Tuminaro, PhD Dissertation (2004)

A cognitive framework for analyzing and describing introductory students' use and understanding of mathematics in physics

J. Tuminaro, Ph.D. Dissertation, E. F. Redish (advisor), (2004). (html TOC and abstract)

Abstract: Many introductory, algebra-based physics students perform poorly on mathematical problem solving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) students simply lack the mathematical skills needed to solve problems in physics, or (2) students do not know how to apply the mathematical skills they have to particular problem situations in physics. While many students do lack the requisite mathematical skills, a major finding from this work is that the majority of students possess the requisite mathematical skills, yet fail to use or interpret them in the context of physics.

In this thesis I propose a theoretical framework to analyze and describe studentsí mathematical thinking in physics. In particular, I attempt to answer two questions. What are the cognitive tools involved in formal mathematical thinking in physics? And, why do students make the kinds of mistakes they do when using mathematics in physics? According to the proposed theoretical framework there are three major theoretical constructs: mathematical resources, which are the knowledge elements that are activated in mathematical thinking and problem solving; epistemic games, which are patterns of activities that use particular kinds of knowledge to create new knowledge or solve a problem; and frames, which are structures of expectations that determine how individuals interpret situations or events.

The empirical basis for this study comes from videotaped sessions of college students solving homework problems. The students are enrolled in an algebra-based introductory physics course. The videotapes were transcribed and analyzed using the aforementioned theoretical framework.

Two important results from this work are: (1) the construction of a theoretical framework that offers researchers a vocabulary (ontological classification of cognitive structures) and grammar (relationship between the cognitive structures) for understanding the nature and origin of mathematical use in the context physics, and (2) a detailed understanding, in terms of the proposed theoretical framework, of the errors that students make when using mathematics in the context of physics.

Atkins, PhD Dissertation (2004)

Analogies as categorization phenomena: Studies from scientific discourse

L. J. Atkins, Ph.D. Dissertation, D. Hammer (advisor), (2004). (html TOC and abstract) (pdf links: Chapter 1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8)

Abstract: Studies on the role of analogies in science classrooms have tended to focus on analogies that come from the teacher or curriculum, and not the analogies that students generate. Such studies are derivative of an educational system that values content knowledge over scientific creativity, and derivative of a model of teaching in which the teacher's role is to convey content knowledge. This dissertation begins with the contention that science classrooms should encourage scientific thinking and one role of the teacher is to model that behavior and identify and encourage it in her students. One element of scientific thinking is analogy. This dissertation focuses on student-generated analogies in science, and offers a model for understanding these. I provide evidence that generated analogies are assertions of categorization, and the base of an analogy is the constructed prototype of an ad hoc category. Drawing from research on categorization, I argue that generated analogies are based in schemas and cognitive models. This model allows for a clear distinction between analogy and literal similarity; prior to this research analogy has been considered to exist on a spectrum of similarity, differing from literal similarity to the degree that structural relations hold but features do not. I argue for a definition in which generated analogies are an assertion of an unexpected categorization: that is, they are asserted as contradictions to an expected schema.

McCaskey, Dancy & Elby, Proceedings of 2003 PER Conference (2004)

Effects on assessment caused by splits between belief and understanding

T. L. McCaskey, M. H. Dancy & A. Elby, in Proceedings of the 2003 Physics Education Research Conference, S. Franklin, J. Marx & K. Cummings (Eds.), 720, p 37-40, Melville, NY: American Institute of Physics (2004). 

Abstract: We performed a new kind of FCI study to get at the differences between what students believe and what they think scientists believe. Students took the FCI in the standard way, and then made a second pass indicating “the answer they really believe” and “the answer they think a scientist would give.” Students split on a large number of the questions, with women splitting more often than men.

Hammer, Enrico Fermi Summer School Proceedings (2004)

The variability of student reasoning, lectures 1-3

D. Hammer, in Proceedings of the Enrico Fermi Summer School, Course CLVI, E. Redish & M. Vicentini (Eds.), Bologna: Italian Physical Society (2004).

Lecture 1: Case studies of children's inquiries, p 279-299.

Abstract: Classroom observations show variability in student reasoning, from young children through adults, even moment-to-moment for the same students in the same class. This varied phenomenology conflicts with views of naïve theories, entrenched conceptions and stages of development as stable attributes. Student knowledge and reasoning is better understood in terms of a manifold ontology of more fine-grained, context sensitive resources. Expectations of variability in student knowledge and reasoning suggest different approaches and objectives in instruction, especially in early science education.
This is the first lecture in a series of three. It introduces the overall agenda and then begins with a series of examples of children’s inquiries to reflect on the beginnings of scientific expertise.

Lecture 2: Transitions, p 301-319.

Abstract:This lecture continues the phenomenology of student reasoning from the first, beginning with brief examples of introductory physics students failing to apply basic logic and common sense. These contrast with the examples from the first lecture of children’s reasoning, but it would be a mistake to interpret the university students’ behavior as evidence that they are not capable of what we saw in elementary students. Rather, students at all ages are capable of reasoning in a variety of ways, and the bulk of this lecture focuses on examples of students shifting in their approaches and ideas over short time scales. Often these shifts follow epistemological prompts from an instructor, suggestions for how students should think about knowledge and learning.

Lecture 3: Manifold cognitive resources, p 321-340.

Abstract: The previous lectures focused on phenomenology: What sorts of occurrences do we see in students’ reasoning? This third and final lecture focuses on ontology: What sorts of things do we attribute to students’ minds? It has become conventional to speak and think in terms of conceptions, naïve theories, and stages of development. These are all attributions of stable properties, and they account well for patterns that can occur in student reasoning. They do not account well, however, for the variability and multiple patterns illustrated in the previous lectures. Research in cognitive science provides an alternative ontology of multiple, fine-grained cognitive resources that are contextsensitive in their activation. This lecture reviews some of that work and draws implications for elementary science education.

Louca, Elby, Hammer & Kagey, Educ Psychologist (2004)

Epistemological resources: Applying a new epistemological framework to science instruction

L. Louca, A. Elby, D. Hammer & T. Kagey, Educational Psychologist, 39(1), p 57-68 (2004). (link to journal article)

Abstract: Most research on personal epistemologies has conceived them as made up of relatively large, coherent, and stable cognitive structures, either developmental stages or beliefs (perhaps organized into theories). Recent work has challenged these views, arguing that personal epistemologies are better understood as made up of finer grained cognitive resources whose activation depends sensitively on context. In this article, we compare these different frameworks, focusing on their instructional implications by using them to analyze a third-grade teacher's epistemologically motivated intervention and its effect on her students. We argue that the resources framework has more predictive and explanatory power than stage- and beliefs-based frameworks do.

Hammer, Elby, Scherr & Redish, Transfer of Learning: Research and Perspectives (2004)

Resources, framing, and transfer

D. Hammer, A. Elby, R. E. Scherr & E. F. Redish, in Transfer of Learning: Research and Perspectives, J. Mestre (Ed.) Information Age Publishing: Greenwich, CT (2005), pp. 89-119.

Abstract: As researchers studying student reasoning in introductory physics, and as instructors teaching courses, we often focus on whether and how students apply what they know in one context to their reasoning in another. But we do not speak in terms of “transfer.” The term connotes to us a unitary view of knowledge as a thing that is acquired in one context and carried (or not) to another. We speak, rather, in terms of activating resources, a language with an explicitly manifold view of cognitive structure. In this chapter, we describe this view and argue that it provides a more firm and generative basis for research.

In particular, our resources-based perspective accounts for why it is difficult, and perhaps unnecessary, to draw a boundary around the notion of “transfer”; provides an analytical framework for exploring the differences between active transfer involving metacognition and passive transfer that “just happens”; helps to explain many results in the transfer literature, such as the rarity of certain kinds of transfer and the ubiquity of others; and provides an ontological underpinning for new views of transfer such as Bransford, Schwartz, and Sears’ (this issue) “preparation for future learning.”

Redish, Proceedings of International School of Physics (2004)

A Theoretical Framework for Physics Education Research: Modeling Student Thinking

E. F. Redish, in Proceedings of the International School of Physics, "Enrico Fermi" Course CLVI, E. F. Redish and M. Vincentini (Eds.) IOS Press, Amsterdam (2004).

Abstract: Education is a goal-oriented field. But if we want to treat education scientifically so we can accumulate, evaluate, and refine what we learn, then we must develop a theoretical framework that is strongly rooted in objective observations and through which different theoretical models of student thinking can be compared. Much that is known in the behavioral sciences is robust and observationally based. In this paper, I draw from a variety of fields ranging from neuroscience to sociolinguistics to propose an over-arching theoretical framework that allows us to both make sense of what we see in the classroom and to compare a variety of specific theoretical approaches. My synthesis is organized around an analysis of the individual’s cognition and how it interacts with the environment. This leads to a two level system, a knowledge-structure level where associational patterns dominate, and a control-structure level where one can describe expectations and epistemology. For each level, I sketch some plausible starting models for student thinking and learning in physics and give examples of how a theoretical orientation can affect instruction and research.