Problem Solving (Cognitive Learning Processes)

Problem-solving doth constitute one of the most paramount forms of cognitive processing, frequently occurring during the course of learning. The subject of problem-solving hath long been a matter of scholarly inquiry—with historical accounts thereof being reviewed in the present section—yet interest in this topic hath waxed significantly with the burgeoning of cognitive theories of learning. Certain theorists regard problem-solving as the keystone process in learning, particularly within domains such as science and mathematics (Anderson, 1993). Albeit the terms “problem-solving” and “learning” are not precisely synonymous, the former doth oft partake in the latter, most especially when learners may exert a degree of self-regulation over their learning, and when the learning involveth challenges and solutions not readily apparent. In the introductory scenario, Miss Meg doth advocate for a greater emphasis upon problem-solving.

A problem exists when there subsists a “situation wherein one endeavoureth to attain a certain goal, and must discover a means by which to arrive thereat” (Chi & Glaser, 1985, p. 229). The problem may consist of answering a question, computing a solution, locating an object, securing employment, instructing a pupil, and suchlike. Problem-solving referreth to the endeavours of individuals to achieve a goal for which they possess no automatic solution.

Irrespective of the content area and its complexity, all problems share certain common characteristics. Problems possess an initial state—that is, the problem-solver's current status or level of knowledge. Problems also possess a goal—what the problem-solver doth aspire to attain. The majority of problems further necessitate that the solver subdivide the goal into subgoals, which, when mastered (usually sequentially), culminate in the attainment of the principal goal. Finally, problems demand the performance of operations (both cognitive and behavioural) upon the initial state and the subgoals, thereby altering the nature of those states (Anderson, 1990; Chi & Glaser, 1985).

Given this definition, not all learning activities incorporate problem-solving. Problem-solving is unlikely to be involved when students' skills become so firmly established that they automatically execute actions to attain their goals, which commonly occurreth with many skills across diverse domains. Problem-solving may likewise be absent in learning of a rudimentary nature (perhaps trivial), wherein students possess a clear understanding of the steps required to learn. This seemeth to be a matter of concern at Nikowsky Middle School, where the pedagogues are concentrating upon the foundational skills requisite for examinations. Simultaneously, students acquire novel skills and discover new applications for skills previously learned; thus, many scholastic activities may, at some juncture during the learning process, involve problem-solving.

Certain historical perspectives on problem-solving are to be examined as a backdrop to current cognitive viewpoints: viz., trial and error, insight, and heuristics.

Trial and Error

Thorndike’s (1913b) research with felines necessitated problem-solving; the problem being how to effect an escape from the cage. Thorndike conceived of problem-solving as a matter of trial and error. The animal was capable of performing certain behaviours within the cage. From this behavioural repertoire, the animal performed a behaviour and experienced the consequences thereof. After a series of random behaviours, the cat made a response which opened the hatch, leading to escape. With repeated trials, the cat made fewer errors prior to executing the escape behaviour, and the time required to solve the problem diminished. The escape behaviour (response) became connected to cues (stimuli) within the cage.

We occasionally employ trial and error to solve problems; we simply perform actions until one proves efficacious. However, trial and error is not reliable and often ineffectual. It can waste time, may never yield a solution, may result in a less-than-ideal solution, and can possess negative effects. In desperation, a pedagogue might employ a trial-and-error approach by attempting different reading materials with Kayla until such time as she begins to read more adeptly. This approach might be effective but might also expose her to materials that prove frustrating, thereby retarding her reading progress.

Insight

Problem-solving is often conceived to involve insight, or the sudden awareness of a likely solution. Wallas (1921) studied eminent problem-solvers and formulated a four-step model as follows:

Wallas’s stages were descriptive and not subjected to empirical verification. Gestalt psychologists also postulated that much human learning was insightful and involved a change in perception. Learners initially considered the ingredients necessary to solve a problem. They integrated these in various ways until the problem was solved. When learners arrived at a solution, they did so suddenly and with insight.

Many problem-solvers report experiencing moments of insight; Watson and Crick had insightful moments in discovering the structure of DNA (Lemonick, 2003). An important educational application of Gestalt theory was in the area of problem-solving, or productive thinking (Duncker, 1945; Luchins, 1942; Wertheimer, 1945). The Gestalt view stressed the rôle of understanding—comprehending the meaning of some event or grasping the principle or rule underlying performance. In contrast, rote memorisation—although used often by students—was inefficient and rarely used in life beyond the confines of the school.

Research by Katona (1940) demonstrated the utility of rule-learning compared with memorisation. In one study, participants were asked to learn number sequences (e.g., 816449362516941). Some learned the sequences by rote, whereas others were given clues to aid learning (e.g., “Think of squared numbers”). Learners who determined the rule for generating the sequences retained them better than those who memorised.

Rules lead to better learning and retention than memorisation, as rules give a simpler description of the phenomenon, so less information must be learned. Furthermore, rules aid in organising material. To recall information, one recalls the rule and then fills in the details. In contrast, memorisation entails recalling more information. Memorisation generally is inefficient, because most situations possess some organisation (Wertheimer, 1945). Problems are solved by discovering the organisation of the situation and the relationship of the elements to the problem solution. By arranging and rearranging elements, learners eventually gain insight into the solution.

Köhler (1926) performed well-known work on problem-solving with apes on the island of Tenerife during the Great War. In one experiment, Köhler placed a banana just out of reach of an ape in a cage; the ape could fetch the banana by using a long stick or by joining two sticks together. Köhler concluded that problem-solving was insightful: Animals surveyed the situation, suddenly “saw” the means for attaining the goal, and tested the solution. The apes’ first problem-solving attempts failed as they tried different ineffective strategies (e.g., throwing a stick at the banana). Eventually, they saw the stick as an extension of their arms and used it accordingly.

In another situation (Köhler, 1925), the animal could see the goal but not attain it without turning away and taking an indirect route. For example, the animal might be in a room with a window and see food outside. To reach the goal, the animal must exit the room via a door and proceed down a corridor that led outside. In going from the pre-solution to the solution phase, the animal might try a number of alternatives before settling on one and employing it. Insight occurred when the animal tested a likely solution.

A barrier to problem-solving is functional fixedness, or the inability to perceive different uses for objects or new configurations of elements in a situation (Duncker, 1945). In a classic study, Luchins (1942) gave individuals problems that required them to obtain a given amount of water using three jars of different sizes. Persons from ages 9 to adult readily learned the formula that always produced the correct amount. Intermixed in the problem set were some problems that could be solved using a simpler formula. Persons generally continued to apply the original formula. Cueing them that there might be an easier solution led some to discover the simpler methods, although many persisted with the original formula. This research demonstrates that when students do not understand a phenomenon, they may blindly apply a known algorithm and fail to understand that easier methods exist. This procedure-bound nature of problem-solving can be overcome when different procedures are emphasised during instruction (Chen, 1999).

Gestalt theory had little to say about how problem-solving strategies are learned, or how learners could be taught to be more insightful. Wertheimer (1945) believed that teachers could aid problem-solving by arranging elements of a situation so that students would be more likely to perceive how the parts relate to the whole. Such general advice may not be helpful for teachers.

Another avenue for problem resolution lies in the employment of heuristics, which are general methods for problem-solving. These methods utilise principles (rules of thumb) that typically conduce to a solution (Anderson, 1990). Polya’s (1945/1957) catalogue of mental operations inherent in problem-solving is as follows:

• Comprehend the problem.
• Formulate a plan.
• Execute the plan.
• Review the outcome.

Comprehending the problem entails posing questions such as, “What is the unknown?” and “What are the givens?”. The construction of a diagram representing the problem and the extant information often proves beneficial. In devising a plan, one endeavours to discover a connection between the data and the unknown. Deconstructing the problem into subgoals is advantageous, as is contemplating a similar problem and its resolution (i.e., employing analogies). The problem may necessitate re-articulation. During the execution of the plan, it is imperative to verify each step to ensure its proper implementation. Reviewing the outcome signifies examining the solution: Is it accurate? Is there an alternative means of attainment?

Bransford and Stein (1984) propounded a comparable heuristic known as IDEAL:

• Identify the problem.
• Define and represent the problem.
• Explore possible strategies.
• Act upon the strategies.
• Review and evaluate the effects of your activities.

The Creative Problem Solving (CPS) model furnishes another instance of a generic problem-solving framework (Treffinger, 1985; Treffinger & Isaksen, 2005). This model encompasses three principal components: comprehending the challenge, generating ideas, and preparing for action (Treffinger, 1995; Treffinger & Isaksen, 2005). Metacognitive components (e.g., planning, monitoring, modifying behaviour) are omnipresent throughout the process.

Comprehending the challenge commences with a general objective or direction for problem-solving. Subsequent to the acquisition of pertinent data (e.g., facts, opinions, concerns), a specific objective or question is formulated. The hallmark of generating ideas is divergent thinking to produce options for attaining the objective. Preparing for action includes scrutinising promising options and seeking sources of assistance and means to surmount resistance.

General heuristics are most efficacious when one is engaged with unfamiliar content (Andre, 1986). They are less effective in a familiar domain, for as domain-specific skills burgeon, students increasingly employ established procedural knowledge. General heuristics possess an instructional advantage: They can aid students in becoming systematic problem-solvers. Albeit the heuristic approach may appear inflexible, there is, in actuality, flexibility in the manner in which steps are executed. For numerous students, a heuristic will be more systematic than their extant problem-solving approaches and will conduce to superior solutions.

Newell and Simon (1972) posited an information processing model of problem-solving that incorporated a problem space with a beginning state, a goal state, and potential solution paths traversing subgoals and necessitating the application of operations. The problem-solver constructs a mental representation of the problem and performs operations to diminish the discrepancy between the beginning and goal states. The process of operating on the representation to discover a solution is known as the search (Andre, 1986).

The initial step in problem-solving is to construct a mental representation. Akin to Polya’s initial step (comprehend the problem), representation necessitates translating known information into a model in memory. The internal representation comprises propositions, and potentially images, in WM. The problem can also be represented externally (e.g., on paper, computer screen). Information in WM activates related knowledge in LTM, and the solver eventually selects a problem-solving strategy. As individuals solve problems, they frequently alter their initial representation and activate new knowledge, particularly if their problem-solving proves unsuccessful. Thus, problem-solving encompasses evaluating goal progress.

The problem representation determines what knowledge is activated in memory and, consequently, how facile the problem is to resolve (Holyoak, 1984). Should solvers incorrectly represent the problem by neglecting to consider all aspects or by imposing excessive constraints, the search process is unlikely to identify a correct solution path (Chi & Glaser, 1985). Irrespective of how lucidly solvers subsequently reason, they will not attain a correct solution unless they construct a new representation. Unsurprisingly, problem-solving training programmes typically devote considerable time to the representation phase (Andre, 1986).

Much like skills, as discussed heretofore, problem-solving strategies may be construed as either general or specific in nature. General strategies lend themselves to application across diverse domains, irrespective of their particular content; specific strategies, conversely, prove efficacious solely within a circumscribed domain. For instance, the decomposition of a complex problem into subordinate components, a practice commonly referred to as subgoal analysis, stands as a general strategy applicable to matters as diverse as the composition of an academic treatise, the selection of a course of academic study, and the determination of a suitable place of abode. Conversely, certain assays conducted for the purpose of classifying specimens within a laboratory environment are, by their very nature, task-specific. The professional development imparted to the instructors under the charge of Mr. Nikowsky, it is presumed, incorporated both general and specific strategies in judicious measure.

General strategies prove most efficacious when one confronts problems for which solutions are not immediately patent. Amongst such strategies, one may cite generate-and-test methodologies, means-ends analysis, analogical reasoning, and brainstorming. General strategies are of diminished utility when applied to content with which one possesses a high degree of familiarity, in which case domain-specific strategies are to be preferred. Some examples of problem solving within the precincts of learning shall be furnished forthwith:

Generate-and-Test Strategy

The generate-and-test strategy commends itself when the corpus of potential problem solutions is limited in extent, permitting each candidate to be assessed for its capacity to attain the desired end (Resnick, 1985). This strategy yields optimal results when multiple solutions can be ranked in terms of their probability of success, and when at least one solution possesses a reasonable likelihood of resolving the problem.

By way of illustration, let us postulate that one enters a chamber, actuates the light switch, and yet the light fails to illuminate. Possible causes include: burnout of the bulb; interruption of the electrical supply; malfunction of the switch; a defect within the lamp socket; tripping of the circuit breaker; failure of a fuse; or a short circuit within the wiring. One would, in all likelihood, generate and test the most probable solution, to wit, replacement of the bulb. Should this fail to rectify the situation, one might then proceed to generate and test alternative, albeit less probable, solutions. Whilst a high degree of familiarity with the subject matter is not strictly indispensable, a certain modicum of knowledge is requisite for the effective deployment of this method. Prior knowledge serves to establish a hierarchy of potential solutions, whilst current knowledge informs the selection of a particular solution. Thus, should one observe an electric utility vehicle within the vicinity of one's abode, one would, with due promptitude, ascertain whether the power supply had been disrupted.

Means–Ends Analysis

In the application of means–ends analysis, one juxtaposes the current state of affairs with the ultimate goal, identifying any discrepancies that obtain between the two (Resnick, 1985). Subgoals are then formulated with the intent of diminishing these discrepancies. Operations are executed to accomplish each subgoal, whereupon the process is iterated until the overarching goal is achieved.

Newell and Simon (1972) conducted extensive investigations into means–ends analysis, culminating in the formulation of the General Problem Solver (GPS), a computer simulation program designed to emulate this process. GPS decomposes a problem into a series of subgoals, each representing a divergence from the present state. The program commences with the most conspicuous divergence and employs operations with the aim of eradicating it. In certain instances, the operations must first address another divergence that is a prerequisite for the more significant one.

Means-ends analysis constitutes a powerful heuristic in the realm of problem-solving. When subgoals are judiciously defined, means-ends analysis is most likely to yield a solution to the problem at hand. One potential drawback, however, lies in the fact that complex problems may tax the capacity of one's working memory (WM), given the need to maintain awareness of multiple subgoals simultaneously. The forgetting of a subgoal invariably impedes the attainment of a solution.

Means-ends analysis may proceed either from the ultimate goal back toward the initial state (a process known as working backward), or from the initial state forward toward the ultimate goal (working forward). In working backward, one commences with the ultimate goal and poses the question: what subgoals must be realised in order to accomplish this goal? One then proceeds to enquire as to the prerequisites for the attainment of each subgoal, and so forth, until the initial state is reached. The successful application of working backward necessitates a fair degree of knowledge within the problem domain, in order to determine the prerequisites for both the ultimate goal and each intermediate subgoal.

The technique of working backward is frequently employed in the proving of geometrical theorems. One commences by assuming the truth of the theorem and then works backward until the fundamental postulates are reached. An example of this approach, applied to a geometrical problem, is furnished in Figure 'Means–ends analysis applied to a geometry problem'. The objective is to determine the measure of angle m. Proceeding backward, students recognise that they must first ascertain the measure of angle n, inasmuch as angle m = 180° less angle n (a straight line subtends an angle of 180°). Continuing to work backward, students further comprehend that, owing to the intersection of parallel lines, the corresponding angle d on line q is equal in measure to angle n. Drawing upon their geometrical knowledge, students deduce that angle d = angle a, which is given as 30°. Ergo, angle n = 30°, and angle m = 180° - 30° = 150°.

As a further illustration of the working backward strategy, let us consider the scenario of an academic paper due in three weeks' time. The final step prior to submission would be to proofread the document (to be undertaken on the day preceding the due date). The penultimate step would entail the typing and printing of the final copy (allowing one day for this task). Prior to this, one would make final revisions (one day), revise the paper itself (three days), and type and print a draft copy (one day). Continuing to work backward, we might allocate five days for the composition of the draft, one day for outlining, three days for library research, and one day for the selection of a topic. This yields a total of seventeen days devoted to the partial undertaking of the paper. Thus, one would need to commence work four days hence.

A second variant of means–ends analysis, known as working forward, is sometimes referred to as hill climbing (Matlin, 2009; Mayer, 1992). The problem solver commences with the current situation and endeavours to modify it in such a manner as to move closer to the ultimate goal. The attainment of the goal typically necessitates a series of such modifications. One potential pitfall lies in the fact that working forward may, on occasion, proceed on the basis of a superficial analysis of the problem. Whilst each step represents an attempt to realise a necessary subgoal, one may readily diverge onto a tangent or arrive at a cul-de-sac, inasmuch as one is typically limited to perceiving only the immediately subsequent step, rather than the broader array of alternatives that may lie ahead (Matlin, 2009).

As an illustration of the working forward strategy, let us consider a group of students in a laboratory setting who are confronted with a collection of jars containing various substances. Their objective is to affix labels to these jars, correctly identifying the contents thereof. To this end, they perform a series of tests on the substances which, if executed correctly, will culminate in a solution. This represents a working forward strategy, inasmuch as each test brings the students progressively closer to their goal of classifying the substances. The tests are ordered, and the results thereof serve to indicate not only what the substances are not, but also what they potentially might be. To forestall the students from embarking upon an erroneous course of action, the instructor meticulously establishes the procedure and ensures that the students possess a clear understanding of how to perform each test.

Another general strategy in problem-solving doth reside in the application of analogical reasoning, whereby one doth draw a parallel 'twixt the problem at hand (the target) and a situation already comprehended (the base or source; Anderson, 1990; Chen, 1999; Hunt, 1989). The problem is tackled within the familiar domain, and thence the solution is related to the problematic circumstance (Holyoak & Thagard, 1997). Analogical reasoning entail accessing the familiar domain’s network within Long-Term Memory, and mapping it upon (relating it to) the problem situation within Working Memory (Halpern, Hansen, & Riefer, 1990). Success hinges upon the structural similarity between the familiar situation and the problem situation, albeit surface features may differ (e.g., one might concern the solar system, the other molecular structures). The subgoals in this approach involve relating the steps in the original (familiar) domain to those in the transfer (problem) area. Students oft employ this analogy method to solve textbook problems, wherein examples are wrought in the text (familiar domain), and students then relate these steps to the problems they must resolve.

Gick and Holyoak (1980, 1983) did demonstrate the power of analogical problem-solving. They did present learners with a difficult medical problem, coupled with a solved military problem as an analogy. Merely presenting the analogical problem did not automatically prompt its usage. However, providing a hint to utilise the military problem for the medical problem did improve problem-solving. Gick and Holyoak also discovered that presenting two analogous stories led to superior problem-solving than but a single story. However, summarising the analog story, presenting the principle underlying the story during reading, or providing a diagram illustrating the problem-solution principle did not enhance problem-solving. These results suggest that, within an unfamiliar domain, students require guidance in utilising analogies, and that multiple examples increase the likelihood of students linking at least one example to the problem awaiting resolution.

For optimal effectiveness, analogical problem-solving demands a sound comprehension of both the familiar and problematic domains. Students oft grapple with utilising analogies to solve problems, even when the solution strategy is highlighted. With inadequate knowledge, students are unlikely to perceive the relation 'twixt the problem and the analogue. Even assuming sound knowledge, the analogy is most prone to failure when the familiar and problem domains are conceptually dissimilar. Learners may comprehend how fighting a battle (the military problem) mirrors fighting a disease (the medical problem), but they may not grasp other analogies (e.g., fighting a corporate takeover attempt).

Developmental evidence doth indicate that, despite its inherent difficulties, children can indeed employ analogical reasoning (Siegler, 1989). Instructing children in analogies—including those with learning disabilities—can improve their subsequent problem-solving abilities (Grossen, 1991). The utilisation of case studies and case-based reasoning can foster analogical thinking (Kolodner, 1997). Effective techniques for employing analogies encompass having the teacher and child verbalise the solution principle underlying both the original and transfer problems, prompting children to recall elements of the original problem’s causal structure, and presenting the two problems such that the causal structures proceed from the most obvious to the least (Crisafi & Brown, 1986). Further suggestions include employing similar original and transfer problems, presenting several similar problems, and utilising pictures to depict causal relations.

This is not to suggest that all children can become experts in employing analogies. The task is difficult, and children oft draw inappropriate analogies. Compared with older students, younger ones require more prompting, are more apt to be distracted by irrelevant perceptual features, and process information with less efficiency (Crisafi & Brown, 1986). Children’s success depends heavily on their knowledge of the original problem and their skill at encoding and making mental comparisons, which exhibit wide individual differences (Richland, Morrison, & Holyoak, 2006; Siegler, 1989). Children learn problem-solving strategies better when they observe and explain them than when they merely observe (Crowley & Siegler, 1999).

Analogical problem-solving proves useful in teaching. Teachers oft find students in their classes whose native tongue is not English. Instructing students in their native language is oft an impossibility. Teachers might relate this problem to teaching students who have difficulty learning. With the latter students, teachers would proceed at a measured pace, employing concrete experiences whenever feasible, and providing ample individual instruction. The same tactics might be attempted with students of limited-English proficiency, whilst simultaneously teaching them English words and phrases that they might keep pace with the other students in class.

This analogy is apt, for students with learning problems and students who speak little English both encounter difficulties within the classroom. Other analogies may be deemed inappropriate. Unmotivated students, too, face learning difficulties. Employing them for the analogy, the teacher might offer the limited-English-proficiency students rewards for learning. This solution is not apt to be effective, as the issue with limited-English-proficiency students is more instructional than motivational.

Cogitation constitutes a general strategem for problem-solving, efficacious in the formulation of putative solutions (Isaksen & Gaulin, 2005; Mayer, 1992; Osborn, 1963). The procedural steps inherent in cogitation are delineated hereunder:

• Define precisely the problem at hand.
• Engender as copious a collection of solutions as is feasible, abstaining from evaluative judgements.
• Establish criteria whereby the merit of prospective solutions may be assessed.
• Apply these established criteria to the selection of the most propitious solution.

Successful cogitation necessitates that participants suspend any inclination to critique nascent notions until such time as all conceivable ideas have been propounded. Furthermore, participants are at liberty to elaborate upon extant ideas, thereby fostering a synergistic effect. Consequently, ideas of an unorthodox or unconventional nature ought to be actively encouraged (Mayer, 1992).

As per analogical problem-solving, the extent of one's knowledge pertaining to the problem domain exerts an influence on the efficacy of cogitation, inasmuch as superior domain knowledge facilitates the generation of a greater multitude of potential solutions and criteria by which to judge their viability. Cogitation may be practiced individually, albeit the interaction of a group typically yields a more comprehensive array of solutions.

Cogitation lends itself admirably to numerous instructional and administrative determinations within educational institutions. It is most beneficent for the generation of a diverse—and, potentially, uniquely novel—range of ideas (Isaksen & Gaulin, 2005). Consider, for instance, a newly instated school principal who discerns a paucity of morale amongst the staff. The staff concur that improved communication is requisite. The grade-level convenes with the principal, and the collective arrives at the following prospective solutions: Institution of a weekly staff meeting, circulation of a weekly (electronic) bulletin, erection of notices upon a bulletin board, convocation of weekly meetings with grade-level leaders (subsequent to which they shall meet with teachers), dissemination of informational missives via electronic mail with regularity, and pronouncements over the public address system. The group formulates two criteria: (a) minimal temporal demand upon the educators and (b) minimal disruption to the conduct of classes. Bearing these criteria in mind, they determine that the principal ought to disseminate a weekly bulletin, dispatch frequent e-mail communications, and convene meetings with grade-level leaders as a collective. Albeit these activities will exact a temporal toll, the assemblies between the principal and grade-level leaders shall be more intently focused than those involving the entirety of the staff.

Problem-solving doth oft partake in the act of learning, albeit the twain concepts be not synonymous in their signification. According to a contemporary view grounded in information processing (Anderson, 1990, 1993, 2000), problem-solving doth encompass the acquisition, retention, and application of production systems. These systems constitute networks of condition–action sequences (rules), wherein the conditions delineate the sets of circumstances that activate the system, and the actions encompass the sets of activities that ensue (Anderson, 1990; Andre, 1986). A production system doth consist of if-then statements.

If statements (the condition) incorporate both the aim and test statements; then statements represent the actions.

Productions be forms of procedural knowledge that incorporate declarative knowledge and the conditions under which these forms are applicable. Productions be represented in Long-Term Memory (LTM) as propositional networks, and are acquired in like manner to other procedural knowledge. Furthermore, productions be organised hierarchically, featuring both subordinate and superordinate productions. For example, to resolve two equations with two unknowns, one doth first represent one unknown in terms of the second unknown (subordinate production), after which one solveth for the second unknown (production) and useth that value to solve for the first unknown (superordinate production).

Productions may be either general or specific. Specific productions apply to content within well-defined areas. In contrast, heuristics constitute general productions, as they apply to diverse content. A means–ends analysis might be represented thusly (Anderson, 1990):

If the goal be to transform the current state into the desired goal state, and 'D' be the largest disparity between the states -> Then set as subgoals:

1. To eliminate the difference 'D'.
2. To convert the resulting state into the goal state.

A second production will then necessitate employment with the if-then statement, “If the goal be to eliminate the difference 'D'.” This sequence persisteth until the subgoals have been identified at a specific level; whereupon domain-specific rules are applied. In brief, general productions are dissected until the level at which domain-specific knowledge is brought to bear. Production systems offer a means of connecting general with specific problem-solving procedures. Other problem-solving strategies (e.g., analogical reasoning) may also be represented as productions.

School learning that is highly regulated may not require problem-solving. Problem-solving is not applicable when students possess a goal and a clear means for its attainment. Problem-solving gains in importance when teachers deviate from lockstep, highly regimented instruction and encourage more original and critical thinking amongst their students. This reflects the endeavours undertaken by the teachers at Nikowsky following their discourse with Meg. There exists a movement within education to foster problem-solving skills in students, and many educators anticipate that this trend will persist. In the interim, students must acquire both general and specific problem-solving strategies to effectively manage the augmented demands associated with learning.

As regards skill acquisition, researchers have discerned disparities 'twixt novice and expert problem solvers (Anderson, 1990, 1993; Bruning et al., 2004; Resnick, 1985). One such disparity pertains to the demands placed upon Working Memory (WM). Expert problem solvers do not activate a surfeit of potentially relevant information; rather, they identify salient features of the problem, relate these to antecedent knowledge, and formulate one or a limited number of potential solutions (Mayer, 1992). Experts attenuate the complexity of problems by segregating the problem space from the broader task environment, which encompasses the domain of facts and knowledge within which the problem is ensconced (Newell & Simon, 1972). Concomitant with the fact that experts exhibit a greater capacity for holding information in WM (Chi, Glaser, & Farr, 1988), this reductive process preserves pertinent information, discards that which is irrelevant, operates within the bounds of WM, and is sufficiently accurate to proffer a solution.

Experts frequently employ a 'working forward' strategy, identifying the problem format and engendering an approach apposite to it (Mayer, 1992). This typically entails dissecting the problem into constituent parts and resolving each sequentially (Bruning et al., 2004). Novice problem solvers, however, often attempt problem solving in a piecemeal fashion, owing, in part, to the inferior organisation of their memories. They may resort to trial and error or endeavour to work backward from the desideratum to the problem givens—an ineffectual strategy if they are unaware of the requisite sub-steps (Mayer, 1992). Their means-ends analyses are oft predicated upon superficial features of problems. In mathematics, novices summon formulae from memory when confronted with word problems, the attempt to store excess information in WM serving only to obfuscate their thinking (Resnick, 1985).

Experts and novices also diverge in background domain-specific knowledge, albeit they appear to be comparably versed in knowledge of general problem-solving strategies (Elstein, Shulman, & Sprafka, 1978; Simon, 1979). Experts possess more extensive and better organised Long-Term Memory (LTM) structures within their area of expertise (Chi et al., 1981). The greater the amount of knowledge experts can utilise in solving problems, the more likely they are to achieve resolution, and the better their memory organisation facilitates efficiency.

Qualitative differences are manifest in the manner in which knowledge is structured in memory (Chi, Glaser, & Rees, 1982). Experts' knowledge is more hierarchically organised. Experts tend to classify problems according to 'deep structure,' whereas novices rely more heavily on surface features (Hardiman, Dufresne, & Mestre, 1989). Intriguingly, training novices to recognise deep features improves their performance relative to that of untrained novices.

Novices typically respond to problems in terms of their presented form; experts, conversely, reinterpret problems to reveal an underlying structure, one that most likely accords with their own LTM network (Resnick, 1985). Novices attempt to translate the given information directly into formulae and solve for the missing quantities. Rather than engendering formulae, experts may initially draw diagrams to clarify the relations amongst problem aspects. They often construct a novel version of the problem. By the time they are prepared to perform calculations, they usually have simplified the problem and execute fewer calculations than novices. Whilst working, experts more vigilantly monitor their performance to assess goal progress and the value of the strategy they are employing (Gagné et al., 1993).

Finally, experts dedicate more time to planning and analysis. They are more circumspect and do not proceed until they have conceived a strategy. Moore (1990) ascertained that experienced teachers devote more time to planning than do their less experienced counterparts, as well as more time exploring unfamiliar classrooms, such planning facilitating easier strategy implementation.

In summation, the disparities 'twixt novice and expert problem solvers are manifold. Compared with novices, experts:

• Possess a greater corpus of declarative knowledge.
• Exhibit a superior hierarchical organisation of knowledge.
• Allocate more time to planning and analysis.
• Recognise problem formats with greater facility.
• Represent problems at a deeper level.
• Monitor their performances more assiduously.
• Comprehend more fully the value of strategic application.

Reasoning, as a matter of course, doth pertain to those mental operations requisite for the generation and evaluation of logical arguments (Anderson, 1990). 'Tis through reasoning that one arriveth at a conclusion, derived from thoughts, perceptions, and assertions (Johnson-Laird, 1999); it involveth the diligent navigation of problems to elucidate the 'whys' and 'wherefores' of occurrences, or to presage events yet to unfold (Hunt, 1989). The skills intrinsic to reasoning encompass clarification, the establishment of a sound basis, inference, and judicious evaluation (Ennis, 1987; Quellmalz, 1987).

Clarification

Clarification requireth the identification and formulation of questions, the analysis of elements, and the definition of terms. These skills involve determining which elements in a given situation are of import, what their significance may be, and how they interrelate. At times, scientific questions are propounded; at others, however, students must needs develop questions, such as 'What constitutes the problem, hypothesis, or thesis?' Clarification doth correspond to the representation phase of problem-solving; students defining the problem to obtain a lucid mental representation. Little productive reasoning doth transpire without a clear statement of the problem.

Basis

A person's conclusions respecting a problem are sustained by information gleaned from personal observations, pronouncements by others, and prior inferences. Judging the credibility of a source is of import. In so doing, one must distinguish 'twixt fact, opinion, and reasoned judgment. Assume that a suspect, armed with a pistol, is apprehended nigh the scene of a murder. That the suspect did possess a pistol when arrested is a fact. Laboratory tests upon the pistol, the bullets, and the victim lead to the reasoned judgment that the pistol was employed in the commission of the crime. One investigating the case might opine that the suspect is the murderer.

Inference

Scientific reasoning proceedeth either inductively or deductively. Inductive reasoning referreth to the development of general rules, principles, and concepts from observation and knowledge of specific examples (Pellegrino, 1985). It requireth determination of a model and its associated rules of inference (Hunt, 1989). People reason inductively when they extract similarities and differences 'twixt specific objects and events, arriving at generalisations, which are tested by applying them to new experiences. Individuals retain their generalisations so long as they prove effective, modifying them when encountering conflicting evidence.

Some of the more common tasks employed to assess inductive reasoning are problems of classification, concept, and analogy. Consider the ensuing analogy (Pellegrino, 1985):

• sugar : sweet :: lemon : ______
• yellow sour fruit squeeze tea

The appropriate mental operations represent a type of production system. Initially, the learner mentally representeth critical attributes of each term in the analogy. She doth activate networks in LTM involving each term, which contain critical attributes of the terms, to include subordinate and superordinate concepts. Next, she doth compare the features of the first pair to determine the link. 'Sweet' is a property of sugar involving taste. She then searcheth the 'lemon' network to determine which of the five features listed correspondeth in meaning to 'lemon' as 'sweet' doth to 'sugar'. Though all five terms are most like to be stored in her 'lemon' network, only 'sour' directly involves taste.

Children commence to display basic inductive reasoning circa age 8. With development, children can reason more swiftly and with more complex material. This doth occur because their LTM networks become more complex and better linked, which, in turn, reduceth the burden on the WM. To help foster inductive thought, teachers might employ a guided discovery approach, wherein children learn diverse examples and attempt to formulate a general rule. For example, children might collect leaves and formulate certain general principles involving stems, veins, sizes, and shapes of leaves from diverse trees. Teachers might pose a problem for students, such as 'Why doth metal sink in water, yet metal ships float?' Rather than instructing students how to solve the problem, the teacher might furnish materials and encourage them to formulate and test hypotheses as they work upon the task. Phye (1997; Klauer & Phye, 2008) discussed effective teaching methods and programmes which have been employed to impart inductive reasoning to students.

Deductive reasoning referreth to the application of inference rules to a formal model of a problem to decide whether specific instances logically follow. When individuals reason deductively, they proceed from general concepts (premises) to specific instances (conclusions) to determine whether the latter follow from the former. A deduction is valid if the premises be true and if the conclusion follow logically from the premises (Johnson-Laird, 1985, 1999).

Linguistic and deductive reasoning processes are intimately linked (Falmagne & Gonsalves, 1995; Polk & Newell, 1995). One type of deduction problem is the three-term series (Johnson-Laird, 1972). For example,

• If Karen is taller than Tina, and ->; (generating the pattern)
• If Mary Beth is not as tall as Tina, then ->; (reaffirming the patern)
• Who is the tallest? => (pattern complete)

The problem-solving processes employed with this problem are similar to those discussed previously. Initially one formeth a mental representation of the problem, such as , . One then worketh forward by combining the propositions ( ) to solve the problem. Developmental factors limit children's proficiency in solving such problems. Children may have difficulty keeping relevant problem information in WM, and may not understand the language employed to express the relationships.

Another type of deductive reasoning problem is the syllogism. Syllogisms are characterised by premises and a conclusion containing the words all, no, and some. The following are sample premises:

• All university professors are lazy. -> (contextual generalisation)
• Some graduate students are not lazy. -> (contextual exclusion)
• No undergraduate student is lazy. -> (disposition of controversy)

A sample syllogism is as follows:

• All the students in Ken’s class are good in math. -> (premise suggestion)
• All students who are good in math will attend college. -> (bias formation)
• (Therefore) All the students in Ken’s class will attend college. -> (general assumption)

Researchers debate what mental processes people employ to solve syllogisms, including whether people represent the information as Venn (circle) diagrams or as strings of propositions (Johnson-Laird, 1985). A production system analysis of syllogisms giveth a basic rule: a syllogism is true only if there be no way to interpret the premises to imply the opposite of the conclusion; that is, a syllogism is true unless an exception to the conclusion can be found. Research needeth to examine the types of rules people apply to test whether the premises of a syllogism allow an exception.

Different views have been proposed to explain the mechanisms of deductive reasoning (Johnson-Laird, Byrne, & Tabossi, 1989). One view holdeth that reasoning proceedeth on the basis of formal rules of inference. People learn the rules (e.g., the modus ponens rule governeth 'if p then q' statements), and then match instances to the rules.

A second, related view postulateth content-specific rules. They may be expressed as productions, such that specific instances trigger the production rules. Thus, a production may involve all cars, and may be triggered when a specific car ('my brand X') is encountered.

A third view holdeth that reasoning dependeth upon semantic procedures that search for interpretations of the premises that are counterexamples to conclusions. According to this view, people construct one or more mental models for the assertions they encounter (interpretations of the premises); the models differ in structure, and are employed to test the logic of the situation. Students may repeatedly re-encode the problem based upon information; thus, deduction largely is a form of verbal reasoning (Polk & Newell, 1995). Johnson-Laird and colleagues (Johnson-Laird, 1999; Johnson-Laird, Byrne, & Schaeken, 1992; Johnson-Laird et al., 1989) have extended this semantic analysis to various classes of inferences (e.g., those involving if, or, and, not, and multiple quantifiers). Further research will also help to determine instructional implications of these theoretical analyses.

Evaluation

Evaluation encompasseth the application of criteria with the purpose of judging the adequacy of a solution proffered for a given problem. In the course of evaluating, students shall address inquiries such as, “Are the data sufficiently ample to resolve the problem at hand?” “Ought I to procure further intelligence?” and “Are my conclusions predicated upon verifiable facts, personal opinions, or reasoned judgements?” Evaluation doth likewise entail the determination of the ensuing course of action—that is to say, the formulation of hypotheses concerning future occurrences upon the presumption that one's problem-solving endeavours have hitherto been veracious.

Deductive reasoning, moreover, may be influenced by content distinct from mere logic. Wason (1966) arrayed four cards (exhibiting A B 2 3) before the participants. They were apprised that each card bore a letter on one face and a numeral on the other, and they were furnished with a conditional rule: “If a card exhibiteth A on one face, then it exhibiteth 2 on the other.” Their task consisted of selecting those cards which necessitated inversion in order to ascertain the veracity of the rule. Albeit the majority of participants elected the A card and a considerable number did likewise select the 2, few elected the 3; albeit, it must be inverted, for should an A reside on the obverse, the rule is thereby rendered false. Upon the alteration of the content to an everyday generalisation (e.g., letter = hue of hair, numeral = colour of eyes, A = blond hair, 2 = azure eyes), the greater number of individuals effected the correct selections (Wason & Johnson-Laird, 1972). These results bear testimony to the significance of refraining from the presumption of generalisation in reasoning, and instead furnishing students with experience in operating upon disparate categories of content.

Metacognitive processes interpose themselves into every facet of scientific reasoning. Learners maintain vigilance over their endeavours to ensure that inquiries are properly framed, that data from adequate sources are both accessible and employed in the derivation of inferences, and that pertinent criteria are engaged in evaluation. The pedagogy of reasoning necessitates instruction in skills and in metacognitive strategies. Cognitive load doth likewise appear consequential. Scientific reasoning is rendered arduous should multiple sources of information demand simultaneous processing, which imposes a strain upon the working memory. Carlson et al. (2003) ascertained that students' scientific performance profited from two procedures conceived to diminish cognitive load: diagrams and instructions that minimised the quantum of information to be processed concurrently.

The correlation betwixt learning and problem-solving doth suggest that pupils may acquire heuristics and strategies, thereby becoming more adept problem-solvers (Bruning et al., 2004). Furthermore, for information to be duly linked in memory, it is most efficacious to integrate problem-solving with academic content (as Mistress Meg did commend in the introductory scenario), rather than to impart problem-solving through independent programmes. Nokes, Dole, and Hacker (2007) did ascertain that instruction in heuristics may be interwoven into classroom pedagogy without detriment to the students' acquisition of content.

Andre (1986) hath enumerated sundry suggestions, derived from both theory and inquiry, that are of utility in training pupils in the arts of problem-solving, most especially as they embody productions within memory.

• Furnish pupils with metaphorical representations. A tangible analogical passage, presented to pupils prior to an instructional passage, doth facilitate comprehension of the target passage.
• Enjoin pupils to verbalise during problem-solving. The verbalisation of thoughts during problem-solving may expedite the solutions to problems and the process of learning.
• Employ queries. Pose to pupils questions requiring them to practice concepts they have learnt; many such questions may prove requisite.
• Provide exemplars. Impart to pupils numerous worked exemplars, demonstrating the application of problem-solving strategies. Pupils may encounter difficulties in discerning how strategies are applicable to various situations.
• Coordinate notions. Demonstrate how productions and knowledge are interrelated, and in what sequence their application might be necessary.
• Employ discovery learning. Discovery learning oft facilitates transfer and problem-solving more efficaciously than expository teaching. Discovery may compel pupils to generate rules from exemplars. The same may be accomplished through expository teaching, albeit discovery may lend itself more readily to certain content (e.g., scientific experiments).
• Furnish a verbal description. Providing pupils with a verbal description of the strategy and its rules for application can prove beneficial.
• Teach learning strategies. Learners may stand in need of assistance in the utilisation of efficacious learning strategies.
• Employ small groups. A number of inquiries have established that small-group learning doth aid in the development of pupils' problem-solving skills. Group members must be held accountable for their learning, and all pupils must partake in the labour.
• Maintain a positive psychological climate. Psychological factors are of import to efficacious problem-solving. Minimise excessive anxiety amongst pupils and aid in fostering a sense of self-efficacy amongst pupils for the betterment of their skills.

Another suggestion for instruction lies in phasing in problem-solving, which may be of particular assistance to pupils with scant experience thereof. This may be achieved through the utilisation of worked exemplars (Atkinson, Renkl, & Merrill, 2003; Renkl & Atkinson, 2003; discussed subsequently in this course's section). Mathematics texts, for instance, oft state a rule or theorem, followed by one or more worked exemplars. Pupils then resolve comparable problems by applying the steps from the worked exemplars (a species of analogical reasoning). Renkl and Atkinson did commend reliance upon exemplars in the early stages of learning, followed by a transition to problem-solving as pupils cultivate their skills. This process doth also serve to minimise demands upon WM, or the cognitive load that learners experience. Thus, the transition might proceed as follows: initially, a complete exemplar is given; then, an exemplar wherein one step is omitted. With each succeeding exemplar, an additional step is omitted until the learners attain independent problem-solving.

Problem-based learning (PBL; Hmelo-Silver, 2004) doth proffer another application for instruction. In this approach, pupils collaborate in groups upon a problem that doth not admit of a single correct solution. Pupils identify what they must know in order to resolve the problem. Teachers act as facilitators, providing assistance but not answers. PBL hath been demonstrated to be efficacious in imparting problem-solving and self-regulation skills, albeit the majority of inquiry hath been conducted in medical and gifted education (Evenson, Salisbury-Glennon, & Glenn, 2001; Hmelo-Silver, 2004). PBL is of utility for the exploration of meaningful problems. As it is time-consuming, teachers must deliberate upon its appropriateness, having regard to the instructional goals.