Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games

Although much is known about the differences between expert and novice problem solvers, knowledge of those differences typically does not provide enough detail to help instructors understand why some students seem to learn physics while solving problems and others do not. A critical issue is how students access the knowledge they have in the context of solving a particular problem. In this paper, we discuss our observations of students solving physics problems in authentic situations in an algebra-based physics class at the University of Maryland. We find that when these students are working together and interacting effectively, they often use a limited set of locally coherent resources for blocks of time of a few minutes or more. This coherence appears to provide the student with guidance as to what knowledge and procedures to access and what to ignore. Often, this leads to the students failing to apply relevant knowledge they later show they possess. In this paper, we outline a theoretical phenomenology for describing these local coherences and identify six organizational structures that we refer to as epistemic games. The hypothesis that students tend to function within the narrow confines of a fairly limited set of games provides a good description of our observations. We demonstrate how students use these games in two case studies and discuss the implications for instruction.


Introduction
Students learning physics at the college level often have considerable difficulty with problem solving despite the fact that problem solving is an integral part of most physics classes. 1 Instructors may assume that these difficulties arise from a lack of mathematical skills, but little evidence has been presented to determine if this is the case.As part of a project to reform introductory algebra-based physics, 2 we have collected extensive data of students learning physics and solving physics problems in a variety of environments.This data includes some familiar but remarkable student behavior, such as • failing to use their personal knowledge in favor of misinterpretations of authority-based knowledge when reasoning in a formal context, • using incorrect qualitative arguments to rebut a qualitative argument even when they know the correct formal argument.These behaviors are often quite robust, with students dramatically ignoring -appearing not even to hear -explicit suggestions from an instructor speaking to them directly.As a result, these behaviors look like what one might crudely describe as "misconceptions of expectations" about how to solve problems.
In order to make sense of this data, we propose a useful way of analyzing students' problem solving behavior in terms of locally coherent goal-oriented activities that we refer to as epistemic games.These games both guide and limit what knowledge students think is appropriate to apply at a given time.Identifying these games provides a way of parsing students' tacit expectations about how to approach solving physics problems.
We work in the context of a theoretical model that allows us to describe the cognitive processes that students use -correctly and incorrectly -in the context of solving physics problems.We build on and extend ideas developed by diSessa, Sherin, and Minstrell 3,4,5,6,7, and by Collins and Ferguson. 8Our theoretical approach fits into the more general theoretical framework we refer to as the resource model. 9,10,11,12In this broad model of student thinking, knowledge elements combine dynamically in associative structures activated by control structures in response to inputs from each other and from the environment.
Our theoretical framework offers researchers and educators a vocabulary (an ontological classification of cognitive structures) and grammar (a description of the relationship between the cognitive structures) to describe students' problem solving (and in particular, their understanding and use of mathematics) in the context of physics. 13Viewing student activity through the lens created by this framework can help researchers and educators understand how teacher-student interactions can more effectively help students develop their own problem solving skills.
In the next section, we give a brief overview of our theoretical framework.In section three, we describe a cognitive model for local coherence in problem solving in physics: epistemic games.In section four, we describe the setting of the study: the student population, the modified instructional environment, and the methodology used to collect and analyze our data.In section five, we use our theoretical model to analyze two student problem-solving sessions.In the final section, we discuss some instructional implications, and present some conclusions.Much of the work described here is taken from the dissertation of Jonathan Tuminaro and more detail can be found there. 14

The Theoretical Framework
Constructivism -the idea that a student constructs new knowledge based largely on what that student already knows -is the dominant paradigm in modern educational theories. 15The teacher's role in the constructivist paradigm is to create environments that help students undertake this construction accurately and effectively.In order to do this, it helps the teacher to know (i) the content and structure of the students' existing knowledge and (ii) how the students use this knowledge to construct new knowledge.There has been considerable direct observational research on the difficulties students have with various items of physics content; 16,17 but to understand how students organize, access, and use their existing knowledge, we need a finer-grained understanding of how students think and respond.We need to know not just that students construct their new knowledge based on what they know, we need some understanding of how students construct new knowledge.To develop such an understanding, we need to know at least some of the basic elements of fundamental cognitive activities and how they are organized.

Previous Research
Research on students' naïve knowledge and on expert/novice differences in problem solving are two topics that are particularly relevant to the current study.In this subsection, we give a brief review of these two areas of research.

Students' Naïve Knowledge
The fact that students bring prior naïve knowledge into a physics class has been well documented in the research literature. 17,18The level of abstraction at which the students' naïve knowledge is described, however, varies considerably.Some researchers describe student knowledge that does not align well with the scientific knowledge we are trying to teach as "misconceptions," "alternative conceptions," or "naïve theories."These researchers assume that students have internally consistent models of how aspects of the world work. 18,19,20,21Others describe the knowledge of beginning students in physics as fragmented and spontaneous. 4,8Both of these approaches contain elements of the truth.Sometimes student knowledge is fragmented, other times it appears coherent.In particular, expert knowledge is often highly coherent.In order to understand the novice-to-expert transition we must have a model that can bridge these two cognitive states.The resource model allows us to do this.

Expert/Novice Differences in Problem Solving
Researchers have studied problem solving in different contexts: problem solving associated with games such as chess, 22 problem solving in mathematics, 23,24,25 and mathematical problem solving in the context of physics. 26,27,28,29,30,31,32There is agreement that there are substantial differences between experts and novices; experts have more knowledge and organize it better.But most attempts to model the differences at a finer scale have focused on creating computer models that would solve problems effectively.Sometimes these models are algorithmic; 33 sometimes they are based on heuristics extracted from expert informants. 34hile these approaches can produce computer software that can carry out some tasks that human experts do, it is not at all clear that they correctly model how a human being learns and functions.(A good summary of the successes and limitations of this approach is given in d'Andrade. 35) Other ap-proaches are phenomenological, using expert heuristics to design learning environments that improve students' problem-solving skills. 36,37,Neither approach helps us understand how students make the transition from novice to expert.

The Resource Model
If our goal is to teach a human being effectively, it is appropriate to build a theoretical model based on our knowledge of the functioning of that system (a human being) and not some other (a computer). 38In order to describe student behaviors, we want to create a model that is sufficiently coarse-grained that it allows us to describe observed behaviors and sufficiently fine-grained that it gives us insight into the mechanisms responsible for those behaviors.The resource model 10,11,12,13 provides such a structure.It is based on a combination of three kinds of scholarship about the functioning human: neuroscience, cognitive science, and behavioral science.It permits us to begin to create a finergrained understanding of student behavior that can bridge the alternative-and fragmented-conception models and can help us develop a more detailed understanding of the novice-toexpert transition.
Researchers in neuroscience, cognitive science, and behavioral science attempt to model human thought at a variety of grain sizes.Much has been learned in all these areas (though one has to be cautious in applying research results at a fine-grained level from neuroscience or cognitive science to real-world situations), but there is still much to be learned, and there is much that is still uncertain about what we know about human thinking.Nevertheless, we can model reasonably safely when the basic structural elements of neuroscience, the well-documented mechanisms of cognitive science, and the observations of real-world (ecological) 39 behavior of real human beings acting in authentic situations all agree and support each other.
Note that in building the resource model we are not attempting to create a fundamental theory of human behavior.Rather, we are developing a theoretical framework or superstructure 11 within which plausible phenomenological models can be created that can help us understand what we see in our classrooms and that are also consistent with what is known about the fundamental mechanisms and operation of the brain.

The Basis of Cognition
A consistent model of cognition that is beginning to emerge from neuroscience and cognitive science is synthesized and documented in many books. 40,41,42,43In this model, cognitive elements of knowledge and memory are represented by networks of connected neurons.When someone recalls or uses the knowledge represented by a particular network, the neurons of the network are activated (increase their firing rate). 44Particular knowledge elements tend to be multi-modal (i.e., to involve activation and interpretation of multiple sensory and interpretive structures) and involve neurons in many parts of the brain. 45Cognitive networks arise from the building of associations among neurons Epistemic Games through synapse growth. 46The association of neurons can vary in strength and increases with repeated associational activations. 47 ,48eural connections can be excitatory or inhibitory. 47This creates the possibility of executive processes that result in the selective activation of some networks and the suppression of others. 49Modern fMRI studies and neurophysiological studies with patients who have brain lesions suggest that the prefrontal cortex is a primary site of a large number of control structures (though they are expected to occur in other parts of the brain as well). 50,51,52he critical elements of this model are the basic elements of knowledge stored in long-term memory, the way those elements are linked, and the way those linked structures are activated in different circumstances.We use the term knowledge element to describe any knowledge stored in long-term memory that activates as a single unit or chunk.We include both declarative and procedural knowledge.We refer to the linking patterns of association as knowledge structures and to the executive function that determines when those structures are activated as control structures.We broadly refer to all the elements of this model as resources.
This model only provides a framework.In order to develop a practical phenomenology, we need to identify various robust patterns of association of knowledge elementsi.e., specific knowledge structures, and demonstrate the value of recognizing these structures.

Basic Knowledge Elements: Compilation
A network corresponding to an element of knowledge becomes robust through practice and experience.For example, one can quickly and easily identify the combination of sensations associated with holding a cup of hot coffee.We effortlessly combine the perception of the pixels (activation of rods and cones) on our retina with the touch, smell, and taste of the coffee into a perception of what appears to be a single object.Neuroscientists call this binding, but we prefer to describe it as compilation. 53Compilation of knowledge elements in associated knowledge structures results in new knowledge elements.They are seen as irreducible by the individual and can be used as a single chunk in working memory. 54The instructional implications of compilation are discussed elsewhere. 55ote that a knowledge element may have a structure and that for some purposes it might be useful to decompose it into finer-grained knowledge elements even when the user sees it as irreducible. 56This is like considering molecules consisting of atoms.For some tightly bound molecules in some situations (e.g., molecules in a gas in kinetic theory) it suffices to consider the molecule as a single functioning unit without substructure.In other circumstances (e.g., situations in which chemical reactions occur or for more weakly bound molecules), it is essential to keep the molecule's structure in terms of atoms in mind.

Patterns of Association: Knowledge Structures
Because cognitive networks are extended and because neurons have large numbers of synapses with other neurons, an individual neuron may be a part of multiple mutually linked knowledge structures.As a result, activation of one network may result in the associated activation of other networks.Patterns of association develop, linking different resources in different situations.Learning occurs as the result of the growth of new synapses that result in changing the topology of existing networks. 57The patterns of association individuals develop may help or hinder them in solving physics problems. 13n this paper, we propose that a useful way to analyze some of the common associational structures in student approaches to physics problem solving is to describe them in terms of locally coherent, goal-oriented activities.We choose to call these epistemic games because of their similarity to the structures proposed by Collins and Ferguson. 9Note that the activities that Collins and Ferguson describe are normative -activities carried out by experts to solve problems.We extend their idea to one that is ethnographic -descriptive of observed student behavior.(Note that some other researchers have also extended their use of the term in this way. 58,59)

Some Specific Resources: Reasoning Primitives and Intuitive Mathematics
A variety of specific relevant resources are available to most students studying physics.These include both knowledge about the physical world and knowledge about mathematics, both intuitive and formal.
Students use a form of intuitive knowledge about physical phenomena and processes that they have learned in their everyday life experiences to make sense of the physical world. 60iSessa 4 proposes that students develop an intuitive sense of physical mechanism from everyday experience.This intuitive sense of physical mechanism arises from the activation and interaction of multiple cognitive resources that diSessa refers to as phenomenological primitives (p-prims).
The name, phenomenological primitives, is used to convey several key aspects of these cognitive structures.The word "phenomenological" reflects the idea that these resources are abstracted from everyday phenomena.For example, closer is stronger could be abstracted from the phenomenon that the closer one is to a fire, the warmer it feels.The word "primitive" reflects the idea that these resources are "irreducible and undetectable" to the user -they are often used as if they were self-explanatory.For example, asked why it is warmer closer to a fire, a student using closer is stronger may respond, "it just is." 61 Because of his focus on the irreducibility of p-prims with respect to the user, diSessa identifies p-prims at differing levels of abstraction: for example, force as mover and abstract balancing.Force as mover involves the very specific concept of an object moving under the influence of a force; whereas abstract balancing involves the general notion that two unspecified influences can be in a state of equilibrium.Because of the specific nature of p-prims like force as Epistemic Games mover, diSessa proposes that there are thousands of p-prims corresponding to the myriad of physical experiences one may have in this complex world.
To reduce the extremely large number of p-prims and to group cognitive structures at their different levels of abstraction, we follow Redish 11 and abstract from p-prims the notion of intuitive pieces of knowledge called reasoning primitives.Reasoning primitives are abstractions of everyday experiences that involve generalizations of classes of objects and influences.In this view a p-prim like force as mover results from mapping an abstract reasoning primitive like agent causes effect into a specific situation that involves forces and motion.The specific agent, in this case, is a force and the effect it causes is movement.When a reasoning primitive is mapped into a specific situation, we refer to it as a facet of that reasoning primitive. 8Agent causes effect could also be mapped into force as spinner, another p-prim identified by diSessa. 4This shows how the notion of reasoning primitives reduces the total number of resources necessary to describe students' previous knowledge about physical phenomena (compared to p-prims).In addition, agent causes effect and abstract balancing both reflect relationships between abstract influences and, therefore, exist at the same level of abstraction.
Another reason to consider the reasoning primitives underlying facets is to understand process components that may be addressable by instruction.If a student is using an appropriate reasoning primitive but has mapped it inappropriately, it may be simple to help the student change the mapping.This more fine-grained theoretical model activates different instructional responses than if one considers a particular pprim to be an irreducible and robust "alternate conception." Students can also activate a variety of resources from their intuitive mathematics knowledge including: intuitive sense of number, 62 counting, ordering, a variety of grounding metaphors, 63 symbolic forms, and interpretive devices. 7ince these mathematical resources do not play a critical role in the examples used here, however, we save their discussion for another publication.(See ref. 15.)

Epistemic Games
Students have a wealth of previous knowledge and ideas that they bring to bear when solving physics problems.To understand and talk about what students are doing, we need a description of the way they organize their resources.
To determine some of these organizational structures, we analyzed 11 hours of video data drawn from about 60 hours of videotapes of groups of students solving homework problems in a reformed algebra-based physics class.The context of the reforms and the methodology of the data collection and analysis are described in section 4. In this section, we describe six locally-coherent organizational control structures that we saw students using in these tapes.Examples of these games drawn from two case studies taken from the video data are given in section 5.
One of the most interesting characteristics of the student behaviors we observed was their local coherence.Over a period of a few minutes to half an hour, we saw students reasoning using a limited set of associated resources.
We can best describe these behaviors by adapting the idea of epistemic game (or, e-game, for short) introduced by Collins and Ferguson. 9Collins and Ferguson define an epistemic game as a complex "set of rules and strategies that guide inquiry."They introduce the idea of epistemic games to describe expert scientific inquiry across disciplines.Students in introductory physics courses are far from experts, so using scientists' approaches to inquiry as a norm by which to describe students' inquiry would not be appropriate.For this reason, we generalize the idea of epistemic games to be descriptive rather than normative.We define an epistemic game to be a coherent activity that uses particular kinds of knowledge and processes associated with that knowledge to create knowledge or solve a problem. 11he activities are 'epistemic' in the sense that students engage in these activities as a means of constructing new knowledge.We use the word 'game' in a very real sense; a particular game (like checkers or chess) is a coherent activity that has ontology components that identify the "things" of the game (players, pieces, and a playing board) and a structure (a beginning and an end, moves, rules) that makes it distinguishable from other activities.Similarly, an e-game has ontological components (concepts, principles, equations) and a structure (starting and ending states, allowed moves, rules).The simplest epistemic game identified by Collins and Ferguson is a familiar one: list making.Every list is implicitly an answer to a question -it builds knowledge to satisfy some goal.Some examples are: "What do I need from the grocery store?"; "What are the fundamental forces of nature?"; and, "What are the constituents of all matter?"Note that the idea of a "game" here -a locally coherent set of behavioral rules for achieving a particular goal -is very general.Some of the behavioral science literature (especially in the opposite extremes of popularizations 64 and mathematical economics 65 ) has used the term game in this way.We focus here on epistemic games -games engaged in for the purpose of creating knowledge.

Ontology of Epistemic Games
Epistemic games have two ontological components: a knowledge base and an epistemic form.An e-game is not simply a structure of a set of associated knowledge; it is an activation of a pattern of activities that can be associated with a collection of resources.The collection of resources that an individual draws on while playing a particular e-game constitutes the knowledge base.For example, to answer a question like, "What are the fundamental forces of nature?" one needs to have some requisite knowledge to list the forces.
The epistemic form is a target structure, often an external representation that helps guide the inquiry during an epistemic game.For example, the epistemic form in the list Epistemic Games making game is the list itself.The list is an external representation that cues particular resources and guides the progression of the inquiry.In some of the games we describe below, the epistemic form could be a written out series of steps, or the derivation of an equation or result.

Structure of Epistemic Games
The structural components of epistemic games include the entry and ending conditions of the game and the moves.
The entry and ending conditions specify the beginning and the ending of the game.As we mentioned above, one may enter into the list making game as a means to answer a question.When solving physics problems, students' expectations about physics problems determine the entry and ending conditions.These expectations can depend on real-time categorizations of physics problems and/or on preconceived notions about the nature of problem solving in physics.Research by Hinsley and Hayes 66 indicates that students can quickly categorize large classes of physics problems very shortly after reading the statement of the problem.In fact, these categorizations can be made after reading the first sentence.(Often these categorizations can be made after reading the first sentence.)The students' ability to very quickly categorize physics problems may stem from their experience with and expectations about physics problem solving.These expectations and categorizations of physics problems affect which epistemic game the students (perhaps tacitly) choose to play.In addition, students' preconceived epistemological stances about problem solving in physics can affect their expectations.If students believe that problem solving in physics involves rote memorization of physics equations, that can affect the strategy they employ (i.e. which e-game they choose to play) and what they believe an answer in physics is (i.e.how they know they are done playing a particular game). 67

Knowledge Base
Cognitive resources associated with the game.

Entry and ending conditions
Conditions for when to begin and end playing a particular game.

Epistemic Form
Target structure that guides inquiry.

Moves
Activities that occur during the course of an e-game.

Table 1: The ontological and structural components of epistemic games
The moves in an e-game are the steps/procedures that occur in the game.In the list-making game the moves may be to add a new item, combine two (or more) items, substitute an item, split an item, and remove an item.As we will see, a critical element of an epistemic game is that playing the game specifies a certain set of moves.What is particularly important about this is not just the moves that are included in the game, but also the moves that are excluded.
Table 1 summarizes the ontological and structural components of epistemic games.

Epistemic Games Students Play in Introductory, Algebra-Based Physics
In this section we discuss some of the epistemic games that account for the different problem-solving strategies seen in our data.We identify six epistemic games that include most of the different problem-solving behaviors we have seen.(See Table 2 .)We do not claim that this list spans all possible problem-solving approaches that could be employed during problem solving in physics and we do not claim to have identified all possible moves within each game.If we had examined a different population of students or a different domain, it is possible that the list of epistemic games would be different, though we expect some of the games identified here to have broad applicability.We present these as examples of the type of structure we are proposing.In the next section, we present two case studies showing how analyzing student behavior in terms of these games helps make sense of what they do and do not do in the context of solving a specific problem.
Each of these games is described in more detail below.For each epistemic game we give a brief introduction and discuss its ontology and structure.Note that some of the games have common moves and one game may look like a subset of another.We identify them as distinct games because they have different ending conditions; students playing different games decide they are "done" when different conditions are met.Section 5 gives an example of students playing each of these games.

List of epistemic games Mapping Meaning to Mathematics Mapping Mathematics to Meaning Physical Mechanism Game Pictorial Analysis Recursive Plug-and-Chug Transliteration to Mathematics Table 2. List of epistemic games identified in our data set
Mapping Meaning to Mathematics: The most intellectually complex epistemic game that we identify is Mapping Meaning to Mathematics.In this game, students begin from a conceptual understanding of the physical situation described in the problem statement, and then progress to a quantitative solution.We identify five basic moves (see Figure (1)): (1)  develop a story about the physical situation, (2) translate quantities in the physical story to mathematical entities, (3) relate the mathematical entities in accordance with the physi-cal story, (4) manipulate symbols, and ( 5) evaluate and interpret the story.
The knowledge base for this game (as with all the games we identify) comes from the set of physics and mathematics resources.In general, however, different resources can be activated during the different moves of the game.During the development of the conceptual story (move 1), reasoning primitives are most often activated.That is, students often rely on their own conceptual understanding to generate this story -not on fundamental physics principles.Translating the conceptual story into mathematical entities (move 2) is difficult for most of the students in our population.Intuitive mathematics knowledge, symbolic forms, and interpretive devices may be activated during this move.Relating the mathematical entities to the physical story (move 3), again is difficult for students in our population, and depends on intuitive mathematics knowledge, symbolic forms, and interpretive devices.Once the physics equations are written, the symbolic manipulations (move 4) often are carried out without a hitch.This is probably because most of our students have had ample practice manipulating symbols.The evaluation of the story (move 5) can occur in many different ways.For example, students may check the solution with a worked example (or solution in the back of the book), students may check their quantitative answer with their conceptual story, or students may check their solution against an iconic example.Note that these evaluations do not necessarily correspond to an expert evaluation or what a teacher would want to see.A superficial similarity to previously seen results may suffice for the student to decide that the end condition of the game has been met.The epistemic form for Mapping Meaning to Mathematics is typically the collection of mathematical expressions that the students generate during moves (2) and ( 3).These expressions lead the direction of the inquiry.Note, however, that the epistemic form is not the entire story in this game.The interpretation (story) that goes with the series of mathe-matical expressions generated, may or may not be explicitly expressed, depending on the instructions for giving a written output and the students' sense of how much "explanation" they are required to provide.
Mapping Mathematics to Meaning: The second most intellectually complex epistemic game that we identify is Mapping Mathematics to Meaning.In this game, students develop a conceptual story corresponding to a particular physics equation.The ontological components of Mapping Mathematics to Meaning are the same as those in Mapping Meaning to Mathematics.In particular, both games involve the same kind of knowledge base (mathematical resources) and the same epistemic form (physics equations).However, the particular resources and physics equations that are used in each game can vary from problem to problem.
In addition, the structural components of the two games are different.In Mapping Meaning to Mathematics, students begin with a conceptual story and then translate it into mathematical expressions.In contrast, in Mapping Mathematics to Meaning students begin with a physics equation and then develop a conceptual story. 68The structural differences between these two games make them distinguishable from each other.We identify four moves in Mapping Mathematics to Meaning (see figure (2)): (1) identify target concepts, (2) find an equation relating the target concepts to other concepts, (3) tell a story using this relationship between concepts, and (4) evaluate story.
Physical Mechanism Game.In the Physical Mechanism Game students attempt to construct a physically coherent and descriptive story based on their intuitive sense of physical mechanism.The knowledge base for this game consists of reasoning primitives.In this game students do not make explicit reference to physics principles or equations.

Tuminaro and Redish 6 Epistemic Games
The ontology of the Physical Mechanism Game is different than in Mapping Meaning to Mathematics and Mapping Mathematics to Meaning.The epistemic form in the latter two games explicitly involves physics equations.In contrast, the epistemic form in the Physical Mechanism Game does not.Although the epistemic form is necessarily different, the same set of resources (intuitive mathematics knowledge, reasoning primitives, symbolic forms, and interpretive devices) may be active in this game as in the previous games.Pictorial Analysis Game.In the Pictorial Analysis Game, students generate an external spatial representation that specifies the relationship between influences in a problem statement.For example, students who make a schematic drawing of a physical situation, a free-body diagram, or a circuit diagram are all playing the Pictorial Analysis Game.In this game, as with all the games previously discussed, the knowledge base consists of all the resources listed above plus some representational translation resources that we do not discuss here.The epistemic form in this game is the distinguishing characteristic.The epistemic form is a schematic or diagram that the students generate.For example, if the students draw a circuit diagram during their inquiry, then that diagram serves as an epistemic form that guides their inquiry.In the same way, a schematic drawing and/or freebody diagram could each serve as a target structure that guides inquiry.
The moves in this game are largely determined by the particular external representation that the students choose.For example, if the students choose to draw a free-body diagram, then one move is to determine the forces that act upon the object in question; whereas, if the students choose to draw a circuit diagram, then one move is to identify the elements (e.g.resistors, capacitors, batteries, etc.).Despite differences that may arise based on the particular external representation chosen, there are three moves that are common to all instantiations of the Pictorial Analysis Game (see figure (4)): (1) determine the target concept, (2) choose an external representation, (3) tell a conceptual story about the physical situation based on the spatial relation between the objects, and (4) fill in the slots in this representation.An example of students who choose to draw a free-body diagram while playing the Pictorial Analysis Game is given in our first case study in section 5. 69 Recursive Plug-and-Chug.In the Recursive Plug-and-Chug Game students plug quantities into physics equations and churn out numeric answers, without conceptually understanding the physical implications of their calculations.
Students do not generally draw on their intuitive knowledge base while playing this game.Instead, they simply identify quantities and plug them into an equation.Consequently, students playing this game rely only on their syntactic understanding of physics symbols, without attempting to understand these symbols conceptually.That is, other cognitive resources (such as intuitive mathematics knowledge, reasoning primitives, symbolic forms, and interpretive devices) are usually not active during this game.
The epistemic form in Recursive Plug-and-Chug is similar or even identical to that in Mapping Meaning to Mathematics and Mapping Mathematics to Meaning.Each game has physics equations as part of the epistemic form, but the resources that are active (i.e.knowledge base) are different.The rules and strategies that are employed in Recursive Plugand-Chug differ from those in Mapping Meaning to Mathematics and Mapping Mathematics to Meaning -even though the epistemic form may be the same.A distinguishing feature of Recursive Plug-and-Chug is the resources that are not activated during this game.
In Recursive Plug-and-Chug, the students first identify a target quantity.This is similar to the first move in Mapping Mathematics to Meaning, but it differs in that here the students only identify the quantity and its corresponding symbol -they do not attempt to understand conceptually what the quantity represents physically as in Mapping Mathematics to Tuminaro and Redish 7 Epistemic Games Meaning.Second, the students identify an equation that relates the target quantity to other quantities, but they do not attempt to create a story that justifies the use of that equation.Third, the students identify which quantities are known and which quantities are unknown.If the target quantity is the only unknown, then they can proceed to calculate the answer.However, if there are additional unknowns, then they must choose a sub-goal and start this process over.Herein lies the 'recursive' nature inherent in this game.Figure (5)  shows a schematic depiction of the moves in this game.The moves in this game are as follows: (1) identify a target quantity, (2) find a solution pattern that relates to the current problem situation, (3) map quantities in the current problem situation into that solution pattern, and (4) evaluate the mapping (see figure (6)).Many students find moves (2) and (3) very tricky.Many times students may find a solution pattern that they think relates to the current problem, when in fact it does not.

The Setting of the Study and Methodology
This study was done as a part of a project carried out at the University of Maryland 3 to determine whether an introductory physics course could serve as a venue to help biology students learn to see science as a coherent process and way of thinking, rather than as a collection of independent facts; and whether this could be achieved within the context of a traditional large-lecture class without a substantial increase in instructional resources.The project adopted reforms that were well-documented to produce conceptual gains and adapted them to create a coherent package that produced epistemological and metacognitive gains.We were able to accomplish this without sacrificing the conceptual gains associated with these reforms. 73ata on the student responses to the modified environment were collected in a variety of ways in order to provide triangulation on the learning process of individual students and evaluations of the overall class results.The learning environments were constructed to encourage students to learn in group discussions taking place both in and out of the classroom.Hundreds of hours of these group discussions were recorded on video and provide the data for this project.In addition, all student homework, quizzes, and exams were scanned before grading.Finally, we gave pre-post conceptual (FCI 74 and FMCE 75 ) and epistemological attitude surveys (modified MPEX 76 ).

Student Population
The students in this study were enrolled in an introductory, algebra-based physics course.They were approximately 60% female; more than 70% were juniors and seniors, about 50% were biological science majors, and about 40% were pre-meds.(There were some year-to-year fluctuations in these numbers.)A particularly interesting statistic for this study is that more than 95% of the students had successfully completed two semesters of calculus, yet they chose to enroll in an algebra-based introductory physics course despite the availability of a calculus-based alternative.Data were collected in 10 semester-long classes over a four-year period from a total of more than 1000 students.

Structure of the modified course
The course had four major structural components.The homework, the lecture, the discussion, and the laboratory were all modified to be non-traditional in some fashion.In addition, we attempted to make all parts of the course coher-

Tuminaro and Redish
8 Epistemic Games ent with each other.We believe that the overall epistemological orientation of the class was responsible, at least in part, for the students' willingness to spend long periods working together on individual problems and for some of the behaviors we observed, such as discussing the physics qualitatively before starting to write equations.We describe here the details of the reforms that are directly relevant to the data presented.Brief descriptions of the other reforms are given in the auxiliary appendix (available online at http://linktobedetermined/aux_appendix.pdf).Homework problems: Problems were regularly assigned and graded.The problems assigned were not traditional endof-chapter textbook exercises.Instead, they included a mix of challenging activities including representation translation problems, context-based reasoning problems, ranking tasks, estimation problems, and essay questions with epistemological content.(For more on these types of problems see Chapter 4 of Redish's Teaching Physics. 77) The instructor (Redish) expected that each problem would take the students about an hour to complete, and he communicated this expectation to the class.In accordance with his expectation, the instructor only assigned about five problems each week.(The specific problems we discuss here are given in the Appendix.)Because these problems were assigned as homework and graded, our observations of students working on these problems gave us an authentic look at how students actually behave in real-world problemsolving situations -as opposed to watching them solve problems artificially posed to them in an interview environment.
Coherence: An important characteristic of the reformed class was the attempt to make the various parts epistemologically oriented and mutually supportive.The instructor and the teaching assistants frequently cross-referenced among homework, lectures, tutorials, and laboratories.Exam questions drew from and mixed information that the students had worked on in each of the class components.

The Course Center
Since the traditional discussion sections were converted to tutorials, the students did not have time to discuss the problems on the homework set with a teaching assistant (TA) during these periods.To close this gap, a room was set up, called the course center, where students could gather to work on the homework problems together.The data reported on here comes from videotaped sessions of students working on homework problems in the course center.
A TA or instructor was available in the course center approximately twenty hours per week.The TA or instructor was present to offer assistance but not to explicitly solve the problems for the students, as is often the case in many traditional recitation sessions.The relevant features of this room were its architecture, the white boards, and the audio-video set up.

Tuminaro and Redish 9 Epistemic Games
Architecture: Many students expect recitation sessions in which a teaching assistant stands at the front of the room and solves problems, while the students frantically copy down the solutions.The architecture of the course center was altered to modify this expectation by removing the front of the room.All the chairs with desk arms were removed, and they were replaced with stools and five long workbenches.(See figure (7) for a schematic lay out.)This seating arrangement did not direct the attention of the students to any one location in the room -as is the case in all lecture halls in which the seating is arranged to face the 'front,' directing attention to the lecturer.The natural focus of attention of a student seated at one of these worktables is the work area in front of them and the students seated across from them.Whiteboards: As a second alteration to the course center, whiteboards were mounted on the walls and the students were provided with dry erase markers.The reason for this was threefold.First, the location of the whiteboards made them difficult to reach for the TAs but easy for the studentsan architectural feature that encourages the students to go to the whiteboards and discourages the TAs from solving the problem for the student or "lecturing" at the whiteboards.Second, the whiteboards facilitate group problem solving.Research on expert and novice problem solving has shown that external representations are a helpful and sometimes necessary tool in the problem-solving process. 25,32The whiteboards offered the students a medium to share their external representations with each other.Third, the whiteboards helped with our research agenda.The students' shared representations on the whiteboards are visible to the video camera.
Audio-video set-up: The course center was equipped with a digital video camera.Microphones were mounted in the middle of two tables to ensure quality audio reception.The video camera was mounted about seven feet above the floor on the wall of the closet across from the tables that were equipped with microphones.The elevation of the camera had three advantages: • Students and staff members walking by the closet did not block the camera; • Students sitting closer to the camera did not block our view of students who sat closer to the wall; and • We had a clear view of what the students wrote on the whiteboards.
Students were encouraged to work at the two tables that could be recorded.At the beginning of a session, the camera was pointed at an occupied table and that microphone connected to the camera.Most students were willing to work at these tables and we have strong evidence that they quickly forgot they were being recorded.

Methodology
The data for this study comes from about 60 hours of videotaped sessions of groups of students solving homework problems in the course center.Sixty hours of video is too much to be analyzed in detail, so we selected promising episodes from the full data set.We looked for episodes rich in articulated student thinking and reasoning, and ones that contained some discussion of mathematical issues (qualitative or quantitative).These selection criteria reduced the data set to about 11 hours of video that were analyzed in detail.
These 11 hours of video were transcribed and analyzed.The games were determined by a semi-phenomenographic approach. 78We identified video segments that appeared to contain students carrying out coherent and consistent activities (whether correct or not).The authors viewed these segments multiple times and identified plausible goals, moves, and exclusions.Hypotheses for specific games were proposed.During weekly meetings of the research team (the authors plus other members of the University of Maryland Physics Education Research Group), the transcription and coding of the episodes were scrutinized and the descriptions of the proposed e-games refined.Finally, two different coders independently analyzed a sample episode in terms of epistemic games, with an inter-rater reliability of 80%.After discussion, the two codings were in complete agreement.The process resulted in the identification of the 6 games described in section 3.

Two Case Studies
We now present two case studies that demonstrate how an analysis in terms of resources and e-games can help make sense of student problem-solving behavior; in particular, why students often do not use what seems to the instructor to be the appropriate resources in a given context.The full transcripts of these episodes are included in the on-line appendix (http://linktobedetermined/aux_appendix.pdf).

Case 1: Building Knowledge Using E-Games
The episode for this case study involves three female students working on an electrostatics problem, which we refer to as the three-charge problem (Appendix, problem #1). 79his episode occurs in the second week of the second semester of a two-semester introductory, algebra-based physics course.All the students in the group had been in the reformed course the first semester and were familiar with its innovative features.In particular, they were familiar with the interaction style between students and teaching assistants in the course center and with the type of homework problems that were assigned in this course.Most importantly, they were cognizant of the fact that the instructor expected the students to spend about an hour on each homework problem -during which time they were expected to generate solutions to the questions that "made sense to them." An "instructor's" solution to the three-charge problem involves straightforward balancing of forces and the use of Coulomb's Law.The parenthetic comment in the problem states there is "no net electrostatic force" acting on charge q 3 .Symbolically, this can be written as . Manipulating this equation, and defining the positive i ˆdirection to be to the right, yields: (1) Canceling similar terms on both sides of the equation and setting q 2 = Q yields the result: q 1 = -4Q.
There are several inferences and steps involved in generating this solution.However, in spite of the multiple steps involved, most experienced physics teachers solve this problem in less than one minute.Some can "see" the answer in a conversational beat and give the correct answer immediately.
The most interesting aspect about the students' approach is that it takes so long.The students work for nearly 60 minutes before arriving at a solution -two orders of magnitude longer than the typical teacher!Why does it take so long?
The typical teacher has a broader mathematical knowledge base (i.e. a larger collection of compiled mathematical resources) and richer collection of problem-solving strategies (i.e. an assortment of epistemic games for solving problems in physics) than most students.For the typical teacher, the problem statement immediately cues the appropriate epistemic game and tightly compiled resources; whereas, the students' mathematical resources do not exist in compiled form.The difference in the teacher and the students' knowledge structure could account for the difference in the speed of the problem solution and demonstrates the power and effectiveness of cognitive compilation. 56he students do not follow a straightforward approach to solving this problem.However, these students' various problem-solving approaches are readily understood in terms of epistemic games.We identify five different epistemic games that are played during this problem-solving session: Physical Mechanism, Pictorial Analysis, Mapping Mathematics to Meaning, Transliteration to Mathematics, and Mapping Meaning to Mathematics.We divide the discussion into segments corresponding to different e-games and refer to these segments as "strips."The names in the transcripts of the strips are gender indicative pseudonyms.

Playing the Physical Mechanism Game
The students' initial attempt to solve this problem follows a less formal path than the instructor's solution outlined above.Throughout this strip the students draw on intuitive reasoning primitives to explain and support their conclu-Tuminaro and Redish 10 Epistemic Games Epistemic Games sions.The students do not activate any formal mathematics or physics principles to support their claims.The reasoning consists almost entirely of facets.This first strip occurs about 7 minutes into the problem-solving process.
Darlene: I'm thinking that the charge q 1 must have it's...negative Q. Alisa: We thought it would be twice as much, because it can't repel q 2 , because they're fixed.But, it's repelling in such a way that it's keeping q 3 there.Bonnie: Yeah.It has to-Darlene: Wait say that.Alisa: Like-q 2 is-q 2 is pushing this way, or attracting-whichever. There's a certain force between two Q, or q 2 that's attracting.Darlene: q 3 .Alisa: But at the same time you have q 1 repelling q 3 .Darlene initiates the conversation by asserting that the charge on q 1 must be "negative Q."The negative sign in this case standing for her realization that q 1 and q 2 will have opposite effects on q 3 .Alisa elaborates on this point by articulating that q 2 exerts an influence on q 3 , which she identifies as a force, that is either repelling or attracting, and that q 1 exerts the opposite influence on q 3 .The semantic content contained in Alisa's explanation can be summarized in the following facet: 'the attractive effect of q 2 on q 3 cancels the repulsive effect of q 1 on q 3 .'The abstract reasoning primitive underlying this facet is canceling.In this case, canceling is an appropriately mapped primitive, because in fact the two forces acting on q 3 do cancel, which results in there being no net electrostatic force on q 3 .From Alisa's initial cursory comment ("we thought [the charge on q 1 ] would be twice as much [as the charge on q 2 ]") it appears that she has the reasoning primitives more is more and balancing activated.That is, since the two influences acting on q 3 balance, q 1 must have more charge because there is more distance between q 1 and q 3 than there is between q 2 and q 3 .
It cannot be confirmed whether Alisa has more is more and balancing activated, because the direction of the conversation changes.Darlene contends with the other students, because it appears she has activated a different reasoning primitive: blocking.The orientation of the charges cues the reasoning primitive of blocking, because q 2 is between q 1 and q 3 .From the superposition principle we know the effect of q 1 on q 3 does not get blocked by the presence of q 2 , so the activation of blocking is an unnecessary distraction.In contrast to the reasoning primitive of canceling that was activated earlier in this strip, blocking does not get mapped into a productive facet for solving this problem.(This is not to say that blocking is 'wrong'; rather, in this particular instance the activation of blocking does not lead to a productive facet.)

Darlene: How is it repelling when it's got this charge in
Bonnie continues Alisa's line of reasoning by explaining why the value of q 1 has to be twice as big as that of q 2 .Alisa: So, we-we were thinking it was like negative two Q or something like that.Bonnie: Yeah.Cause it has to be like big enough to push away.Darlene: Push away q 3 .Bonnie: Yeah, which we-which I figured out negative two.Darlene: Cause it's twice the distance away than q 2 is?Bonnie: Yeah.Darlene: I agree with that.
It appears that Alisa draws on overcoming when she explains that 'Like if it's bigger, than q 2 it can still [have an effect]' and Bonnie restates this as '[q 1 ] has to be like big enough to push away [q 3 ].'That is, q 1 has to have enough charge to overcome the influence of q 2 .The tacit conclusion from this assertion is that the charge of q 1 must have a larger magnitude than that of q 2 .This is particularly interesting since Alisa later shows (see below) that she understands Coulomb's law and superposition.But in the context of Physical Mechanism she generates an (incorrect) argument in support of her sense that the force from both source charges must be included using reasoning primitives and facets.This is a clear example of the e-game she is playing limiting the responses that she considers appropriate within the context of this particular game.
Bonnie and Darlene quantify this conclusion by using the reasoning primitive more is more and the symbolic form dependence (which has the symbol template = […x…]) to assert that the charge on q 1 has to be twice the magnitude of q 2 .More is more and dependence get mapped into the facet twice the distance is twice the charge.Bonnie's argument stays within the rules of the local e-game.Because physical mechanism does not include moves that access formal knowledge, they do not invoke the formal knowledge that says blocking is irrelevant.We will see later that they (and Alisa in particular) indeed do have the relevant formal knowledge.
The students' problem-solving activities during this entire strip have the ontology of Physical Mechanism.While playing this game the students draw on their intuitive knowledge base rather than their formal knowledge to support their claims.During this strip the students use various reasoning Epistemic Games primitives and do not mention any formal mathematics or physics principles.The epistemic form in the Physical Mechanism Game involves a coherent, physical description that is either verbal or imagistic.These students are actively seeking physical causes for the effects that are described in the problem.
Playing this game helps the students become oriented to this problem, but the solution to this problem necessarily involves physics equations (in particular Coulomb's Law).Since Physical Mechanism does not include mathematical expressions or equations, it cannot ultimately lead them to the correct answer.In the next strip, a comment from the TA helps them reframe the problem, which activates other resources they have, and cues them to play another epistemic game.

Playing the Pictorial Analysis Game
In the last strip we saw the students making sense of the problem by using their intuitive reasoning primitives in the context of Physical Mechanism game.At the end of the strip, the students appear to have difficulty focusing their collective attention.Although they achieve the correct conclusion (which forces are relevant to consider), a question asked by one of them causes the others to begin the discussion anew, losing the ground they had gained through the discussion. 80This happens twice.To assist them, the TA (Tuminaro) offers a suggestion.
Darlene: I think they all have the same charge.Bonnie: You think they all have the same charge?Then they don't repel each other.Darlene: Huh? Bonnie: Then they would all repel each other.Darlene: That's what I think is happening.Bonnie: Yeah, but q 3 is fixed.If it was being repelled-Alisa: No, it's not.q 3 is free to move.Bonnie: I mean, q 3 is not fixed.That's what I meant.Darlene: Right.Bonnie: So, like... Darlene: So, the force of q 2 is pushing away with is only equal to d. Bonnie: Yeah, but then... Darlene: These two aren't moving.Bonnie: Wouldn't this push it somewhat?Alisa: Just because they're not moving doesn't mean they're not exerting forces.Darlene: I know.Alisa: What do you think?TA: Can I make a suggestion?Darlene: Uh-huh.TA: You guys are talking about like a lot of forces and stuff.And, one thing I've suggested in previous semesters, if you write it down and say, what forces do you think are acting here, you can all talk about it.Darlene: Where did the marker go?TA: That's a suggestion-a general suggestion-that I might make.
In the first few lines above, it seems as though the students take a step back.Earlier, they appeared to have established the major aspect of the problem: two influences act on q 3 , which exactly cancel each other.In this strip, the students restate the set up of the problem ("these two are moving") and recite remembered facts ("just because they're not moving doesn't mean they're not exerting forces").While these things are important to keep straight, this discussion does not appear to push the problem-solving process forward.
The suggestion to write on the whiteboards has two effects on the students.First, it nudges them into playing a different epistemic game, pictorial analysis. 81Second, the introduction of this new epistemic game and a new e-form reframes the students' interactions and helps them focus their collective attention and clarifies their communication.
Alisa attempts to make an external representation of this problem on the white board while Bonnie and Darlene offer their assistance: Darlene: You're trying to figure out what q 1 is, right?Bonnie: Oh, yeah.Alisa: Because this [q 3 ] is in equilibrium, there's some force... Darlene: Pulling it that way and some force pulling exequally back on it.Bonnie: Yeah.Alisa: And, they're equal?Bonnie: Yes.Darlene: Same with up and down.Not that that matters, really.Bonnie: We'll just stick with... Darlene: Horizontal.Bonnie: Yeah, one dimension.
In this strip the students are deciding which features mentioned in the problem should be included in their diagrama move within Pictorial Analysis.The structure of this game is similar to Physical Mechanism; however, the ontological components of Physical Mechanism and Pictorial Analysis are different.The epistemic form in Pictorial Analysis involves a coherent, physical description and an external representation; the epistemic form for Physical Mechanism only involves a coherent, physical description.
The external representation generated in the Pictorial Analysis epistemic game activates additional resources in the students, which help them better understand this problem.In particular, the students draw on the interpretive device of physical change to conclude that q 1 and q 2 have to have opposite charges.
Alisa: So, maybe this is pushing... Darlene: That's [q 2 ] repelling and q 1 's attracting?Bonnie: Yeah, it's just that whatever q 2 is, q 1 has to be the opposite.Right?Bonnie claims that the charge on q 1 has to be the opposite of q 2 , but the other students do not initially agree, despite the fact that this claim was agreed upon in the context of a different game in the previous strip.Bonnie's suggestion to verify, or falsify, her claim involves the interpretive strategy of physical change.That is, she considers the affect of an actual physical alteration to the system ("OK, like what if they were both positive?").From this move the students almost immediately conclude that the charges on q 1 and q 2 must be different, or else q 3 would go 'zooming' away, since both forces would push in the same direction.
Switching to Pictorial Analysis turns out to be a very effective strategy for this group of students.By decomposing the forces in space and creating an external representation, they are able to physically justify why q 1 and q 2 have to have opposite charge.This strip also illustrates that the students' problem does not stem from lack of knowledge or skills; rather, the epistemic game the students play in their initial approach (Physical Mechanism) does not help them adequately articulate the physical relationship between the charges.The external representation they collectively generate in Pictorial Analysis cues resources they already possess (physical change), which helps them make progress on this problem (i.e., conclude with confidence that q 1 and q 2 have opposite charges).
Although the students' external representation and conclusion marks progress, they have yet to solve the problem.In fact, they have not even identified the necessary physics principle (Coulomb's Law), which is what happens in the next strip.

Playing the Mapping Mathematics to Meaning Game
So far the students have drawn a diagram representing which forces act and in what direction, and they have concluded that q 1 and q 2 have opposite charges.However, they have not yet solved the problem.In this strip we see Alisa spontaneously reframe the problem-solving process by drawing on a new set of resources: formal mathematics knowledge.
Alisa: Are we going to go with that?[That q 1 is twice as big as q 2 and opposite.] Bonnie Alisa is not only attempting to introduce an equation, she is negotiating a shift in how this problem is being viewedasking the group to play a new epistemic game.All the previous reasoning relied on intuitive reasoning primitives, without any explicit reference to physics principles or equations.Alisa's introduction of Coulomb's Law is the first mention of a physics principle during this entire problemsolving process.In addition, it's the first time anyone explicitly makes reference to an equation ("when you cover a distance you use it in the denominator as the square").Alisa's use of formal physics principles and explicit reference to equations is (tacitly) asking the other students to play Mapping Mathematics to Meaning.and force are identified as the relevant concepts in this problem.Second, she identifies Coulomb's Law, F = kq 1 q 2 /r 2 , as an equation that relates the target concept to other concepts.Third, she develops a story using this relationship between concepts: "When you're covering a distance you're using it in the denominator as the square."Fourth, she evaluates the validity of her story by referencing a previous problem.She acknowledges that her intuitive reasoning had failed her on the previous problem, which justifies for her the need for Coulomb's Law on this problem: "I thought that like if you move this a little bit to the right the decrease for this would make up for the increase for this.But, then we decided it didn't."Alisa's use of Coulomb's Law is significant progress on this problem, but all the other students don't know how to apply this new piece of information.In fact, the introduction of Coulomb's Law cues Darlene to play a new (and counterproductive) epistemic game.

Playing the Transliteration to Mathematics Game
Although it appears the students are making progress on this problem, they take a detour and attempt to use another problem as a prototype for solving this problem.Alisa has suggested that Coulomb's Law is an important concept.It appears that Darlene does not initially know how to apply this new information.She attempts to find a different problem that uses Coulomb's Law in its solution, and then map the solution pattern from the other problem to the Three-Charge Problem.The problem that Darlene identifies as using Coulomb's Law in the solution is the two-charge problem (Appendix, problem #3).Darlene: No, no with the q and four q and all that, you know how there was this question that asked when you move the charges three times further apart than they originally were, what the resulting force is.Alisa: OK.Darlene: And, you said it was-we said it was four-the charge would be like q, or nine, but it would get three times as far apart.Why it's not three I don't understand, but that's all right.So-Alisa: Well, 'cause in the equation you square this-the distance between them.Like if you're multiplying by three... Darlene: Oh! So, I would think this one would be q over four-negative q over four.Cause it's twice as far away, opposite charge.Does that make sense?Alisa: But, then it's a smaller charge than this.Bonnie: Yeah.Alisa: So, I don't understand how it would be pushing three or pulling three whatever it's doing.

Darlene
In the Force-Distance Two-Charge Problem, the students had found that if the force between two charges for a given distance is F, tripling the distance results in a force between the two charges that is decreased by a factor of nine (see Appendix, problem #3), in compliance with Coulomb's Law.Darlene is attempting to match the quantities in the Three Charge Problem with quantities from the Force-Distance Charge Problem, so the solution pattern can be transferred; i.e. she is playing the Transliteration to Mathematics epistemic game.
One piece of evidence that Darlene is playing Transliteration to Mathematics comes when she says, "Why it's not three I don't understand, but that's all right."Darlene is explicitly indicating that she does not understand the previous problem, but conceptual understanding is not a move in the Transliteration to Mathematics epistemic game.All that is important is that the problems have enough similar features that the solution from one problem can be transferred to the other.
Darlene's metacognitive statement ("Why it's not three I don't understand, but that's all right.")stands in stark contrast to Alisa's meta-cognitive statement ("I thought that like if you move this a little bit to the right the decrease for this would make up for the increase for this.").Darlene simply admits she does not understand and slavishly transfers the solution pattern from the previous problem anyway.In contrast, Alisa's metacognitive statement leads to her justification for using Coulomb's Law.
Darlene's Transliteration to Mathematics approach doesn't help her with the Three-Charge Problem.She says, "If you increase the distance that they are from each other it's decreasing by the same amount."The problem with Darlene's approach is that she is unaware of the two meanings that she attributes to the pronoun 'it.'In the previous problem the pronoun stands for 'force,' so that the statement would read, "If you increase the distance that they are from each other, then the force is decreasing by the same amount."However, Darlene tacitly maps this into the statement, "If you increase the distance that they are from each other, then the charge is decreasing by the same amount."The Transliteration to Mathematics game is not helpful in this case because force and charge are not related to distance in the same way in Coulomb's Law.The charge has to be found from the balance of two forces.This is not to say that the Transliteration to Mathematics game is wrong.Rather, it does not work in this situation because of Darlene's inappropriate mapping of force and charge.The components (ontology and structure) of Transliteration to Mathematics can also be found as a part of the richer Mapping Mathematics to Meaning game -just as is true for Recursive Plug-and-Chug, but the goals (perceived endstates) of the games differ.

Playing the Mapping Meaning to Mathematics Game
In this strip the students finally come to the solution of the problem.Alisa summarizes her final solution as the other students listen.Alisa's problem-solving activities follow the Mapping Meaning to Mathematics epistemic game (see figure ( 9)).First, she develops a conceptual story describing the physical situation.This conceptual story relies heavily on the reasoning primitives of balancing.
TA: What did you do there?Alisa: What did I do there?TA: Yeah, can I ask?Alisa: All right, so because this isn't moving, the two forces that are acting on it are equal: the push and the pull.Alisa correctly maps 'force' as the two influences that balance in this physical situation.Second, Alisa uses the idea of identity (a symbolic form, 7 with the symbol template = …) to translate her conceptual story into mathematical expressions: Alisa: So, the F-I don't know if this is the right F symbol-but, the F q 2 on q 3 is equal to this (see Equation 2).And, then the F q 1 on q 3 is equal to this (see Equation 3), because the distance is twice as much, so it would be four d squared instead of d squared.(3) Alisa explains why she wrote the charge on q 1 as 'xQ,' by drawing on the reasoning primitive of scaling (another symbolic form, 7 this one with the syntax x ).

Alisa: And, then I used xQ like or you can even do-
yeah-xQ for the charge on q 1 , because we know in some way it's going to be related to Q like the big Q we just got to find the factor that relates to that.
In the third step in the Mapping Meaning to Mathematics, Alisa relates the mathematical entities that she derived in step 2 with her conceptual story that she developed in step 1:

Alisa: Then, I set them equal to each other…
Fourth, she manipulates the mathematical expression to arrive at the desired solution: Alisa: … and I crossed out like the q 2 and the k and the d squared and that gave me Q equals xQ over four.And, then Q equals four Q, so x would have to be equal to four.That's how you know it's four Q.
Fifth, the other students evaluate Alisa's problem-solving approach and conclusion.Bonnie: Well, shouldn't it be-well equal and opposite, but... Alisa: Yeah, you could stick the negative.Bonnie: Yeah.Darlene: I didn't use Coulomb's equation, I just-but it was similar to that.Bonnie: That's a good way of proving it.Darlene: Uh-huh.Bonnie: Good explanation.

Alisa: Can I have my A now?
Darlene and Bonnie accept Alisa's approach is "a good way of proving it."In fact, Alisa must realize that this is a good way to prove this, since she self-evaluates her solution and asks for an "A now."First, this population of students often works on problems within an invisible and often unstated web of constraining expectations we refer to as epistemic games.These expectations limit the resources these students are willing to use at a given time in the context of a given task.This limitation is in itself neither good nor bad.One cannot access all one's knowledge at any given time.One must restrict to a set of tools appropriate to the given task.
Second, a problem does arise when a student's perception of the tools appropriate to the task are not what we intend as instructors.If we only consider the epistemic form -a derivation, say, of an equation or result -and not the intuitive sense-making the instructor expects to go on behind it, we can be misled as to what the students are doing and the students can misinterpret what we are trying to teach.They wind up learning to play the wrong game.
The study reported in this paper focuses on students at the university level and most of them were upper division students who had already taken many science courses at the university.As a result, we make no claims concerning the origin of the games the students were playing.However, we often saw students playing games that were clearly learned somewhere.Students playing recursive plug-and-chug, for example, had a very strong sense of what they thought they were supposed to be doing -both the goals of their local activities and what they ought to be doing to get there.We can well imagine the students being taught to "identify the variable to be found," to "find an equation containing that variable," and so forth.This strongly suggests that we need to be aware that when we are teaching our students processes that can produce effective results in situations with a particular limited class of problems to be solved, we may also unintentionally be teaching them to play particular epistemic games without helping them to develop a good sense of when those games might (or might not) be appropriate.Such instruction might help students get through the vicissitudes of a particular course but might have unintended negative consequences at later stages in the students' education.

Implications for Instruction
Far too often in physics instruction, physics teachers focus on the content and the answers to physics problems rather than on what it is they really want their students to learnhow to think about physics and how to solve problems effectively.When we choose our content as learning Newton's or Kirchoff's Laws, when we give students a syllabus consist-ing of particular chapters in an encyclopedic text, and when we permit them to take a card consisting of all the "necessary equations" into an exam, we are sending an unintended message that what matters in physics are the equations and answers rather than the processes of generating and evaluating them.
Physics teachers generally know that this is not the case and may stress understanding and reasoning in their lectures.But students may not understand what this means.They may ignore the derivations and reasoning the teacher presents, box the final results in their notes for memorization, and ignore the process that generated them.
Research in physics education documents that students bring understandings of the physical world into their physics class that may contradict and confound their attempt to make sense of what is being taught.This can lead them to emphasize memorization -which in some environments works even when you have little understanding of what you are memorizing.
Physics teachers often have the sense that "problem solving is where you learn to actually do physics" -where students should learn the process and sense-making that exemplifies good physical thinking.We assign many problems and in our lectures we model the processes we want our students to follow.But until we have a good understanding of what kinds of thinking the students need to activate for solving problems, it is difficult to understand why some students seem to get so little out of solving large numbers of problems. 83s experts, our knowledge exists in compiled and wellstructured form, whereas the students' knowledge does not.As instructors, we may not be aware of all the knowledge and reasoning that goes into solving a problem, if we can see the solution easily and quickly.Decomposing students' problem-solving sessions in terms of epistemic games and resources allows us to "see" and examine the knowledge and reasoning that is involved in solving a problem in more detail.With increased understanding of the knowledge and reasoning involved in a seemingly simple problem, we can begin to develop teaching environments and interventions that more effectively and efficiently cue the appropriate resources and epistemic games.This in turn could help students become better and more efficient problem solvers.
The study reported on here gives two examples of how cognitive modeling helps increase our understanding of what our students need to learn.The specific resources and games we describe are not the only ones students use or play.Our intention is to introduce a new kind of structure for analyzing students' thoughts on problem solving.In addition, we have studied a particular population in a particular course.Further research is needed to determine whether the games that we have observed to be common among our students occur commonly among other student populations.Although it is true that, in the end, an epistemic game lives in the head of an individual student, and, therefore, each student may possess a unique set of games, we expect that, since many stu-dents experience similar learning environments, a limited set of games will prove useful in describing the behavior of large numbers of students.
Our focus has been on structures in the cognitive model of the individual student, but it is clear that two additional factors play essential roles and also require further research.First, the student's decisions (tacit or conscious) about which games to play have a critical role.Second, the interaction of the students in their group games is extremely important and the structures proposed here could be of considerable help in understanding a group's negotiation of how to approach and solve a problem.We expect that the description of student problem solving behavior in terms of epistemic games will be useful both in understanding how to teach strategies and metacognition in problem solving and in analyzing group behavior in the context of problem solving.

Fig. ( 1
Fig. (1): Schematic diagram of some moves in the epistemic game Mapping Meaning to Mathematics

Fig. ( 2
Fig. (2): Schematic diagram of some moves in the epistemic game Mapping Mathematics to Meaning.

Fig.( 3
Fig.(3): Schematic diagram of some moves in the epistemic game Physical Mechanism The structure of the Physical Mechanism Game is similar to the first move in Mapping Meaning to Mathematics -both involve the development of a conceptual story.However, we can distinguish the two because the Physical Mechanism Game represents a separate, coherent unit of student activities; it has a different endstate.In Mapping Meaning to Mathematics, after move (1) students go on to move (2), then move (3), etc.After (1) creating and (2) evaluating the conceptual story developed in the Physical Mechanism Game (see figure (3)) students decide they are done.The activities that follow this game do not cohere with the conceptual story -in direct contrast with the activities that follow move (1) in Mapping Meaning to Mathematics.Pictorial Analysis Game.In the Pictorial Analysis Game, students generate an external spatial representation that specifies the relationship between influences in a problem statement.For example, students who make a schematic drawing of a physical situation, a free-body diagram, or a circuit diagram are all playing the Pictorial Analysis Game.
Fig. (4): Schematic diagram of some moves in the epistemic game Pictorial Analysis

Fig. ( 5
Fig. (5): Schematic diagram of some moves in the epistemic game Recursive Plug-and-Chug Transliteration to Mathematics.Research on problem solving indicates that students often use worked examples to develop solutions to novel problems. 70,71Transliteration to Mathematics is an epistemic game in which students use worked examples to generate a solution without developing a conceptual understanding of the worked example."Transliterate" means "to represent (letters or words) in the corresponding characters of another alphabet." 72In the Transliteration to Mathematics game students map the quantities from a target problem directly into the solution pattern of an example problem.

Fig. ( 6 )
Fig. (6): Schematic diagram of some moves in the epistemic game Transliteration to Mathematics Because students use the symbolism in this game without conceptual meaning, usually only resources associated with the syntactic structure of equations are active during this game.The solution pattern of the target example serves as

Fig. ( 7 )
Fig. (7): Top-view of the lay-out of the course center.

Fig. ( 8
Fig. (8): Schematic map of Alisa's moves within Mapping Mathematics to MeaningAlisa's discussion follows all the moves within Mapping Mathematics to Meaning (see figure(8)).First, the distance

Case 2 :
Playing Recursive Plug-and-ChugIn our second case study, a group of three students is attempting to solve the Air Pressure problem (Appendix, Problem #4).The episode occurs near the end of the first semester of the class.The problem is a fairly straightforward estimation problem in which the students have to come up with numerical data on their own from their personal experience.There has been such a problem on nearly every homework assignment and on every exam throughout the semester.Nonetheless, some students bring into the class what appearTuminaro and Redish 15 Epistemic GamesWhat We Learn from These Case StudiesThese two studies are reasonably typical of the 11 hours of video that we have studied of students in algebra-based physics class authentically solving physics problems.They demonstrate two points.