Description:

Co-Principal Investigators: Alibali, Martha W.; Rogers, Timothy

Purpose: Educators often see that students have difficulty applying the lessons taught in school to new contexts, particularly in mathematics. For instance, students' knowledge of mathematics often seems like a collection of disconnected facts and procedures and is insufficient for mastering mathematic skills. To foster students' acquisition of robust and transferable knowledge, it is not clear how much of a focus should be placed on efficient retrieval of arithmetic facts or on rich understanding of the quantitative relationships that underlie arithmetic. Different forms of instruction lead students to form different memory models (i.e., representations that students form during learning) that, in turn, support different kinds of transfer. The goal of this project is to understand how malleable factors of instructional practice support fluent and transferable learning. Researchers will explore a set of malleable factors linking forms of arithmetic practice to specific memory models for the purpose of identifying a range of outcomes of mathematics instruction, highlighting distinct goals for mathematics education, and providing a set of design principles for achieving those outcomes and goals.

Project Activities: The research plan is divided into four phases. The research team will first develop and fine tune the instructional tasks and materials through a series of small-scale pilot studies. Then, the team will conduct a series of experiments to explore the effects of a set of malleable factors on the formation of different memory models, and consequently, on the patterns of transfer to novel problems and problem types. Researchers will then conduct experiments designed to investigate whether particular memory models, which may seem more desirable, are more difficult to learn and take longer to acquire than other models. In the final phase of the project, researchers will conduct a classroom study to address whether the instructional practices identified in the laboratory can be used to promote efficient learning and transfer in a classroom setting.

Products: The products of this project will be information about which malleable factors of mathematics instruction support fluency and transfer. Peer-reviewed publications will also be produced.

Structured Abstract

Setting: The classroom study will take place in three second-grade classrooms at public elementary schools in a college town in Wisconsin and a small town in Connecticut. The remainder of the research will be conducted in research laboratories at the University of Wisconsin in Madison.

Sample: For the classroom study, approximately 75 second-grade students from three classrooms will participate. Approximately 300 second-grade students will participate in the laboratory experiments. Approximately 300 undergraduate students from the University of Wisconsin will participate in the laboratory experiments.

Intervention: This project will identify malleable factors that promote different kinds of transfer when learning arithmetic. This information will inform the future development of instructional methods to promote particular learning goals, such as fluency and transfer.

Research Design and Methods: The research plan consists of four phases. In Phase 1, researchers will fine-tune the arithmetic practice tasks that will be used in the formal experiments using a series of small pilot studies. In Phase 2, the research team will conduct a set of experiments to explore how a set of malleable factors of arithmetic instruction affects the formation of different memory models which, in turn, yield different patterns of transfer to novel problems and problem types. In Phase 3, researchers will use a set of experiments to investigate whether there is a tradeoff between the speed (and difficulty) of forming a particular memory model and transfer performance and to identify which combinations of malleable factors lead to the broadest transfer with modest amounts of practice. Finally, Phase 4 will be a classroom study in three second-grade classrooms that assesses whether the instructional practices identified in the laboratory will be effective in a classroom setting. Experiments in Phases 2 and 3 will include both children (learning base 10 addition) and adults (learning base 8 addition). All experiments in Phases 2, 3, and 4 will be between-subjects, meaning each participant will be randomly assigned to one condition, and assignment to condition will occur at the individual student level.

Control Condition: For the laboratory experiments, the control condition will vary as a function of the research question being addressed. For the classroom study, students will be randomly assigned to one of three groups receiving instruction about arithmetic, each designed to promote a different type of memory model.

Key Measures: The key measures will be researcher-designed assessments that will measure student performance on practiced and transfer problems. The classroom study will also include teacher-designed assessments of mathematics knowledge typical for the classrooms involved in the study. The team will also collect demographic information as well as parent/self-assessed math attitudes and abilities.

Data Analytic Strategy: Analyses will consist of standard analysis of variance models to address whether particular treatments lead to certain patterns of problem-solving fluency and transfer and regression analyses to look for potential moderators (e.g., high versus low achievers) in response to the malleable factors of instructional practice.