Understanding Coordinated Sets of Resources: An Example from Quantum Tunneling Michael C. Wittmann Physics Education Research Laboratory, Department of Physics and Astronomy The University of Maine, Orono, ME;
[email protected]. In studying student reasoning about quantum physics in the context of tunneling through a barrier, we observe that students commonly use several reasoning resources in conjunction with one another. Our data is gathered in individual student interviews, ungraded quizzes, diagnostic surveys, and examination questions. We believe that solely a microscopic perspective on the individually used reasoning resources is too narrow to help us understand student reasoning. We also believe that students do not have a coherent, robust (macroscopic) concept of tunneling that can be described as a coordination class. To account for our data, we introduce a mesoscopic description of a coordinated set of resources. We describe a possible coordinated set in quantum tunneling, complete with a readout strategy and net of associated resources and mathematical forms, that a student uses in favor of another possible coordinated set, the resources and forms of which he has available but which he seems not to read out of the given situation. Introduction In this paper, we describe investigations into student thinking about quantum tunneling. The data we are gathering is helping us refine models of student reasoning, specifically the manner in which phenomenological primitives (p-prims) (diSessa, 1993), resources (Hammer, 2000), and mathematical forms (Sherin, 1996) combine to form coordination classes (diSessa & Sherin, 1998). We present a single detailed description of student reasoning about quantum tunneling to show how a mesoscopic model (sitting between a microscopic resources or p-prims model and macroscopic coordination class model of concept) can help interpret student reasoning in learning situations. We begin with a brief outline of our theoretical framework, then describe our data sources and methods of analysis, and give a single detailed example of student reasoning which can be described at a mesoscopic level. A brief discussion of the implications of our work concludes the paper. Theoretical model We model physics reasoning in terms of small-scale reasoning elements and describe how those reasoning elements may coordinate into knowledge structures for organization in memory. For the purposes of this paper, we do not describe issues of metacognition and conceptual change, though we have extended this model to include these issues. Several researchers have introduced small-scale models of reasoning which act as building blocks of analytical, procedural, conceptual, and epistemological reasoning (Hammer and Elby, 2002; Elby and Hammer, 2001; Hammer, 2000; Tirosh, Staby, and Cohen, 1998; Sherin, 1996; diSessa, 1993; Minstrell, 1992; Minsky, 1985). We use Hammer’s general term “resources” to refer to any of these constructs (Hammer, 2000). As an example of a conceptual resource, consider the idea of part-for-whole, where a part of an object (or system) can represent the whole of the system (Wittmann, 2002). This idea is natural in the context of politics (a president or queen represents a country), corporate descriptions (a charismatic Chief Executive Officer - CEO - represents a company), or literature (as described by the term synecdoche). In physics, we use part-for-whole when we represent the motion of an object by the motion of its center of mass. Other classic examples of resources include ideas such as closer means stronger or effects die away. Resources serve an adaptive function in thinking by uniting a variety of specific experiences (e.g., experiences in which effects die away) into a general statement. Reasoning can then be described in terms of resources acting in working memory as useful “blocks” of data. Most problem solutions require the coordination of many resources. Several models of resource coordination have been described in literature that bridges physics education research with cognitive science (e.g., diSessa and Sherin, 1998; Minsky, 1985). “Coordination classes” (diSessa and Sherin, 1998) are one such model. Coordination classes have been proposed as one kind of large-scale concept, but built up from several small-grain elements. A coordination class contains
two types of resources: “readout” resources that organize sensory information, and reasoning resources that form causal links. For example, if an object falls towards the ground in front of a moving car, we may observe (read out) in particular that the object is small, orange, and moving slowly. It might be only a leaf. Reasoning resources (perhaps small things have small effects, or slow-moving things have small impact) are primed and activated along with the readout resources. A combination of resources is coordinated (most likely subconsciously) to create a single behavior or result (for example, do not swerve to miss the leaf – but swerve to miss a small but quickly falling grey stone). Though the coordination classes model of concept has great power in describing student reasoning, we find it incomplete. We do not know how students coordinate information, how large a coordination class is, nor how to measure learning in terms of a coordination classes model. The goal of this paper is to show how student thinking about quantum tunneling may help us refine the approach to modeling student reasoning suggested by the coordination classes model. Data sources and methods of interpretation The primary data described in this paper was gathered at The University of Maryland. A final examination question was given to two classes in quantum physics for engineering students at the University of Maryland (n=13 and n=11). This question is analyzed in more detail in the next section. We will focus on a single student’s response and use other data to support our statements. We have used individual demonstration interviews with students both before and after they have studied quantum tunneling. While recognizing that individual quotes from interviews can be misleading or incomplete (Wittmann and Scherr, 2003), we will use interview quotes to clarify points made in our analysis below. Interviews carried out by Jeffrey T. Morgan with second (n=4) and fourth year students (n=2, interviewed both before and after instruction) at The University of Maine were transcribed and annotated according to gestures and important sketches drawn by students. Their content was analyzed with an emphasis on student reasoning about the physics and the resources students use in making sense of tunneling (Wittmann and Morgan, 2003). Additional written data came from several sources. Morgan developed a diagnostic survey on tunneling at the University of Maine based on his interview results. Students in their second (n=15) and fourth years (n=4) of instruction answered the surveys one semester after they had completed instruction on quantum physics. The multiple-choice survey results were analyzed with the help of a spreadsheet to discover trends in the data (Morgan and Wittmann, 2003). Student responses to an examination question about tunneling Student responses to a single examination question often reveal common themes among responses that indicate the complexity of reasoning in a student learning new material. The final examination question shown in Figure 1 was given to engineering students who had completed instruction on quantum physics. Students were asked to interpret typical quantities (probability, wave function, and energy) in a situation typical to a textbook problem (a particle incident on a square barrier). Student performance on this question has been reported (Redish, Wittmann, and Steinberg, 2000) with an emphasis on how students perform better Consider a beam of electrons with energy E0 incident in a modified instruction I E II III from the left (x < 0) on a potential barrier of height U class (Redish, Steinberg, and width a (see energy diagram to the right). U and Wittmann, 1999Three regions are indicated on the energy diagram E0 as I, II, and III. present). x a. Sketch the shape of the total wave function of an In this paper, we a electron in regions I, II, and III in the diagram to the describe one student y I II III right. Explain how you arrived at your answer. response in detail and b. Write equations for the wave function in each of the support our interpretation x regions in the diagram. Leave normalization constants of it through reference to a unspecified. additional data. In the c. Compare the energy of electrons found in regions I next section, we analyze and III. Explain how you arrived at your answer. the student’s responses in terms of reasoning Figure 1: Final examination quantum tunneling resources and question
mathematical forms to show details of the student’s thinking. When asked to describe the energy of a particle that tunneled through the barrier compared to one that had not, the student gave the common “energy loss” answer. He wrote, “[the particles] in region 3 ‘lost’ energy while tunneling through the barrier.” This answer has been common throughout much of our research, independent of the specific phrasing of the question, the level of schooling of the student, and the specific context of the question. Students regularly believe that tunneling particles lose energy. For example, a third year engineering student stated, “[the particle] collides and loses energy in the barrier.” In survey results at The University of Maine, roughly 3/4 of second year students who completed a course in quantum physics stated that there was energy loss for a particle tunneling through a barrier. At the University of Maryland, roughly 2/3 of students described energy loss in a barrier after completing only traditional instruction. The student had been asked to sketch the wave function of the particles. His sketch is shown in Figure 2. The axis around which the wave function oscillates is higher in region I (the incoming wave) than in region III. In the barrier (region II), the student sketched an exponential decay (though both exponential increase and decay are mathematically possible). The axis around which the wave Figure 2: Student sketch and mathematical equations function oscillates shifts with the exponential decay term in the barrier. This sketch is extremely common in all our investigations. For example, a second year student in an interview sketched the wave function shown in Figure 3. Note that the student included the energy barrier in her sketch even though she had not been asked for it. The student’s mathematical equations (with unspecified normalization constants) are Figure 3: Interview student sketch shown in Figure 2. The student has generally correct elements which are partially consistent with the sketch of the wave function. In regions I and III, the wave function oscillates sinusoidally. In region II, the wave function is described by exponential decay and increase. Only one term exists for the wave function in region III, since it describes the outgoing wave. These are all correct, but the student has not made the axis of his function consistent with the mathematics. Also, the wavelength of the wave function he has sketched (which correctly does not change from region I to region III) does not match the answer he gave about energy loss (for which a longer wavelength would have been expected in region III). The wave function sketched in figure 3 shows a different situation, namely that smaller amplitude waves have shorter wavelengths. For further results on this topic, see Ambrose et al., 1999 and Vokos et al., 2000. A coordinated set of resources to describe student reasoning We can describe this student’s responses in terms of reasoning resources and mathematical forms to create a coordinated set of ideas that the student brings into play when thinking about tunneling. In this section, we first name the individual resources we can use to describe the student’s thinking. Then we show how these responses can be understood as a coordinated set of only some and not all ideas visible in the student’s response. The student interprets axis height as indicating the energy of the particle. He could be using the resource compelling visual attribute (Elby, 2000) in terms of axis placement to help interpret a piece of the system. When describing the energy lost in the barrier, he could be using dying away (diSessa, 1993), where resistance of some sort leads to a lessening of some effect. (A second year student in an interview described the effects die away resource in terms of a snowball passing through a snowbank). The graphical focus on only the exponential term of the mathematical equation is consistent with the dying away resource, and can also be described in terms of the mathematical form exponential decay.
Additional resources are visible in the student’s responses. Another compelling visual attribute is the wavelength of the wave function, which the student has drawn such that the energy is the same on either side of the barrier. The mathematical form of exponential increase is mathematically available to the student, but not used in his graph. This student has given three consistent resources and others which are inconsistent. The student may be interpreting the potential barrier as a resistive element by reading out that “things going through other things lose energy.” (Another student describing a similar problem spoke of a snowball slowing down as it passed through a snowbank.) Based on this readout, he may find consistency (and reinforcement of his thinking) in using the axis-height compelling visual attribute to describe the energy dying away while emphasizing the exponential decay mathematical term. The specific readout that barriers cause loss leads the student to choose some resources linked to energy, and not others. We note that linking barriers cause loss to probability would be perfectly appropriate, and we find that most students answer questions about the probability of finding particles in region III very well. The student seems to link his readout of the problem to energy and energy loss instead. The other resources seen in the student’s responses might have been triggered had the student used the idea that nothing changes with regard to the energy of a particle tunneling through a potential barrier. Then, the consistency of the wavelength in regions I and III might have been noted by the student saying that there is no energy loss in the barrier. Also, by considering the exponential increase term in the mathematical equation, he might have called into question his interpretation of the axis height. We suggest that the student’s readout of barriers cause loss was linked inappropriately to a net of ideas about energy, which kept him from attending to other correct details in his response. To a first approximation, we refer to the readout and the coordination of resources carried out by the student as a coordinated set of resources in tunneling. The set consists of a possible readout strategy (barriers cause loss) and a few (but not many) connected reasoning resources. The resources are chosen in favor of other possible resources in a given setting because they have been triggered by the specific readout. A different readout (nothing changes in the energy when passing through a barrier) is counterintuitive for students learning quantum physics and thus less likely to trigger the appropriate resources for understanding the situation. Discussion In this paper, we have used a specific model of student reasoning to analyze a single student’s performance on a final examination question. The student’s responses are consistent with additional research results showing that students have difficulties answering many questions about tunneling. When combining the results into a description of the student’s thinking about quantum tunneling, we find that a single readout may trigger only a few resources (and keep a student’s attention from others) to guide the student’s reasoning. To remain consistent with the idea of coordination classes while addressing a much smaller cognitive structure, we refer to the readout and the resources used by the student as a coordinated set. Though our data are only for a single student, we have additional data from other students to support the construct of a coordinated set. We believe that the idea of a coordinated set of resources is a productive tool for understanding spontaneous reasoning and the process of learning as knowledge is coordinated into ever larger sets of relevant, linked information. Acknowledgements The research described in this paper was begun while at the University of Maryland while part of the research group led by Edward F. Redish. Research has continued at The University of Maine as part of a Ph.D. project carried out by Jeffrey T. Morgan. I thank Rachel E. Scherr for her valuable contributions in clarifying several of the cognitive issues described in this paper. References Ambrose, B.S., Heron, P.R.L., Vokos, S., McDermott, L.C. (1999) Student Understanding of Light as an Electromagnetic Wave: Relating the Formalism to Physical Phenomena. American Journal of Physics, 67:10, 891-98.
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