9 Development of relational thinking: An information-processing perspective -- Introduction -- Relational thinking as a building block of human cognition -- A componential analysis of matrix completion problem-solving -- Extending the componential analysis approach to other tasks and populations -- Concluding remarks: a componential model of the development of relational thinking -- References -- Part III Improving education -- 10 Translating knowledge of children's thinking to improve education -- Convincing education stakeholders that we must think about what students are thinking -- Middle school students' knowledge and encoding of problem features -- Developing effective classroom tools based on cognitive science principles -- Algebra by example -- Infusing knowledge of children's thinking into teacher preparation -- Cognitive science in teacher preparation -- Summary and conclusion -- References -- 11 The effects of retrieval practice on fraction arithmetic knowledge -- Introduction -- Benefits of retrieval practice -- Students' difficulties with fraction arithmetic -- Current studies -- Experiment 1 -- Method -- Results -- Discussion -- Experiment 2 -- Method -- Results -- Discussion -- General discussion -- Notes -- References -- 12 Studying the understanding of arithmetic with rational numbers -- Introduction -- Part 1: working on something important, and keeping an open mind -- Working on something important -- Keeping an open mind -- Part 2: understanding of fraction arithmetic -- Study 1: do students understand direction of effect? (Siegler & Lortie-Forgues, 2015) -- Study 2: do teachers understand direction of effect? (Siegler & Lortie-Forgues, 2015) -- Study 3: do students understand the direction of effect of decimal arithmetic operations? (Lortie-Forgues & Siegler, 2016) -- Future directions -- References