Adult Learning (K. P. Cross)

Overview:

Cross (1981) presents the Characteristics of Adults as Learners (CAL) model in the context of her analysis of lifelong learning programs. The model attempts to integrate other theoretical frameworks for adult learning such as andragogy ( Knowles ), experiential learning ( Rogers ), and lifespan psychology.

The CAL model consists of two classes of variables: personal characteristics and situational characteristics. Personal characteristics include: aging, life phases, and developmental stages. These three dimensions have different characteristics as far as lifelong learning is concerned. Aging results in the deterioration of certain sensory-motor abilities (e.g., eyesight, hearing, reaction time) while intelligence abilities (e.g., decision-making skills, reasoning, vocabulary) tend to improve. Life phases and developmental stages (e.g., marriage, job changes, retirement) involve a series of plateaus and transitions which may or may not be directly related to age.

Situational characteristics consist of part-time versus full-time learning, and voluntary versus compulsory learning. The administration of learning (i.e., schedules, locations, procedures) is strongly affected by the first variable; the second pertains to the self-directed, problem-centered nature of most adult learning.

Scope/Application:

The CAL model is intended to provide guidelines for adult education programs. There is no known research to support the model.

Example:

Consider three adults: a nursing student, a new parent, and a middle-aged social worker about to take a course on child development. Each of these individuals differs in age (20,30,40) and life/developmental phases (adolescent/searching, young/striving, mature/stable). They also differ in terms of situational characteristics: for the nursing student, the course is full-time and compulsory, for the parent, it is part-time and optional; for the social worker it is part-time but required. According to the CAL model, a different learning strategy might be necessary for these three individuals to accommodate the differences in personal and situational characteristics.

Principles:

1. Adult learning programs should capitalize on the experience of participants.

2. Adult learning programs should adapt to the aging limitations of the participants.

3. Adults should be challenged to move to increasingly advanced stages of personal development.

4. Adults should have as much choice as possible in the availability and organization of learning programs.

Andragogy (M. Knowles)

Overview:

Knowles' theory of andragogy is an attempt to develop a theory specifically for adult learning. Knowles emphasizes that adults are self-directed and expect to take responsibility for decisions. Adult learning programs must accommodate this fundamental aspect.

Andragogy makes the following assumptions about the design of learning: (1) Adults need to know why they need to learn something (2) Adults need to learn experientially, (3) Adults approach learning as problem-solving, and (4) Adults learn best when the topic is of immediate value.

In practical terms, andragogy means that instruction for adults needs to focus more on the process and less on the content being taught. Strategies such as case studies, role playing, simulations, and self-evaluation are most useful. Instructors adopt a role of facilitator or resource rather than lecturer or grader.

Scope/Application:

Andragogy applies to any form of adult learning and has been used extensively in the design of organizational training programs (especially for "soft skill" domains such as management development).

Example:

Knowles (1984, Appendix D) provides an example of applying andragogy principles to the design of personal computer training:

1. There is a need to explain why specific things are being taught (e.g., certain commands, functions, operations, etc.)

2. Instruction should be task-oriented instead of memorization -- learning activities should be in the context of common tasks to be performed.

3. Instruction should take into account the wide range of different backgrounds of learners; learning materials and activities should allow for different levels/types of previous experience with computers.

4. Since adults are self-directed, instruction should allow learners to discover things for themselves, providing guidance and help when mistakes are made.

(See computers for further discussion of this topic).

Principles:

1. Adults need to be involved in the planning and evaluation of their instruction.

2. Experience (including mistakes) provides the basis for learning activities.

3. Adults are most interested in learning subjects that have immediate relevance to their job or personal life.

4. Adult learning is problem-centered rather than content-oriented.

Experiential Learning (C. Rogers)

Overview:

Rogers distinguished two types of learning: cognitive (meaningless) and experiential (significant). The former corresponds to academic knowledge such as learning vocabulary or multiplication tables and the latter refers to applied knowledge such as learning about engines in order to repair a car. The key to the distinction is that experiential learning addresses the needs and wants of the learner. Rogers lists these qualities of experiential learning: personal involvement, self-initiated, evaluated by learner, and pervasive effects on learner.

To Rogers, experiential learning is equivalent to personal change and growth. Rogers feels that all human beings have a natural propensity to learn; the role of the teacher is to facilitate such learning. This includes: (1) setting a positive climate for learning, (2) clarifying the purposes of the learner(s), (3) organizing and making available learning resources, (4) balancing intellectual and emotional components of learning, and (5) sharing feelings and thoughts with learners but not dominating.

According to Rogers, learning is facilitated when: (1) the student participates completely in the learning process and has control over its nature and direction, (2) it is primarily based upon direct confrontation with practical, social, personal or research problems, and (3) self-evaluation is the principal method of assessing progress or success. Rogers also emphasizes the importance of learning to learn and an openness to change.

Roger's theory of learning evolved as part of the humanistic education movement (e.g., Patterson, 1973; Valett, 1977).

Scope/Application:

Roger's theory of learning originates from his views about psychotherapy and humanistic approach to psychology. It applies primarily to adult learners and has influenced other theories of adult learning such as Knowles and Cross. Combs (1982) examines the significance of Roger's work to education. Rogers & Frieberg (1994) discuss applications of the experiential learning framework to the classroom.

Example:

A person interested in becoming rich might seek out books or classes on ecomomics, investment, great financiers, banking, etc. Such an individual would perceive (and learn) any information provided on this subject in a much different fashion than a person who is assigned a reading or class.

Principles:

1. Significant learning takes place when the subject matter is relevant to the personal interests of the student

2. Learning which is threatening to the self (e.g., new attitudes or perspectives) are more easily assimilated when external threats are at a minimum

3. Learning proceeds faster when the threat to the self is low

4. Self-initiated learning is the most lasting and pervasive.

Information Processing Theory (G. Miller)

Overview:

George A. Miller has provided two theoretical ideas that are fundamental to cognitive psychology and the information processing framework.

[image]

The first concept is "chunking" and the capacity of short term memory. Miller (1956) presented the idea that short-term memory could only hold 5-9 chunks of information (seven plus or minus two) where a chunk is any meaningful unit. A chunk could refer to digits, words, chess positions, or people's faces. The concept of chunking and the limited capacity of short term memory became a basic element of all subsequent theories of memory.

The second concept is TOTE (Test-Operate-Test-Exit) proposed by Miller, Galanter & Pribram (1960). Miller et al. suggested that TOTE should replace the stimulus-response as the basic unit of behavior. In a TOTE unit, a goal is tested to see if it has been achieved and if not an operation is performed to achieve the goal; this cycle of test-operate is repeated until the goal is eventually achieved or abandoned. The TOTE concept provided the basis of many subsequent theories of problem solving (e.g., GPS) and production systems.

Scope/Application:

Information processing theory has become a general theory of human cognition; the phenomenon of chunking has been verified at all levels of cognitive processing.

Example:

The classic example of chunks is the ability to remember long sequences of binary numbers because they can be coded into decimal form. For example, the sequence 0010 1000 1001 1100 1101 1010 could easily be remembered as 2 8 9 C D A. Of course, this would only work for someone who can convert binary to hexadecimal numbers (i.e., the chunks are "meaningful").

The classic example of a TOTE is a plan for hammering a nail. The Exit Test is whether the nail is flush with the surface. If the nail sticks up, then the hammer is tested to see if it is up (otherwise it is raised) and the hammer is allowed to hit the nail.

Principles:

1. Short term memory (or attention span) is limited to seven chunks of information.

2. Planning (in the form of TOTE units) is a fundamental cognitive process.

3. Behavior is hierarchically organized (e.g., chunks, TOTE units).

Multiple Intelligences (H. Gardner)

Overview:

The theory of multiple intelligences suggests that there are a number of distinct forms of intelligence that each individual possesses in varying degrees. Gardner proposes seven primary forms: linguistic, musical, logical-mathematical, spatial, body-kinesthetic, intrapersonal (e.g., insight, metacognition) and interpersonal (e.g., social skills).

According to Gardner, the implication of the theory is that learning/teaching should focus on the particular intelligences of each person. For example, if an individual has strong spatial or musical intelligences, they should be encouraged to develop these abilities. Gardner points out that the different intelligences represent not only different content domains but also learning modalities. A further implication of the theory is that assessment of abilities should measure all forms of intelligence, not just linguistic and logical-mathematical.

Gardner also emphasizes the cultural context of multiple intelligences. Each culture tends to emphasize particular intelligences. For example, Gardner (1983) discusses the high spatial abilities of the Puluwat people of the Caroline Islands, who use these skills to navigate their canoes in the ocean. Gardner also discusses the balance of personal intelligences required in Japanese society.

The theory of multiple intelligences shares some common ideas with other theories of individual differences such as Cronbach & Snow, Guilford, and Sternberg .

Scope/Application:

The theory of multiple intelligences has been focused mostly on child development although it applies to all ages. While there is no direct empirical support for the theory, Gardner (1983) presents evidence from many domains including biology, anthropology, and the creative arts and Gardner (1993a) discusses application of the theory to school programs. Gardner (1982, 1993b) explores the implications of the framework for creativity (see also Marks-Tarlow, 1995).

Example:

Gardner (1983, p 390) describes how learning to program a computer might involve multiple intelligences:

"Logical-mathematical intelligence seems central, because programming depends upon the deployment of strict procedures to solve a problem or attain a goal in a finite number of steps. Linguistic intelligence is also relevant, at least as long as manual and computer languages make use of ordinary language...an individual with a strong musical bent might best be introduced to programming by attempting to program a simple musical piece (or to master a program that composes). An individual with strong spatial abilities might be initiated through some form of computer graphics -- and might be aided in the task of programming through the use of a flowchart or some other spatial diagram. Personal intelligences can play important roles. The extensive planning of steps and goals carried out by the individual engaged in programming relies on intrapersonal forms of thinking, even as the cooperation needed for carrying a complex task or for learning new computational skills may rely on an individual's ability to work with a team. Kinesthetic intelligence may play a role in working with the computer itself, by facilitating skill at the terminal..."

Principles:

1. Individuals should be encouraged to use their preferred intelligences in learning.

2. Instructional activities should appeal to different forms of intelligence.

3. Assessment of learning should measure multiple forms of intelligence.

Script theory (R. Schank)

Overview:

The central focus of Schank's theory has been the structure of knowledge, especially in the context of language understanding. Schank (1975) outlined contextual dependency theory which deals with the representation of meaning in sentences. Building upon this framework, Schank & Abelson (1977) introduced the concepts of scripts, plans and themes to handle story-level understanding. Later work (e.g., Schank, 1982,1986) elaborated the theory to encompass other aspects of cognition.

The key element of conceptual dependency theory is the idea that all conceptualizations can be represented in terms of a small number of primative acts performed by an actor on an object. For example, the concept, "John read a book" could be represented as: John MTRANS (information) to LTM from book, where MTRANS is the primative act of mental transfer. In Schank's theory, all memory is episodic, i.e., organized around personal experiences rather than semantic categories. Generalized episodes are called scripts -- specific memories are stored as pointers to scripts plus any unique events for a particular episode. Scripts allow individuals to make inferences needed for understanding by filling in missing information (i.e., schema).

Schank (1986) uses script theory as the basis for a dynamic model of memory. This model suggests that events are understood in terms of scripts, plans and other knowledges structures as well as relevant previous experiences. An important aspect of dynamic memory are explanatory processes (XPs) that represent sterotyped answers to events that involve analomies or unusual events. Schank proposes that XPs are a critical mechanism of creativity .

Scope/Application:

Script theory is primarily intended to explain language processing and higher thinking skills. A variety of computer programs have been developed to demonstrate the theory. Schank (1991) applies his theoretical framework to story telling and the development of intelligent tutors. Shank & Cleary (1995) describe the application of these ideas to educational software.

Example:

The classic example of Schank's theory is the restaurant script. The script has the following characteristics:

Scene 1: Entering
S PTRANS S into restaurant, S ATTEND eyes to tables, S MBUILD where to sit, S PTRANS S to table, S MOVE S to sitting position

Scene 2: Ordering
S PTRANS menu to S (menu already on table), S MBUILD choice of food, S MTRANS signal to waiter, waiter PTRANS to table, S MTRANS 'I want food' to waiter, waiter PTRANS to cook

Scene 3: Eating
Cook ATRANS food to waiter, waiter PTRANS food to S, S INGEST food

Scene 4: Exiting
waiter MOVE write check, waiter PTRANS to S, waiter ATRANS check to S, S ATRANS money to waiter, S PTRANS out of restaurant

There are many variations possible on this general script having to do with different types of restaurants or procedures. For example, the script above assumes that the waiter takes the money; in some restaurants, the check is paid to a cashier. Such variations are opportunities for misunderstandings or incorrect inferences.

Principles:

1. Conceptualization is defined as an act or doing something to an object in a direction.

2. All conceptualizations can be analyzed in terms of a small number of primative acts.

3. All memory is episodic and organized in terms of scripts.

4. Scripts allow individuals to make inferences and hence understand verbal/written discourse.

5. Higher level expectations are created by goals and plans.

Situated Learning (J. Lave)

Overview:

Lave argues that learning as it normally occurs is a function of the activity, context and culture in which it occurs (i.e., it is situated). This contrasts with most classroom learning activities which involve knowledge which is abstract and out of context. Social interaction is a critical component of situated learning -- learners become involved in a "community of practice" which embodies certain beliefs and behaviors to be acquired. As the beginner or newcomer moves from the periphery of this community to its center, they become more active and engaged within the culture and hence assume the role of expert or old-timer. Furthermore, situated learning is usually unintentional rather than deliberate. These ideas are what Lave & Wenger (1991) call the process of "legitimate peripheral participation."

Other researchers have further developed the theory of situated learning. Brown, Collins & Duguid (1989) emphasize the idea of cognitive apprenticeship: "Cognitive apprenticeship supports learning in a domain by enabling students to acquire, develop and use cognitive tools in authentic domain activity. Learning, both outside and inside school, advances through collaborative social interaction and the social construction of knowledge." Brown et al. also emphasize the need for a new epistemology for learning -- one that emphasizes active perception over concepts and representation. Suchman (1988) explores the situated learning framework in the context of artificial intelligence.

Situated learning has antecedents in the work of Gibson (theory of affordances) and Vygotsky (social learning). In addition, the theory of Schoenfeld on mathematical problem solving embodies some of the critical elements of situated learning framework.

Scope/Application:

Situated learning is a general theory of knowledge acquisition . It has been applied in the context of technology-based learning activities for schools that focus on problem-solving skills (Cognition & Technology Group at Vanderbilt, 1993). McLellan (1995) provides a collection of articles that describe various perspectives on the theory.

Example:

Lave & Wenger (1991) provide an analysis of situated learning in five different settings: Yucatec midwives, native tailors, navy quartermasters, meat cutters and alcoholics. In all cases, there was a gradual acquisition of knowledge and skills as novices learned from experts in the context of everyday activities.

Principles:

1. Knowledge needs to be presented in an authentic context, i.e., settings and applications that would normally involve that knowledge.

2. Learning requires social interaction and collaboration.

Social Learning Theory  (A. Bandura)

Overview:

The social learning theory of Bandura emphasizes the importance of observing and modeling the behaviors, attitudes, and emotional reactions of others. Bandura (1977) states: "Learning would be exceedingly laborious, not to mention hazardous, if people had to rely solely on the effects of their own actions to inform them what to do. Fortunately, most human behavior is learned observationally through modeling: from observing others one forms an idea of how new behaviors are performed, and on later occasions this coded information serves as a guide for action." (p22). Social learning theory explains human behavior in terms of continuous reciprocal interaction between cognitive, behavioral, an environmental influences. The component processes underlying observational learning are: (1) Attention, including modeled events (distinctiveness, affective valence, complexity, prevalence, functional value) and observer characteristics (sensory capacities, arousal level, perceptual set, past reinforcement), (2) Retention, including symbolic coding, cognitive organization, symbolic rehearsal, motor rehearsal), (3) Motor Reproduction, including physical capabilities, self-observation of reproduction, accuracy of feedback, and (4) Motivation, including external, vicarious and self reinforcement.

Because it encompasses attention, memory and motivation, social learning theory spans both cognitive and behavioral frameworks. Bandura's theory improves upon the strictly behavioral interpretation of modeling provided by Miller & Dollard (1941).  Bandura’s work is related to the theories of Vygotsky and Lave which also emphasize the central role of social learning.

Scope/Application:

Social learning theory has been applied extensively to the understanding of aggression (Bandura, 1973) and psychological disorders, particularly in the context of behavior modification (Bandura, 1969). It is also the theoretical foundation for the technique of behavior modeling which is widely used in training programs. In recent years, Bandura has focused his work on the concept of self-efficacy in a variety of contexts (e.g., Bandura, 1997).

Example:

The most common (and pervasive) examples of social learning situations are television commercials. Commercials suggest that drinking a certain beverage or using a particular hair shampoo will make us popular and win the admiration of attractive people. Depending upon the component processes involved (such as attention or motivation), we may model the behavior shown in the commercial and buy the product being advertised.

Principles:

1. The highest level of observational learning is achieved by first organizing and rehearsing the modeled behavior symbolically and then enacting it overtly. Coding modeled behavior into words, labels or images results in better retention than simply observing.

2. Individuals are more likely to adopt a modeled behavior if it results in outcomes they value.

3. Individuals are more likely to adopt a modeled behavior if the model is similar to the observer and has admired status and the behavior has functional value.

Attribution Theory (B. Weiner)

Overview:

Attribution theory is concerned with how individuals interpret events and how this relates to their thinking and behavior. Heider (1958) was the first to propose a psychological theory of attribution, but Weiner and colleagues (e.g., Jones et al, 1972; Weiner, 1974, 1986) developed a theoretical framework that has become a major research paradigm of social psychology. Attribution theory assumes that people try to determine why people do what they do, i.e., attribute causes to behavior. A person seeking to understand why another person did something may attribute one or more causes to that behavior. A three-stage process underlies an attribution: (1) the person must perceive or observe the behavior, (2) then the person must believe that the behavior was intentionally performed, and (3) then the person must determine if they believe the other person was forced to perform the behavior (in which case the cause is attributed to the situation) or not (in which case the cause is attributed to the other person).

Weiner focused his attribution theory on achievement (Weiner, 1974).  He identified ability, effort, task difficulty, and luck as the most important factors affecting attributions for achievement. Attributions are classified along three causal dimensions: locus of control, stability, and controllability. The locus of control dimension has two poles: internal versus external locus of control. The stability dimension captures whether causes change over time or not. For instance, ability can be classified as a stable, internal cause, and effort classified as unstable and internal. Controllability contrasts causes one can control, such as skill/efficacy, from causes one cannot control, such as aptitude, mood, others' actions, and luck.

Attribution theory is closely associated with the concept of motivation. It also relates the work done on scripts and inferencing done by Schank.

Scope/Application:

Weiner’s theory has been widely applied in education, law, clinical psychology, and the mental health domain. There is a strong relationship between self-concept and achievement. Weiner (1980) states: "Causal attributions determine affective reactions to success and failure. For example, one is not likely to experience pride in success, or feelings of competence, when receiving an ‘A’ from a teacher who gives only that grade, or when defeating a tennis player who always loses...On the other hand, an ‘A’ from a teacher who gives few high grades or a victory over a highly rated tennis player following a great deal of practice generates great positive affect." (p.362). Students with higher ratings of self-esteem and with higher school achievement tend to attribute success to internal, stable, uncontrollable factors such as ability, while they contribute failure to either internal, unstable, controllable factors such as effort, or external, uncontrollable factors such as task difficulty. For example, students who experience repeated failures in reading are likely to see themselves as being less competent in reading.  This self-perception of reading ability reflects itself in children's expectations of success on reading tasks and reasoning of success or failure of reading.  Similarly, students with learning disabilities seem less likely than non-disabled peers to attribute failure to effort, an unstable, controllable factor, and more likely to attribute failure to ability, a stable, uncontrollable factor.

Lewis & Daltroy (1990) discuss applications of attribution theory to health care. An interesting example of attribution theory applied to career development is provided by Daly (1996) who examined the attributions that employees held as to why they failed to receive promotions.

Example:

Attribution theory has been used to explain the difference in motivation between high and low achievers. According to attribution theory, high achievers will approach rather than avoid tasks related to succeeding because they believe success is due to high ability and effort which they are confident of. Failure is thought to be caused by bad luck or a poor exam, i.e. not their fault. Thus, failure doesn't affect their self-esteem but success builds pride and confidence. On the other hand, low achievers avoid success-related chores because they tend to (a) doubt their ability and/or (b) assume success is related to luck or to "who you know" or to other factors beyond their control. Thus, even when successful, it isn't as rewarding to the low achiever because he/she doesn't feel responsible, i.e., it doesn't increase his/her pride and confidence.

Principles:

1. Attribution is a three stage process: (1) behavior is observed, (2) behavior is determined to be deliberate, and (3) behavior is attributed to internal or external causes.

2. Achievement can be attributed to (1) effort, (2) ability, (3) level of task difficulty, or (4) luck.

3. Causal dimensions of behavior are (1) locus of control, (2) stability, and (3) controllability.

Cognitive Dissonance  (L. Festinger)

Overview:

According to cognitive dissonance theory, there is a tendency for individuals to seek consistency among their cognitions (i.e., beliefs, opinions). When there is an inconsistency between attitudes or behaviors (dissonance), something must change to eliminate the dissonance. In the case of a discrepancy between attitudes and behavior, it is most likely that the attitude will change to accommodate the behavior.

Two factors affect the strength of the dissonance: the number of dissonant beliefs, and the importance attached to each belief. There are three ways to eliminate dissonance: (1) reduce the importance of the dissonant beliefs, (2) add more consonant beliefs that outweigh the dissonant beliefs, or (3) change the dissonant beliefs so that they are no longer inconsistent.

Dissonance occurs most often in situations where an individual must choose between two incompatible beliefs or actions. The greatest dissonance is created when the two alternatives are equally attractive. Furthermore, attitude change is more likely in the direction of less incentive since this results in lower dissonance. In this respect, dissonance theory is contradictory to most behavioral theories which would predict greater attitude change with increased incentive (i.e., reinforcement).

Scope/Application:

Dissonance theory applies to all situations involving attitude formation and change. It is especially relevant to decision-making and problem-solving.

Example:

Consider someone who buys an expensive car but discovers that it is not comfortable on long drives. Dissonance exists between their beliefs that they have bought a good car and that a good car should be comfortable. Dissonance could be eliminated by deciding that it does not matter since the car is mainly used for short trips (reducing the importance of the dissonant belief) or focusing on the cars strengths such as safety, appearance, handling (thereby adding more consonant beliefs). The dissonance could also be eliminated by getting rid of the car, but this behavior is a lot harder to achieve than changing beliefs.

Principles:

1. Dissonance results when an individual must choose between attitudes and behaviors that are contradictory.

2. Dissonance can be eliminated by reducing the importance of the conflicting beliefs, acquiring new beliefs that change the balance, or removing the conflicting attitude or behavior.

Constructivist Theory  (J. Bruner)

Overview:

A major theme in the theoretical framework of Bruner is that learning is an active process in which learners construct new ideas or concepts based upon their current/past knowledge. The learner selects and transforms information, constructs hypotheses, and makes decisions, relying on a cognitive structure to do so. Cognitive structure (i.e., schema, mental models) provides meaning and organization to experiences and allows the individual to "go beyond the information given".

As far as instruction is concerned, the instructor should try and encourage students to discover principles by themselves. The instructor and student should engage in an active dialog (i.e., socratic learning). The task of the instructor is to translate information to be learned into a format appropriate to the learner's current state of understanding. Curriculum should be organized in a spiral manner so that the student continually builds upon what they have already learned.

Bruner (1966) states that a theory of instruction should address four major aspects: (1) predisposition towards learning, (2) the ways in which a body of knowledge can be structured so that it can be most readily grasped by the learner, (3) the most effective sequences in which to present material, and (4) the nature and pacing of rewards and punishments. Good methods for structuring knowledge should result in simplifying, generating new propositions, and increasing the manipulation of information.

In his more recent work, Bruner (1986, 1990, 1996) has expanded his theoretical framework to encompass the social and cultural aspects of learning as well as the practice of law.

Scope/Application:

Bruner's constructivist theory is a general framework for instruction based upon the study of cognition. Much of the theory is linked to child development research (especially Piaget ). The ideas outlined in Bruner (1960) originated from a conference focused on science and math learning. Bruner illustrated his theory in the context of mathematics and social science programs for young children (see Bruner, 1973). The original development of the framework for reasoning processes is described in Bruner, Goodnow & Austin (1951). Bruner (1983) focuses on language learning in young children.

Note that Constructivism is a very broad conceptual framework in philosophy and science and Bruner's theory represents one particular perspective. For an overview of other Constructivist frameworks, see http://carbon.cudenver.edu/~mryder/itc_data/constructivism.html.

Example:

This example is taken from Bruner (1973):

"The concept of prime numbers appears to be more readily grasped when the child, through construction, discovers that certain handfuls of beans cannot be laid out in completed rows and columns. Such quantities have either to be laid out in a single file or in an incomplete row-column design in which there is always one extra or one too few to fill the pattern. These patterns, the child learns, happen to be called prime. It is easy for the child to go from this step to the recognition that a multiple table , so called, is a record sheet of quantities in completed mutiple rows and columns. Here is factoring, multiplication and primes in a construction that can be visualized."

Principles:

1. Instruction must be concerned with the experiences and contexts that make the student willing and able to learn (readiness).

2. Instruction must be structured so that it can be easily grasped by the student (spiral organization).

3. Instruction should be designed to facilitate extrapolation and or fill in the gaps (going beyond the information given).

Phenomenonography (F. Marton & N. Entwistle)