Workers in science and technology-related fields use proportional reasoning extensively when making sense of quantitative data. Mathematics instruction in middle school and high school often places a corresponding emphasis on ratios and proportions, however many students still have difficulty reasoning about product and ratio quantities in introductory physics classes. This project is developing curricular materials to strengthen the ability of students to reason in the context of the topics regularly covered in introductory physics. These materials employ "invention instruction," an approach shown to be effective in facilitating mathematical reasoning. In an invention task, students are given a "job" that they complete by inventing a quantity to characterize a set of physical situations and make meaningful comparisons. The tasks are sequenced so students can start by reasoning about ratios and proportions in a familiar, everyday context, and they progress toward more abstract physical quantities for which physicists commonly use the same type of reasoning. These invention sequences are designed to highlight the similarity of the reasoning required. The project workers are developing invention sequences for use in both high school and introductory college classes and are measuring their effectiveness in developing students' content knowledge and reasoning ability with more abstract quantities. In parallel, they are also conducting basic research into how students are actually using proportions in various settings. This work is contributing to our understanding of how and why students struggle with reasoning about abstract quantities in introductory physics and provides instructional approaches that more efficiently develop reasoning skills for maturing students of science.
The goals of this collaborative project between Western Washington, Rutgers, and New Mexico State Universities were to (1) better understand student thinking about ratio and proportion in physics; (2) develop methods for assessing student reasoning about proportions; and (3) design and test instruction intended to promote flexible reasoning about ratios in physics. We developed a series of multiple-choice questions that diagnose students’ facility and flexibility with proportional relationships, called the PRAT (Proportional Reasoning Aptitude Test). The diagnostic is based on how proportions are used in physics, prior research in student difficulties with proportions, student interviews, and analysis of open-ended versions of the questions. Questions fall into six categories: Recognizing ratio as an appropriate measure; verbal interpretation of ratios; construction of ratios to characterize physical systems; applying ratios to make quantitative predictions; translating between different representations of direct proportions; and reasoning about non-direct relationships. The PRAT questions are available for use both as a diagnostic pretest and to measure the effect of instruction on student ratio fluency. University students at all levels struggle with the PRAT questions. For these questions, success rates drop when variables are substituted for numbers. In addition, comparing isomorphic questions in different contexts, success rates drop in unfamiliar contexts. Our results suggest that while students may have an algorithmic understanding of ratio and proportion, many haven’t developed the requisite fluency with their use for making sense of ratio quantities in physics. To develop fluency with proportions, and to more firmly link physics concepts with their mathematical representation, we developed and classroom-tested a series of invention tasks for introductory physics. Based on similar tasks developed by Dan Schwartz and colleagues, invention tasks require that students develop an index to characterize a physical situation. For example, students might be shown different archery targets, each with arrows embedded at various places, and asked to come up with a way to compare the skill of the archers. Typically, these exercises are used before formal instruction covering the topic at hand. Schwartz has shown that by grappling with invention beforehand (even without a correct end product), students are better prepared to understand the defined quantity when it is presented (compared to their classmates who spend an equal amount of time participating in a discussion about the topic). An additional benefit is that repeated practice with such tasks promotes a view of science as generated knowledge rather than as transmitted information. For a variety of topics spanning the introductory course, we developed invention sequences, typically 2 or 3 tasks that are presented in series, starting with familiar (often non-physics) contexts and proceeding to similar situations in physics. Follow-on questions reinforce the similarity of procedure and reasoning between the tasks in the sequence, allowing students to make analogies to guide their understanding of new concepts and to develop their creativity using mathematics in unfamiliar situations. We refined the invention sequences based on classroom observations. The invention sequences are available to instructors in a password-protected website. Based on promising results at all three institutions, we pre- and post-tested more than 600 students in two courses for freshmen engineering students at Rutgers: EAP1, for students with a pre-calculus math placement, and AP1, for students with a calculus placement. The EAP1 students (with 40% of its students from underreprepresented ethnic groups) used invention instruction throughout the semester, and the AP1 students did not. We used PRAT questions, conceptual questions from standard mechanics inventories, and attitude questions from the CLASS (Colorado Learning Attitudes about Science Survey). Students in the EAP1 course did as well as the AP1 students on a subset of the conceptual questions; on all other measures they outperformed the AP1 students. Remarkably, the CLASS gains for EAP1 students were the largest that we are aware of for a course with an enrollment > 30 (and were especially strong in categories measuring mathematical reasoning and problem solving sophistication). CLASS gains were negative for AP1 students in every category. When we compared EAP1 student performance on the Force Concept Inventory to that of students who took this course before invention tasks were introduced (with no other changes made), we saw a 10% improvement in student gain. Our research on proportional reasoning fluency has allowed us to develop diagnostic measurement tools, and has led to a more nuanced understanding of student thinking. Use of curriculum developed on the basis of this research has improved students’ mathematical reasoning, conceptual understanding, and attitudes about science. This broader impact of our project lies in the potential to improve student reasoning and attitudes about science, and in the interest generated in developing invention task sequences to promote similar gains in chemistry, mathematics, and earth science. Our assessment of students graduating from low SES school districts indicates that this research focus and these materials have potential to help close performance and representation gaps.
