A Problem-Solving Template for Integrating ...

A Problem-Solving Template for Integrating Qualitative and Quantitative Physics Instruction Author(s): Janice M. Fink and Gary J. Mankey Source: The Journal of General Education, Vol. 59, No. 4 (2010), pp. 273-284 Published by: Penn State University Press Stable URL: http://www.jstor.org/stable/10.5325/jgeneeduc.59.4.0273 Accessed: 29-08-2015 20:07 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/10.5325/jgeneeduc.59.4.0273?seq=1&cid=pdf-reference# references_tab_contents You may need to log in to JSTOR to access the linked references.

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A Problem-Solving Template for Integrating Qualitative and Quantitative Physics Instruction

Janice M. Fink and Gary J. Mankey

It is the charge of many physics professors to deliver service courses to beginning engineering and science students. A vast amount of research has been performed to stimulate the delivery of better introductory physics instruction (Hsu, Brewe, Foster, & Harper, 2004). One can find various strategies outlined to improve conceptual understanding and problem-solving abilities. For instance, a recent trend is for instructors to establish learning goals and then construct a rubric of assessment methods to track student progress. Introductory science courses often emphasize coverage more than methods (Keller, 2002). However, physics is a problem-solving discipline, and one could argue that problem solving and questioning should be the standard goal of all education. Science education reform has focused on efforts to deliver a better product through various methods, with particular emphasis on increasing active, experiential, hands-on activities; peer and collaborative work groups; increased faculty–student interaction; and continual assessment (Stage & Kinze, 2009). The wide range of available tools and methods is testament to the magnitude of the challenge faced by the instructor in an environment where the question “How do students solve problems?” can lead to ambiguous conclusions. The goal of all instruction should include demonstrating that reasoning facilitates learning and showing that reasoning is an essential element of all science. jge: the journal of general education, Vol. 59, No. 4, 2010 Copyright © 2011 The Pennsylvania State University, University Park, PA.


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Another important factor that must be considered is that grading practices have a significant impact on student behavior, so a better question to ask may be, “How can we grade problem solutions to facilitate learning problem solving?” In the framework of a learner-centered approach to teaching, regular, timely, and thorough assessments of student learning are a key component of the curriculum. Implementing a learner-centered approach involves four stages: (a) defining learning outcomes, (b) matching assessments to outcomes, (c) identifying baselines and progression toward benchmarks, and (d) documenting the process using a quality enhancement plan matrix. The first two steps in the process have led to our problem-solving template, which is designed to enhance critical thinking and problem-solving skills. It combines key aspects of conceptual and problem-solving instruction as related to the resource model of cognitive development. This idea grew out of an effort to combine learner-centered, goal-oriented teaching strategies; conceptual physics instruction; and problem-solving instruction. The central theme is that practice is essential for the development of physics knowledge, and continual feedback must be provided to ensure that the practice is done right. We recognized the need for a tool to guide the steps of problem solving and thereby successfully impart the thought patterns involved. The gradual development of problem-solving skills can be readily accomplished through diligence. The grading scheme of the template facilitates integration of the knowledge structures needed for students to become expert problem solvers, which is rewarded by consistent grading practices on homework, quizzes, and examinations.

Qualitative Versus Quantitative Recent comments in The Physics Teacher brought to light some of the reservations and overgeneralizations associated with changing well-established methods of instruction (Lasry, Finklestein, & Mazur, 2009; Sobel, 2009a, 2009b). It is becoming increasingly evident that the traditional methods of expert lecture and demonstrations followed by examinations have limited effectiveness, and change is, therefore, necessary and inevitable. Students, parents, administrators, and governing bodies expect ever more detailed tracking of student progress, and there is an across-the-board call for action plans to improve student success. However, evidence suggests that even after large expenditures of time and effort in transitioning to new methods, only marginal gains in student achievement are obtained (Cummings, Marx, Thornton, & Kuhl, 1999; Hoellwarth, Moelter, & Knight, 2005; Kohl, Kuo, & Ruskell, 2008). Moreover, there seems to be a divide between those who support increased conceptual understanding and those who strive to develop increased problem-solving skills. 274




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The process of improving problem-solving skills has been addressed by a number of researchers, and most agree that practice is essential for this to occur. It seems obvious that the process of rote learning—following a procedure without understanding it—has limited effectiveness. In fact, evidence suggests that even after solving a large number of textbook problems, students still have conceptual difficulties with physics because they tend to organize qualitative and quantitative knowledge in different categories (Kim & Park, 2002). Even the commonly accepted notion of end-of-chapter problems serving as practice for problem solving has been pointedly questioned. This topic was investigated in several papers listed in a “Resource Letter” in the American Journal of Physics, which contains an extensive collection of references concerning research on problem solving (Hsu et al., 2004).

Practice Makes Perfect Assigning and grading problem sets as homework assignments has long been an established practice in physics instruction. As early as 1941, an essay by C. J. Lapp (1941) illustrated the importance of regularly grading homework in improving student success. The recent advent of computer-graded homework seems to have relieved instructors of the task of reviewing mounds of homework from large-enrollment classes, but research into the effectiveness of computergraded assignments shows that they have little positive influence on student success (Bonham, Beichner, & Deardorff, 2001). This is surprising because we are often led to the misconception that animations, numerical randomization of problems, and other features and benefits made possible with the computer should lead to an improvement of student learning with electronic homework. However, currently available computer homework problems have limitations because important information used by the student in solving the problem is not tracked. The lack of detailed tracking of student problem-solving techniques can lead to a misalignment of goals between student and instructor. Whereas the student’s goal is simply to provide the correct answer, the instructor’s goal is for the student to develop an understanding of the fundamental principles and how to apply them in a logical and consistent manner in order to provide the correct answer (Van Heuvelen, 1991). Today, there are many arguments against the time-honored strategy of assigning large amounts of homework and relying on the process of memorization to teach physics. Although Lapp’s essay clearly demonstrated the gain in understanding associated with grading assigned work, it emphasized the concept that “practice makes perfect, but only if you do it right” (Johnson, 2001, p. S2). Incorporating a problem-solving template, clearly specifying stages in the thought process, facilitates high-quality practice of problem-solving technique. Integrating Qualitative and Quantitative Physics Instruction 275



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The guide can also be used to improve conceptual understanding of basic physics principles, as it allows easy assessment of the results of problem-solvingbased learning.

Problem Solving It is imperative for physics instructors to realize that at the beginning of higher education, the average introductory student does not possess the same problem-solving skills as a professional scientist. A model must be developed to encourage novice problem solvers to become expert at the skill. With that said, the distinction between a novice and expert is difficult to explain, and quantifying when one has moved from one to the other group may be dependent on the method of measurement. Students often say, “If I only had the formula for the answer, I could get it right every time.” This is a typical new problem solver’s expectation—focusing on calculating an answer rather than solving the problem. In How to Solve It, Polya (2009) explains the steps an expert mathematician follows when solving problems: (1) understand the problem, (2) devise a plan, (3) carry out the plan, and (4) look back. Popular physics texts also have similar approaches. For example, Serway and Jewett’s (2008) approach is as follows: (1) conceptualize, (2) categorize, (3) analyze, and (4) finalize; Young and Freedman’s (2008) approach is to (1) identify, (2) set up, (3) execute, and (4) evaluate; and Tipler and Mosca’s (2007) approach is to (1) picture, (2) solve, and (3) check. Each of these approaches involves segmenting and sequencing the method to enable smaller, tractable segments to be performed. From a cognitive perspective, practicing a consistent method assists in organization of knowledge, which is a necessary component in the context of the resources model of student thinking. There is a danger that Polya’s techniques could be misapplied as linear models of problem solving adhering to a series of rigid steps, which is inconsistent with genuine problem solving (Lederman, 2009). Lederman prescribes requiring multiple approaches to the problem solution, to bridge the gap between conceptual physics and problem-solving methods. Many physics problems can be solved by both novices and experts by simply applying a set of “tricks” that are acquired as a result of experience and analogies to other problems. Polya’s stages, and our problem-solving template, allow students to learn these shortcuts on their own, so eventually they will rely on these instincts rather than applying a rigid linear method to every problem. This will serve students well when they are faced with new problems that are similar to those already confronted. Training

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by use of defined stages leading to recognizable solutions fosters the kind of associative thinking processes characteristic of expert problem solvers. The problem-solving template also serves as an aid for grading, because the student responses are organized in a manner that is easily evaluated. Measures of conceptual, as well as problem-solving, abilities are present in the template. In addition, different sections of the template can be tied to general goals for learner-centered assessment. An accepted method of measuring conceptual understanding involves giving conceptual tests. Some educators attempt to separate the conceptual from the problem-solving parts of physics instruction. There are potential problems with this idea, pertaining to how students’ cognitive resources are activated in terms of patterns of associations. Engineering students in particular tend to categorize conceptual and quantitative problems differently, and this may lead to inconsistencies (Sabella & Redish, 2007). A recent study of knowledge organization and activation in students concluded not only that students react differently to conceptual versus quantitative questions but also that the order in which the questions are given can lead to “entrapment.” This can occur when a particular sequence of knowledge structures is activated that does not contain all of the information needed to solve the problem, and “students may not be able to access the needed knowledge, even if they possess it” (Sabella & Redish, 2007, p. 1028). One purpose of the first part of the template is to facilitate connecting the conceptual part of the problem with the quantitative part. In addition, the final and arguably most important part of the template reinforces the conceptual understanding once more. Although this teaching model may not avoid entrapment altogether, it recognizes the possibility of entrapment and attempts to alleviate it.

Implementation Students are asked to follow guidelines in homework by placing headings on the separate sections of their solutions. These headings form a template, whereby knowledge is organized and the process of finding the solution is followed in a rational and consistent manner. The students are asked to label each part of their solution as “Given,” “Find,” “Relevant Equations,” “Detailed Sketch,” “Symbolic Solution,” “Numeric Solution,” “Dependencies,” “Dimensions,” and “Order of Magnitude.” Each of these areas can be found in the Assessment column of Table 1. The template outlines a path for homework, in-class quizzes, and examination problems. This has the benefit of consistency, as the students are given immediate feedback on their mastery of problem-solving methods throughout the course, not just on the examinations.

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table 1 Outcomes and Assessments in the Template Learning Outcome

Assessment

Course Text Step

Relevant Equations and Set Up

Identify

2.1. Identifying the known and unknown variables in a physics problem. [comprehension]

Given and Find

Identify

2.2. Describing a physical situation with a diagram. [knowledge, analysis]

Sketch

Set Up

2.3. Recognizing the relevant formulae needed to solve a physics problem. [comprehension, knowledge]

Symbolic Solution

Execute

2.4. Formulating the solution of a physics problem. [analysis, synthesis]

Symbolic and Numeric Solution

Execute

2.5. Analyzing the accuracy of a result. [evaluation]

Dimensional Analysis

Evaluate

2.6. Evaluating the plausibility of a result. [evaluation]

Dependencies and Order of Magnitude

Evaluate

1.1. Recognizing physics concepts that involve developing mathematical models of ordinary phenomena, such as weights and measures, moving objects, and forces. [knowledge, evaluation, analysis]

note: In this example, the course text used was Young and Freedman, 2008.

Three problems are assigned for each class session. Paper solutions are handed in upon arrival, and the teaching assistant grades one or two of these problems while a lecture is being delivered for the first half hour. After the grading, additional time is devoted to addressing difficulties encountered in the handed-in work. Finally, there is a quiz about one of the assigned problems— this is again graded in situ to “close the loop.” The template used for homework and quizzes greatly simplifies the in situ grading process. A quiz every day with instant feedback is similar to the method called Justin-Time Teaching (jitt [Rozycki, 1999]). However, jitters use the Web to collect homework and examine the responses before class. Recent data suggest that there is little or no correlation between student grades on Web-based homework and final grades in the course. This may be because student responses are limited to a number or formula, and their process of problem solving is not assessed. The problem-solving template promises to track the process and will help to assess the development of problem-solving skills essential for doing physics. 278




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Learner-Centered Goals As an example of the assessment function, the template serves to measure progress in achieving this set of teaching goals associated with 100-level physics courses, namely: Physics 100-Level Learner-Centered Content 1. Anticipated Learning Outcomes for 100- and 200-Level Courses 1.1. Recognizing physics concepts that involve developing mathematical models of ordinary phenomena, such as weights and measures, moving objects, and forces. [knowledge, evaluation, analysis] 1.2. Knowing the scientific method and the process of critically evaluating scientific information. [knowledge, comprehension, evaluation] 2. Anticipated Learning Outcomes for This Course 2.1. Identifying the known and unknown variables in a physics problem. [comprehension] 2.2. Describing a physical situation with a diagram. [knowledge, analysis] 2.3. Recognizing the relevant formulae needed to solve a physics problem. [comprehension, knowledge] 2.4. Formulating the solution of a physics problem. [analysis, synthesis] 2.5. Analyzing the accuracy of a result. [evaluation] 2.6. Evaluating the plausibility of a result. [evaluation] Next to each of the outcomes in brackets are active verbs that label the cognitive process the outcome is targeted to develop. The completed process requires the development of direct and indirect assessment measures to track progress from an established baseline through the progression of time, where actions are taken to improve assessment results in the goal-oriented plan. The template is an example of a direct assessment measure, where graded work must be frequently and immediately returned to the students as feedback, to facilitate learning. Each of the desired learning outcomes is addressed on the problem-solving template. In the example above, the template was graded by assigning a two-point value to each of the first four categories, 2.1–2.4, and a one-point value to each of the remaining two, 2.5 and 2.6, so that each problem was worth ten points. The template is used for homework, in-class quizzes, and examination problems. This method has the added advantage of consistency, and the students are given immediate feedback on their mastery of problem-solving methods throughout the course, not just on the examinations. Integrating Qualitative and Quantitative Physics Instruction 279



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Deconstructing the Process The template contains different boxes for separate parts of the problem-solving process. In addition, parts of the template were further categorized to emphasize the four steps used by Young and Freedman (2008), because this was the current text being used for the course. To maintain alignment with the traditional view that proper quantitative evaluation of a result is the most important aspect of solving a physics problem, the “execute” step of the four-part process was given additional weight for grading purposes, counting as a full 40 percent of the total. An important part of the process is giving partial credit for the parts of the solution that students learn along the way, even if the correct answer is not achieved on initial attempts. Eventually all the parts will come together, as students find their way to the end of the problem-solving path. Another important stage of the template that is noticeably missing from calculationbased examinations is to “evaluate.” It has been argued that reflecting on the result of a problem solution is the most important ingredient in developing expertise in problem-solving skills. This reflection enables the student to explore tendencies, discover how the simple method of dimensional analysis may often lead to the same result, and think about the reasonableness of a numeric result. As for the primary learning outcomes for 100- and 200-level physics courses, it can be argued that the solution in its entirety met these goals. But with particular emphasis being placed on conceptual physics nowadays, the first listed goal can be covered by the ability of the student to make an accurate, detailed sketch of the physical situation. Indeed, a properly described “setup” of a problem should be rich in conceptual physics. Simply drawing an accurate representation of the various forces, fields, velocities, accelerations, and so on in a physics problem constitutes a qualitative or conceptual solution of the problem. Introducing this part of the problem as an essential ingredient before the symbolic and numeric solution may stimulate the formation of globally coherent knowledge structures, which are a characteristic of expert problem solvers.

Knowledge Organization In a case study of engineering physics students, it was suggested that the order and nature of questions given in a problem that necessitated the activation of more than one type of physics knowledge, force and energy, clearly demonstrated that student knowledge exhibited patterns of local coherence (Sabella &

280




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Redish, 2007). Local coherence was defined as the students’ perception that only strongly related sets of knowledge work together. However, for many problems, a global coherence characterized by the integration of separate concepts such as force and energy is necessary to formulate a complete solution. The lack of a broader coherence was tested through a problem that could be solved by using either dynamic or kinematic methods. Students who chose one method were found to be unwilling to even consider the integration of concepts. More importantly, it was concluded that they tend to form isolated sets of knowledge associated with qualitative and quantitative reasoning. Formulating instructional materials that integrate the qualitative and quantitative aspects of problem solving is necessary to activate the connections linking these two important types of reasoning. The “set up” and “evaluate” sections of the problem-solving template attempt to link qualitative reasoning with the quantitative solution. For set up, a detailed sketch of the physical situation is required. The process of making diagrams and sketches of the physical situation is thought to be an essential ingredient of learning to think like a physicist (Van Heuvelen, 1991). It helps the student to expand on the problem and inhibits the formation of primitive formula-centered problem-solving strategies. In order to make an accurate and descriptive sketch of the physical situation, a basic qualitative understanding is necessary. This qualitative information can then be used as a foundation for the formulation of a more general problem solution. The “look back” or “evaluate” sections of the template also include a strong conceptual component. The students are asked to examine the solution for consistency in three categories: dependencies, dimensions, and order of magnitude. In the dependencies section, the expectation is to examine what will happen if the variables, such as mass, force, and charge, take on extreme values, going to zero or infinity. Of course, the dimensional analysis also requires a conceptual component, because various physical quantities must be dimensionally consistent. Finally, the order of magnitude estimate gauges the reasonableness of the numeric solution.

Doing the Math The effectiveness of this approach may be difficult to gauge because the preponderance of textbook and examination questions usually fall into separate categories of qualitative versus quantitative. Successful integration of these two categories into a single problem-solving exercise requires grading of each component with knowledge of this intent. Only by grading each component with equal vigor can we hope to effect students’ integration of conceptual and




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quantitative knowledge. By recognizing the difficulties students face when dealing with physics problems, progress can be made toward instructing students to build coherent knowledge structures that cue on appropriate physical situations. Most physics students and instructors would agree that the “execute” part of the problem solution is the most difficult to complete. With proper identification and setup of the problem, all that remains is for the student to “do the math.” The template clearly distinguishes between symbolic and numeric solutions to the problem. Such a distinction has been discussed in the literature, with one study (Lapp, 1940) concluding that including both types of solutions in a problem-solving exercise would produce the most significant increase in student achievement. Complete separation of the symbolic and numeric solutions is often difficult, even to the point that many solutions manuals of introductory textbook problems contain a mix, with many suffering from premature numerical substitution in even the simplest of algebraic expressions. Most physicists will agree that the problem-solving style used by solutions manuals constitutes what is commonly referred to as a “one-liner,” where the perceived goal is simply to arrive at the correct numerical result—a practice that we also agree is shortsighted and contrary to the goal of developing expert problem-solving skills. The existence of readily available solutions manuals to textbook problems has prompted some physics professors to abandon the problems given in the textbook and either make up different, more challenging problems or assign from sources other than the textbook. Careful consideration must be given to the acceptable amount of numerical substitution to be allowed in the symbolic solution. Enough symbols must remain in the derived formula to perform a dependency analysis of the symbolic solution. Over the course of the semester when the sample problem-solving template was used, with a total of over one hundred assigned problems, only two derived solutions required some numerical substitution. These problems required numerical substitution as systems of four or five linear equations in circuit analysis. Although a purely symbolic solution to a problem of this type is possible, extracting dependencies for variation of component values was still possible by examining the initial matrix. These findings further emphasize the importance of guiding students through a process of formulating a symbolic solution first, as most introductory physics problems have solutions that can be expressed as simple algebraic forms. Vigilance must be maintained, since many novice problem solvers believe the preconceived notion that the physical (symbolic) solution is separated from the mathematical (numeric) solution. As physicists, the “plug and chug” numerical evaluation of a symbolic expression is

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generally considered to have little value. For example, Lapp (1940) stated that a nonphysicist can use the physical solution to arrive at the numeric solution with no fundamental knowledge of the problem at hand, so the forgone conclusion is that “a computer who knows nothing about physics” can do the same.

“The” Solution Given the arguments above, the least significant part of the problem is the one that results in the greatest reward to those students who can develop alternate strategies for computing the correct number in a problem that is graded solely on the accuracy of a numeric result. However, perfect performance is incompatible with problem solving (Lederman, 2009). That is to say, all errors are seen as mistakes in this paradigm, and the approach used in obtaining the numeric result is either all wrong or all right. It is more important for us to teach skills necessary for self-evaluation of a result than it is for us to arbitrarily assign a zero or one to a single response. This is the purpose of the final part of the template. Developing the disposition to look back is very hard to accomplish with students, despite the fact that looking back may be the most important part of problem solving. In addition, this part of the process promises to have the greatest impact on eliminating common misconceptions about physical situations. This approach explicitly addresses the findings that students do not automatically acquire conceptual understanding in the process of learning to solve problems (Hoellwarth et al., 2005). The problem-solving template assists the instructor in identifying how to meld these two important parts of scientific thinking by grounding each step of the process on learner-centered goals.

References Bonham, S., Beichner, R., & Deardorff, D. (2001). Online homework: Does it make a difference? Physics Teacher, 39, 293–96. Cummings, K., Marx, J., Thornton, R., & Kuhl, D. (1999). Evaluating innovation in studio physics. American Journal of Physics Supplement, 67(S1), S38–S44. Hoellwarth, C., Moelter, M., & Knight, R. (2005). A direct comparison of conceptual learning and problem solving ability in traditional and studio style classrooms. American Journal of Physics, 73(5), 459–62. Hsu, L., Brewe, E., Foster, T., & Harper, K. (2004). Resource letter RPS-1: Research in problem solving. American Journal of Physics, 72(9), 1147–56. Johnson, M. (2001). Facilitating high quality student practice in introductory physics. American Journal of Physics Supplement, 69(S1), S2–S11. Keller, G. (2002). Using problem-based and active learning in an interdisciplinary science course for non-science majors. JGE: The Journal of General Education, 15(4), 272–81.




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Kim, E., & Park, S. (2002). Students do not overcome conceptual difficulties after solving 1000 traditional problems. American Journal of Physics, 70(7), 759–65. Kohl, P., Kuo, V., & Ruskell, T. (2008). Documenting the conversion from traditional to studio physics formats at the Colorado School of Mines: Process and early results. PER Conference Series, 1064, 135–38. Lapp, C. J. (1940). The effectiveness of mathematical versus physical solutions in problem solving in college physics. American Journal of Physics, 8(4), 241–43. Lapp, C. J. (1941). The effectiveness of problem solving in producing achievement in college physics. American Journal of Physics, 9(4), 239–41. Lasry, N., Finklestein, N., & Mazur, E. (2009). Are most people too dumb for physics? Physics Teacher, 47(7), 418–22. Lederman, E. (2009). Journey into problem solving: A gift from Polya. Physics Teacher, 47(2), 94–97. Polya, G. (2009). How to solve it: A new aspect of mathematical method. New York: Ishi Press. Rozycki, W. (1999). Just-in-time teaching. Research and Creative Activity, Indiana University, 22(1), 1–4. Sabella, M., & Redish, E. (2007). Knowledge organization and activation in physics problem solving. American Journal of Physics, 75(11), 1017–29. Serway, R., & Jewett, J., Jr. (2008). Physics for scientists and engineers (7th ed.). Belmont, Calif.: Thompson. Sobel, M. (2009a). Physics for the non-scientist: The middle way. Physics Teacher, 47(6), 346–49. Sobel, M. (2009b). Response to “Are Most People Too Dumb for Physics?” Physics Teacher, 47(2), 422–23. Stage, F., & Kinze, J. (2009). Reform on undergraduate science, technology, engineering, and mathematics: The classroom context. JGE: The Journal of General Education, 58(2), 85–105. Tipler, P., & Mosca, G. (2007). Physics for scientists and engineers (6th ed.). New York: W. H. Freeman and Co. Van Heuvelen, A. (1991). Learning to think like a physicist: A review of research-based instructional strategies. American Journal of Physics, 59(10), 891–97. Young, H., & Freedman, R. (2008). University physics (12th ed.). Upper Saddle River, N.J.: Pearson.

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