Scientific Reasoning Research Institute - Topic

Transfer of Learning

root — Thu, 14 Feb 2008 04:24:39 +0000

How people "transfer" knowledge from the learning context to other contexts

Transfer of learning is among the most important problems in education today. Students, especially in science, are all too often unable to apply what they learn to novel contexts both in, and outside the classroom. Making headway on the transfer problem is important since all of education is predicated on the premise that what is taught in one course will be used in relevant situations in other courses, as well as out of school and in the workplace. Research suggests, however, that lack of transfer is pervasive and persistent, and that promoting more transfer is a difficult enterprise given the complexity of factors that affect it.

Transfer has attracted recent attention from both cognitive scientists and science education researchers. A recent conference, sponsored by the National Science Foundation and organized by Jose Mestre, brought together cognitive scientists, science education researchers and NSF program officers to discuss primary issues, identify the research base, and propose a future research agenda for transfer. The report from this conference has been disseminated by NSF through a report, Transfer of Learning: Issues and Research Agenda.

Neural Networks

root — Fri, 11 Jan 2008 23:18:34 +0000

A perspective on neural network dynamics, complexity, and cognition

by:

Ian Beatty

Neural network (NN) modeling has developed as a major component of science's attempt to understand the brain. The fundamental question is, how do the brain's formidable information-processing abilities emerge from the self-organizing behavior of a collection of relatively simple neurons? NN modelers aspire to develop artificial systems with some brain-like abilities. Much of the progress in NN modeling has been made by physicists, who have extensive experience formulating and analyzing complicated mathematical models [1-4].

I. Introduction

A neural network is an information-processing system consisting of a collection of simple processing elements or "nodes." If the network has real neurons for nodes, it is a "biological neural network" (for example, the brain); otherwise it is an "artificial neural network" (ANN). The elements are interconnected so that the input to each node is determined by the outputs of some or all of the other nodes, and the whole accomplishes some useful "computation." In contrast with traditional electronic circuits, the nodes in a NN are generally approximately identical, and the collection's "program" is encoded in the architecture (pattern of connectivity) and in the strengths of the inter-node connections. Such an approach to computation is often called "parallel distributed processing" [5].

In neural network modeling, one constructs a mathematical model of such a system and attempts to answer questions like "What behavior is this model capable of producing?" and "How does the displayed behavior depend on the details of the model and its history?" Research usually involves both formal analytic derivation and computer simulation.

Much of the current work in NN modeling attempts to answer specific questions about the practical applicability of ANNs, such as "What is the most efficient algorithm for training an ANN to categorize input patterns into desired classes?" [6] Research also commonly seeks to illuminate the biological basis of brain functioning by matching neural network behavior to observable brain behavior, for example by asking "How does the ability of a network to recall stored 'memories' degrade as inter-node connections are destroyed?" [7]

More fundamentally, research with NN models can address questions like "How do sophisticated-seeming patterns of organization and behavior emerge from the interactions of simple elements acting on limited or local information, without any master blueprint or directing agent?" In the context of this latter sort of question, NNs are seen as merely one example of a class of "complex systems" [8] which includes ecosystems composed of interdependent and co-evolving species [9-11], economies composed of cooperating and competing agents [12], and embryos undergoing morphogenesis, composed of differentiating and structure-forming cells [13]. Research in NN modeling is therefore intimately linked to research in so-called "complex systems theory" or "complexity theory." This is appropriate, since the human brain is the most impressive example we know of sophisticated and organized behavior emerging from the operations of many simple interacting elements.

Complex systems theory is a recent interdisciplinary field of research with contributions from biologists, ecologists, economists, computer scientists, chemists, and physicists [14-16].

Perhaps the most profound "emergent property" of the human brain, and to a lesser extent the brains of other species, is the ability to construct (explicitly or unconsciously) models of itself and its environment. Loosely speaking, "modeling" as used here means the process of observing sensory data, discovering regularities of various sorts within the data stream, inferring general rules from these correlations, and using predictions of these general rules to guide behavior. The brain in fact seems to "folds back" this process upon itself, by abstracting and observing its general rules and rule-making with a higher-level description, discovering regularities at this level, inferring meta-level rules, and so forth for level upon level. It remains to be seen whether a neural network model can be constructed which demonstrates the emergent property of modeling its environment in this way.

If NN models are to demonstrate such sophisticated emergent behavior, we clearly must progress beyond the rigid and limited models currently being studied. We must seek models which allow a wider range of behavior if we are to find less trivial emergent and self-organizational properties. In so doing we will likely sacrifice the use of many conceptual approaches and analytic techniques currently in use, so we must attempt to develop new ways of conceptualizing and analyzing NN behavior. Viewing NN from a complex systems perspective may aid us to do this.

II. Selected Background

A. Neural Network Dynamics

One can designate a subset of the nodes of a NN as "input nodes," whose states are externally controlled, and another subset as "output nodes," whose states represent the model's observables. One can then attempt to encode an input-output mapping into the network's connection strengths such that when any particular set of input states is applied, the dynamics of the network generates the desired output states at the output nodes. Many tasks we would like NNs to perform, such as pattern recognition and similar input classification tasks, can be represented as an input state to output state mapping, so much of NN research concerns so-called "rule learning" [17, 18]. Networks designed for rule learning are generally (but not necessarily) arranged in a layered, "feed-forward" architecture, in which information propagates in one direction through the network. Such an architecture has no feedback loops anywhere, so that the states of all nodes are determined and stable as soon as signals from the input nodes have had time to propagate through the network. This type of network has no nontrivial dynamics at all, and research is almost exclusively concerned with developing and characterizing algorithms for setting the connection strengths by "training" a network on a set of representative input/output mappings [6, 19-22].

In 1981, J.J. Hopfield, a physicist by training, presented the first example of a NN model with nontrivial dynamics (state evolution) and predictable, useful behavior [4]. As an added attraction, his model was biologically plausible (if oversimplified) and quite tractable to formal analysis. Through both mathematical analysis and computer simulation, he demonstrated that a network of two-state nodes with symmetric bidirectional connections between all node pairs and with a simple threshold function for each node's updating rule always evolves to a stable fixed-point attractor. Furthermore, he presented a simple algorithm for determining the link strengths that would encode any desired set of node states ("patterns") as attractor states.

As a result, he showed that the network can be used as an "associative memory": the link strength rule can be used to encode a set of patterns representing memories, and if the network is presented with an "input pattern" initial state sufficiently close in state space to one of the stored patterns--say, the stored pattern plus noise or minus missing sections--the network state evolves until it stabilizes at the stored pattern, thus "recalling" the memory from partial information. The network's dynamics is less trivial than that of simple rule-learning input-output networks because the system's location in state space evolves dynamically, traveling from the initial state to a stable fixed-point attractor at the bottom of the appropriate basin of attraction.

Because the Hopfield model is formally identical to an Ising spin glass [1], Hopfield's work attracted many physicists from the study of random magnetic systems to NN modeling, with the result that a great deal was learned about the behavior of NNs. In particular, by casting NN models into a form addressable by statistical mechanics, one can derive results indicating the existence and nature of "phase transitions" between useful and useless network behavior as a function of various network parameters; the existence of such phase transitions has strong implications for the maximum number of patterns which can be stored in a network [23].

If one allows asymmetric connections in a NN model, dynamics more complicated than evolution to a fixed-point attractor can result: the network trajectory can limit to state cycles or chaotic attractors [24]. Herz et al. [25, 26] have shown that if the Hopfield model is extended so that the connections between nodes each have a characteristic signal propagation time, and if the propagation times are chosen so that a suitable distribution of times are represented in the network, then a straightforward extension of Hopfield's connection strength rule allows the network to encode sequences or cycles of node-state patterns. Static patterns can still be stored as fixed-point attractors, and pattern cycles are stored as limit cycles of the dynamics. The existence of limit cycle attractors is made possible by the fact that when encoding temporal information, the connection strength rule produces asymmetric connections.

B. Dynamical Systems and Emergent Computation

We have been describing NN models in terms of their regime of dynamical behavior: trivial (almost immediate stability), evolution to a fixed-point attractor, evolution to a limit-cycle attractor, or evolution to the discrete-state equivalent of a strange (chaotic) attractor. In complex systems theory, the prevailing view is that a system's regime of dynamical behavior largely determines the system's capacity to process information, perform computations, and generally exhibit sophisticated behavior and self-organization of complex structure [14]. Langton, Packard, and others have argued that cellular automata (CA) with nontrivial computational abilities, including all those that have been proven computation-universal, lie at the "edge of chaos" (EOC), a sort of critical line in CA parameter-space between CA with static or periodic asymptotic behavior and CA with chaotic asymptotic behavior [27-32]. This EOC has many of the properties of a second-order phase transition, including power-law behavior in spatial and temporal correlations and divergent transient times ("critical slowing down").

Preliminary results suggest that, in general, complex systems which demonstrate sophisticated behavior such as information-processing, adaptation to environmental changes, and self-organization of complex structural interrelationships exist at the EOC in an appropriately defined space of dynamical behavior. This is an intuitively reasonable statement. Static or periodic systems demonstrate simple behavior which is relatively robust against external influences. Chaotic systems are extremely sensitive to any sort of noise and are effectively random in their behavior. Only systems on an extended transient trajectory are capable of evolving in a way that encodes significant information. Moreover, strong parallels can be drawn between the behavior of dynamical systems and the properties of computational systems [33-36], and these also point to the significance of the EOC.

The behavior typical of systems at the EOC, evolving along extended transients, is termed "complex," as opposed to static, periodic, or chaotic. Such behavior is typically characterized by the formation of quasi-stable, interacting structures localized in time and space. The analysis of complex behavior is one of the outstanding problems of complex systems theory.

Most NN models studied to date operate in the static or periodic regime of behavior, and encode information in the location of the attractors in state space and the shapes of their basins of attraction; the few exceptions are models whose purpose is to study the existence and nature of chaotic behavior in a NN [24, 37]. Models with static or periodic behavior can be carefully tuned to provide desirable behavior, by "sculpting" state space behavior via connection strengths so that attractors and basin boundaries are in useful locations. However, such models are inherently limited in the sophistication of the behavior they can model. If we wish to investigate the possibility of designing networks which can self-organize and develop increasingly deep behavior, particularly the ability to interact with a changing environment and "learn" more than simple input/output mapping or pattern recall, we should consider networks with complex, EOC dynamics.

Designing or analyzing the behavior of a system at the EOC, however, is difficult; no general techniques have been developed. One usually discovers such a system by fortuitous observation, and then develops a phenomenological description of the system in question in terms of the various observed types of local metastable structures and their interactions. One example of this approach is the study of the John Conway's "Game of Life" CA, and in particular the proof that this CA is capable of constructing a Turing machine and thus of supporting universal computation [38, 39]. Another example is economics, where one begins with a general economic theory and then spends most of one's time classifying exceptions to the theory [12]. For surface waves in shallow water, an example drawn from nonlinear dynamics, one can study solitons and their interactions; in this case, however, the inverse scattering transform allows a complete theory of nonlinear, interacting modes [40].

Mitchell, Crutchfield, et al. have had some success using genetic algorithms to "evolve" CA systems capable of performing a specified computational task [32, 33, 41]. They found that genetic algorithm methods "discovered" CA rules which operated at the edge of chaos, and which used propagating and interacting "particles" (localized, stable structures) to transmit information and perform computation.

C. Modeling Emergent Modeling

Crutchfield, Mitchell, and their associates have suggested a framework, the "calculi of emergence," for discussing the modeling capabilities of adaptive dynamical systems with computational abilities [34]. In this framework, one considers a dynamical system as an "adaptive agent" which monitors a data stream representing measurements of its environment, and which attempts to represent or encode the data stream as efficiently as possible by discovering regularities within it. One classifies the modeling abilities of an agent by the computational class of its regularity-detecting algorithm; the computational classes used are an extension of those developed in formal computation theory, e.g. "finite stack automaton," "universal Turing machine," etc. The complexity of the data stream, as perceived by the agent, is defined to be a measure of the computational resources required under the agent's current computational scheme.

In this framework, the most significant aspect of "emergent" modeling capabilities is the ability of an agent to "discover" or "innovate" a more powerful modeling algorithm (computational class) when it has insufficient computational resources to model its environment with its current algorithm. Crutchfield does not discuss any particular dynamical system nor propose specific mechanisms by which a system might innovate a new algorithm and develop the required structures; his intent is to provide a language for describing and analyzing the process. One might hope to apply this framework to the modeling abilities of neural networks.

References

(It has recently come to our attention that the reference list for this essay is incomplete, and is missing numbers 35 through 41. Unfortunately, the earliest archived copy we have (1999) is similarly incomplete. Since the piece is over ten years old, and the field has evolved dramatically since then, we don't see much point in investing the time to reconstruct the original list. Apologies for any inconvenience. — IDB, 2005-05-27.)

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Mitchell, M., P.T. Hraber, and J.P. Crutchfield, Revisiting the edge of chaos: Evolving cellular automata to perform computations. Complex Systems, 1993. 7: p. 89-130.
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Early PER Findings

root — Fri, 11 Jan 2008 23:05:57 +0000

Early results from Physics Education Research, relevant to physics instruction

by:

Jose Mestre

This document is quite old (early 1990s) and mentions nothing about all the relevant research since then. It's also lacking references. Caveat emptor.

The areas of cognitive research we will focus on are: (1) the prevalence and virulence of misconceptions; (2) the differences between the ways that experts and novices store domain-specific knowledge and solve problems; (3) the importance of goal-free activities; and (4) the effects of "meta-communicating" with students about the learning process. Each area has critically affected the development of our approach, and therefore, each area is reviewed to help you understand the construction of our materials and how they should be implemented.

Misconceptions

An important area of cognitive research is the study of misconceptions. Results show that misconceptions: (1) are extremely common; (2) are not easily displaced; (3) can be found (even) among experts; and (4) hinder understanding. It appears that people continually and unconsciously build models of how the world operates. The human brain seeks patterns and quickly establishes categories. Patterns of experience are put into models, but often, these models are based on insufficient experience. Furthermore, these models, misconceptions included, affect how later experiences are interpreted.

Thus, demonstrations that teachers show to students often can reinforce misconceptions rather than dislodge them. Later, especially if their models have been (reasonably) successful at getting the right answer, students are extremely reluctant to abandon them. When educators are not aware of the models that students are using to interpret experiences, answer questions, and solve problems, it can be a shock and a mystery when previously successful students get wrong answers. In fact, misconceptions may have been there all along, but they were not manifested by the interaction and communication between the student and teacher.

Expert-novice differences

The following table summarizes some of the differences between experts and novices that cognitive research has studied and revealed. We believe that one of the tasks of good curriculum materials is to encourage beginners to think more like experienced problem-solvers.

**Table 1:** A summary of expert-novice differences
	Experts	Novices
Knowledge Characteristics	Large store of domain-specific knowledge	Sparse knowledge set
	Knowledge richly interconnected and hierarchically structured	Disconnected and amorphous structure
	Integrated multiple representations	Poorly formed and unrelated representations
Problem-Solving Behavior	Conceptual knowledge impacts problem-solving	Problem-solving largely independent of concepts
	Performs qualitative analysis	Manipulates equations
	Uses forward-looking concept-based strategies	Uses backward-looking means-ends techniques

Goal-free vs. goal-directed questions

The prevailing approach to teaching physics has been to lecture, assign homework problems, and give exams. Although teachers usually emphasize the importance of concepts, students generally do not understand concepts, nor do they use them to solve problems. One reason might be cognitive overload, in which students are so focused on the answers, and are paying so much attention to the questions asked, that they can not also focus on problem-solving skills and do not notice the patterns intended by the teacher; they never see the connections and the repetitions among the assigned problems.

Homework and exam questions in the prevailing approach are goal-directed, and it is believed that asking open-ended, goal-free questions reduces cognitive overload. This allows students to devote greater cognitive resources to other activities, such as reflection, and leads to improved understanding of subject matter. It is recommended, therefore, that fewer goal-directed questions (but not none) should be given to students, and that more conceptual questions should be asked.

Effects of meta-communication on learning

Meta-communication is defined to be the process of communicating information and knowledge other than content. It usually involves higher level skills, such as: different ways to organize the material; how people think; how people solve problems; what to study in a textbook; how to read and extract information from textual material, etc. Meta-communication is intended to help students learn how to learn, by helping them learn how they learn best and by pointing out the relevant issues along the way.

In a recent study, we have shown that students in a traditional college engineering physics sequence can learn how to write qualitative strategies before they solve problems. Although students struggled with the task, by the end of one semester, many were able to write "expert-like" strategies that correctly identified: (1) the principle they would use to solve a particular problem; (2) the justification and applicability of the principle to the problem; and (3) the way in which the principle would be applied to the problem.

Our own experience confirms that people must be actively engaged in the structuring of knowledge for it to be useful for solving problems and understanding content. Meta-communication is the way in which teachers encourage (sometimes reluctant) students to consciously construct their own self-consistent models of how the world operates.

We believe that educational materials designed with reference to these research results will likely be more effective than materials designed otherwise. To help organize these findings, and to represent them in a useful way, we have developed a model for the acquisition, storage, and utilization of knowledge. Our model is consistent not only with research findings but also with our own reflections on the problem-solving process.

Knowledge Structure

root — Fri, 11 Jan 2008 21:09:07 +0000

A qualitative model for the storage of domain-specific knowledge and its implications for problem-solving

_Extracted from a booklet accompanying a workshop for high school science teachers._

by Robert J. Dufresne, William J. Leonard, and William J. Gerace

Our group at UMass [PERG] has developed a cognitive model that helps us to represent the differences in the ways that experts and novices store and use content knowledge. The Model also helps us to target specific areas on which novices need to concentrate in order to become better problem-solvers. However, the Model is only one representation of the structure of knowledge, and it is constantly evolving. Thus, it must not be taken too literally. Its usefulness comes from its ability to provide a concrete manifestation, however imperfect, of the ways in which experts and novices think. The Model helps us to discuss:

the storage of domain-specific knowledge;
expert- and novice-like problem-solving behavior;
the hierarchical structure of an expert's knowledge store;
misconceptions;
the effects of goal-free and goal-directed questions; and
the meta-communication process.

General structure of knowledge

In this workshop---and in our approach to physics in general---there are 3 basic themes:

Particular types of knowledge and knowledge structures are needed for proficient problem-solving. Much of this knowledge is conceptual in nature, as opposed to operational or procedural, and powerful knowledge structures necessarily involve conceptual elements. The presence of conceptual elements in the knowledge structures is the key to having a "deeper understanding" of physics.
Particular types of cognitive processes are required for the acquisition of conceptual knowledge and the construction of useful knowledge structures.
It is possible to design activities that promote these desirable cognitive processes. In many cases, these activities are simply actualizations of the cognitive processes themselves. (This point will, hopefully, become clearer later.)

We will elaborate on each of these themes before focusing on specific examples.

Theme 1: What do students need to know and how should what they know be structured for efficient problem-solving?

We begin by identifying some of the various types of knowledge that students need to know:

Conceptual knowledge, such as the concept of momentum or energy, or that the velocity of an object changes when it accelerates, or that the gravitational potential energy of an object decreases as it falls.
Factual knowledge, such as the value of the gravitational constant g, the radius of the moon, or the density of iron.
Representational knowledge, such as how to draw and use graphs.
Strategic knowledge, such as the ability to recognize the applicability of a concept, such as, momentum is conserved when there are no external forces, or that energy is conserved when there are no non-conservative forces.
Meta-cognitive knowledge, for example, the awareness of underlying assumptions, or that an answer should be checked by solving the problem a different way.
Self knowledge, such as knowing one's likely sources of mistakes, or knowing that one should be more procedural when solving problems.
Operational knowledge, such as how to take the cross product or dot product of two vectors, or how to take the determinant of a matrix, or how to draw a free-body diagram.
Procedural knowledge, such as when to use conservation of energy (i.e., when all forces are conservative), or when to specify a coordinate system (e.g., when finding potential energy), or when to draw a free-body diagram (e.g., when applying Newton's Laws).
Problem-state knowledge, which are the features of a problem used for deciding how to solve it. Examples are: knowing that there are no external forces in a particular problem, or that there are no non-conservative forces in the problem, or that an object is at rest initially, or that the object is on an incline.

These types of knowledge need to be organized and structured for efficient use when problem-solving. In order to discuss the organizational and structural aspects of knowledge, we have found it convenient to broadly classify these types into three general categories. We call these three groups: Conceptual Knowledge, Operational and Procedural Knowledge, and Problem-State Knowledge. In Fig. 1, these three general categories are shown in a representation of how experts store content knowledge.

Fig. 1: A representation of an expert's structure of knowledge

The expert has a rich clustering of concepts, in which each concept is related to many other concepts, and the relationships between concepts are clearly understood. Concepts are arranged hierarchically using umbrella concepts to more tightly relate them. In fact, umbrella concepts are used to group elements within each of the three categories. The expert has a large store of problem-state knowledge, including much information about which principles apply to particular situations. The expert also has a large store of equations, operations, and procedures (EOPs) that can be quickly accessed.

The links between each pair of categories are very strong: Problem states are strongly linked to concepts and to EOPs, which are themselves strongly linked to each other. The same umbrella concepts are used to group concepts, problem states, and EOPs. Therefore, for any particular problem, concepts can be used to decide the appropriateness and applicability of equations, and the utility of specific operations and procedures.

(One must be very careful here not to be too rigid in one's thinking, because it is easy to disagree about which of these categories should be used to classify a particular element of knowledge, or whether it belongs in only one category. Our purpose is simply to have a mechanism for visualizing the several levels of association that can occur between elements.)

We tend to use the term linking to mean a formed association between two elements of the same or different knowledge types, and the term clustering to refer to associations between several elements or clusters. What makes this kind of discussion very difficult is that there is a kind of iterative process going on here: A very strong bond between three items---a conceptual cluster, the recognition of the circumstances making the concept applicable, and the procedures needed to apply the concept---forms a new type of knowledge element which we put into Strategic Knowledge (a fourth category). This new knowledge element is what some refer to as a schema and often involves problem-state knowledge as well. Since the knowledge element is conceptual in nature, it becomes replicated (i.e., repeated) in the conceptual bubble.

If we take a closer look at the conceptual bubble, we see some of the specific types of conceptual knowledge, for example, Representational Knowledge, Strategic Knowledge, Meta-Cognitive Knowledge, and Knowledge of Basic Concepts. This is represented in Fig. 2. (Other types of conceptual knowledge are not shown.)

Fig. 2: Types of conceptual knowledge

How is a novice's structure of knowledge different from an expert's? As represented in Fig. 3, novices generally have a poor clustering of concepts. Many links are inappropriate; others are non-existent. Some of the inappropriate links are extremely strong, which leads to misconceptions. Novices generally do not use umbrella concepts to group elements. They have a small store of problem situations, in which surface features are used to cluster them together and to decide how problems should be solved. They are familiar with a relatively large number of equations, but they often remember them incorrectly or need to look them up in order to use them. They have been taught operations and procedures, but they are not yet proficient at them. Therefore, it cannot be said that they "know" them.

Fig. 3: A representation of the novice's structure of knowledge

Links between concepts and EOPs are weak or non-existent. Links between concepts and problem situations are also weak. Thus, a novice cannot analyze a problem and cannot decide the appropriateness of particular equations. The links between problem situations and EOPs are relatively strong, but the links are based primarily on the quantities that the equations have in common with the givens and explicit unknowns of the problem.

Theme 2: In what kinds of cognitive processes must students engage to develop appropriate knowledge structures?

To answer this question, we must first understand how experts and novices solve problems differently.

Novices are generally unsuccessful when they try to solve typical problems in physics. Using the Model to represent the process, how does a typical novice solve a problem? Because the strongest links in the novice's structure of knowledge are between problem situations and equations, reading a problem immediately suggests equations involving the quantities (known and unknown) explicitly given in the problem. Without determining the applicability of those equations, and without trying to think of other equations that might involve the same quantities, the novice usually starts to manipulate the most familiar equations until the unknown can be solved for. The most recent equation covered in class is the most easily accessible and the most quickly recalled. Thus, the novice looks for and (if "successful") finds only one way to solve a problem and usually stops, without investigating other possibilities and without analyzing the problem situation. The equations found this way are often inappropriate because novices often don't use concepts to justify their application. Even if students are driven to invoke concepts by analyzing problems beforehand, their links to EOPs are generally too weak to be useful for problem-solving. Also, links between concepts and problem situations are uni-directional, so analogies are not particularly useful: Novices can't use analogies to solve problems because they can't identify which of the problems they have already solved are conceptually similar to the one they are currently trying to solve. Instead, novices use surface features to establish "similarity" and try to solve new problems based on their similarity to the surface features of problems they've already solved.

Because experts classify problems and EOPs according to the same umbrella concepts, they can often go directly from problem situations to appropriate equations, operations, and/or procedures. Because the links between different categories are strong, difficult problems (ones for which a direct link between problem states and EOPs does not yet exist) can be solved by consciously invoking concepts, thereby indirectly connecting problem states to the appropriate EOP(s). Because the links between concepts and problem situations are bi-directional, analogies are an extremely useful problem-solving tool for experts. Finally, experts usually have more than one approach to solving any particular problem.

Here is a summary of the major differences between experts and novices:

Novices have a poor clustering of concepts, which often leads to misconceptions. Experts have a rich clustering of concepts, problem situations, equations, procedures, and operations, which leads to improved problem-solving ability.
Novices usually have only one way of solving a particular problem, whereas experts often can find more than one way. Therefore, the expert can attempt to resolve inconsistencies when they occur and check answers, while novices are unaware that inconsistencies exist and cannot check their answers.
Novices often use equation manipulation and seldom use concept-based strategies to get an answer. The expert uses concepts and analogies to suggest several methods of solution and plans a strategy for finding the correct answer.
Novices often fail to get the right answer, and when they do get the right answer it can easily be for the wrong reason. When the novice gets the right answer for the wrong reason, misconceptions are reinforced and become even harder to overcome. An expert usually gets the right answer and can explain why the answer is correct.

There are a variety of cognitive processes beneficial for helping novices develop a concept-based problem-solving approach, which we divide into three categories: Analysis Processes, Reasoning Processes, and Meta-Cognitive Processes.

Analysis processes

Problem analysis, such as constructing a problem representation.
Conceptual analysis, such as using concepts to determine the qualitative behavior of physical objects or to form a strategy.
Strategic analysis, such as identifying and justifying physics principles relevant to a problem situation.
Representational analysis, such as exploring different representations of a problem.
Complex constructive analysis, such as decomposing a complex situation into simpler ones.

Reasoning processes

Comparing and contrasting, such as identifying how items, situations, or conditions are similar and/or different.
Interpreting, for example, using the shape of a plot of position vs. time to estimate the acceleration of the object.
Special and limiting cases, that is, exploring extreme and/or known conditions.
Prototype and counter-examples, for example, generating archetypical categories.
Generalization, that is, recognizing the salient features of a circumstance or situation.

Meta-cognitive processes

Reflection, that is, self-directed review of purpose, goals, effects of experience, etc.
Meta-communication, which is conscious participation in establishing and refining lines of communication with the teacher and other students, and in deciding the goals of learning.
Self-evaluation, such as evaluating one's performance, or identifying reasons that difficulties were encountered while solving a problem.

These and other processes are encouraged by our curriculum materials. The specific types of activities to do this are presented in the next section.

Theme 3: What types of learning activities or experiences promote these beneficial cognitive processes?

The following activities can be used by teachers to stimulate the cognitive processes needed to develop a conceptual understanding of physics:

Use multiple representations. A representation may be linguistic, abstract, symbolic, pictorial, or concrete. Using many different representations for the same knowledge, and having students translate between representations, helps the student to inter-relate knowledge types and relate the knowledge to physical experience. It encourages the formation of links between knowledge elements and promotes a rich clustering of knowledge.
Make forward and backward references. Concepts require a long time to be formed. Thus, you cannot wait for students to completely learn one topic before moving on to the next. By making forward references, you prepare the student for new material. By making backward references, you associate new material with established (or partially established) material, thus making knowledge interwoven and interconnected, rather than linear.
Explore extended contexts. Concepts can be extremely context dependent and do not become globally useful until they are abstracted. Investigating a broad context of applicability helps the student to refine and abstract concepts. It also avoids incorrect or oversimplified generalizations.
Compare and contrast. Essential to the process of structuring (or re-structuring) knowledge is the classification and inter-relation of knowledge elements. Comparisons and contrasts sensitize students to categories and relationships, and helps students perceive the commonalities and distinctions needed to organize their knowledge store.
Categorize and classify. In parallel with comparisons and contrasts, students must be aware of categories and classification systems. Students must also practice creating and recognizing categorization systems. By requiring students to classify items, to choose names for their categories, and to explain their system, we can help students re-structure their knowledge store.
Predict & Show (inadequacy of old model). Carefully selected demonstrations and experiments can be used to bring out inconsistencies in student models. Students should be shown a set-up or experimental apparatus and should be asked to predict what will happen when something is done. It is important that students make predictions beforehand, thus making them aware of their own model. Students will consider alternate conceptions only if their own fails. Requiring students to use their models and showing them how their models are inconsistent or inadequate will prepare them to create better (though still their own) models.
Explain (summarize, describe, discuss, define, etc.). Standard problems seldom tell the teacher what students don't understand. Even when students get a problem right, there can still be confusion about the applicability of the equations used. Requiring students to explain how they will solve a problem exposes misunderstandings and misconceptions, and helps students reorganize their knowledge store. In addition, students seldom see in standard demonstrations and experiments what experts see. Students should explain and discuss what they think they've seen (during Predict & Show, for example), so that the teacher can interact with the students' models. Furthermore, the process of explaining (or summarizing, describing, discussing, etc.) helps students become aware of their own models as well as the models of other students.
Generate multiple solutions. Efficient problem-solving cannot occur unless students choose from a set of valid solution paths. By solving problems in more than one way, students learn to prioritize elements of their Strategic Knowledge.
Plan, justify, and strategize. Very few relationships in physics are always valid. To avoid equation manipulation, students should be asked to plan (and then explain) how they will solve problems. Students must learn how to determine which concepts are relevant (and which are irrelevant) for any particular problem situation and how to implement the relevant concepts to solve that problem. Having students generate their own strategies helps them to learn how concepts are used to solve problems.
Reflect (evaluate, integrate, extend, generalize, etc.) After completing most activities, students benefit from looking back on what they've done. What patterns have they perceived? What general rules can be constructed? Other types of activities give students the pieces needed to create a coherent picture of physics, but some sort of reflective activity is usually needed to "put the pieces together".
Meta-communicate about the learning process. To learn physics (or any other complex subject), students must become self-invested. They must be exposed to other people's (teacher's and student's) models. They must be warned that precision in communication is essential; they must be informed of common pitfalls and misinterpretations; and they must be told that they should re-structure their knowledge. Students must learn how they learn best.

Physics Education Research

root — Mon, 03 Dec 2007 20:07:33 +0000

Formal, rigorous research into the teaching, learning, understanding, and application of physics knowledge

Although the field of Physics Education Research (PER) has some overlap with Education, Cognitive Science, Psychology, Computer Science, and other disciplines, it has its own distinct identity, concerns, perspectives, and approaches.

SRRI's Physics Education Research Group (PERG) does PER.

Wikipedia has an entry on Physics Education Research.

Question Driven Instruction

root — Sun, 25 Nov 2007 05:15:17 +0000

Question-driven instruction (QDI) means having students wrestle with rich, meaty, meaningful questions and problems as a context for sense-making and a vehicle for learning, not just as assessments.

(We'll be writing more about this soon, but we needed a placeholder page here now!)

TEFA

root — Sun, 28 Oct 2007 04:05:35 +0000

Technology-Enhanced Formative Assessment

by:

Ian Beatty

Technology-Enhanced Formative Assessment (TEFA) is a pedagogical approach for using classroom response technology to conduct effective, interactive, student-centered instruction in classes with anywhere from a dozen to hundreds of students. It has been tested in multiple science disciplines, and to a lesser extent in mathematics and social sciences, at both university and secondary-school levels¹.

TEFA honors four key values: question-driven instruction, dialogical discourse, formative assessment, and meta-level communication. Separately, each of these is of well-established value to science instruction². Integrated into an interlocking whole as TEFA, they provide a powerful, flexible instructional approach.

Honoring question-driven instruction means having students wrestle with rich, meaty, meaningful questions and problems as a context for sense-making and a vehicle for learning, not just as assessments. Honoring dialogical discourse means involving students in extensive discussions in which multiple points of view and ways of thinking, including ones not anticipated by the teacher, are sought, explored, and compared; it also means helping students practice speaking the "social language" of the discipline being taught rather than focusing exclusively on content. Honoring formative assessment means continually probing and modeling students' knowledge and thinking, adjusting instruction accordingly, and providing students with individualized, prescriptive feedback to guide their learning efforts. Honoring meta-level communication means explicitly and implicitly addressing and discussing instructional communication, thinking and learning, and the instructional narrative (along with course content) in order to help students re-frame their learning activities and participate more productively and consciously in the learning process.

TEFA implements question-driven instruction, dialogical discourse, and formative assessment by structuring large chunks of class time around an iterative question cycle, a pattern that can be altered and embellished as required. The basic cycle consists of posing a question to the class; allowing students a few minutes to ponder it alone or in small groups; collecting students' answers; presenting a summary of the answers; eliciting students' justifications for their choices; moderating a class discussion to develop, challenge, compare, contrast, and integrate the ideas raised; and providing appropriate wrap-up or closure. In a typical hour-long class, three or four related question cycles might be used to develop students' understanding of a set of related concepts.

A classroom response system (CRS) facilitates the TEFA question cycle. Although the fundamentals of TEFA can be practiced without technological aid, using a CRS makes a surprising difference to the quality of the results³.

¹ Feldman, A. and Capobianco, B. (2003), Real-time formative assessment: A study of teachers’ use of an electronic response system to facilitate serious discussion about physics concepts, a paper presented at the Annual Meeting of the American Educational Research Association, April 23, Chicago, IL.

² Bell, B. and Cowie, B. (2001), The characteristics of formative assessment in science education, Science Education 85 (5) 536–553. Black, P. and Wiliam, D. (1998), Assessment and classroom learning, Assessment in Education: Principles, Policy & Practice 5 (1) 7–74. Bonwell, C. C. and Eison, J. A. (1991), Active learning: Creating excitement in the classroom, ASHE-ERIC Higher Education Report No. 1, ERIC Clearinghouse on Higher Education. Mortimer, E. F. and Scott, P. H. (2003), Meaning Making in Secondary Science Classrooms, Open University Press. Bransford, J. D., Brown, A. L., and Cocking, R. R. (1999), How People Learn: Brain, Mind, Experience, and School, National Academy Press, Washington, D.C.

³ Dufresne, R. J., Gerace, W. J., Leonard, W. J., Mestre, J. P., and Wenk, L. (1996), Classtalk: A classroom communication system for active learning, Journal of Computing in Higher Education 7 3–47.

Collaborative Group Techniques

root — Sun, 28 Oct 2007 03:54:17 +0000

Methods for teaching via small-group cooperative learning work

This document has been excerpted from Supplement A of the "Teacher's Guide to accompany Minds•On Physics: Motion" (cf. Minds•On Physics).

Introduction

Collaborative groups and cooperative learning refer to a variety of structured classroom management techniques and grading systems developed and studied by Aronson, Johnson & Johnson, Kagan, Slavin, and others since the early 1970s. These terms usually do not refer to loosely structured group work in which students are told simply to "work together" on a problem or assignment. To emphasize the difference between unstructured group work and collaborative group work, groups are usually referred to as teams. Collaborative structures are content-free, and thus can be used in a variety of contexts.

Studies have shown that in well structured cooperative groups, students consistently learn many different subjects better than students in traditionally structured classrooms. Cooperative learning also has a number of psychological and social benefits, such as being exposed to other points-of-view, learning how to cooperate, having more positive feelings about school, having more positive feelings about themselves and others, and wanting their classmates to do well.

Studies have shown also that all students benefit academically from cooperative learning. Successful students show modest gains in performance and achievement, while historically unsuccessful students usually show tremendous gains when taught using cooperation as the primary motivator. Cooperative grouping lets students organize their thoughts in a less threatening context than whole-class discussions, and prepares students for sharing their thoughts with the class. Also relevant for Minds•On Physics is that students can make progress on exercises they would not be able to attempt alone.

Getting started with cooperative learning

Because students often lack collaborative group skills, it is essential to begin the school year with activities designed to target interaction skills and team building within the class. Students need to learn how to listen to other students, and to analyze and interpret what they are saying. Students must learn, for example, how to encourage others in their group to participate, how to ask questions, how to manage dominant personalities, how to monitor and modify the group dynamic, and how to communicate effectively. Unless these skills are targeted early in the year, cooperative learning is likely to fail.

Therefore, the focus of instruction at the beginning of the year should be on developing group skills, rather than on physics. This investment of time will yield huge dividends later in the year. For example, have students sit in a circle and have volunteers define what they think science is all about. Then require the person sitting on his/her left (or right, whichever you choose) to paraphrase the definition. Be sure to tell students the structure of the activity beforehand, and have the class discuss and reflect on the activity immediately afterwards. Another effective structure is to have a team of three or four students work on a problem together -- a problem from algebra, for example, that they should already know how to solve -- and have three or four other students observe the interactions. Afterwards, have everyone discuss what happened, and what didn't happen, as the inner group solved the given problem. (This is sometimes called Fishbowl.)

Some common collaborative group structures

There are literally hundreds of cooperative structures and dozens of books available to help teachers incorporate cooperative learning into their classrooms. The structures listed and described here are believed to be particularly useful with the Minds•On Physics materials and approach.

Fishbowl. Teams of three or four work on a problem or exercise. At the same time, other teams of three or four observe the first teams. In particular, the first teams work on seeking other points-of-view, listening to and paraphrasing ideas, and other communication skills while solving the given problem. The second teams focus their attention on the team dynamic and make sure they are prepared to discuss how well or poorly the first teams worked together to solve the problem. (There is sometimes the tendency of the second teams to focus on the problem rather than the team dynamic.) After some time (even if every team has not finished the problem), the class discusses what happened and what didn't happen during the activity.
Pairs Check. Teams of four work in pairs on a set of exercises. First one member works on a problem, while the second member coaches. Then the second member works on a problem while the first coaches. Pairs then check their answers with members of the same team. After all problems, inconsistencies, etc. are resolved, the process is repeated for subsequent exercises.
Pairs Check II. This is the same as Pairs Check, except that students do exercises individually beforehand. One student explains his/her answer on a question to another student, and they discuss it. Then, they reverse roles for the next question. Answers are agreed upon before sharing with the whole team.
Teams Check. Teammates help each other understand answers to exercises, so that any member of the team may be called upon to answer any one of the questions.
Jigsaw. If there is reading material (such as background) to be digested before doing an activity, split it up into 3 or 4 self-contained parts. Divide the class into the same number of Reading Groups, with one member from each team. Give one part of the reading to each team to digest and to prepare to explain to their team. Then rearrange the students so that each team has someone who has read one of the self-contained parts, and have each student teach his/her part of the reading to the rest of the team.
Think-Pair-Share. Students think about each question, pair off and discuss the question with a classmate, and share their answers with the class.
Think-Pair-Square. This is the same as Think-Pair-Share, except that students share their answers with members of another pair.
Word Webbing. As a team or individually, open-ended or with concepts provided by the teacher, students construct a concept map within a specified domain. If done in teams, each member should have a different color of pen.
Team Product. Students work together, but each has a primary role within the team. Some favorite roles are:
- Manager (to keep the team on task);
- Reader (to read aloud the question being answered by the team);
- Encourager (to make sure everyone participates);
- Checker (to make sure everyone understands);
- Writer (to record results and to make sure everyone agrees);
- Artist (if needed to prepare the presentation); and
- Presenter (if needed to explain the team's answer to the rest of the class).
The most important roles are the Manager, Checker, and Writer. (In other words, teams of three with these three roles are the most common.) In order to accommodate some of the other roles, students may take on two of them at the same time.
Blackboard Share. Teams share their answers with the class and get feedback. This can be done on posterboards or transparencies instead.
Roving Reporter. When a team gets stuck, one member is allowed to roam the room looking for ideas and reports back to the team.
Two-Box (or Two-Column) Induction. The teacher puts items into one of two boxes (usually on the blackboard) without telling students what the criteria are for sorting the items. As the teacher adds items to the two boxes, students (standing in teams) discuss the items and possible categories. When a team decides that they know how the sort was done, they sit down without revealing their answer. (This is a non-disruptive way of letting the teacher know how the individual teams are doing.) When all teams are seated, there are three different options:
- Ask each team to add an item to each box, and let the other teams evaluate and comment on the choices;
- Present additional items to the class, and ask teams to decide which box each item belongs in.
- Have teams describe their categories.

Deciding which structure should be used

Some structures are more compatible with certain activities or instructional goals than others. For instance, Fishbowl is good for developing skills; Pairs Check and Jigsaw are good for learning new material; and Word Webbing and Two-Box Induction are good for relating concepts. Also, do not introduce too many new structures too quickly; usually about one new structure per week is recommended.

Other advice

It is usually a good idea to have the details of a cooperative group activity worked out before class. You should know how students are to divide themselves into teams (e.g., by assignment, by drawing lots, or by personal preference); how many students should be on each team (2, 3, or 4, usually); what the team composition should be (heterogeneous or homogeneous; all male, all female, or mixed); which questions each team should work on; when the activity is officially over; how to bring the activity to closure; and how to grade the activity (if at all). Also, it's important that everyone has an active role within each team, and that there are "sponge" activities that teams can work on if they finish earlier than other teams. Finally, hang in there. It takes some perseverance for both students and teachers to get collaborative groups to work effectively, but the rewards are definitely worth the effort.

References

The following books and articles should help teachers incorporate collaborative learning. In particular, Cooperative Learning by Spencer Kagan has an entire chapter devoted to resources, including books on theory, research, and methods, manipulatives, video tapes, newsletters, and the names and addresses of some cooperative learning organizations.

Aronson, E., Blaney, N., Stephan, C., Sikes, J. & Snapp, M. (1978). The Jigsaw Classroom. Beverly Hills, CA: Sage Publications.
Johnson, R.T. & Johnson, D.W. (1991). Learning Together and Alone: Cooperative, Competitive, and Individualistic Learning (3rd ed.). Boston, MA: Allyn & Bacon.
Johnson, R.T., Johnson, D.W. & Holubec, E.J., Eds. (1987). Structuring Cooperative Learning: Lesson Plans for Teachers. Edina, MN: Interaction Book Company.
Johnson, R.T., Johnson, D.W. & Holubec, E.J. (1993). Circles of Learning: Cooperation in the Classroom. Edina, MN: Interaction Book Company
Johnson, R.T., Johnson, D.W. & Smith, K.A. (1991). Active Learning: Cooperation in the College Classroom. Edina, MN: Interaction Book Company.
Heller, P., Keith, R. & Anderson, S. (1992). Teaching problem solving through cooperative grouping. Part 1: Group versus individual problem solving. American Journal of Physics, 60, 627-636.
Heller, P. & Hollabaugh, M. (1992). Teaching problem solving through cooperative grouping. Part 2: Designing problems and structuring groups. American Journal of Physics, 60, 637-644.
Kagan, S. (1990). The structural approach to cooperative learning. Educational Leadership, 47(4): 12-15.
------ (1996). Cooperative Learning. San Clemente, CA: Kagan Cooperative Learning.
Sharan, S., Hare, P., Webb, C.D. & Hertz-Lazarowitz, S., Eds. (1980). Cooperation in Education. Provo, UT: Brigham Young University Press.
Slavin, R. (1990). Cooperative Learning: Theory, Research, and Practice. Englewood Cliffs, NJ: Prentice-Hall.

Constructivism

root — Sun, 28 Oct 2007 03:16:16 +0000

A philosophical viewpoint on how the mind forms and modifies its understanding of reality

According to Ernst von Glasersfeld, our resident philosopher and expert on constructivism:

Constructivism was introduced in the modern era by Jean Piaget as a way of thinking about cognition and knowledge, not as a metaphysical theory about what might exist. It's fundamental tenet is: The mind organizes the world by organizing itself (Piaget 1937).

The "radical" version of constructivism was developed independently by Heinz von Foerster (1981) and Ernst von Glasersfeld (1984). Knowledge, from this perspective, is not a representation of "objective" facts, but a compendium of concepts, conceptual relationships, and rules that have proven useful in domesticating our experiential world.

Foerster, H. von (1981) Observing systems. Seaside, California: Intersystems Publications.

Glasersfeld, E. von (1984) An introduction to radical constructivism, in P. Watzlawick (ed.) The invented reality. New York: Norton. German original, 1981.

Piaget, J. (1971) The construction of reality in the child; New York: Basic Books. French original, 1937.

For a great deal more about radical constructivism — both more extensive and more authoritative than we can provide here — we direct you to the extensive writings on the Radical Constructivism web site.

(The essay that follows is lacking in "scholarly" background, citations, etc. The reason is that it was pulled out of a very specific context. We present it here in the spirit of "perhaps you will find it useful.")

Constructivism and Science Education

In what way is a constructivist view of science education different from other views? The answer lies in the tenets of constructivist philosophy, which assert that all knowledge is constructed as a result of cognitive processes within the human mind. While this may appear to be a harmless enough statement, many find (so-called) radical constructivism somewhat unpalatable.

Classroom Response Systems

root — Sun, 28 Oct 2007 02:37:41 +0000

In-class technology for engagement, interactivity, and formative assessment

by:

Ian Beatty

A classroom response system is technology that allows an instructor to present a question or problem to the class; allows students to enter their answers into some kind of device; and instantly aggregates and summarizes students' answers for the instructor, usually as a histogram. Most response systems provide additional functionality. Some additional names for this class of system (or for subsets of the class) are classroom communication system (CCS), audience response system (ARS), voting machine system, audience feedback system, and -- most ambitiously -- CATAALYST system (for "Classroom Aggregation Technology for Activating and Assessing Learning and Your Students' Thinking").

UMPERG has been teaching with and researching classroom response systems since 1993. We find that the technology has the potential to transform the way we teach science in large lecture settings. CRSs can serve as catalysts for creating a more interactive, student-centered classroom in the lecture hall, thereby allowing students to become more actively involved in constructing and using knowledge. CRSs not only make it easier to engage students in learning activities during lecture but also enhance the communication among students, and between the students and the instructor. This enhanced communication assists the students and the instructor in assessing understanding during class time, and affords the instructor the opportunity to devise instructional interventions that target students' needs as they arise.

UMass has a support page for PRS users.