#### Date on Master's Thesis/Doctoral Dissertation

5-2010

#### Document Type

Doctoral Dissertation

#### Degree Name

Ph. D.

#### Department (Legacy)

Department of Teaching and Learning

#### Committee Chair

Ronau, Robert N.

#### Author's Keywords

Rational numbers; Probability; Algebra; Geometry

#### Subject

Mathematics--Study and teaching; Common fallacies

#### Abstract

In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if misunderstood, create persistent difficulties across content areas: rational number meaning, additive/multiplicative structures, absolute/relative comparison, variable meaning, and spatial reasoning misconceptions. Probability instruction naturally provides concrete, authentic experiences that engage students with abstract mathematical concepts, establish relationships between mathematical topics, and connect inter-related problem solving strategies. The intervention consisted of five probability lessons about counting principles, randomness, independent and dependent event probability, and probability distributions. The unit lasted approximately two weeks. This study used mixed methodology to analyze data from a randomly assigned sample of students from an untreated control group design with a switching replication. Document analysis was used to examine patterns in student responses to items on the mathematics knowledge test. Multiple imputation was used to account for missing data. Structural equation modeling was used to examine the causal structure of content area misconceptions. Item response theory was used to compute item difficulty, item discrimination, and item guessing coefficients. Generalized hierarchical linear modeling was used to explore the impact of item, student, and classroom characteristics on incorrect responses due to misconceptions. These analyses resulted in 7 key findings. (1) Content area is not the most effective way to classify mathematics misconceptions; instead, five underlying misconceptions affect all four content areas. (2) Mathematics misconception errors often appear as procedural errors. (3) A classroom environment that fosters enjoyment of mathematics and value of mathematics are associated with reduced misconception errors. (4) Higher mathematics self confidence and motivation to learn mathematics is associated with reduced misconception errors. (5) Probability misconceptions do not have a causal effect on rational numbers, algebra, or geometry misconceptions. (6) Rational number misconceptions do not have a causal effect on probability, algebra, or geometry misconceptions. (7) Probability instruction may not affect misconceptions directly, but it may help students develop skills needed to bypass misconceptions when solving difficult problems.

#### Recommended Citation

Rakes, Christopher R., "Misconceptions in rational numbers, probability, algebra, and geometry." (2010). *Electronic Theses and Dissertations.* Paper 1176.

https://doi.org/http://dx.doi.org/10.18297/etd/1176