Search

Professional problem-solving practice in physics and engineering relies on mathematical sense making--reasoning that leverages coherence between formal mathematics and conceptual understanding. A key question for physics education is how well current instructional approaches develop students' mathematical sense making. We introduce an assessment paradigm that operationalizes a typically unmeasured dimension of mathematical sense making: use of calculations on qualitative problems and use of conceptual arguments on quantitative problems. Three assessment items embodying this calculation-concept crossover assessment paradigm illustrate how mathematical sense making can positively benefit students' problem solving, leading to more efficient, insightful, and accurate solutions. These three assessment items were used to evaluate the efficacy of an instructional approach focused on developing students' mathematical sense making skills. In a quasi-experimental study, three parallel lecture sections of first-semester, introductory physics were compared: two mathematical sense making sections, one with an experienced instructor and one with a novice instructor, and a traditionally taught section, as a control group. Compared to the control group, mathematical sense making groups used calculation-concept crossover approaches more often and gave more correct answers on the crossover assessment items, but they did not give more correct answers to associated standard problems. In addition, although students' postcourse epistemological views on problem-solving coherence were associated with their crossover use, they did not fully account for the differences in crossover approach use between the mathematical sense making and control groups. These results demonstrate a new assessment paradigm for detecting a typically unmeasured dimension of mathematical sense making and provide evidence that a targeted instructional approach can enhance engagement with mathematical sense making in introductory physics.

[This paper is part of the Focused Collection on Curriculum Development: Theory into Design.] Common models of curricular development in physics education research (PER) have typically involved a hierarchical relationship between researchers and students, where researchers lead the design and testing of curriculum for students. We draw from work in students as partners and related fields in order to codesign curriculum in partnership with students. Such work has the potential to disrupt typical hierarchical relationships and interactions between students and faculty by involving students in the process of making curricular decisions. We invited undergraduate students to participate in a partnership to codesign a set of curricular materials for topics in quantum mechanics that students often struggle with. Four undergraduate students, one PER graduate student, and one PER faculty member met for a series of codesign meetings. We collected videotapes of the meetings, written artifacts, and meeting reflections. This paper presents a fine-grained analysis of one interaction in which researchers attempted to create space for students to contribute to decision making about how the collaboration should proceed. Through analyzing the complex dynamics of how participants negotiated decision-making space, including characterizing the types of decisions that were made, we describe how access to those decisions were opened up or cut off, and how those decisions contested or reaffirmed participants' roles. Working towards partnership is a complex and messy process: attempts to open up space for some forms of decision making closed off access to other forms of decision making. In some ways, the interactions between the participants also reified the traditional student and faculty roles that the partnership had intended to disrupt. Through closely analyzing these dynamics, we aim to self-critically reflect on the challenges and tensions that emerge in codesign partnerships. We discuss our own areas for growth and speak to implications for more responsible partnerships.

Background: While there has been increasing recognition of the importance of attending to students' views about what counts as knowing and learning a STEM field, surveys that measure these "epistemological" beliefs are often used in ways that implicitly assume the fields, e.g., "physics," to be a single domain about which students might have sophisticated or naïve beliefs. We demonstrate this is not necessarily the case and argue for attending to possible differences in students' epistemological beliefs across different sub-domains of physics. In modern physics and quantum mechanics courses for engineering and physics students, we administered a set of modified Colorado Learning Attitudes about Science Survey (CLASS) items. Each selected item was turned into two items, with the word "physics" changed to "classical physics" in one and "quantum physics" in the other. Results: We found significant splits between students' survey responses about classical vs. quantum physics on some items, both pre- and post-instruction. In classical physics, as compared to quantum physics, students were more likely to report the salience of real-world connections and the possibility of combining mathematical and conceptual reasoning during problem solving. Conclusions: These findings suggest that attending to sub-domain specificity of students' beliefs about physics can be fruitful and ought to influence our instructional choices.

Energy is an abstract science concept, so the ways that we think and talk about energy rely heavily on ontological metaphors: metaphors for what kind of thing energy is. Two commonly used ontological metaphors for energy are "energy as a substance" and "energy as a vertical location." Our previous work has demonstrated that students and experts can productively use both the substance and location ontologies for energy. In this paper, we use Fauconnier and Turner's conceptual blending framework to demonstrate that experts and novices can successfully blend the substance and location ontologies into a coherent mental model in order to reason about energy. Our data come from classroom recordings of a physics professor teaching a physics course for the life sciences, and from an interview with an undergraduate student in that course. We analyze these data using predicate analysis and gesture analysis, looking at verbal utterances, gestures, and the interaction between them. This analysis yields evidence that the speakers are blending the substance and location ontologies into a single blended mental space.

Many science education researchers have argued that learners' commitment to a substance (matter-based) ontology impedes the learning of scientific concepts that scientists typically conceptualize as processes or interactions, such as force, electric current, and heat. By this account, students' tendency to classify these entities as substances or properties of substances leads to robust misconceptions, and instruction should steer novices away from substance-based reasoning. We argue that substance-based reasoning can contribute to the learning and understanding of these very same physics concepts. Our case study focuses on a group of elementary school science teachers in our professional development program. Starting from substance-based metaphors for gravity, the teachers build a sophisticated explanation for why objects of different masses fall with the same acceleration. We argue that, for conceptual, epistemological, and affective reasons, instructional interventions should focus on tapping these productive substance-based resources when they arise rather than attempting to suppress them.

Much research in engineering and physics education has focused on improving students' problem-solving skills. This research has led to the development of step-by-step problem-solving strategies and grading rubrics to assess a student's expertise in solving problems using these strategies. These rubrics value "communication" between the student's qualitative description of the physical situation and the student's formal mathematical descriptions (usually equations) at two points: when initially setting up the equations, and when evaluating the final mathematical answer for meaning and plausibility. We argue that (i) neither the rubrics nor the associated problem-solving strategies explicitly value this kind of communication "during" mathematical manipulations of the chosen equations, and (ii) such communication is an aspect of problem-solving expertise. To make this argument, we present a case study of two students, Alex and Pat, solving the same kinematics problem in clinical interviews. We argue that Pat's solution, which connects manipulation of equations to their physical interpretation, is more expert-like than Alex's solution, which uses equations more algorithmically. We then show that the types of problem-solving rubrics currently available do not discriminate between these two types of solutions. We conclude that problem-solving rubrics should be revised or repurposed to more accurately assess problem-solving expertise. (Contains 2 tables, 1 figure and 4 footnotes.)

Researchers have argued against deficit-based explanations of students' difficulties with mathematical sense-making, pointing instead to factors such as epistemology. Students' beliefs about knowledge and learning can hinder the activation and integration of productive knowledge they have. Such explanations, however, risk falling into a "deficit trap"--substituting a concepts/skills deficit with an epistemological one. Our interview-based case study of a freshman engineering major, "Jim," explains his difficulty solving a physics problem (on hydrostatic pressure) in terms of his epistemology, but avoids a deficit trap by modeling the dynamics of his epistemological stabilities and shifts in terms of fine-grained cognitive elements that include the seeds of epistemological expertise. Specifically, during a problem-solving episode in the interview, Jim reaches and sticks with an incorrect answer that violates common sense. We show that Jim has all the mathematical skills and physics knowledge he would need to resolve the contradiction. We argue that his difficulty doing so stems in part from his epistemological views that (i) physics equations are much more trustworthy than everyday reasoning, and (ii) physics equations do not express meaning that tractably connects to common sense. For these reasons, he does not view reconciling between common sense and formalism as either necessary or plausible to accomplish. But Jim's in-the-moment shift to a more sophisticated epistemological stance highlights the seeds of epistemological expertise that were present all along: he does see common sense as connected to formalism (though not always tractably so), and in some circumstances, this connection is both salient and valued. (Contains 1 figure, 1 table, and 3 notes.)

The authors appreciate Professor Slotta's responding to their critique (Slotta, this issue). For their part, they believe that Professor Slotta has misinterpreted aspects of their position. In this commentary, the authors clarify two particular points. First, they explain their use of "static ontologies," which they maintain applies. Second, they respond to the question of how their dynamic view could account for evidence of stabilities. In addition, they take the opportunity to note differences in methodology that underlie much of this debate. (Contains 3 footnotes.)

In a series of well-known papers, Chi and Slotta (M. T. H. Chi, 1992, 2005; M. T. H. Chi & J. D. Slotta, 1993; M. T. H. Chi, J. D. Slotta, & N. de Leeuw, 1994; J. Slotta & M. T. H. Chi, 2006; J. D. Slotta, M. T. H. Chi, & E. Joram, 1995) have contended that a reason for students' difficulties in learning physics is that students think about concepts as "things" rather than as "processes" and that there is a significant barrier between these 2 ontological categories. We contest this view, arguing that expert and novice reasoning often and productively traverses ontological categories. We cite examples from everyday, classroom, and professional contexts to illustrate this. We agree with Chi and Slotta that instruction should attend to learners' ontologies, but we find that these ontologies are better understood as dynamic and context dependent rather than as static constraints. To promote 1 ontological description in physics instruction, as suggested by Slotta and Chi, could undermine novices' access to productive cognitive resources that they bring to their studies and inhibit their transition to the dynamic ontological flexibility required of experts. (Contains 15 footnotes.)

Abstract: Problem solving in groups can be rich with tension for students. This tension may arise from conflicting approaches (conceptual and/or epistemological) and/or from conflict emerging in the social relations among group members. Drawing on video records of undergraduate students working collaboratively, we use three cases to illustrate the multifaceted ways in which conflict arises—combining conceptual, epistemological, and socioemotional dynamics—and a specific way of managing the tension that can emerge from the multifaceted conflict, “taking an escape hatch.” An escape hatch is a set of discourse moves through which participants close the conversational topic, thereby relieving tension, but before a conceptual resolution is achieved. We describe how epistemological twists and turns can be recruited as a means of managing the strong emotions experienced by the students, showing the coupling of emotion and epistemology in students’ conceptual sense making during group work. We help to provide the groundwork necessary for instructors to notice, understand, and respond to one way in which conceptual–epistemological–social–emotional aspects of interaction are coupled in the emergence of tension, rather than narrowly targeting instructional moves based on only conceptual or epistemological considerations. Instead, instructors should often respond to—and help students become aware of—the emotional component of peer interactions and its entanglement with the “cold cognitive” conceptual and epistemological components. [ABSTRACT FROM AUTHOR]