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

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fall Spring

Pro

porti

on C

orre

ct

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.