«David P. Maloney Indiana University Purdue University Fort Wayne Fort Wayne, IN 46805 Abstract: An overview of the research on solving tasks commonly ...»
An Overview of Physics Education
Research on Problem Solving
David P. Maloney
Indiana University Purdue University Fort Wayne
Fort Wayne, IN 46805
An overview of the research on solving tasks commonly used as
“problems” in introductory physics courses is presented as an introduction
to this domain of physics education research. This overview, which is not
intended to be exhaustive, describes many aspects of the complex topic of
investigating human problem solving to help identify issues of potential interest to researchers and instructors. The article identifies links between research in physics education and more general research on problem solving, and provides useful references. The article ends with a dozen open questions which the author believes deserve answers.
Getting Started in Physics Education Research 1 Maloney Research in Problem Solving
1. Introduction Investigating humans engaged in problem solving is a very diverse and complex endeavor, and narrowing the focus to investigating students engaged in problem solving in introductory physics only reduces the scope a little. Early general problem-solving research established several aspects/facets of problem solving that apply to all domains. However, how these general aspects play out within a particular domain depends on the domain, and problem solving within a specific domain has features that are particular to that domain. Because there are many interactions between the problem-solving processes people use and their domain knowledge, presenting research on problem solving cannot be done in a nice, clean, sequential manner. Consequently, deciding how to structure this introductory overview presented the author with a rather challenging problem.
The result, after much thought and struggle, is the following organization:
(A) General background on research on problem solving; (B) Aspects of how individuals solve physics problems, which consists of three parts— problem-solving steps, problem-solving strategies and problem-solving states of mind (this framework is borrowed from the ThinkFun® Game Club website: www.thinkfungameclub.com.); (C) Aspects of “teaching problem solving”; and (D) Miscellaneous aspects. Finally, we take stock of where we are in understanding problem solving in physics and identify some possible future research questions.
It is important to point out two things here at the beginning. First, even though the focus of this presentation is problem solving in physics, a number of the studies that are described do not involve physics.
Nonetheless these studies, because problem solving is a concern for many fields and shares characteristics in those fields, do tell us useful things that can be applied to problem solving in physics.
The second thing to be aware of is that this presentation is not intended to be in any way exhaustive. If one is interested in pursuing research in this area it would be useful to consult the review by Maloney1 and the resource letter by Hsu, Brewe, Foster, and Harper2 to get a better perspective on what work has been done in physics. In addition, consulting problemsolving reviews derived from other perspectives3-5 is critical for placing research on physics problem solving in the broader context of problemsolving research.
2. General Background
2.1 The Difficulty with Defining “Problem” As stated above, problem solving involves a very diverse and complex set of processes, and one of the difficulties with interpreting the problemsolving literature is comparing and contrasting the activities associated with people working the spectrum of tasks that are called problems. As Adams points out, “The study of problem solving is almost impossible if you try to look at it as one thing that a person does. It has many facets and to study these it’s useful to isolate and identify the individual facets.”6 There are many definitions for problem in the literature and while they share a number of elements, they are not identical. In many studies the authors never provide a definition of what they are calling a problem, rather it is taken for granted that the reader understands and agrees with the researchers that the tasks presented to the subjects were problems for the subjects. This approach is “problematic.” One useful general definition of a problem is given by Hayes: whenever there is a gap between where you are now and where you want to be, and you don’t know how to find a way to cross that gap, you have a problem.7 It is useful to notice that this definition implies that tasks or situations are not in themselves problems. The problem arises when an individual interacts with the task or situation. Different people interacting with the same task/situation might not all find it to be a problem. The skills and knowledge an individual brings to a situation play a major role in whether that individual thinks of a situation as a problem.
While the Hayes description provides a general definition of a problem, it is actually of limited usefulness for comparing studies that have been done. The reason for this is that anything from an ill-defined, complex, multi-step task to a one-step knowledge recall true/false question can be considered a problem under this definition. However, while some of the processes employed by an individual responding to these two tasks might be the same, there will also be large differences in the reasoning used to deal with them, or with conceptual tasks versus computational tasks. This means that identifying clearly and explicitly what tasks/situations are being used to produce the interactions that qualify as problems for the subjects in a study is very important.
Getting Started in Physics Education Research 3 Maloney Research in Problem Solving As an example of the variation in the nature of the reasoning that two different types of problems require, consider the following two physics tasks. In task A the mass of an object and the magnitudes and directions of two forces acting on the object are given, and the goal is to find the acceleration of the object. In task B the mass of an airplane, its landing speed and the magnitude of the net (braking) force it can have are given along with the length of the runway, and a question is posed: Can the plane land safely? These two tasks share a number of characteristics, but a specific difference is how the goal of the task is identified. For the first task the goal is to find a specific numerical value. The simplest way to achieve this goal is to use the equation which involves the given values and the unknown. No real analysis of the physical system and its behavior is needed. In contrast, in the second task the solver has to first determine how to use the given information to find something that will enable him/her to answer the question. In this case someone would need to think about what physical principles/relationships are relevant and which one(s) will lead to being able to answer the question. Other task formats (problem types), such as context rich problems,8 Jeopardy problems,9 or ranking tasks10 require other and/or additional reasoning processes.
The wide range of tasks that qualify as problems for novices, who by definition have little domain specific knowledge for a field of study, is part of what makes investigating problem solving a complex domain of study. Adding to the complexity is the fact that problems can be classified in a variety of ways, such as conceptual or quantitative, or well-defined versus ill-defined. For the latter contrast, there is actually a spectrum along which problems can be placed.11 Using the definition of problem given above, there are three components of a problem: the initial state, the goal state, and procedures to eliminate the gap between them. A very well-defined problem would have all three aspects explicitly identified. For example, end-of-chapter numerical exercises fit this description of well-defined since the initial state (the given values), the final state (the quantity to be found) and the procedures to be used are all roughly specified. The procedures to be used are not exactly specified, but these tasks are often labeled with the section to which they relate and doing that specifies the procedures indirectly.
(Actually one might argue that such tasks do not qualify as problems under the Hayes definition.) Problems that have one, or more, of these three features not explicitly identified then fall further along the spectrum toward ill-defined. The ill-defined end of the spectrum has situations where the problem solver may not even be sure there is a problem; such Getting Started in Physics Education Research 4 Maloney Research in Problem Solving tasks require explicit identification and definition of the problem, i.e., the initial state, goal state, and nature of the gap, before an attempt can be made to solve them. The vast majority of academic problems are welldefined while real-world problems tend to be ill-defined to various extents.
In light of these aspects of the problem-solving domain, anyone studying the literature in this domain should take care to identify what tasks/situations the authors are calling problems and whether such identification is reasonable for the subjects involved in the study.
2.2 Types of Problems
Since the nature of the tasks/situations that can be problems for people is so broad, it is useful, at least at times, to have ways to classify and distinguish among those that share characteristics. Not surprisingly, considering the previously mentioned broad range of tasks that can be problems, there are various ways to classify problems. Perhaps the most obvious way is on the basis of the knowledge domain involved, i.e., physics or chemistry problems. However, there are other ways to classify problems that can be useful if we are interested in gaining insight into students’ solving approaches and abilities.
One general dichotomous classification scheme has insight problems as one category and what might be called non-insight, or systematic, problems as the other category.12 Systematic problems are those that are amenable to solution by persistent application of known procedures. An example would be the popular Sudoku puzzles for which even the most challenging versions can be solved by systematic application of the guess and test heuristic. (A heuristic is a general problem-solving procedure that one typically employs when he/she doesn’t have specific actions available.
Much more will be said about heuristics below.) In contrast, insight problems require, somewhere along the way, a breakthrough shift in thinking—the Aha moment—to solve.13 The identification of these two types of problems immediately raises a number of questions such as: Does experience with systematic problems promote skill at solving insight problems in a domain? Does solving one insight problem facilitate solution of others? We will return to these questions later.
We have already mentioned above the classification of problems along a spectrum from well-defined to ill-defined. Another classification scheme which has some relation to the well-defined/ill-defined spectrum was
developed by Johnstone.14 This scheme for problem types is based on status for the solver of three components: data given, method to be used, and goal. These clearly correspond to the three parts, initial state, procedures for reducing the gap and goal state identified in the Hayes
definition. The eight resulting problem types are:
Looking at this scheme it is pretty clear that the vast majority of tasks that students encounter in academic situations are types 1 to 3, but real world problems are more commonly types 4 to 8. How do we help students develop the skills they need to tackle those types of problems?
Jonassen has developed a general typology of problems that includes 11 groups. The different types are: (1) logic problems, (2) algorithms, (3) story problems, e.g., the typical end-of-chapter word problems found in science, technology, engineering and mathematics (STEM) textbooks, (4) rule-using problems, (5) decision making problems (which usually require that problem solvers select maximal solutions from a set of alternative solutions based on a number of selection criteria), (6) troubleshooting, (7) diagnosis-solution problems, (8) strategic performance, (9) policy analysis, (10) design, and (11) dilemmas (problems which involve social and/or ethical conflicts).4 He contends that his scheme “represents a developmental theory of problem solving” and that “How discrete each kind of problem is, and whether additional kinds of problems exist, is not certain.”11 Clearly not all of these categories are involved in scientific problem solving, but several—e.g., algorithms, story problems and ruleusing problems—just as clearly are part of science.
Jonassen also argues that problems can be encountered as either discrete items or as aggregates. He contends that the latter are more common in real work contexts.11 To what extent experience with one type of problem—troubleshooting—provides useful learning for dealing with a different type—story problems—is very much an open question.
2.3 The Difficulty with “Problem-solving Skills” The difficulties described above about defining problems carries over into discussions about “problem-solving skills”. Teaching “problem-solving skills” is fairly commonly cited as a major goal of physics, or mathematics or chemistry, instruction. However, determining what this means, and whether it can actually be done, is another matter. Even a rather quick exploration of the problem-solving literature brings to light the fact that there is a difficulty with even identifying what qualifies as a problemsolving skill. While many researchers, not to mention teachers, strongly believe that there are general problem-solving skills, those skills are always applied to, and with, knowledge from some specific domain.
Consequently, identifying skills that operate in the same way in all domains is very difficult.