Elements of a cognitive model of physics problem solving: Epistemic games

 
PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 3, 020101 共2007兲

              Elements of a cognitive model of physics problem solving: Epistemic games

                                            Jonathan Tuminaro and Edward F. Redish*
                      Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
                                        共Received 13 January 2007; published 6 July 2007兲
                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.

             DOI: 10.1103/PhysRevSTPER.3.020101                                       PACS number共s兲: 01.40.Fk, 01.40.Ha

                    I. INTRODUCTION                                    eral theoretical framework we refer to as the resource
                                                                       model.9–12 In this broad model of student thinking, knowl-
   Students learning physics at the college level often have           edge elements combine dynamically in associative structures
considerable difficulty with problem solving despite the fact          activated by control structures in response to inputs from
that problem solving is an integral part of most physics               each other and from the environment.
classes.1 Instructors may assume that these difficulties arise            Our theoretical framework offers researchers and educa-
from a lack of mathematical skills, but little evidence has            tors a vocabulary 共an ontological classification of cognitive
been presented to determine if this is the case. As part of a          structures兲 and grammar 共a description of the relationship
project to reform introductory algebra-based physics,2 we              between the cognitive structures兲 to describe students’ prob-
have collected extensive data of students learning physics             lem solving 共and in particular, their understanding and use of
and solving physics problems in a variety of environments.             mathematics兲 in the context of physics.13 Viewing student
These data include some familiar but remarkable student be-            activity through the lens created by this framework can help
havior, such as 共i兲 failing to use their personal knowledge in         researchers and educators understand how teacher-student in-
favor of misinterpretations of authority-based knowledge               teractions can more effectively help students develop their
when reasoning in a formal context; 共ii兲 using incorrect               own problem-solving skills.
qualitative arguments to rebut a qualitative argument even                In the next section, we give a brief overview of our the-
when they know the correct formal argument. These behav-               oretical framework. In Sec. III, we describe a cognitive
iors are often quite robust, with students dramatically                model for local coherence in problem solving in physics:
ignoring—appearing not even to hear—explicit suggestions               epistemic games. In Sec. IV, we describe the setting of the
from an instructor speaking to them directly. As a result,             study: the student population, the modified instructional en-
these behaviors look like what one might crudely describe as           vironment, and the methodology used to collect and analyze
“misconceptions of expectations” about how to solve prob-              our data. In Sec. V, we use our theoretical model to analyze
lems.                                                                  two student problem-solving sessions. In the final section,
   In order to make sense of these data, we propose a useful           we discuss some instructional implications, and present some
way of analyzing students’ problem solving behavior in                 conclusions. Much of the work described here is taken from
terms of locally coherent goal-oriented activities that we re-         the dissertation of Jonathan Tuminaro and more detail can be
fer to as epistemic games. These games both guide and limit            found there.14
what knowledge students think is appropriate to apply at a
given time. Identifying these games provides a way of pars-                          II. THEORETICAL FRAMEWORK
ing students’ tacit expectations about how to approach solv-
ing physics problems.                                                     Constructivism—the idea that a student constructs new
   We work in the context of a theoretical model that allows           knowledge based largely on what that student already
us to describe the cognitive processes that students use—              knows—is the dominant paradigm in modern educational
correctly and incorrectly—in the context of solving physics            theories.15 The teacher’s role in the constructivist paradigm
problems. We build on and extend ideas developed by                    is to create environments that help students undertake this
diSessa, Sherin, and Minstrell3–7 and by Collins and                   construction accurately and effectively. In order to do this, it
Ferguson.8 Our theoretical approach fits into the more gen-            helps the teacher to know 共i兲 the content and structure of the

1554-9178/2007/3共2兲/020101共22兲                                  020101-1                            ©2007 The American Physical Society
JONATHAN TUMINARO AND EDWARD F. REDISH                                       PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

students’ existing knowledge and 共ii兲 how the students use         approaches are phenomenological, using expert heuristics to
this knowledge to construct new knowledge. There has been          design learning environments that improve students’
considerable direct observational research on the difficulties     problem-solving skills.36,37
students have with various items of physics content;16,17 but         None of these approaches, however, help us understand
to understand how students organize, access, and use their         how students make the transition from novice to expert. In
existing knowledge, we need a finer-grained understanding          order to make progress on these issues we have to understand
of how students think and respond. We need to know not just        how novices and experts approach problems, and we have to
that students construct their new knowledge based on what          have effective ways of talking about and describing the dif-
they know, we need some understanding of how students              ferences. In this paper, we identify one concept, the
construct new knowledge. To develop such an understanding,         epistemic game, that we hope will provide some first steps
we need to know at least some of the basic elements of             toward obtaining a more detailed analysis of the novice-
fundamental cognitive activities and how they are organized.       expert transition.

                     A. Previous research                                                B. Resource model
                                                                       If our goal is to teach a human being effectively, it is
   Research on students’ naïve knowledge and on expert and
                                                                   appropriate to build a theoretical model based on our knowl-
novice differences in problem solving are two topics that are
                                                                   edge of the functioning of that system 共a human being兲 and
particularly relevant to the current study. In this subsection,
                                                                   not some other 共a computer兲.38 In order to describe student
we give a brief review of these two areas of research.             behaviors, we want to create a model that is sufficiently
                                                                   coarse-grained that it allows us to describe observed behav-
                 1. Students’ naïve knowledge                      iors and sufficiently fine-grained that it gives us insight into
                                                                   the mechanisms responsible for those behaviors. The re-
    The fact that students bring prior naïve knowledge into a
                                                                   source model9–12 provides such a structure. It is based on a
physics class has been well documented in the research
                                                                   combination of three kinds of scholarship about the function-
literature.16,17 The level of abstraction at which the students’
                                                                   ing human: neuroscience, cognitive science, and behavioral
naïve knowledge is described, however, varies considerably.
                                                                   science. It permits us to begin to create a finer-grained un-
Some researchers describe student knowledge that does not
                                                                   derstanding of student behavior that can bridge the
align well with the scientific knowledge we are trying to
                                                                   alternative- and fragmented-conception models and can help
teach as “misconceptions,” “alternative conceptions,” or
                                                                   us develop a more detailed understanding of the novice-to-
“naïve theories.” These researchers assume that students
                                                                   expert transition.
have internally consistent models of how aspects of the
                                                                       Researchers in neuroscience, cognitive science, and be-
world work.18–21 Others describe the knowledge of begin-
                                                                   havioral science attempt to model human thought at a variety
ning students in physics as fragmented and spontaneous.3,7
                                                                   of grain sizes. Much has been learned in all these areas
Both of these approaches contain elements of the truth.
                                                                   共though one has to be cautious in applying research results at
Sometimes student knowledge is fragmented, other times it
                                                                   a fine-grained level from neuroscience or cognitive science
appears coherent. In particular, expert knowledge is often
                                                                   to real-world situations兲, but there is still much to be learned,
highly coherent. In order to understand the novice-to-expert
                                                                   and there is much that is still uncertain about what we know
transition we must have a model that can bridge these two
                                                                   about human thinking. Nevertheless, we can model reason-
cognitive states. The resource model allows us to do this.
                                                                   ably safely when the basic structural elements of neuro-
                                                                   science, the well-documented mechanisms of cognitive sci-
        2. Expert-novice differences in problem solving            ence, and the observations of real-world 共ecological兲39
                                                                   behavior of real human beings acting in authentic situations
    Researchers have studied problem solving in different
                                                                   all agree and support each other.
contexts: problem solving associated with games such as
                                                                       Note that in building the resource model we are not at-
chess,22 problem solving in mathematics,23–25 and math-
                                                                   tempting to create a fundamental theory of human behavior.
ematical problem solving in the context of physics.26–32
                                                                   Rather, we are developing a theoretical framework or
There is agreement that there are substantial differences be-
                                                                   superstructure10 within which plausible phenomenological
tween experts and novices; experts have more knowledge
                                                                   models can be created that can help us understand what we
and organize it better 共i.e., so that relevant knowledge is
                                                                   see in our classrooms and that are also consistent with what
easily primed for activation兲. But most attempts to model the
                                                                   is known about the fundamental mechanisms and operation
differences at a finer scale have focused on creating com-
                                                                   of the brain.
puter models that would solve problems effectively. Some-
times these models are algorithmic;33 sometimes they are
                                                                                         1. Basis of cognition
based on heuristics extracted from expert informants.34
While these approaches can produce computer software that             A consistent model of cognition that is beginning to
can carry out some tasks that human experts do, it is not at       emerge from neuroscience and cognitive science is synthe-
all clear that they correctly model how a human being learns       sized and documented in many books.40–43 In this model,
and functions. 共A good summary of the successes and limi-          cognitive elements of knowledge and memory are repre-
tations of this approach is given in d’Andrade.35兲 Other           sented by networks of connected neurons. When someone

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ELEMENTS OF A COGNITIVE MODEL OF PHYSICS…                                    PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

recalls or uses the knowledge represented by a particular           in which chemical reactions occur or for more weakly bound
network, the neurons of the network are activated 共increase         molecules兲, it is essential to keep the molecule’s structure in
their firing rate兲.44 Particular knowledge elements tend to be      terms of atoms in mind.
multimodal 共i.e., to involve activation and interpretation of
multiple sensory and interpretive structures兲 and involve                   3. Patterns of association: Knowledge structures
neurons in many parts of the brain.45 Cognitive networks
arise from the building of associations among neurons                  Because cognitive networks are extended and because
through synapse growth.46 The association of neurons can            neurons have large numbers of synapses with other neurons,
vary in strength and increases with repeated associational          an individual neuron may be a part of multiple mutually
activations.47,48                                                   linked knowledge structures. As a result, activation of one
    Neural connections can be excitatory or inhibitory.46 This      network may result in the associated activation of other net-
creates the possibility of executive processes that result in the   works. Patterns of association develop, linking different re-
selective activation of some networks and the suppression of        sources in different situations. Learning occurs as the result
others.49 Modern fMRI studies and neurophysiological stud-          of the growth of new synapses that result in changing the
ies with patients who have brain lesions suggest that the           topology of existing networks.57 The patterns of association
pre-frontal cortex is a primary site of a large number of con-      individuals develop may help or hinder them in solving
trol structures 共though they are expected to occur in other         physics problems.12
parts of the brain as well兲.50–52                                      In this paper, we propose that a useful way to analyze
    The critical elements of this model are the basic elements      some of the common associational structures in student ap-
of knowledge stored in long-term memory, the way those              proaches to physics problem solving is to describe them in
elements are linked, and the way those linked structures are        terms of locally coherent, goal-oriented activities. We choose
activated in different circumstances. We use the term knowl-        to call these epistemic games because of their similarity to
edge element to describe any knowledge stored in long-term          the structures proposed by Collins and Ferguson.8 Note that
memory that activates as a single unit or chunk. We include         the activities that Collins and Ferguson describe are
both declarative and procedural knowledge. We refer to the          normative—activities carried out by experts to solve prob-
linking patterns of association as knowledge structures and to      lems. We extend their idea to one that is ethnographic—
the executive function that determines when those structures        descriptive of observed student behavior. 共Note that some
are activated as control structures. We broadly refer to all the    other researchers have also extended their use of the term in
elements of this model as resources.                                this way.58,59兲
    This model only provides a framework. In order to de-
velop a practical phenomenology, we need to identify vari-           4. Some specific resources: Reasoning primitives and intuitive
ous robust patterns of association of knowledge elements,                                     mathematics
i.e., specific knowledge structures, and demonstrate the value         A variety of specific relevant resources are available to
of recognizing these structures.                                    most students studying physics. These include both knowl-
                                                                    edge about the physical world and knowledge about math-
           2. Basic knowledge elements: Compilation                 ematics, both intuitive and formal.
                                                                       Students use a form of intuitive knowledge about physical
   A network corresponding to an element of knowledge be-           phenomena and processes that they have learned in their ev-
comes robust through practice and experience. For example,          eryday life experiences to make sense of the physical
one can quickly and easily identify the combination of sen-         world.60 DiSessa3 proposes that students develop an intuitive
sations associated with holding a cup of hot coffee. We ef-         sense of physical mechanism from everyday experience. This
fortlessly combine the perception of the pixels 共activation of      intuitive sense of physical mechanism arises from the activa-
rods and cones兲 on our retina with the touch, smell, and taste      tion and interaction of multiple cognitive resources that
of the coffee into a perception of what appears to be a single      diSessa refers to as phenomenological primitives 共p-prims兲.
object. Neuroscientists call this binding, but we prefer to            The name, phenomenological primitives, is used to con-
describe it as compilation.53 Compilation of knowledge ele-         vey several key aspects of these cognitive structures. The
ments in associated knowledge structures results in new             word “phenomenological” reflects the idea that these re-
knowledge elements. They are seen as irreducible by the             sources are abstracted from everyday phenomena. For ex-
individual and can be used as a single chunk in working             ample, closer is stronger could be abstracted from the phe-
memory.54 The instructional implications of compilation are         nomenon that the closer one is to a fire, the warmer it feels.
discussed elsewhere.55                                              The word “primitive” reflects the idea that these resources
   Note that a knowledge element may have a structure and           are “irreducible and undetectable” to the user—they are often
that for some purposes it might be useful to decompose it           used as if they were self-explanatory. For example, asked
into finer-grained knowledge elements even when the user            why it is warmer closer to a fire, a student using closer is
sees it as irreducible.56 This is like considering molecules        stronger may respond, “it just is.”61
consisting of atoms. For some tightly bound molecules in               Because of his focus on the irreducibility of p-prims with
some situations 共e.g., molecules in a gas in kinetic theory兲 it     respect to the user, diSessa identifies p-prims at differing
suffices to consider the molecule as a single functioning unit      levels of abstraction: for example, force as mover and ab-
without substructure. In other circumstances 共e.g., situations      stract balancing. Force as mover involves the very specific

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JONATHAN TUMINARO AND EDWARD F. REDISH                                        PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

concept of an object moving under the influence of a force;        drawn from two case studies taken from the video data are
whereas abstract balancing involves the general notion that        given in Sec. V.
two unspecified influences can be in a state of equilibrium.           One of the most interesting characteristics of the student
Because of the specific nature of p-prims like force as mover,     behaviors we observed was their local coherence. Over a
diSessa proposes that there are thousands of p-prims corre-        period of a few minutes to half an hour, we saw students
sponding to the myriad of physical experiences one may             reasoning using a limited set of associated resources.
have in this complex world.                                            We can best describe these behaviors by adapting the idea
    To reduce the extremely large number of p-prims and to         of epistemic game 共or, e-game, for short兲 introduced by Col-
group cognitive structures at their different levels of abstrac-   lins and Ferguson.8 Collins and Ferguson define an epistemic
tion, we follow Redish10 and abstract from p-prims the no-         game as a complex “set of rules and strategies that guide
tion of intuitive pieces of knowledge called reasoning primi-      inquiry.” They introduce the idea of epistemic games to de-
tives. Reasoning primitives are abstractions of everyday           scribe expert scientific inquiry across disciplines. Students in
experiences that involve generalizations of classes of objects     introductory physics courses are far from experts, so using
and influences. In this view a p-prim like force as mover          scientists’ approaches to inquiry as a norm by which to de-
results from mapping an abstract reasoning primitive like          scribe students’ inquiry would not be appropriate. For this
agent causes effect into a specific situation that involves        reason, we generalize the idea of epistemic games to be de-
forces and motion. The specific agent, in this case, is a force    scriptive rather than normative. We define an epistemic game
and the effect it causes is movement. When a reasoning             as:
primitive is mapped into a specific situation, we refer to it as
                                                                        “a coherent activity that uses particular kinds of
a facet of that reasoning primitive.7 Agent causes effect could
                                                                        knowledge and processes associated with that knowl-
also be mapped into force as spinner, another p-prim identi-
                                                                        edge to create knowledge or solve a problem.”10
fied by diSessa.3 This shows how the notion of reasoning
primitives reduces the total number of resources necessary to          The activities are “epistemic” in the sense that students
describe students’ previous knowledge about physical phe-          engage in these activities as a means of constructing new
nomena 共compared to p-prims兲. In addition, agent causes            knowledge. We use the word “game” in a very real sense; a
effect and abstract balancing both reflect relationships be-       particular game 共like checkers or chess兲 is a coherent activity
tween abstract influences and therefore exist at the same          that has ontology components that identify the “things” of
level of abstraction.                                              the game 共players, pieces, and a playing board兲 and a struc-
    Another reason to consider the reasoning primitives un-        ture 共a beginning and an end, moves, rules兲 that makes it
derlying facets is to understand process components that may       distinguishable from other activities. Similarly, an e-game
be addressable by instruction. If a student is using an appro-     has ontological components 共concepts, principles, equations兲
priate reasoning primitive but has mapped it inappropriately,      and a structure 共starting and ending states, allowed moves,
it may be simple to help the student change the mapping.           rules兲. The simplest epistemic game identified by Collins and
This more fine-grained theoretical model activates different       Ferguson is a familiar one: list making. Every list is implic-
instructional responses than if one considers a particular         itly an answer to a question—it builds knowledge to satisfy
p-prim to be an irreducible and robust “alternate concep-          some goal. Some examples are: “What do I need from the
tion.”                                                             grocery store?”; “What are the fundamental forces of na-
    Students can also activate a variety of resources from         ture?”; and, “What are the constituents of all matter?”
their intuitive mathematics knowledge, including intuitive             Note that the idea of a game here—a locally coherent set
sense of number,62 counting, ordering, a variety of grounding      of behavioral rules for achieving a particular goal—is very
metaphors,63 symbolic forms, and interpretive devices.6            general. Some of the behavioral science literature 共especially
Since these mathematical resources do not play a critical role     in the opposite extremes of popularizations64 and mathemati-
in the examples used here, however, we save their discussion       cal economics65兲 has used the term game in this way. We
for another publication 共see Ref. 14兲.                             focus here on epistemic games—games engaged in for the
                                                                   purpose of creating knowledge. Note that we are not claim-
                                                                   ing that students choose to play these games consciously or
                  III. EPISTEMIC GAMES
                                                                   can articulate what game they are playing. We are describing
   Students have a wealth of previous knowledge and ideas          their behavior, not their knowledge of their own behavior.
that they bring to bear when solving physics problems. To
understand and talk about what students are doing, we need a
                                                                                   A. Ontology of epistemic games
description of the way their resources are organized.
   To determine some of these organizational structures, we           Epistemic games have two ontological components: a
analyzed 11 h of video data drawn from about 60 h of vid-          knowledge base and an epistemic form. An e-game is not
eotapes of groups of students solving homework problems in         simply a structure of a set of associated knowledge; it is an
a reformed algebra-based physics class. The context of the         activation of a pattern of activities that can be associated
reforms and the methodology of the data collection and             with a collection of resources. The collection of resources
analysis are described in Sec. IV. In this section, we describe    that an individual draws on while playing a particular e-game
six locally coherent organizational control structures that we     constitutes the knowledge base. For example, to answer a
saw students using in these tapes. Examples of these games         question like, “What are the fundamental forces of nature?”

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ELEMENTS OF A COGNITIVE MODEL OF PHYSICS…                                        PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

                              TABLE I. The ontological and structural components of epistemic games.

              Ontological components                                 Structural components

                 Knowledge             cognitive resources                Entry and            conditions for when to
                      base             associated with the                   ending            begin and end playing a
                                       game                               conditions           particular game

                  Epistemic            target structure that                  Moves            activities that occur
                       form            guides inquiry                                          during the course of an
                                                                                               e-game

one needs to have some requisite knowledge to list the               Table I summarizes the ontological and structural compo-
forces.                                                              nents of epistemic games.
   The epistemic form is a target structure, often an external
representation that helps guide the inquiry during an                  C. Epistemic games students play in introductory, algebra-
epistemic game. For example, the epistemic form in the list                                 based physics
making game is the list itself. The list is an external repre-           In this section we discuss some of the epistemic games
sentation that cues particular resources and guides the pro-         that account for the different problem-solving strategies seen
gression of the inquiry. In some of the games we describe            in our data. We identify six epistemic games that include
below, the epistemic form could be a written out series of           most of the different problem-solving behaviors we have
steps, or the derivation of an equation or result.                   seen 共see Table II兲. 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
               B. Structure of epistemic games                       have identified all possible moves within each game.68 If we
    The structural components of epistemic games include the         had examined a different population of students or a different
entry and ending conditions of the game and the moves. The           domain, it is possible that the list of epistemic games would
entry and ending conditions specify the beginning and the            be different, though we expect some of the games identified
ending of the game. As we mentioned above, one may enter             here to have broad applicability. We present these as ex-
into the list making game as a means to answer a question.           amples of the type of structure we are proposing. In the next
When solving physics problems, students’ expectations about          section, we present two case studies showing how analyzing
physics problems determine the entry and ending conditions.          student behavior in terms of these games helps make sense
These expectations can depend on real-time categorizations           of what they do and do not do in the context of solving a
of physics problems and/or on preconceived notions about             specific problem.
the nature of problem solving in physics. Research by Hins-              Each of these games is described in more detail below.
ley and Hayes66 indicates that students can quickly catego-          For each epistemic game we give a brief introduction and
rize large classes of physics problems very shortly after read-      discuss its ontology and structure. Note that some of the
ing the statement of the problem. In fact, these                     games have common moves and one game may look like a
categorizations can be made after reading the first sentence.        subset of another. We identify them as distinct games be-
The students’ ability to very quickly categorize physics prob-       cause they have different ending conditions; students playing
lems may stem from their experience with and expectations            different games decide they are “done” when different con-
about physics problem solving. These expectations and cat-           ditions are met. Section V gives an example of students play-
egorizations of physics problems affect which epistemic              ing each of these games.
game the students 共perhaps tacitly兲 choose to play. In addi-                           1. Mapping Meaning to Mathematics
tion, students’ preconceived epistemological stances about
problem solving in physics can affect their expectations. If            The most intellectually complex epistemic game that we
students believe that problem solving in physics involves            identify is Mapping Meaning to Mathematics. In this game,
rote memorization of physics equations, that can affect the
strategy they employ 共i.e., which e-game they choose to                   TABLE II. List of epistemic games identified in our data set.
play兲 and what they believe an answer in physics is 共i.e., how
they know they are done playing a particular game兲.67                List of epistemic games
    The moves in an e-game are the steps and procedures that
occur in the game. In the list-making game the moves may             Mapping Meaning to Mathematics
be to add a new item, combine two 共or more兲 items, substi-           Mapping Mathematics to Meaning
tute an item, split an item, and remove an item. As we will          Physical Mechanism Game
see, a critical element of an epistemic game is that playing         Pictorial Analysis
the game specifies a certain set of moves. What is particu-          Recursive Plug-and-Chug
larly important about this is not just the moves that are in-        Transliteration to Mathematics
cluded in the game, but also the moves that are excluded.

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JONATHAN TUMINARO AND EDWARD F. REDISH                                     PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

                                                                     FIG. 2. Schematic diagram of some moves in the epistemic
                                                                  game Mapping Mathematics to Meaning.

   FIG. 1. Schematic diagram of some moves in the epistemic
game Mapping Meaning to Mathematics.                              suffice for the student to decide that the end condition of the
                                                                  game has been met.
students begin from a conceptual understanding of the physi-         The epistemic form for Mapping Meaning to Mathematics
cal situation described in the problem statement, and then        is typically the collection of mathematical expressions that
progress to a quantitative solution. We identify five basic       the students generate during moves 2 and 3. These expres-
moves 共see Fig. 1兲: 共1兲 develop a story about the physical        sions lead the direction of the inquiry. Note, however, that
situation, 共2兲 translate quantities in the physical story to      the epistemic form is not the entire story in this game. The
mathematical entities, 共3兲 relate the mathematical entities in    interpretation 共story兲 that goes with the series of mathemati-
accordance with the physical story, 共4兲 manipulate symbols,       cal expressions generated may or may not be explicitly ex-
and 共5兲 evaluate and interpret the story.                         pressed, depending on the instructions for giving a written
    The knowledge base for this game 共as with all the games       output and the students’ sense of how much “explanation”
we identify兲 comes from the set of physics and mathematics        they are required to provide.
resources. In general, however, different resources can be
activated during the different moves of the game. During the                   2. Mapping Mathematics to Meaning
development of the conceptual story 共move 1兲, reasoning               The second most intellectually complex epistemic game
primitives are most often activated. That is, students often      that we identify is Mapping Mathematics to Meaning. In this
rely on their own conceptual understanding to generate this       game, students develop a conceptual story corresponding to a
story—not on fundamental physics principles. Translating          particular physics equation. The ontological components of
the conceptual story into mathematical entities 共move 2兲 is       Mapping Mathematics to Meaning are the same as those in
difficult for most of the students in our population. Intuitive   Mapping Meaning to Mathematics. In particular, both games
mathematics knowledge, symbolic forms, and interpretive           involve the same kind of knowledge base 共mathematical re-
devices may be activated during this move. Relating the           sources兲 and the same epistemic form 共physics equations兲.
mathematical entities to the physical story 共move 3兲, again is    However, the particular resources and physics equations that
difficult for students in our population, and depends on in-      are used in each game can vary from problem to problem.
tuitive mathematics knowledge, symbolic forms, and inter-             In addition, the structural components of the two games
pretive devices. Once the physics equations are written, the      are different. In Mapping Meaning to Mathematics, students
symbolic manipulations 共move 4兲 often are carried out with-       begin with a conceptual story and then translate it into math-
out a hitch. This is probably because most of our students        ematical expressions. In contrast, in Mapping Mathematics
have had ample practice manipulating symbols. The evalua-         to Meaning students begin with a physics equation and then
tion of the story 共move 5兲 can occur in many different ways.      develop a conceptual story.69 The structural differences be-
For example, students may check the solution with a worked        tween these two games make them distinguishable from each
example 共or solution in the back of the book兲, students may       other.
check their quantitative answer with their conceptual story,          We identify four moves in Mapping Mathematics to
or students may check their solution against an iconic ex-        Meaning 共see Fig. 2兲: 共1兲 identify target concepts, 共2兲 find an
ample. Note that these evaluations do not necessarily corre-      equation relating the target concepts to other concepts, 共3兲
spond to an expert evaluation or what a teacher would want        tell a story using this relationship between concepts, and 共4兲
to see. A superficial similarity to previously seen results may   evaluate story.

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ELEMENTS OF A COGNITIVE MODEL OF PHYSICS…                                     PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

   FIG. 3. Schematic diagram of some moves in the epistemic
game Physical Mechanism.

                 3. Physical Mechanism Game

    In the Physical Mechanism Game students attempt to con-
struct 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 prin-
ciples or equations.
    The ontology of the Physical Mechanism Game is differ-
ent than in Mapping Meaning to Mathematics and Mapping
Mathematics to Meaning. The epistemic forms in the latter              FIG. 4. Schematic diagram of some moves in the epistemic
two games explicitly involve physics equations. In contrast,        game Pictorial Analysis.
the epistemic form in the Physical Mechanism Game does
not. The form in this case is a story—a description of the
                                                                    diagram during their inquiry, then that diagram serves as an
mechanism of what is happening in terms of physical prin-
                                                                    epistemic form that guides their inquiry. In the same way, a
ciples. Although the epistemic form is different, the same set
                                                                    schematic drawing and/or free-body diagram could each
of resources 共intuitive mathematics knowledge, reasoning
                                                                    serve as a target structure that guides inquiry.
primitives, symbolic forms, and interpretive devices兲 may be
                                                                        The moves in this game are largely determined by the
active in this game as in the previous games.
                                                                    particular external representation that the students choose.
    The structure of the Physical Mechanism Game is similar
                                                                    For example, if the students choose to draw a free-body dia-
to the first move in Mapping Meaning to Mathematics—both
                                                                    gram, then one move is to determine the forces that act upon
involve the development of a conceptual story. However, we
                                                                    the object in question whereas if the students choose to draw
can distinguish the two because the Physical Mechanism
                                                                    a circuit diagram, then one move is to identify the elements
Game represents a separate, coherent unit of student activi-
                                                                    共e.g., resistors, capacitors, batteries, etc.兲. Despite differences
ties; it has a different end state. In Mapping Meaning to
                                                                    that may arise based on the particular external representation
Mathematics, after move 1 students tend to go on to move 2,
                                                                    chosen, there are three moves that are common to all instan-
then move 3, etc. After 共1兲 creating and 共2兲 evaluating the
                                                                    tiations of the Pictorial Analysis Game 共see Fig. 4兲: 共1兲 de-
conceptual story developed in the Physical Mechanism
                                                                    termine the target concept, 共2兲 choose an external represen-
Game 共see Fig. 3兲 students decide they are done. The activi-
                                                                    tation, 共3兲 tell a conceptual story about the physical situation
ties that follow this game do not cohere with the conceptual
                                                                    based on the spatial relation between the objects, and 共4兲 fill
story—in direct contrast with the activities that follow move
                                                                    in the slots in this representation. An example of students
1 in Mapping Meaning to Mathematics.
                                                                    who choose to draw a free-body diagram while playing the
                                                                    Pictorial Analysis Game is given in our first case study in
                   4. Pictorial Analysis Game                       Sec. V70
    In the Pictorial Analysis Game, students generate an ex-
                                                                                      5. Recursive Plug-and-Chug
ternal spatial representation that specifies the relationship be-
tween influences in a problem statement. For example, stu-              In the Recursive Plug-and-Chug Game students plug
dents who make a schematic drawing of a physical situation,         quantities into physics equations and churn out numeric an-
a free-body diagram, or a circuit diagram are all playing the       swers, without conceptually understanding the physical im-
Pictorial Analysis Game.                                            plications of their calculations.
    In this game, as with all the games previously discussed,           Students do not generally draw on their intuitive knowl-
the knowledge base consists of all the resources listed above       edge base while playing this game. Instead, they simply
plus some representational translation resources. The               identify quantities and plug them into an equation. Conse-
epistemic form in this game is the distinguishing character-        quently, students playing this game rely only on their syntac-
istic. The epistemic form is a schematic or diagram that the        tic understanding of physics symbols, without attempting to
students generate. For example, if the students draw a circuit      understand these symbols conceptually. That is, other cogni-

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JONATHAN TUMINARO AND EDWARD F. REDISH                                     PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

                                                                                                    FIG. 5. Schematic diagram of
                                                                                                 some moves in the epistemic
                                                                                                 game Recursive Plug-and-Chug.

tive resources 共such as intuitive mathematics knowledge,             Because students use the symbolism in this game without
reasoning primitives, symbolic forms, and interpretive de-        conceptual meaning, usually only resources associated with
vices兲 are usually not active during this game.                   the syntactic structure of equations are active during this
   The epistemic form in Recursive Plug-and-Chug is simi-         game. The solution pattern of the target example serves as
lar or even identical to that in Mapping Meaning to Math-         the epistemic form for the Transliteration to Mathematics
ematics and Mapping Mathematics to Meaning. Each game             game.
has physics equations as part of the epistemic form, but the         The moves in this game are as follows: 共1兲 identify a
resources that are active 共i.e., knowledge base兲 are different.   target quantity, 共2兲 find a solution pattern that relates to the
The rules and strategies that are employed in Recursive Plug-     current problem situation, 共3兲 map quantities in the current
and-Chug differ from those in Mapping Meaning to Math-            problem situation into that solution pattern, and 共4兲 evaluate
ematics and Mapping Mathematics to Meaning—even                   the mapping 共see Fig. 6兲. Many students find moves 2 and 3
though the epistemic form may be the same. A distinguishing       very tricky. Many times students may find a solution pattern
feature of Recursive Plug-and-Chug is the resources that are      that they think relates to the current problem, when in fact it
not activated during this game.                                   does not.
   In Recursive Plug-and-Chug, the students first identify a
target quantity. This is similar to the first move in Mapping          IV. SETTING OF THE STUDY AND METHODOLOGY
Mathematics to Meaning, but it differs in that here the stu-         This study was done as a part of a project carried out at
dents only identify the quantity and its corresponding            the University of Maryland2 to determine whether an intro-
symbol—they do not attempt to understand conceptually
what the quantity represents physically as in Mapping Math-
ematics to Meaning. Second, the students identify an equa-
tion 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 cal-
culate the answer. However, if there are additional un-
knowns, then they must choose a subgoal and start this pro-
cess over. Herein lies the “recursive” nature inherent in this
game. Figure 5 shows a schematic depiction of the moves in
this game.

               6. Transliteration to Mathematics
    Research on problem solving indicates that students often
use worked examples to develop solutions to novel
problems.71,72 Transliteration 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
共letters or words兲 in the corresponding characters of another
alphabet.”73 In the Transliteration to Mathematics game stu-
dents map the quantities from a target problem directly into         FIG. 6. Schematic diagram of some moves in the epistemic
the solution pattern of an example problem.                       game Transliteration to Mathematics.

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ELEMENTS OF A COGNITIVE MODEL OF PHYSICS…                                   PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

ductory physics course could serve as a venue to help biol-        ties including representation translation problems, context-
ogy students learn to see science as a coherent process and        based reasoning problems, ranking tasks, estimation prob-
way of thinking, rather than as a collection of independent        lems, and essay questions with epistemological content. 共For
facts; and whether this could be achieved within the context       more on these types of problems see Chap. 4 of Ref. 78.兲
of a traditional large-lecture class without a substantial in-         The instructor 共Redish兲 expected that each problem would
crease in instructional resources. The project adopted re-         take the students about an hour to complete, and he commu-
forms that were well-documented to produce conceptual              nicated this expectation to the class. In accordance with his
gains and adapted them to create a coherent package that           expectation, the instructor only assigned about five problems
produced epistemological and metacognitive gains. We were          each week. 共The specific problems we discuss here are given
able to accomplish this without sacrificing the conceptual         in Appendix B.兲 Because these problems were assigned as
gains associated with these reforms.74                             homework and graded, our observations of students working
    Data on the student responses to the modified environ-         on these problems gave us an authentic look at how students
ment were collected in a variety of ways in order to provide       actually behave in real-world problem-solving situations—as
triangulation on the learning process of individual students       opposed to watching them solve problems artificially posed
and evaluations of the overall class results. The learning en-     to them in an interview environment.
vironments were constructed to encourage students to learn
in group discussions taking place both in and out of the                                    2. Coherence
classroom. Hundreds of hours of these group discussions
were recorded on video and provide the data for this project.          An important characteristic of the reformed class was the
In addition, all student homework, quizzes, and exams were         attempt to make the various parts epistemologically oriented
scanned before grading. Finally, we gave pre-post conceptual       and mutually supportive. The instructor and the teaching as-
共FCI75 and FMCE76兲 and epistemological attitude surveys            sistants frequently cross-referenced among homework, lec-
共modified MPEX77兲.                                                 tures, tutorials, and laboratories. Exam questions drew from
                                                                   and mixed information that the students had worked on in
                                                                   each of the class components.
                    A. Student population
   The students in this study were enrolled in an introduc-
tory, algebra-based physics course. They were approximately                               C. Course center
60% female; more than 70% were juniors and seniors, about             Since the traditional discussion sections were converted to
50% were biological science majors, and about 40% were             tutorials, the students did not have time to discuss the prob-
pre-meds. 共There were some year-to-year fluctuations in            lems on the homework set with a teaching assistant 共TA兲
these numbers.兲 A particularly interesting statistic for this      during these periods. To close this gap, a room was set up,
study is that more than 95% of the students had successfully       called the course center, where students could gather to work
completed two semesters of calculus, yet they chose to enroll      on the homework problems together. The data reported on
in an algebra-based introductory physics course despite the        here come from videotaped sessions of students working on
availability of a calculus-based alternative. Data were col-       homework problems in the course center.
lected in ten semester-long classes over a 4-yr period from a         A TA or instructor was available in the course center ap-
total of more than 1000 students.                                  proximately 20 h per week. The TA or instructor was present
                                                                   to offer assistance but not to explicitly solve the problems for
              B. Structure of the modified course                  the students, as they often do in many traditional recitation
    The course had four major structural components. The           sessions. The relevant features of this room were its archi-
homework, the lecture, the discussion, and the laboratory          tecture, the white boards, and the audio-video setup.
were all modified to be nontraditional in some fashion. In
addition, we attempted to make all parts of the course coher-                              1. Architecture
ent with each other. We believe that the overall epistemologi-
                                                                      Many students expect recitation sessions in which a teach-
cal orientation of the class was responsible, at least in part,
                                                                   ing assistant stands at the front of the room and solves prob-
for the students’ willingness to spend long periods working
                                                                   lems, while the students frantically copy down the solutions.
together on individual problems and for some of the behav-
                                                                   The architecture of the course center was altered to modify
iors we observed, such as discussing the physics qualitatively
                                                                   this expectation by removing the front of the room. All the
before starting to write equations. We describe here the de-
                                                                   chairs with desk arms were removed, and they were replaced
tails of the reforms that are directly relevant to the data pre-
                                                                   with stools and five long workbenches. 共See Fig. 7 for a
sented. Brief descriptions of the other reforms are given in
                                                                   schematic layout.兲 This seating arrangement did not direct
Appendix A.
                                                                   the attention of the students to any one location in the
                                                                   room—as is the case in all lecture halls in which the seating
                    1. Homework problems
                                                                   is arranged to face the “front,” directing attention to the lec-
   Problems were regularly assigned and graded. The prob-          turer. The natural focus of attention of a student seated at one
lems assigned were not traditional end-of-chapter textbook         of these worktables is the work area in front of them and the
exercises. Instead, they included a mix of challenging activi-     students seated across from them.

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JONATHAN TUMINARO AND EDWARD F. REDISH                                       PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

                                                                   lems in the course center. Sixty h 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 共qualitative or quantita-
                                                                   tive兲. These selection criteria reduced the data set to about
                                                                   11 h of video that were analyzed in detail.
                                                                       These 11 h of video were transcribed and analyzed. The
                                                                   games were determined by a semiphenomenographic
                                                                   approach.79 We identified video segments that appeared to
                                                                   contain students carrying out coherent and consistent activi-
                                                                   ties 共whether correct or not兲. The authors viewed these seg-
                                                                   ments multiple times and identified plausible goals, moves,
      FIG. 7. Top view of the layout of the course center.         and exclusions. Hypotheses for specific games were pro-
                                                                   posed. During weekly meetings of the research team 共the
                        2. Whiteboards                             authors plus other members of the University of Maryland
                                                                   Physics Education Research Group兲, the transcription and
   As a second alteration to the course center, whiteboards        coding of the episodes were scrutinized and the descriptions
were mounted on the walls and the students were provided           of the proposed e-games refined. Finally, two different cod-
with dry erase markers. The reason for this was threefold.         ers independently analyzed a sample episode in terms of
First, the location of the whiteboards made them difficult to      epistemic games, with an inter-rater reliability of 80%. After
reach for the TAs but easy for the students—an architectural       discussion, the two codings were in complete agreement. The
feature that encourages the students to go to the whiteboards      process resulted in the identification of the six games de-
and discourages the TAs from solving the problem for the           scribed in Sec. III.
student or “lecturing” at the whiteboards. Second, the white-
boards facilitate group problem solving. Research on expert
and novice problem solving has shown that external repre-                             V. TWO CASE STUDIES
sentations are a helpful and sometimes necessary tool in the
problem-solving process.24,31 The whiteboards offered the             We now present two case studies that demonstrate how an
students a medium to share their external representations          analysis in terms of resources and e-games can help make
with each other. Third, the whiteboards helped with our re-        sense of student problem-solving behavior; in particular, why
search agenda. The students’ shared representations on the         students often do not use what seems to the instructor to be
whiteboards are visible to the video camera.                       the appropriate resources in a given context. The full tran-
                                                                   scripts of these episodes are included in Appendix A.
                     3. Audio-video setup
   The course center was equipped with a digital video cam-                A. Case 1: Building knowledge using e-games
era. Microphones were mounted in the middle of two tables
to ensure quality audio reception. The video camera was               The episode for this case study involves three female stu-
mounted about 7 ft above the floor on the wall of the closet       dents working on an electrostatics problem, which we refer
across from the tables that were equipped with microphones.        to as the three-charge problem 共Appendix B, Problem 1兲.80
The elevation of the camera had three advantages:                  This episode occurs in the second week of the second semes-
                                                                   ter of a two-semester introductory, algebra-based physics
   • students and staff members walking by the closet did          course. All the students in the group had been in the re-
not block the camera;                                              formed course the first semester and were familiar with its
   • students sitting closer to the camera did not block our       innovative features. In particular, they were familiar with the
view of students who sat closer to the wall; and                   interaction style between students and teaching assistants in
   • we had a clear view of what the students wrote on the         the course center and with the type of homework problems
whiteboards.                                                       that were assigned in this course. Most importantly, they
                                                                   were cognizant of the fact that the instructor expected the
   Students were encouraged to work at the two tables that         students to spend about an hour on each homework
could be recorded. At the beginning of a session, the camera       problem—during which time they were expected to generate
was pointed at an occupied table and that microphone con-          solutions to the questions that “made sense to them.”
nected to the camera. Most students were willing to work at           An “instructor’s” solution to the three-charge problem in-
these tables and we have strong evidence that they quickly         volves straightforward balancing of forces and the use of
forgot they were being recorded.                                   Coulomb’s law. The parenthetic comment in the problem
                                                                   states there is “no net electrostatic force” acting on charge
                       D. Methodology                              q3. Symbolically, this can be written as F  ជ         ជ
                                                                                                                 q2→q3 + Fq1→q3 = 0.
   The data for this study come from about 60 h of video-          Manipulating this equation, and defining the positive î direc-
taped sessions of groups of students solving homework prob-        tion to be to the right, yields

                                                             020101-10
ELEMENTS OF A COGNITIVE MODEL OF PHYSICS…                                      PHYS. REV. ST PHYS. EDUC. RES. 3, 020101 共2007兲

                      ជ
                      F           ជ                                 Darlene:        Wait say that.
                        q2→q3 = − Fq1→q3 ,
                                                                    Alisa:          Like— q2 is— q2 is pushing this way, or
                                                                                    attracting—whichever. There’s a certain
                      kq2q3       kq1q3                                             force between two Q, or q2 that’s
                         2 î = −       î.                  共1兲
                       d          共2d兲2                                             attracting.
Canceling similar terms on both sides of the equation and           Darlene:        q 3.
setting q2 = Q yields the result q1 = −4Q.                          Alisa:          But at the same time you have q1 repelling
    There are several inferences and steps involved in gener-                       q 3.
ating this solution. However, in spite of the multiple steps            Darlene initiates the conversation by asserting that the
involved, most experienced physics teachers solve this prob-        charge on q1 must be “negative Q.” The negative sign in this
lem in less than 1 min. Some can “see” the answer in a              case standing for her realization that q1 and q2 will have
conversational beat and give the correct answer immediately.        opposite effects on q3. Alisa elaborates on this point by ar-
    The most interesting aspect about the students’ approach        ticulating that q2 exerts an influence on q3, which she iden-
is that it takes so long. The students work for about 45 min        tifies as a force, that is either repelling or attracting, and that
before arriving at a solution—two orders of magnitude               q1 exerts the opposite influence on q3. The semantic content
longer than the typical teacher! Why does it take so long?          contained in Alisa’s explanation can be summarized in the
The typical teacher has a broader mathematical knowledge            following facet: “the attractive effect of q2 on q3 cancels the
base 共i.e., a larger collection of compiled mathematical re-        repulsive effect of q1 on q3.” The abstract reasoning primi-
sources兲 and richer collection of problem-solving strategies        tive underlying this facet is canceling. In this case, canceling
共i.e., an assortment of epistemic games for solving problems        is an appropriately mapped primitive, because in fact the two
in physics兲 than most students. For the typical teacher, the        forces acting on q3 do cancel, which results in there being no
problem statement immediately cues the appropriate                  net electrostatic force on q3.
epistemic game and tightly compiled resources; whereas, the             From Alisa’s initial cursory comment 共“we thought 关the
students’ mathematical resources do not exist in compiled           charge on q1兴 would be twice as much 关as the charge on q2兴”兲
form. The difference in the teacher and the students’ knowl-        it appears that she has the reasoning primitives more is more
edge structure could account for the difference in the speed        and balancing activated. That is, since the two influences
of the problem solution and demonstrates the power and ef-          acting on q3 balance, q1 must have more charge because
fectiveness of cognitive compilation.55                             there is more distance between q1 and q3 than there is be-
    The students do not follow a straightforward approach to        tween q2 and q3.
solving this problem. However, these students’ various                 It cannot be confirmed whether Alisa has more is more
problem-solving approaches are readily understood in terms          and balancing activated, because the direction of the conver-
of epistemic games. We identify five different epistemic            sation changes. Darlene contends with the other students,
games that are played during this problem-solving session:          because it appears she has activated a different reasoning
Physical Mechanism, Pictorial Analysis, Mapping Math-               primitive: blocking.
ematics to Meaning, Transliteration to Mathematics, and
Mapping Meaning to Mathematics. We divide the discussion            Darlene:       How is it repelling when it’s got this charge
into segments corresponding to different e-games and refer                         in the middle?
to these segments as “strips.” The names in the transcripts of      Alisa:         Because it’s still acting. Like if it’s bigger
the strips are gender-indicative pseudonyms.                                       than q2 it can still, because they’re fixed.
                                                                                   This isn’t going to move to its equilibrium
           1. Playing the Physical Mechanism Game                                  point. So, it could be being pushed
   The students’ initial attempt to solve this problem follows                     this way.
a less formal path than the instructor’s solution outlined          Darlene:       Oh, I see what you’re saying.
above. Throughout this strip the students draw on intuitive         Alisa:         Or, pulled. You know, it could be being pulled
reasoning primitives to explain and support their conclu-                          more, but it’s not moving.
sions. The students do not activate any formal mathematics
                                                                    Darlene:       Um-huh.
or physics principles to support their claims. The reasoning
consists almost entirely of facets. This first strip occurs about
7 min into the problem-solving process.                                The orientation of the charges cues the reasoning primi-
                                                                    tive of blocking, because q2 is between q1 and q3. From the
Darlene:       I’m thinking that the charge q1 must have            superposition principle we know the effect of q1 on q3 does
               it’s…negative Q                                      not get blocked by the presence of q2, so the activation of
Alisa:         We thought it would be twice as much, be-            blocking is an unnecessary distraction. In contrast to the rea-
               cause it can’t repel q2, because                     soning primitive of canceling that was activated earlier in
               they’re fixed. But, it’s repelling in such a way     this strip, blocking does not get mapped into a productive
               that it’s keeping q3 there.                          facet for solving this problem. 共This is not to say that block-
                                                                    ing is “wrong;” rather, in this particular instance the activa-
Bonnie:        Yeah. It has to—
                                                                    tion of blocking does not lead to a productive facet.兲

                                                              020101-11
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