Cognitive Learning Processes: Instructional Applications

Worked examples, as briefly discussed heretofore in our course, present problem solutions in a stepwise fashion, often accompanied by diagrams. They delineate an expert's problem-solving paradigm for scholars to scrutinise ere they embark upon emulation.

Worked examples are reflective of Anderson's ACT-R theory (Lee & Anderson, 2001) and are particularly apposite for complex forms of learning, such as algebra, physics, and geometry (Atkinson et al., 2000, 2003). Applying the novice-expert model, researchers have ascertained that experts typically focus on deeper (structural) aspects of problems, whereas novices more frequently address surface features. Practice alone is less efficacious in fostering skills than practice coupled with worked examples (Atkinson et al., 2000).

Worked examples appear most advantageous for students in the nascent stages of skill acquisition, as opposed to proficient learners refining skills. Its applicability is distinctly manifest in the four-stage model of skill acquisition within the ACT-R framework (Anderson, Fincham, & Douglass, 1997). In stage 1, learners employ analogies to relate examples to problems to be solved. In stage 2, they develop abstract declarative rules through practice. During stage 3, performance becomes swifter and smoother as aspects of problem solution become automatised. By stage 4, learners retain in memory numerous types of problems and can promptly retrieve the appropriate solution strategy when confronted with a problem. The employment of worked examples is best suited for stage 1 and early stage 2 learners. During later stages, individuals benefit from practice to hone their strategies, albeit even at advanced stages, the study of expert solutions can prove helpful.

A cardinal instructional issue pertains to the integration of example components, such as diagram, text, and aural information. It is imperative that a worked example not overburden the learner's WM, which simultaneous presentation from multiple sources can engender. Stull and Mayer (2007) discovered that providing graphic organisers (akin to worked examples) yielded superior problem-solving transfer compared to allowing learners to construct their own. The latter task may have engendered excessive cognitive load. Further evidence suggests that worked examples can mitigate cognitive load (Renkl, Hilbert, & Schworm, 2009).

Research corroborates the prediction that dual presentation facilitates learning more effectively than single-mode presentation (Atkinson et al., 2000; Mayer, 1997). This result aligns with dual-coding theory (Paivio, 1986), with the caveat that excessive complexity is undesirable. Similarly, examples interspersed with subgoals aid in the creation of deep structures and facilitate learning.

A key point is that examples encompassing multiple presentation modes should be unified such that learners' attention is not bifurcated across nonintegrated sources. Aural and verbal explanations ought to indicate the aspect of the example to which they pertain, lest learners be compelled to search on their own. Subgoals should be clearly labelled and visually isolated in the overall display.

A secondary instructional issue concerns the sequencing of examples. Research supports the conclusions that two examples are superior to a single one, that varied examples are preferable to two of the same type, and that intermixing examples and practice is more effective than a lesson presenting examples followed by practice problems (Atkinson et al., 2000). Gradually fading out worked examples in an instructional sequence is associated with enhanced student transfer of learning (Atkinson et al., 2003).

Chi, Bassok, Lewis, Reimann, and Glaser (1989) found that students who furnished self-explanations whilst studying examples subsequently attained higher levels compared with students who did not self-explain. Presumably, the self-explanations aided students in comprehending the deep structure of the problems, thereby encoding it more meaningfully. Self-explanation is also a form of rehearsal, and the benefit of rehearsal on learning is well established. Thus, students should be encouraged to self-explain whilst studying worked examples, such as by verbalising subgoals.

Another issue is that worked examples can engender passive learning since learners may process them superficially. Incorporating interactive elements, such as prompts or gaps that learners must complete, conduces to more active cognitive processing and learning (Atkinson & Renkl, 2007). Animations are likewise helpful (Wouters, Paas, & van Merriënboer, 2008).

In summary, sundry features, when incorporated with worked examples, aid learners in creating cognitive schemas to facilitate subsequent achievement. These instructional strategies are best employed during the early stages of skill learning. Through practice, the initial cognitive representations should evolve into the refined schemas employed by experts.

Suggestions for using worked examples in instruction

• Present examples in close proximity to problems students shall solve.
• Present multiple examples exhibiting different types of problems.
• Present information in different modalities (aural, visual).
• Indicate subgoals in examples.
• Ensure that examples present all information requisite to solve problems.
• Teach students to self-explain examples, and encourage self-explanations.
• Allow sufficient practice on problem types, so students refine skills.

The art of writing doth reflect many of the cognitive processes discussed within this section of our course. Good writers are not merely born, but diligently developed; effective instruction is critical for the cultivation of writing skills (Graham, 2006; Harris, Graham, & Mason, 2006; Scardamalia & Bereiter, 1986; Sperling & Freedman, 2001).

Contemporary models scrutinise the mental processes of writers as they engage in divers aspects of writing (Byrnes, 1996; de Beaugrande, 1984; Graham, 2006; Mayer, 1999; McCutchen, 2000). A research goal is to define expertise. By comparing expert writers with novices, researchers identify how their mental processes diverge (Bereiter & Scardamalia, 1986).

Flower and Hayes (1980, 1981a; Hayes, 1996; Hayes & Flower, 1980) formulated a model that reflects the general problem-solving framework developed by Newell and Simon (1972). Writers define a problem space and perform operations on their mental representation of the problem to attain their goals. Key components of this model are the rhetorical problem, planning, organising, goal setting, translating, and reviewing.

The rhetorical problem encompasseth the writer’s topic, intended audience, and goals. The rhetorical problem for students often is well defined. Teachers assign a term paper topic, the audience is the teacher, and the goal (e.g., to inform, to persuade) is provided; however, the rhetorical problem is ne'er defined completely by someone other than the writer. Writers interpret problems in their own ways.

The writer’s Long-Term Memory (LTM) plays a crucial role. Writers differ in their knowledge of the topic, audience, and mechanics (e.g., grammar, spelling, punctuation). Writers knowledgeable about their topics include fewer irrelevant statements but more auxiliary statements (designed to elaborate upon main points) compared with less knowledgeable writers (Voss, Vesonder, & Spilich, 1980). Differences in declarative knowledge affect the quality of writing.

Planning involves forming an internal representation of knowledge to be used in composing. The internal representation generally is more abstract than the actual writing. Planning includes several processes such as generating ideas by retrieving relevant information from memory or other sources. These ideas may be well formed or fragmentary.

There are wide individual differences in planning. Children’s writing typically resembles “knowledge telling” (McCutchen, 1995; Scardamalia & Bereiter, 1982). They often follow a “retrieve and write” strategy by accessing LTM with a cue and writing what they know. Children do little planning and reviewing and much translating. Whereas older writers also retrieve content from LTM, they do it as part of planning, after which they evaluate its appropriateness prior to translating. Children’s retrieval and translating are integrated in seamless fashion (Scardamalia & Bereiter, 1986).

Young children produce fewer ideas than older ones (Scardamalia & Bereiter, 1986). They benefit from prompting (e.g., “Can you write some more?”). Englert, Raphael, Anderson, Anthony, and Stevens (1991) shewed that fourth and fifth graders’ writing improved when they were exposed to teachers who modelled metacognitive components (e.g., which strategies were useful, when and why they were useful) and when they were taught to generate questions during planning. Older and better writers make greater use of internal prompts. They search relevant topics in LTM and assess knowledge before they begin composing. Teachers can foster idea generation by cueing students to think of ideas (Bruning et al., 2004).

Organising is conveyed through cohesion among sentence parts and coherence across sentences. Cohesive devices tie ideas together with pronouns, definite articles, conjunctions, and word meanings. Young children have more difficulty with cohesion, but unskilled writers of any age use cohesion less well. Developmental differences also are found in coherence. Young and poor writers have difficulty linking sentences with one another and with the topic sentence (McCutchen & Perfetti, 1982).

A major subprocess is goal setting. Goals are substantive (what the writer wants to communicate) and procedural (how to communicate or how points should be expressed). Good writers often alter their goals based on what they produce. Writers have goals in mind prior to writing, but as they proceed, they may realise that a certain goal is not relevant to the composition. New goals are suggested by actual writing.

The primary goal of skilled writers is to communicate meaning, whereas poor writers often practice associative writing (Bereiter, 1980). They may believe the goal of writing is to regurgitate everything they know about the topic; order is less important than inclusiveness. Another goal of less-skilled writers is to avoid making errors. When asked to critique their own writing, good writers focus on how well they communicated their intentions, whereas poor writers cite surface considerations (e.g., spelling, punctuation) more often.

Translation refers to putting one’s ideas into print. For children and inexperienced writers, translating often overburdens Working Memory (WM). They must keep in mind their goal, the ideas they wish to express, and the necessary organisation and mechanics. Good writers concern themselves less with surface features during translation; they focus more on meaning and correct surface problems later. Poor writers concentrate more on surface features and write more slowly than good writers. Better writers take stylistic and surface considerations into account when they pause during writing. Poorer writers benefit when they read what they have written as they prepare to compose.

Reviewing consists of evaluating and revising. Reviewing occurs when writers read what they have written as a precursor to further translation or systematic evaluation and revision (Flower & Hayes, 1981a; Hayes & Flower, 1980). During reviewing, writers evaluate and modify plans and alter subsequent writing.

These processes are important because writers may spend as much as 70% of their writing time pausing (Flower & Hayes, 1981), much of which is spent on sentence-level planning. Writers reread what they have written and decide what to say next. These bottom-up processes construct a composition a section at a time. When such building up is accomplished with the overall plan in mind, the composition continues to reflect the writers’ goals.

Poor writers typically depend on bottom-up writing. Whilst pausing, good writers engage in rhetorical planning not directly linked to what they have produced. This type of planning reflects a top-down view of writing as a problem-solving process; writers keep an overall goal in mind and plan how to attain it or decide that they need to alter it. Planning includes content (deciding what topic to discuss) and style (deciding to alter the style by inserting an anecdote). This planning subsumes sentence-level planning and is characteristic of mature writers (Bereiter & Scardamalia, 1986).

Children may do little revising without teacher or peer support (Fitzgerald, 1987). Students benefit from instruction designed to improve the quality of their writing. Fitzgerald and Markham (1987) gave average sixth-grade writers instruction on types of revisions: additions, deletions, substitutions, and rearrangements. The teacher explained and modelled each revision strategy, after which students worked in pairs (peer conferences). Instruction improved students’ knowledge of revision processes and their actual revisions. Beal, Garrod, and Bonitatibus (1990) found that teaching third- and sixth-grade children a self-questioning strategy (e.g., “What is happening in the story?”) led to significantly greater text revising.

Evaluation skills develop earlier than revision skills. Even when fourth graders recognise writing problems, they may not successfully correct them as often as 70% of the time (Scardamalia & Bereiter, 1983). When children correct problems, poor writers revise errors in spelling and punctuation, whereas better writers revise for stylistic reasons (Birnbaum, 1982).

Given the complexity of writing, the course of skill acquisition is better characterised as the development of fluency rather than automaticity (McCutchen, 1995). Automatic processes become routinized and require few attentional or WM resources, whereas fluent processes—although rapid and resource efficient—are thoughtful and can be altered “online.” Good writers follow plans but revise them as they write. Were this process automatic, writers’ plans—once adopted—would be followed without interruption. Although component skills of writing (i.e., spelling, vocabulary) often become automatic, the overall process does not.

Mathematics hath proven a fertile ground for cognitive and constructivist inquiries (Ball, Lubienski, & Mewborn, 2001; National Research Council, 2000; Newcombe et al., 2009; Schoenfeld, 2006; Voss et al., 1995). Investigators have explored the manner in which learners construct knowledge, the distinctions between expert and novice practitioners, and the efficacy of diverse instructional methodologies (Byrnes, 1996; Mayer, 1999; Schoenfeld, 2006). The amelioration of instruction doth bear import, considering the difficulties encountered by a multitude of students in the acquisition of mathematical understanding.

A distinction is typically drawn between mathematical computation (employment of rules, procedures, and algorithms) and concepts (problem-solving and strategic application). Computational and conceptual problems necessitate the student's implementation of productions entailing rules and algorithms. The salient disparity betwixt these two categories resides in the explicitness with which the problem doth convey to the student the operations to be performed. The ensuing examples serve as computational problems.

• Solve for x and y.
• What be the length of the hypotenuse of a right triangle, its sides equalling 3 and 4 inches?

Albeit the students are not explicitly instructed as to the appropriate course of action in problems 2 and 3, recognition of the problem's format and a familiarity with established procedures shall guide them toward the performance of the correct operations.

Now, contrast the aforementioned problems with the following:

• Alex possesses 20 coins, a composition of dimes and quarters. Should the quarters be dimes, and the dimes quarters, he would then possess 90 cents more than his current holdings. How much currency doth Alex presently own?
• If a passenger train doth require twice the duration to traverse a freight train, subsequent to its initial overtaking, as it doth require for the two trains to pass when travelling in opposing directions, by what factor doth the passenger train exceed the velocity of the freight train?
• When engaging in hill-walking, Shana doth maintain an average velocity of 2 mph whilst ascending, and 6 mph whilst descending. If she proceeds uphill and downhill, allotting no duration to the summit, what shall be her average velocity for the entirety of the journey?

These word problems, whilst not explicitly delineating the necessary actions, nonetheless require computations of no greater difficulty than those encountered in the initial set. Solving word problems involves recognition of their characteristic formats, generation of appropriate productions, and execution of the required computations.

This assertion is not intended to imply that conceptual expertise surpasses computational proficiency, albeit Rittle-Johnson and Alibali (1999) discovered that conceptual understanding exerted a greater influence upon procedural knowledge than the inverse. Deficiencies in either domain doth engender complications. Comprehension of the problem-solving method, coupled with an inability to execute the requisite computations, shall yield incorrect solutions, as shall computational proficiency bereft of conceptual understanding.

Computation

The earliest computational faculty exercised by children is that of counting (Byrnes, 1996; Resnick, 1985). Children enumerate objects upon their digits and within their minds, employing a strategic methodology (Groen & Parkman, 1972). The sum model entails the setting of a hypothetical counter at zero, the counting in of the first addend in increments of one, and subsequently the counting in of the second addend to arrive at the solution. For the problem “2 + 4 = ?” children might count from 0 to 2, and then count out 4 more. A more efficacious strategy is to set the counter at the first addend (2), and then count in the second addend (4) in increments of one. Yet more efficacious is the min model: Set the counter at the larger of the two addends (4), and then count in the smaller addend (2) in increments of one (Romberg & Carpenter, 1986).

These types of invented procedures are successful. Children and adults oft construct procedures to solve mathematical problems. Errors generally are not random but rather reflect buggy algorithms, or systematic mistakes in thinking and reasoning (Brown & Burton, 1978). Buggy algorithms reflect the constructivist assumption that students form procedures based on their interpretation of experiences. A common mistake in subtraction is to subtract the smaller number from the larger number in each column, regardless of direction, as follows:

• 53 - 27 = 34
• 602 - 374 = 472

Mathematical bugs probably develop when students encounter new problems and incorrectly generalise productions. In subtraction without regrouping, for example, students subtract the smaller number from the larger one column by column. It is easy to see how they could generalize this procedure to problems requiring regrouping. Buggy algorithms are durable and can instil in students a false sense of self-efficacy, perhaps because their computations produce answers.

Another source of computational difficulties is poor declarative knowledge of number facts. Many children do not know basic facts and show deficiencies in numerical processing (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007). Until facts become established in LTM through practice, children count or compute answers. Speed of fact retrieval from memory relates directly to overall mathematical achievement in students from elementary school through college (Royer, Tronsky, Chan, Jackson, & Marchant, 1999). Computational skill improves with development, along with WM and LTM capabilities (Mabbott & Bisanz, 2003).

Many difficulties in computation result from using overly complex but technically correct productions to solve problems. Such procedures produce correct answers, but because they are complex, the risk of computational errors is high. The problem 256 divided by 5 can be solved by the division algorithm or by successively subtracting 5 from 256 and counting the number of subtractions. The latter procedure is technically correct but inefficient and has a high probability of error.

Learners initially represent computational skill as declarative knowledge in a propositional network. Facts concerning the different steps (e.g., in the algorithm) are committed to memory through mental rehearsal and overt practice. The production that guides performance at this stage is general; for example: “If the goal is to solve this division problem, then apply the method the teacher taught us.” With added practice, the declarative representation changes into a domain-specific procedural representation and eventually becomes automated. Early counting strategies are replaced with more-efficient rule-based strategies (Hopkins & Lawson, 2002). At the automatic stage, learners quickly recognize the problem pattern (e.g., division problem, square root problem) and implement the procedure without much conscious deliberation.

The resolution of problems necessitates that students, in the first instance, accurately represent the matter at hand, encompassing both the information provided and the objective sought, and subsequently elect and implement a problem-solving strategy (Mayer, 1985, 1999). The transmutation of a problem from its linguistic expression into a mental representation frequently presents a challenge (Bruning et al., 2004). The more abstract the idiom employed, the greater the impediment to textual comprehension, and correspondingly, the lower the probability of a solution being achieved (Cummins, Kintsch, Reusser, & Weimer, 1988). Students encountering difficulties in comprehension demonstrate diminished recall of information and attenuated performance. This observation is particularly pertinent to younger pupils, who grapple with the translation of abstract linguistic representations.

Translation, furthermore, mandates robust declarative and procedural knowledge. The solution to the aforementioned problem concerning Alex and his assemblage of twenty coins necessitates the cognizance that dimes and quarters constitute coinage, that a dime embodies one-tenth ($0.10) of a dollar, and that a quarter represents one-fourth ($0.25) of the same. This declarative knowledge must be conjoined with a procedural understanding that dimes and quarters function as variables, such that the summation of the number of dimes and the number of quarters equals twenty.

One causative factor in the superior translational acumen of experts resides in the enhanced organisation of their knowledge within long-term memory; the organisation mirroring the intrinsic structure of the subject matter (Romberg & Carpenter, 1986). Experts tend to disregard superficial attributes of a problem, instead analysing it with reference to the operations requisite for its solution. Novices, conversely, are more readily influenced by surface-level features. Silver (1981) ascertained that adept problem solvers categorised problems contingent upon the process indispensable for their solution, whereas less proficient problem solvers exhibited a propensity to group problems exhibiting analogous content (e.g., money, trains).

Beyond disparities in problem translation and categorisation, experts and novices diverge also in their productions (Greeno, 1980). Novices frequently adopt a strategy of working backward, commencing with the objective and regressing towards the given data. This heuristic proves valuable in the nascent stages of learning, when learners have acquired a modicum of domain knowledge but lack the competence to swiftly discern problem formats.

In contradistinction, experts commonly proceed forward. They identify the problem's classification and elect the appropriate production to effect a solution. Hegarty, Mayer, and Monk (1995) discovered that successful problem solvers employed a problem model approach, translating the problem into a mental model wherein the numerical values within the problem statement were correlated with their respective variable names. Conversely, less successful solvers exhibited a greater likelihood of employing a direct translation approach, combining the numerical values within the problem with arithmetic operations prompted by keywords (e.g., the association of addition with the keyword “more”). The latter strategy is superficial and predicated upon surface features, whereas the former is more intimately associated with meanings.

Experts cultivate sophisticated procedural knowledge for the categorisation of mathematical problems according to type. Problems encountered in secondary school algebra can be broadly assigned to approximately twenty general categories, encompassing motion, current, coins, and interest/investment, inter alia (Mayer, 1992). These categories may be aggregated into six principal groups. For instance, the amount-per-time group encompasses problems pertaining to motion, current, and work. These problems are amenable to solution via the general formula: amount = rate time. The attainment of expertise in mathematical problem-solving hinges upon the correct classification of a problem into its appropriate group and the subsequent application of the pertinent strategy. The verbalisation of steps in problem-solving facilitates the development of proficiency (Gersten et al., 2009).

Many theorists do maintain that constructivism doth represent a viable model for the explication of how mathematics is learned (Ball et al., 2001; Cobb, 1994; Lampert, 1990; Resnick, 1989). Mathematical knowledge is not passively absorbed from the environment, but rather is constructed by individuals as a consequence of their interactions. This construction process doth also include children’s inventing of procedures that incorporate implicit rules.

The following unusual example doth illustrate rule-based procedural invention. Some time ago I was working with a teacher to identify children in her class who might benefit from additional instruction in long division. She named several students and said that Tim also might qualify, but she was not sure. Some days he wrought his problems correctly, whereas other days his work was incorrect and made no sense. I gave him problems to solve and asked him to verbalize whilst working because I was interested in what children thought about whilst they solved problems. This is what Tim said: “The problem is 17 divided into 436. I start on the side of the problem closest to the door . . .” I then knew why on some days his work was accurate and on other days it was not. It depended on which side of his body was closest to the door!

The process of constructing knowledge beginneth in the preschool years (Resnick, 1989). Geary (1995) distinguished biologically primary (biologically based) from biologically secondary (culturally taught) abilities. Biologically primary abilities are grounded in neurobiological systems that have evolved in particular ecological and social niches and that serve functions related to survival or reproduction. They should be seen cross-culturally, whereas biologically secondary abilities should show greater cultural specificity (e.g., as a function of schooling). Furthermore, many of the former should be seen in very young children. Indeed, counting is a natural activity that preschoolers do without direct teaching (Gelman & Gallistel, 1978; Resnick, 1985). Even infants may be sensitive to different properties of numbers (Geary, 1995). Preschoolers show increasing numerical competence involving the concepts of part–whole additivity and changes as increases/decreases in quantities. Conceptual change proceedeth quickly during the elementary years (Resnick, 1989). Teaching children to use schematic diagrams to represent word problems doth facilitate problem solving (Fuson & Willis, 1989).

Mathematical competence doth also depend on sociocultural influence (Cobb, 1994). Vygotsky (1978) stressed the role of competent other persons in the zone of proximal development (ZPD). In contrast to the constructivist emphasis on cognitive reorganizations among individual students, sociocultural theorists advocate cultural practices—especially social interactions (Cobb, 1994). The sociocultural influence is incorporated through such activities as peer teaching, instructional scaffolding, and apprenticeships.

Research doth support the idea that social interactions are beneficial. Rittle-Johnson and Star (2007) found that seventh graders’ mathematical proficiency was enhanced when they were allowed to compare solution methods with partners. Results of a literature review by Springer, Stanne, and Donovan (1999) showed that small-group learning significantly raised college students’ achievement in mathematics and science. Kramarski and Mevarech (2003) found that combining cooperative learning with metacognitive instruction (e.g., reflect on relevant concepts, decide on appropriate strategies to use) raised eighth graders’ mathematical reasoning more than either procedure alone. In addition to these benefits of cooperative learning (Stein & Carmine, 1999), the literature on peer and cross-age tutoring in mathematics reveals that it is effective in raising children’s achievement (Robinson, Schofield, & Steers-Wentzell, 2005). Coordination of the constructivist and sociocultural perspectives is possible; students can develop knowledge through social interactions but then idiosyncratically construct uses of that knowledge.

Cognitive and constructivist learning processes find application in fundamental forms of instruction, yet their significance is amplified in intricate learning scenarios. The cultivation of competence within an academic domain necessitates a comprehension of the facts, principles, and concepts pertinent to that domain, in conjunction with general strategies applicable across diverse domains, and specific strategies tailored to each individual domain. Scholarly investigations have delineated numerous disparities between experts and novices within any given field of study.

Conditional knowledge encompasses the understanding of when and why to employ declarative and procedural knowledge. Mere possession of knowledge regarding what to do and how to do it does not assure triumph. Students are compelled to comprehend the temporal and contextual utility of knowledge and procedures. Conditional knowledge is most plausibly retained within Long-Term Memory (LTM) as propositions interlinked with other declarative and procedural knowledge constructs. Metacognition pertains to the deliberate and conscious superintendence of mental activities. Metacognition subsumes knowledge and monitoring activities purposed to ensure the successful execution of tasks. The development of metacognition commences circa ages five to seven and persists throughout the duration of formal schooling. An individual's metacognitive awareness is contingent upon task, strategy, and learner variables. Learners derive benefit from instruction pertaining to metacognitive activities.

Concept learning entails higher-order processes involved in the formation of mental representations of the critical attributes inherent to categories. Contemporary theories lay emphasis upon the analysis of features and the formulation of hypotheses concerning concepts (feature analysis), as well as the generation of generalised images of concepts that incorporate solely certain defining characteristics (prototypes). Prototypes may be employed to classify typical instances of concepts, whereas feature analysis may be utilised for instances of lesser typicality. Models of concept acquisition and pedagogy have been propounded, and motivational processes are likewise implicated in conceptual change.

Problem solving comprises an initial state, a desired goal, subgoals, and operations executed to attain the aforementioned goal and subgoals. Researchers have scrutinised the mental processes of learners engaged in problem solving and the distinctions between experts and novices. Problem solving has been perceived as reflective of trial and error, insight, and heuristics. These general methodologies may be applied to academic content. As individuals accrue experience within a domain, they acquire knowledge and production systems, or sets of rules to strategically apply to the fulfilment of objectives. Problem solving mandates the formation of a mental representation of the problem and the application of a production to resolve it. In instances of well-defined problems wherein potential solutions may be ordered in terms of likelihood, a generate-and-test strategy proves efficacious. For problems of greater difficulty or lesser definition, means-ends analysis is employed, necessitating either working backward or forward. Alternative problem-solving strategies encompass analogical reasoning and brainstorming.

Transfer constitutes a complex phenomenon. Historical perspectives encompass identical elements, mental discipline, and generalisation. From a cognitive standpoint, transfer entails the activation of memory structures and transpires when information is interlinked. Distinctions are drawn between near and far, literal and figural, and low-road and high-road transfer. Certain forms of transfer may occur spontaneously, yet much is conscious and entails abstraction. Furnishing students with feedback pertaining to the utility of skills and strategies augments the likelihood of transfer.

Technology continues to ascend in significance within learning and instruction. Two domains that have witnessed rapid augmentation are computer-based learning environments and distance learning. Applications encompassing computer-based environments incorporate computer-based instruction, games and simulations, hypermedia/multimedia, and e-learning. Distance learning transpires when instruction originates in a single location and is transmitted to students at one or more remote sites. Interactive capabilities facilitate two-way feedback and synchronous discussions. Distance learning frequently entails online (Web-based) asynchronous instruction, and courses may be organised utilising a blended model (a combination of face-to-face and online instruction). Research substantiates the benefits of technology upon metacognition, deep processing, and problem solving. Future innovations shall culminate in enhanced accessibility and interactive capabilities.

Inclusions pertaining to the principles summarised within this lesson encompass worked examples, writing, and mathematics. Worked examples present problem solutions in a step-by-step manner and frequently include accompanying diagrams. Worked examples incorporate numerous features that facilitate learners' problem-solving endeavours. Writing necessitates composing and reviewing. Experts orchestrate text around a communicative goal of conveying meaning and maintain this objective in mind throughout the review process. Novices exhibit a tendency to transcribe what they can recollect concerning a topic, rather than concentrating upon their overarching objective. Children demonstrate early mathematical competence through counting. Computational skills mandate algorithms and declarative knowledge. Students frequently overgeneralise procedures (erroneous algorithms). Students acquire knowledge of problem typologies through experience. Experts recognise these typologies and apply the appropriate productions to resolve them (working forward). Novices operate backward by applying formulae that encompass quantities stipulated within the problem.