The importance of mathematics and well-chosen examples when preparing students for future learning in simple physics [electronic resource]
Keywords: Education.

Scientific explanations require levels of both precision and generality that are not characteristic of everyday thinking. For example, an equation such as F = ma can be used to make specific predictions across a wide range of situations. How can science instruction help students to appreciate the utility and importance of precise general explanations? Furthermore, how can this instruction promote transfer to new contexts? In this dissertation, I begin with a review of literature on transfer, with a particular focus on transfer from solutions that are sufficient in one context but inadequate for a future context. Measures of Preparation for Future Learning (PFL) are noted as useful ways to assess the transfer of students' scientific thinking from one problem to the next. I draw on ideas of information sufficiency from Perceptual Learning (Gibson & Gibson, 1959) to systematically design instructional materials for learning general explanations in science. I suggest two possible ways to encourage students to appreciate the utility of precise general explanations: selecting a comprehensive set of examples and providing a task orientation. In turn, I hypothesize that these instructional approaches will lead students to be better prepared to learn in future learning contexts. In the current research, community college students learned about two-factor physics problems in three studies. In these physics problems, two variables, such as mass and velocity or mass and distance, must be multiplicatively combined to create an intensive quantity, such as momentum or torque. Students worked with contrasting cases showing inelastic collisions problems on the initial learning task and with a balance scale contrasting cases activity second. The balance task served as a PFL transfer assessment. Studies 1 and 3 tested a hypothesis about information sufficiency by comparing two versions of learning materials for the inelastic collisions task. Some students received materials that showed the main effects of each variable (mass and velocity) and the interaction of the two (mass x velocity). These students were likely to produce a multiplicative strategy, which is a precise and general explanation for the interaction between the mass and speed variables. Other students used materials that only isolated the main effects of each variable (mass and velocity). They created qualitative, less precise solutions. Materials showing systematic variation led to greater learning than those that under-sampled the variation in service of simplification. Studies 2 and 3 addressed whether a prompt to "use math" would encourage students to make precise general explanations by co-opting their epistemologies about math. The prompt was marginally successful at influencing students to make additive and multiplicative solutions on the collisions task, especially in combination with the insufficient main effects only materials. The prompt did not impact solutions with the more complicated materials, suggesting that it did not contribute to learning over and above the selection of cases. Still, explicit instructions to mathematize in science problem solving may be helpful, especially when it is impossible to systematically sample the relevant variation. All three studies examined how initial learning on the collisions task influenced future learning on the transfer balance scale task. Students who received informationally sufficient materials on the collisions were better prepared for future learning from informationally sufficient materials on the balance scale task. In contrast, students who received insufficient materials negatively transferred, bringing imprecise, non-general strategies to the balance scale task despite the sufficiency of these new materials. These students performed at lower levels than a control condition that approached the balance scale task without completing the collisions activity. Providing simplified materials to make it easier for students in the short run may have negative consequences in the long run. Implications for the selection of examples and design of instructional prompts are discussed. Finally, I draw parallels between the participants who created strategies to be transferred within these studies and students who leave school with knowledge that is soon-to-be-insufficient in broader contexts. I conclude that future work must help students with insufficient, but correct-at-the-time understanding learn to learn from more complex information in the future.