Algebra in Preschool: Emerging Understanding of Patterns in Four-Year-Olds

Bethany Rittle-Johnson, Emily R. Fyfe, Laura E. McLean & Katherine L. McEldoon

ABSTRACT

BACKGROUND

METHOD

DISCUSSION

REFERENCES

CONSTRUCT MAP

Preschoolers spontaneously engage in a form of early algebraic thinking—patterning. We assessed four-year-old children’s pattern knowledge on three occasions (N = 66). Children could duplicate and extend patterns, and some showed a deeper understanding of patterns by being able to abstract patterns (i.e., create the same kind of pattern using new materials). A small proportion of the children had explicit knowledge of pattern units. Error analyses indicated that some pattern knowledge was apparent before children were successful on items. Overall, findings indicate that young children are developing an understanding of patterns, a key component of algebraic knowledge, before school entry.

• Modern mathematics has been defined as the science of patterns (Steen, 1998).

• Young children spontaneously engage in patterning activities (Ginsburg, Lin, Ness & Seo, 2003).

• Knowledge of patterns is a central algebraic topic in consensus documents on mathematics education (e.g., NCTM, 2000).

• However, recent National Mathematics Advisory Panel (2008) recommended greatly reducing emphasis on patterning in curriculum.

Participants: 66 4-year-olds (4.0 to 5.3 years at 1st assessment)Design: Assessed in Fall and Spring of school yearAssessment: 10 items, each targeted at 1 of 4 levels of construct map (dropped 1 item). Sample Patterns:

Sample Tasks:Duplicating – Level 1 Extending – Level 2

Abstracting – Level 3 Pattern Unit – Level 4

ERRORS IN FALL

LEVEL SKILL SAMPLE TASKLevel 4: Pattern unit recognition

Identifies the pattern unit.

“What is the smallest tower you could make and still keep the same pattern as this?”

Level 3: Pattern abstraction

Translates patterns into new patterns with same structural rule.

“I made a pattern with these blocks. Please make the same kind of pattern here, using these cubes” (using new colors and shapes).

Level 2: Pattern extension

Extends patterns at least one pattern unit.

“I made a pattern with these blocks. Finish my pattern here the way I would.”

Level 1: Pattern duplication

Duplicates patterns. “I made a pattern with these blocks. Please make the same kind of pattern here.”

RELIABILITY &VALIDITYGood internal consistency:

Time 1 α=.82 Time 2 α=.84Good stability:

Time 1 – Time 2: r(63) = .74Time 1 – Time 3: r(63) = .58

Strong Content Validity: Items rated as important (rating of 3) to essential (rating of 5), with a mean rating of 4.5. Strong Construct Validity Items were at the expected level of difficulty and children were evenly distributed across levels – see Wright Map.

• Describe 4-year-olds’ knowledge of repeating patterns. • Understanding the pattern unit

(i.e., the sequence that repeats over and over).

• Use a construct modeling approach (Wilson, 2005) to develop reliable and valid measure.• Develop and test a construct map

– a representation of the continuum of knowledge that people are thought to progress through.

GOALS

Children Logits Itemsxxxxx | SmallestTower_AAB

  2xxxxxxx |

  1.5xxxx |

  1 Memory_ABBxxxxx |

  0.5xxxxxxxxxxxx | AbstractColor_ABB; AbstractColor_AAB; AbstractColor_AABB

  0 AbstractShape_AABB  |

xxxxxxxxx -0.5  | Extend_AABB  -1

xxxxxxx | Extend_ABB  -1.5  |  -2  |

xxxxxxxxxxxxx -2.5  |  -3 Duplicate_AABB

WRIGHT MAP – FALL

Error Type Example for ABB pattern

% Used across trials

% Children who used

Correct ABBABB 42 82Partial Correct ABBAAB 15 68Wrong Pattern AB ABABABAB 10 41Wrong Pattern Other

AABBAABB6 38

Sort AAAABBBB 9 41Random Order ABBAA 11 45Off Task Made a tower 6 23

IMPROVEMENTS FALL TO SPRING

Level Fall Spring

Level 4: Pattern unit recognition .12 .16

Level 3: Pattern abstraction .31 .60

Level 2: Pattern extension .47 .71

Level 1: Pattern duplication .77 .95

Note: Used a new cross-classified IRT model to handle sample size around 50 (Cho & Rabe-Hesketh, 2011; Hofman & De Boeck, 2011).

Some pattern knowledge apparent before successful on items

Large improvements in proportion correct on Level 1, 2 & 3 items

• 4-year-olds can go beyond simple pattern tasks• Many move beyond duplicating and

extending patterns• Can abstract pattern and recreate

with new materials (although not doing this in school!)

• Young children have more than number knowledge • Children are paying attention to

structure in the world• Repeating patterns may support

algebraic reasoning• We have a good measure for assessing

this knowledge• Construct map and assessment

captures shifts in knowledge over year of preschool.

• Construct modeling approach is powerful

• 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.

• Ginsburg, H. P., Lin, C.-l., Ness, D., & Seo, K.-H. (2003). Young american and chinese children's everyday mathematical activity. Mathematical Thinking and Learning, 5(4), 235-258.

• 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.

• National Mathematics Advisory Panel (2008). Foundations of Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

• NCTM (2000). Principles and standards for school mathematics. Reston, Va: NCTM.

• Steen, L. A. (1988). The science of patterns. Science, 240(4852), 611-616.• Wilson, M. (2005). Constructing measures: An item response modeling

approach. Mahwah, NJ: Lawrence Erlbaum Associates.

Based on Clements and Sarama (2009)