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Saturday, December 27, 2014

Problem-Based Learning

Problem-Based Learning
n  Learning that is situated around an event, case, problem, or scenario.
Five Strategies for Using PBL
1.     The Problem as a Guide: the problem is presented in order to gain attention prior to presenting the lesson.
2.     The Problem as an Integrator or Test: the problem is presented after readings are completed and/or discussed -- these are used to check for understanding.
3.     The Problem as an Example: the problem is integrated into the material in order to illustrate a particular principle, concept or procedure.
4.     The Problem as a Vehicle for Process: the problem is used to promote critical thinking whereby the analysis of how to solve it becomes a lesson in itself.
5.     The Problem as a Stimulus for Authentic Activity: the problem is used to develop skills necessary to solve it and other problems -- skills can include physical skills, recall of prior knowledge, and metacognitive skills related to the problem solving process. A form of authentic assessment of the skills and activity necessary in the content domain.

Design PBL Instruction:
    1. Task Analysis: analysis must take place not only within the content domain but should also determine the actual setting where the learning will be authentic.
    2. Problem Generation: The problems must be constructed so they include the concepts and principles that are relevant and they must be set in a real context.
Learning Sequence:
    1. Collaborative Analysis session where groups work together to solve the problem.
    2. Self-directed Learning where the students identify the information and resources that are necessary to solve the problem.

n  The instructor in PBL only acts as a facilitator to learning, instead of a transmitter of the necessary information.
n  Assessment: assessment of learning must occur within the context of the problems and should be in the form of both self assessment and peer assessment.

Cognitive apprenticeships are situated within the social constructivist paradigm. They suggest students work in teams on projects or problems with close scaffolding of the instructor. Cognitive apprenticeships are representative of Vygotskian "zones of proximal development" in which student tasks are slightly more difficult than students can manage independently, requiring the aid of their peers and instructor to succeed. 

          Scaffolding refers to the role played by parents, teachers and others by which children acquire their knowledge and skills (Wood et al, 1976).
          As a task becomes more familiar to the child and more within its competence, so those who provide the scaffold leave more and more for the child to do until it can perform the task successfully.
          In this way, the developing thinker does not have to create cognition ‘from scratch’: there are others available who have already ‘served’ their apprenticeship.
Zone of Proximal Development
The theory of the "Zone of Proximal Development" (ZPD) is a term coined by Vygotsky to refer to the:

            ‘level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers…..What children can do with the assistance of others might be in some sense even more indicative of their mental development than what they can do alone’ (Vygotsky, 1978).
          "Proximal" simply means "next". He observed that when children were tested on tasks on their own, they rarely did as well as when they were working in collaboration with an adult. It was by no means always the case that the adult was teaching them how to perform the task, but that the process of engagement with the adult enabled them to refine their thinking or their performance to make it more effective. Hence, for him, the development of language and articulation of ideas was central to learning and development.  The common-sense idea which fits most closely with this model is that of "stretching" learners.

Representation of knowledge, Prototype Theory, Schemas, Scripts, ACT-R Model, Squire’s Non-declarative Knowledge, PDP Model

Representation of knowledge
Knowledge representation promote exploration of a wide range of phenomena and offer the strength of converging operationsthe use of multiple approaches and techniques to address a problem. The way in which knowledge is represented profoundly influences how effectively knowledge can be manipulated for performing any number of cognitive tasks.
The fundamental unit of symbolic knowledge is the conceptan idea about something that provides a means of understanding the world. One way to organize them is by means of categories. A category is a group of items into which different objects or concepts can be placed that belong together because they share some common features, or because they are all similar to a certain prototype.

·      Categories
·      Prototypes
·      Schemas
·      Scripts
·      Acquisition of declarative & procedural knowledge

Earlier models were based on behaviorist theory. “ Stimulus-response association theory was proposed by Clark Hull (1920). He argued that we learn to associate a particular response (the concept) with a variety of stimuli that define the concept.”
Jerome Bruner formulated a concept formation theory that involved cognitive processes, i.e. hypothesis testing about a concept by making guesses about which attributes are essential for defining the concept. Concept attainment according to Bruner et al. 1967:233) is "the search for and listing of attributes that can be used to distinguish exemplars from nonexemplars of various categories"
Eleanor Rosch (1978) suggested that the natural concepts in everyday life are learned through examples rather than abstract rules.
Anderson's Adaptive Control of Thought (ACT) theory suggests that long-term memory is an interconnect network of propositions (facts of concepts). Only a subset of interconnected propositions can be activated and more connected propositions are easier to retrieve. A concept that has many connections is elaborated.
“ Tennyson & Cocchiarella (1986) suggest a model for concept teaching that has three stages: (1) establishing a connection in memory between the concept to be learned and existing knowledge, (2) improving the formation of concepts in terms of relations, and (3) facilitating the development of classification rules. This model acknowledges the declarative and procedural aspects of cognition.”
“ Klausmeier (1974) suggests four levels of concept learning: (1) concrete - recall of critical attributes, (2) identity - recall of examples, (3) classification - generalizing to new examples, and (4) formalization - discriminating new instances.”

Using Our Minds
        Knowing that…Declarative knowledge
        Knowing how…Procedural knowledge
Declarative Knowledge
         Stored in Concepts
        A mental representation of an item and associated knowledge and beliefs (cat, tools, furniture)
When Do We Use Concepts?
         Create categories
         Make inferences
         Combine to form complex thoughts
         For communication
Organizing Structures of Declarative Knowledge
         Concept  --Unit of symbolic knowledge
         Category  --Rule used to organize concepts
         Schemas -- Framework used to organize concepts

Different Types of Concepts
         Natural Concept
        Occur naturally (e.g. plants, trees, cats)
         Artifact Concept
        Created by humans (e.g., hammers, computers)
         Ad Hoc Concepts
        Created individually to suit a need (things you need to be happy, things you do to please parents)
Different Theories on Concept Organization
         Defining Features (Classical View)
         Hierarchically semantic networks

Categorization and Concepts
o  Basic cognitive function is to categorize
n  Use experience to aid in future behavior and decision-making
o  Cognitive economy
o  Concepts
n  Mental representation of a category serving multiple functions
Functions of Concepts
o  Classification
n  Determine category membership
o  Understanding, making predictions, inference
n  Once classified one can then understand its relevant parts, know how to interact with it, infer other properties
o  Explanation and Reasoning
n  For example, of others’ behavior
o  Learning
n  New entities compared to and understood in terms of old and provide feedback for modification
o  Communication
n  Shared concepts and categorization allow for easier expression of ideas to others
o  Categories
n  Collection of objects, attributes, or actions, etc.
o  List of concepts
o  Hierarchy
n  Set of entities or examples picked out by the concept
o  How is experience distilled? How are functional relations established?
n  Category learning
o  How is knowledge represented in a category?
n  Structure
n  Schema
o  General knowledge structure that integrates objects, attributes, and actions into a cohesive representation
n  Script
n  Sequence
o  How do we use categorical knowledge?
o  Determining the category membership of various things (objects, properties, abstractions etc.)
o  Allows for treating otherwise discriminable entities as similar
n  Similarity as the organizing principle for categories and categorization
Types of Categories
n  Abstract vs. Concrete
o  Love vs. Mammal
n  Hierarchical vs. Non
o  Mammal vs. woman
n  Feature-Based Categories: A Defining View 
All those features are then necessary (and sufficient) to define the category. This means that each feature is an essential element of the category.  Together, the features uniquely define the category; they are defining features (or necessary attributes):  For a thing to be an X, it must have that feature. Otherwise, it is not an X.

Ex. Bachelor (male, unmarried, adult)  The features are each single necessary.  If one feature is absent, the object cannot belong to the category.  The three features are jointly sufficient.  If a person has all three features, then he is automatically a bachelor.

Prototype Theory: A Characteristic View  Prototype Theory – grouping of things together not by their defining features but rather by their similarity to an averaged model of the category.  Prototype is an abstract average of all objects in the category we have encountered before.  Crucial are characteristic features, which describe (characterize or typify) the prototype but are not necessary for it. They are commonly present in typical examples of concepts, but they are not always present.
Ex. Prototype of a game Prototype of a bird (robin or ostrich)  Whereas a defining feature is shared by every single object in a category, a characteristic feature need not be.
Classical concepts are categories that can be readily defined through defining features, such as bachelor.  Fuzzy concepts are categories that cannot be so easily defined, such as game or death.
Exemplars are typical representatives of a category.  Ex. Birds, we might think not only of the prototypical songbird, which is small, flies, builds nest, sings, and so on. We also might think of exemplars for birds of prey, for large flightless birds, for medium-sized waterfowl, and so on.

         Schemas are models of the external world based on past experience
         Schemas for concepts underlying situations, events, or sequences of actions
         Abstraction that allows particular objects or events to be assigned to general categories
         Organize our knowledge
         May include other schemas
         Help in encoding, storage, and recall
         Allows us to make inferences

Schemas – mental framework for organizing knowledge. It creates a meaningful structure of related concepts.
 A cognitive structure that organizes related concepts and integrates past events.  Ex. Kitchen (tells us the kind of things we might find in a kitchen and where we might find them)
Schemas have several characteristics that ensure wide flexibility in their use.  Schemas can include other schemas. Ex. A schema for animals includes a schema for cows, a schema for apes, and so on.  Schemas encompass typical, general facts that can vary slightly from one specific instance to another.  Schemas can vary in their degree of abstraction.

         Type of schema about events
         Structure captures general information about routine events
        Eating in a restaurant, attending a movie, a visiting a doctor’s office
         Scripts have typical roles
        (Customers, waiter, cook), (ticket vendor, patrons, refreshments), (doctor, nurse, patient)
         When we hear or read about a scripted event, our knowledge of the entire script is activated
         We can fill in or infer the scenes and actions that are not explicitly mentioned

Script contains information about the particular order in which things occur.  Ex. Restaurant script (coffee shop)  Props: tables, a menu, food, a check, and money  Roles to be played: a customer, a waiter, a cook, a cashier, and an owner.  Opening conditions for the script: the customer is hungry, and he or she has money  Scenes: entering, ordering, eating, and exiting  A set of results: the customer has less money; the owner has more money; the customer is no longer hungry; and sometimes the customer and the owner are pleased.

Jargon – specialized vocabulary commonly used within a group, such as a profession or a trade.  Imaging studies reveal that the frontal and parietal lobes are involved in the generation of scripts. The generation of scripts requires a great deal of working memory. Further script generation involves the use of both temporal and spatial information.
Scripts enable us to use a mental framework for acting in certain situations when we must fill in apparent gaps within a given context.

Acquisition of declarative & procedural knowledge
Representing Procedural Knowledge
         Serial Processing
        Linear sequence of operations
        Create using production rules
         If – then rules
        If sliding on ice then pump the brakes
         Tasks may take multiple rules
        Organized into routines and subroutines
ACT-R Model
         Combines declarative and procedural knowledge in a model
         Declarative knowledge is represented in structures called chunks defined by its type and slots
        Type represents concepts or categories (e.g., dogs) and slots as category attributes (e.g., color or size)
         Anderson (1980)
        Cognitive Stage
         Consciously think about steps to complete task
        Associative Stage
         Practice the procedure
        Autonomous Stage
         Skill has become automatic
Squire’s Non-declarative Knowledge
         Procedural knowledge
         Associative conditioning
        Classical and operant conditioning
         Simple nonassociative knowledge

Support for Squire’s Taxonomy
         Basil Ganglia damage
         Examine Parkinson’s and early Huntington disease patients
         No apparent amnesia (declarative memory intact)
         Problems with procedural memory
        Perceptual motor learning
        Habits & skills
         Just one example of variety of studies with humans and animals have supported Squire’s taxonomy

Two Types of Priming
         Semantic priming
        Meaning is primed
        Remember Nurse-Doctor study?
         Repetition priming
        Prior exposure primes same items seen later

Connectionist Model
         Parallel processing
        Multiple operations occur simultaneously
         Parallel Distributed Processing (PDP) models
        Goal is to model information as it is represented in the brain

The PDP Model
         The representation of information is distributed
         Knowledge for specific things are not stored explicitly, but stored in the activations of patterns among units 
         Learning occurs with changes in connection strength by experience
         Units send excitatory and inhibitory signals to other unit

Reasoning: Types- Inductive, Deductive, Propositional, Syllogistic, Analogical, Categorical, Approaches- The componential, Rules/ Heuristics, Mental models

Reasoning: Types- Inductive, Deductive, Propositional, Syllogistic, Analogical, Categorical
Reasoning is the process of using existing knowledge to draw conclusions, make predictions, or construct explanations. 
Reasoning - a specific type of thinking
Def: cognitive processes by which people start with information and transform info to reach conclusions

l  Arguments generally are divided into two types: deductive and inductive.
l  Deductive arguments? Valid or invalid?
l  Inductive arguments? Strong or weak?

Types of Reasoning
Deductive Reasoning – starting with some pieces of information and making a logical conclusion
            Reasoning from the general to the specific
            ex. 1. to graduate from State University, you have to have at least a C average (general statement)
                  2. Stacie is graduating from State University
                  a. Therefore, Stacie has at least a C average. (specific statement)
If premises are true, then the conclusions will be true (deductive validity)
      Deductive Reasoning – A type of logic in which one goes from a general statement to a specific instance.
      The classic example
All men are mortal. (major premise)
Socrates is a man. (minor premise)
Therefore, Socrates is mortal. (conclusion)
The above is an example of a syllogism.

Inductive Reasoning – arrive at a conclusion that is probably true based on evidence
Reasoning from the specific to the general
Ex. 1. Richard attended State U for 4 years
      2. Richard is now the vice president of a bank
      a. Richard probably graduated from State U
Do we know that Richard definitely graduated? No, but he probably did
Conclusions cannot be certain, there can only be stronger or weaker beliefs in the conclusions (inductive strength)

Propositional Reasoning
Propositional reasoning is a form of deductive reasoning, which goes from the general to the specific. Through deductive reasoning one can state with absolute certainty that a conclusion is either true or false. In that sense it is similar to syllogistic reasoning, which involves syllogisms, where a conclusion is drawn or evaluated based on two or more premises. Propositional reasoning on the other hand involves propositions, which are sentences that are either true or false (Galotti, 2008), e.g. “Mark likes football” or “Berlin is the capital of Germany”. These propositions are abbreviated to letters, such as p or q. By using logical connectives single propositions can be combined into more complicated ones. Logical connectives include: ¬ (‘not’) for negations, & (‘and’) for conjunctions, v (‘or’) fordisjunctions, and → (‘If…, then…’) for conditionals (Galotti, 2008). For instance, the sentence “If it is raining, then I get wet” can be rephrased as p → q , where p stands for ‘rain’ and q stands for ‘getting wet’.

Syllogistic reasoning is concerned with using syllogisms to draw conclusions from premises.
      Syllogism: An argument composed of two statements or premises (the major and minor premises), followed by a conclusion.
      For any given set of premises, if the conclusion is guaranteed, the arguments is said to be valid.
      If the conclusion is not guaranteed (at least one instance in which the conclusion does not follow), the argument is said to be invalid.
All math teachers are over 7 feet tall.
    Mr. D. is a math teacher.
    Therefore, Mr. D is over 7 feet tall.
      The argument is valid, but is certainly not true.

Analogical Reasoning
Analogical reasoning is a method of processing information that compares the similarities between new and understood concepts, then uses those similarities to gain understanding of the new concept. It is a form of inductive reasoning because it strives to provide understanding of what is likely to be true, rather than deductively proving something as fact. This method can be used by both children and adults as a way to learn new information or as part of a persuasive argument.
Analogical reasoning is based on the brain’s ability to form patterns by association. The brain may be able to understand new concepts more easily if they are perceived as being part of a pattern. If a new concept is compared to something the brain already knows, it may be more likely that the brain will store the new information more readily.
The field of science also uses this type of reasoning, but it is used for coming up with new concepts rather than for persuasion. Scientists will often compare a proven scientific process with an unproven one to form hypotheses to base new research on. They may reason that because two processes are similar in one way, they may be more likely to have more things in common.
Psychologists often focus on the cognitive aspects of reasoning. They may perform research to determine how and why the brain retains information through analogies. Psychologists may also study the differences between how children and adults use them.
l  A form of deductive argument.
l  Also called syllogism.
l  Consists of two or more premises that precede the conclusion.
l  Reasoning conclusions about the properties of individuals from more general premises that concern all the members of the relevant categories.
l  Example:
                        All whales live in water (Premise)
                        All fish live in water, too (Premise)
                        All fish must be whales. (Conclusion)
l  If the conclusion of an argument is not guaranteed by the truth of the premises then the syllogism is not valid.
l  An invalid argument involving categories is called a categorical fallacy.
l  In a categorical syllogism, each of the premises states a relationship between the middle term and one of the others.
l  Categorical syllogism features:
l  Every proposition is in standard categorical form
l  There are three terms.
l  The major premise is listed first, the minor is listed second.
l  Each term is used in the same sense throughout the argument.
l  Example:
            All German cars are reliable (major premise)
            All BMW are German cars (minor premise)
            All BMW will be reliable (conclusion – derived from relationship between minor premise and major premise)

Approaches- The componential, Rules/ Heuristics, Mental models
Major programmatic approaches to the study of reasoning are classified into three types: the componential approach, the rules/heuristics approach, and the mental models/search approach.

The Componential Approach
The componential approach studies reasoning by analyzing a task into its component cognitive processes. A computer metaphor may be useful here. Those of you who have done computer programming know that programs can be built from subroutines, each of which performs a very specific function (such as sorting a list of numbers or adding a list of numbers). To understand how reasoning works, we need to figure out if analogous mental subroutines of reasoning exist and, if so, what they are, when and for how long each is executed, and the chances of each one’s running without error.
To illustrate, let’s return to the sample analogy given at the beginning of the chapter: Washington is to one as Jefferson is to what? Sternberg (1977a, 1977b, 1986a, 1986b) studied people’s performance on such problems extensively. He argued that to fill in the blank, we must perform several mental subroutines or, to put it more formally, execute a number of component cognitive processes. First we must encode each of the terms; that is, we must read the words Washington, one, and Jefferson. We then must recognize these terms, retrieving from memory the meaning of each term and mentally representing these meanings. Next we must infer the relationship between the first two terms (often called the A and the B terms)—in this case, Washington and one. One relationship that comes immediately to mind is that Washington was the first president of the United States. The next step is to map the A term and the C term (here, Jefferson)—that is, find a relationship between them. Jefferson was also a president of the United States. In the next step, we apply the relationship previously found between the A and the B terms onto the C term, remembering (if we recall our U.S. history) that Jefferson was the third president. Thus our answer to the analogy would be three. Sometimes analogies are provided in multiple-choice format, and in those cases the answers don’t always fit.
In Sternberg’s theory, each component has associated with it several parameters that determine, for instance, the probability that it will be used, the amount of time it will take to execute, and the difficulty of executing it.
Sternberg’s method of estimating these parameters was quite clever. He presented participants with a number of different verbal and pictorial analogies (Sternberg, 1977a) on a tachistoscope, and each trial consisted of two parts: (a) precueing and (b) presentation of the full analogy. Figure 12-7 presents examples.
During precueing, participants saw either a blank field (no cues), the A term of the analogy only, the A and the B terms of the analogy only, or the A, B, and C terms of the analogy. Sternberg (1977a) compared the amounts of time it took participants to decide if the full analogy was true or false on trials with no cues to the amounts of time on trials in which cues had first appeared. For example, when participants had been precued with the A term, Sternberg reasoned that they had been able to encode this term; thus they should be (and were) faster to respond to the full analogy. Let’s say it took participants, on average, 2 seconds to respond to the full analogy, but only 1.8 seconds when the A term was precued. That 0.2-second difference presumably reflects the time it takes to encode the A term. Similarly, if participants were precued with the A and B terms and then took only 1 second to respond to the full analogy, we could infer that the other 1 second was the time it took to encode the A term, encode the B term, and infer a relationship between them.

Sternberg and his colleagues have studied other inductive reasoning tasks and have presented componential models of each. Each componential model identifies the mental processes (such as encoding or comparing) that are used in each task. From these studies, Sternberg has argued that the componential approach to the study of reasoning will reveal important insights into what reasoning is and how it can be improved (Sternberg, 1986a).
In later work, Sternberg (1983, 1984) distinguished among three kinds of components used in reasoning. Components that consist of individual cognitive processes are called performance components and include those given earlier. Metacomponents are “executive” processes used in the planning and monitoring of a task. For example, metacomponents select which performance components will be used and in what order. Knowledge acquisition components are used whenever we acquire new information. These include things such as selective encoding (sifting relevant from irrelevant information), selective combination of previously encoded information, and selective comparison.
Knowledge acquisition components have to do with how we learn or pick up new information. Sternberg (1986a) argued that people differ in the ways in which they learn: Given the same  exposure to the same novel situation, two people may acquire different kinds and amounts of information from it. The mental processes we use in learning are knowledge acquisition components.

Sternberg’s (1986a) book gives advice and examples to provide practice with performance components, metacomponents, and knowledge acquisition components. Many errors in reasoning tasks appear to stem from problems in encoding (a performance component). Many of the problems in reasoning noted earlier (such as problems with premise interpretations, alteration of premise meanings, or failure to consider all possibilities) arise because reasoners fail to encode enough relevant information. Moreover, complicated syntactic expressions (those with many terms, those with negatives) presumably take more processing resources to encode, leaving fewer resources available for other components of reasoning. Other problems in reasoning could be described in terms of differences in using metacomponents. A reasoner who plans her approach to a problem and monitors his or her performance is less likely to be biased or to make other kinds of errors.

The Rules/Heuristics Approach
One idea is that the rules of logic are the same rules we use to draw conclusions Most modern psychologists reject the strong version of this idea but agree that people use “mental logics” or systems of inference rules to draw conclusions. Using this rules/heuristics approach, these researchers make an analogy between mental logics and grammars: Both are systems of rules to which we have only implicit access.
Different researchers describe slightly different sets of inference rules. Generally, rules take the form “(premises) → (conclusion).” Here’s a specific example (from Braine, 1978): “p q; p therefore q.” The idea is that when given information, people try to match it to one of these rules and use the rules to draw appropriate conclusions.
These proposals are not the only ones that view reasoning in terms of mental rules. Work by philosophers on practical logic also follows a rules approach. The goal of practical logic is to teach people to avoid fallacies or errors that occur in real-life arguments. Box 12-4 presents some common fallacies.
The key issue for the rules explanation of reasoning is how people figure out when and what rules apply. Braine (1990) proposed the existence of abstract rules that we use in all situations. Patricia Cheng and her colleagues rejected the idea of abstract rules, instead proposing sets of rules that are sensitive to the context. The idea here is that different rules are called to mind in different situations.
One example is the permission schema, made up of four rules:
Rule 1. If the action is to be taken, then the precondition must be satisfied.
Rule 2. If the action is not to be taken, then the precondition need not be satisfied.
Rule 3. If the precondition is satisfied, then the action may be taken.
Rule 4. If the precondition is not satisfied, then the action must not be taken.

The rules approach to reasoning is particularly effective at explaining content effects in reasoning.  The explanation goes as follows:
Presumably, different contents “cue” different sets of rules, although exactly how this process works is not well understood. It may be that personal experience facilitates this cueing, so that people are more likely to reason correctly with premises about drinking ages simply because their own experiences cause them to interpret the situation in terms of a permission rule.
Cheng et al. (1986) have reported success in teaching people to recognize and use pragmatic reasoning rules correctly after only brief periods of practice. This suggests that people quickly learn to use inference rules as a guide to processing information on certain tasks. Similarly, the existence of logic courses in colleges and universities suggests that rules of logic can be taught. The hope is, of course, that people who learn to use a set of inferencerules in one situation will transfer their understanding of the rules to new circumstances.

The Mental Models Approach
Proponents of the mental models approach deny that reasoning consists of using special-purpose rules of inference and that reasoning involves specialpurpose cognitive processes. Philip Johnson-Laird (1982, 1983), a major spokesperson for the models approach, argued that the processes we use to draw conclusions are also the ones we use to comprehend language.
Reasoning, for Johnson-Laird, consists of constructing mental models to depict the premises. Effective reasoning occurs when the reasoner checks to be sure his or her first idea of what the conclusion might be is assessed by an attempt to construct alternative models consistent with the premises but inconsistent with the hypothesized conclusion.
To explore Johnson-Laird’s approach, consider the following syllogism:
Some of the scientists are parents. All of the parents are drivers.” Figure 12-8 offers one interpretation of how these premises might be mentally modeled for this relatively easy-to-solve reasoning problem. Scientists are depicted as people holding a flask; drivers, as people standing next to a car; and parents, as people holding a child. The diagram indicates that some scientists are drivers but (possibly) some other scientists aren’t drivers (those shown in faded lines) and, also possibly, some drivers aren’t parents (also rendered in faded lines). Notice that the two scientists in the middle of the diagram (the ones who aren’t “optional”) are drivers, leading to the necessarily true conclusion, “Some of the scientists are drivers.”
Consider another of Johnson-Laird’s syllogisms, this one more difficult to work with: “All of the beekeepers are artists. None of the chemists are beekeepers.” You might try this one yourself before reading on. Figure 12-9(A) depicts the model most people generate first. Notice that no individual is both a chemist and a beekeeper nor both a chemist and an artist. This depiction would lead one to conclude, “None of the chemists are artists.” However, if they keep at it, people may discover other possible depictions, such as the one shown in Figure 12-9(B), where one artist is a chemist. This depiction means the preceding conclusion cannot be true. At this point, a reasoner who had constructed both models might conclude, “Some of the chemists are not artists.” Again, however, another possibility exists, the one depicted in Figure 12-9(C). Here, all the chemists are artists, so the last conclusion cannot be valid, either.
Is there no valid conclusion, then? In fact, there is. The one statement true of all three models is “Some of the artists are not chemists.” In particular, the beekeeper/artists, necessarily depicted in each model, are not chemists. One problem with the mental models framework is specifying what information models contain and what information is omitted. Notice, for instance, that in Figures 12-8 and 12-9 we did not specify any physical, ethnic, or philosophical
information about the people depicted. How much information the reasoner chooses to represent and how this decision affects performance are issues that remain to be investigated.
The construction of a mental model can be considered a creative act. Perkins (1985) argued that—contrary to stereotype—model building (and therefore good reasoning) relies on imagination. The more imaginative the process, the more likely a reasoner is to generate potential counterexamples and avoid drawing hasty conclusions. Interestingly, this view links reasoning with other kinds of thinking, helping to explain the apparent links among reasoning, problem solving, and decision making. In the mental models approach, errors in reasoning derive from several possible sources. One is the failure to construct relevant models. If the premises are not presented in an optimal order (for example, in a syllogism, in the order A-B, B-C), it is harder to construct an integrated representation of both premises that accurately depicts all the relevant information. If there is a great deal of extraneous information in the premises, mental resources may be diverted from the processes needed to selectively represent the essential information. A second source of error is the failure to assess the implications of all the models found. For instance, in the previous example, someone might have decided
that no conclusion relating artists and chemists was valid, overlooking the one relation shown in all three models. A final and important source of error is the failure to search for and construct enough models. This accounts for the findings described earlier—namely, that people often fail to consider enough of the possibilities allowed by any set of premises.