recurrent activity of young children that is central to early mathematics education (NCTM, 2000).
• Working with repeating patterns (e.g., ABBABB) helps children learn to make generalizations important for algebra (Papic et al., 2011).
• Evidence is currently limited on the growth of different repeating pattern skills (Clements & Sarama, 2009; Rittle-Johnson et al., in press).
Method
Goals • Examine the relative difficulty
of different repeating pattern skills for preschoolers.
• Develop and test a construct map (Wilson, 2005) that represents the continuum of repeating pattern knowledge that preschoolers are thought to progress through.
• Investigate changes in preschoolers’ repeating pattern knowledge over time.
• Cho, S.-J., & Rabe-Hesketh, S. (2011). Alternating imputation posterior estimation of models with crossed random effects. Computational Statistics and Data Analysis, 55, 12-25.
• Clements, D. H., & Sarama, J. (2009). Other content domains. Learning and teaching early math: The learning trajectories approach (pp. 189-202). New York: Routledge.
• Hofman, A., & De Boeck, P. (2011). Distributional assumptions and sample size using crossed random effect models for binary data: A recovery study based on the lmer function.
• NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers’ mathematical patterning. Journal for Research in Mathematics Education, 42, 237-268.
• Rittle-Johnson, B., Fyfe, E. R., McLean, L. E., & McEldoon, K. L. (in press). Emerging understanding of patterning in four-year-olds. Journal of Cognition and Development.
• Wilson, M. (2005). Constructing measures: An item response modeling approach. Mahwah, NJ: Lawrence Erlbaum Associates.
Participants: 64 preschoolers (4.0 to 5.3 years in Fall). Design: Given brief pattern practice, and then assessed in Fall and Spring of school year. Assessment: 10 items, each targeted at 1 of 4 levels of the construct map (dropped one Level 4 item). Sample Tasks: Level 1: Duplicate_AABB Level 2: Extend_ ABB
Level 3:AbstractColor_AABB Level 4: SmallestTower_AAB
Wright Map – Fall
Error type Example for ABB pattern
% Used across trials
% Children who used
Fall Spring Fall Spring Correct ABBABB 46 62** 80 98* Partial Correct ABBAAB 12 19 60 75 Wrong Pattern AB ABABAB 9 10 43 38
Wrong Pattern Other AABBAABB 6 2** 42 14**
Sort AAAABBBB 11 3** 48 17** Random Order ABBAA 10 3** 42 17** Off Task Made a tower 6 2* 20 9*
References
Improvements Over Time
Error Analysis
Wright Map – Spring
• 4-year-olds gain more accurate knowledge of repeating patterns over the preschool year. - Many advance beyond
duplication and extension. - Learning to abstract
patterns – although not being instructed in school!
- May provide a foundation for early algebraic thinking.
• Frequency of less sophisticated errors decreased over time.
• Construct map and assessment captured shifts in pattern knowledge over the preschool year.
• Future research should identify sources of change in pattern knowledge.
- Better understand relevant cognitive mechanisms underlying preschoolers’ patterning skills.
- Further investigate links between patterns and mathematics learning.
Conclusions
* p < .05. **p < .01.
• Large improvements in proportion correct on Level 1, 2, & 3 items.
Children Logits Items 3
xxx | 2.5 |
xxxxxx 2 SmallestTower_AAB | Memory_ABB
xxxxxxxxxx 1.5 |
xxxxxx 1 | 0.5 | Extend_AABB; AbstractShape_AABB 0 AbstractColor_ABB |
xxxxxxxx -‐0.5 AbstractColor_AABB | AbstractColor_AAB -‐1
xxxxxx | Extend_ABB xxxxxx -‐1.5
| -‐2 |
-‐2.5 Duplicate_AABB xxxxxxxxxxxxx |
-‐3 |
-‐3.5
0
0.1
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0.4
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Fall Spring
Pro
porti
on C
orre
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Pattern Knowledge by Level
Level 1: Duplication Level 2: Extension Level 3: Abstraction Level 4: Pattern Unit
*
**
Construct Map Level Skill
Level 4: Pattern unit recognition
Identifies the pattern unit
Level 3: Pattern abstraction
Translates patterns into new patterns with same structural rule
Level 2: Pattern extension
Extends patterns at least one pattern unit
Level 1: Pattern duplication Duplicates patterns
Based on Clements and Sarama (2009)
Note: Used a new cross-‐classified IRT model to handle sample size around 50 (Cho & Rabe-‐Hesketh, 2011; Hofman & De Boeck, 2011).
• Items directly across from a child (X) are expected to be solved 50% of the Tme.
• The greater the distance of an item below a child (X), the greater the expectaTon that the child will solve that item.
• The greater the distance of an item above a child (X), the greater the expectaTon that the child will not solve that item.
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