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(Page 18)
them as a set of procedures for thc design of instruction. He outlined these in a near-incomprehcnsible treatise which he entitled Mathetks and sub-titled The Technology ojEducation (Gilbert 1961). In this he lays down a set of rules forthe analysis oflearning tasks, and the construction of appropriate training exercises. His rules are somewhat strict and were originally based more on reasoning than on experirnental evidence. Some of his suggestions do however seem to be bornc out by.subsequent experimcntation. In particular his three-class classification of learning tasks into chains, discriminations and generalizations has been orvalue to traincrs.
Gilbert Suggests an analysis of the performance required in terms of stimulus and rcsponse, to identify whether the performance ls made up of chains, discriminations or generalizations. This 'prescription of the mastery performance' then undergoes several further analyscs, in which it is compared with the performance of typical trainees, the most common errors they perform, other behavioun they have previously mastered which are similar and rnay therefore help or hinder the acquisition of the new performance, and so on.
Many writers of instructional materials have found Gilbert's work stimulating. Although few use his techniques in their totality, many have adopted S—R notation as a technique of analysis, and the 'Demonstrate, Prompt, Release' model as a basis for exercise design. Some have found they can adapt these techniques todeai witli almost any instructional task, while others find that the three main behaviour constructs of chain, discrimination and generalization are insufficicnt to deal with all forms oflearning tasks. Creative behaviour in particular does not seem to be very well described by stimulus-response techniques. Rtaders who feel this way may find the classification suggested by Robert Gagne more attractive.
Robert Gagne (1965) suggested a hierarchical list of eight categories oflearning. 'f The list is hierarchical in the sense that it proceeds from very simple conditioning-type learning, up to complex learning, such as is involved in problem solving.
1. Signal learning
This may be equated with the Pavlovian conditiored response. The subject learns that a given event is the signal for another eveiu, as the dinner beli was the signal for Pavlov's dogs' dinner.
2. Stimulus-response learning
This is differentiated from signal learning in that the response is not a generalized emotional one, but a very precisc act.
3. Chaining
Chaining is the type oflearning we have already described when discussing Gilbert's work.
4. Verbal chaining
Gagne says 'verbal association might well be classified as only a subvariety of chaining . . . But because these chains are verbal and because they explain the remarkable versatility of human processes, verbal association has some unique characteristics'.
5. Discrimination learning
This^is Uie same category as Gilbert's muitiple discriminations.
6. Concept learning
This may be compared to the 'generalization' of mathetics. In this formof learging a stimulus is classified in terms of its abstract properties, as shape, position, number, etc.
7. Rule learning
In a formal sense, a rule is a chain of two or more concepts. The simplest type /       of rule may be 'If A, then B' -eg 'If a (German) feminine noun (concept A) then the feminine article (concept B)'.
8. Problem-solving
Once a human being has acquired some rules he can combine these rules into a great variety of higher order rules. In doing this he can use what he already knows to sohe problems which are new to him (though they may or may not be so to other people).

1.2.2 The emergence of behavioural objectives (page 20)
At  about the same time that Gilbert was working on his treatise, another American psychologist, Bob Mager (1962), was writing a book in praiseof behavioural objectives. 11 is based on the simple inference that if onedefines learning as a change in behaviour, then the teacher may bc wise to define theaims or objectives of his lesson in terms of the behaviour patterns he wishes toestablish. This is useful, because it irr.plies a student-orientated, learning-orientated approach, rather than one based on subject matter coverage. It is also useful because once an objective is defined in behavioural terms (ie the student should bc able to perform tasks A, B and C), then it is simplicity itself to design an appropriatc test situation to evaluate the method of teaching: we simply get the student to attempt A, B and C. Of course, the crunch is to have the objectives defined in behavioural terms. It is not as easy as it sounds, espccially in traditional schoolroom subjects which are sometimes taught more through tradition, than for anv^ practical end purpose. The problem perhaps is not so much in statingan objective as in stating the right one. We shall come back to this lateron.
The essential ingredients in a beiiavioural objective, according to Mager, are:
1.A statement of what the student should be able to do at the end ofthc learning session (the terminal behaviour).
2.The conditions under which he should be able to exhibit ihe terminal behaviour.
3.The Standard to which he should be able to perform (the criteria).
For example: 'The student (1) should be able to find the square root of any number, (2) using tables ofsquare roots or logarithm tables, and'(3) getting theanswer correct to 3 significant figures 9 times out of 10.'
Mager popularized the precise statement of objectives for programmed instruction. Later his approach became more widely applied to the design of any form of instructional material. After all, as Mager put it, 'ifyou don't know where you're heading you'll probably end up someplace else'. However, the concept of objectives stated in performance terms is ofa much earlier date. The term has of course always been used in 'he context of education, but often in a general sense, such as *to provide equal opportunities for all* oi 'to create an environment for self-development'. Gradually it became usual to discriminate between sach general aims and more detailed objectives.
It also became usual to think of three categories ofobjeclryes:
1.   Cognitive
objectives   what the student should know or be able to do.
2.   Affective
objectives  what the student should feel and should be prepared to dc.
3.   Psychomotor
objectives   physical skills that the student should develop.
One person who used these three categories and then developed sub-divisions was B S BIooti. Bloom's taxonomy of the cognitive objectives of education published in 1956 became (and still is) a Standard handbook for many concerned with curriculum planning and instructional design. Much later (in 1964), the secohd velutne, giving a suggested classification of affective objectives, was published. Psychomotor objectives have not yet been dealt with so thoroughly. The table in Figure 1.14 deseribes the categories of objectives in the cognitive and affective domains, suggested by Bloom et al. (1956) and Krathwohl et al. (i964).
Notice again that these are arranged as a hierarchy in which the lower levels are prerequisites to the higher levels. To this extcnt the categories of Bloom's taxonomy are similar to Gagne's categories of learning tasks. Indeed, as far as the cognitive domain is concerned, the similarity is very pronounced, as can be seen by examining a few examples of the application of Bloom's taxonomy.
1. Knowledge (page 22)
This is considered thc lowcst level of cogfiitive objective. To demonstrate the attainment of objectives at this level, students would be expected to do such tasks as: namt the parts of an object;point out a certain object; state a definition; recognize a phenomenon when he sees it. (Gagne would classify most of these as examples of stimulus-response leaming or as learning of verbal chains.)
2. Comprehension
Bloom identifies this level by signs of 'undcrstanding' on the part 6f the studeiit - such signs as: seltcling an example of a particular phenomenon; giving reasonsfor a phenomenon; classijying an object into a category; contrasting tv/o objects/phcnomena. (These seem to be examples of multiple diserimination and simple concept learning in Gagne's terms.)
3. Application
This level is charaeterized by the student's abiliiy to apply theoretical statements and generalizations to real situatioDS, for example: caleulak a maihematical result; perform a task; use a particular set of rules and procedures; prtdict the result of a proposed course of aetion. (There seems to be much in common here with what Gagne terms 'rule-following'.)
4. Analysis
Being able to compare and cov'rasl altematives; iustijy ihe adoption of certain procedures; break down a problem into its components.
5. Synthcsis
Being able to stlect among alternative courses of aetion; organizc the components of a problem; derivc a solutien tc a problem. (Analysis and synthesis are both involved in Gagne's deseription of problem-solving activity.)
6. Evaluation
This is, according to Bloom, the highest level ofeognitive objective. Itis charaeterized by such activity as being able to judge the value of a particular block of knowledge; arguc for or against a proposal; defend or criticiz* a particular viewpoint. (Although some evaluation is involved in problem-solving, Bloom's concept of evaluation seems to go further into thc realms of intellectual activity than does Gagne's classification of learning categories.)
There are, however, some important differences between Bloom's and Gagne's classifications:
(a) Ifyou look at Figure 1.14 it is quite obvious that Bloom is suggestinga continuum in the development of a set of objectives froni the simple and concrete to thc comp!ex and abstraet (note the parallel here with well-used instructional cliches). The six levels are like milestones on the way to perfect accomplishment, rather than watertight categories. Gagne's hierarchy on the other hand has specific and exclusive charaeteristies defined for each category of learning, particularly at the     lower levels.
(b) Bloom's taxonomy merely sets out to classify objectives, and, by stating them in terms of observable student behaviour, to indicate appropriate types of test questions. It does not attempt to formulate general rules abpul how one should teach in order to achieve particular objectives. ■Gagne's hierarchy, on the other hand, was constructed with this latter aim in mind. Each learning category is not only associated with particular conditions for testing but also with conditions for learning (both external - how the teacher should plan the instructional event -and internal - the state of readiness of the learner).
There have been several other early attempts to build up a comprehensive model for preparing objectives, eg Miller (1962), and for the classification of types of learning, eg Merrill (1971), Tennyson and Merrill (1971).

7.2.3 The cognitive/developmental viewpoint (PAGE 23)
The influence of Jerome Bruner on teaching (particuiarly elementary school mathematics in the USA) has been iminense. He is probably the foremost proponent of the discovery approach in mathematical education. However, he is not by any means the inventor of the discovery approach. This concept was well known in mathematics education at the beginning of the century (Young, 1906). Bruner's approach (1966) to discovery learning is characterized by threestages, which he calls enactive, iconic and symbolic. These stages are firmly based on the developmental psychology ofjean Piaget. Piaget was perhaps the most proHfic researcher in developmental psychology. His interests have centred on the study and definition of the stages of cognitive development of the child. I shall not reiterate herc the Piagetian stages of cognitive development. This is available in many other works (Piaget, 1957, 1965). I shall concentrate on the characteristics of Piaget's view of the growth of intelllgence as they may relate to the processof instruction.
Piaget views the development of intelligence as part of the more general process of biological development. Gailagher (1964) has suggested five major thcmes running through Piaget's work.
1.Continuous and progressivc changes take place in the structures of behaviour and thought ir. the developing child.
2.Successive structures make their appearance in a fixed order.
3.The nature of accommcdation (adaptive change to outercircumstances) suggests that the rate of development is, to a considerable degree, a funclion ofthe child's encounters with his environment.
4.Thought processes are conceived to originate through a process of iuternalizing actions. Intelligence increases as thought processes are looscned from their basis in perception and action and tlicreby become reversible, transitive, associative, and soon.
5.A close relationship exists between thought processes and propcrties of formal logic.         .
Piaget observes that only in man can intelligence develop to the point where the domain of ideas and symbols can serve as the environmental source of disequilibrium. That is, we can construct intellectual universes, for example, 'transintuitive spaces, which can stimulate our own cognitive growlh as surely as the confrontation by a baby with the problem of reaching his pacifier can lead to new insights of equilibria on his part' (Shulman and Keislar, l966).
Piaget has written little specifically direeted at problems of education. He has repeatedly disavowed any expertise in the pedagogical domain. Yet, either directly or through such interpreters as Bruner, his influence has been strongly felt.
Piaget's emphasis upon action as a prerequisite ofthe internalization of cognitive operations has stimulated the focus upon direct manipulation of concrete materials in the early grades. His deseription of cognitive development occurring through autoregulation has reinforeed tendencies to emphasize pupil-initiated, problem-solving aetivities. Much ofthe work of such practical innovators as Z P Dienes (1960, 1964) and such theoreticians as Bruner is directly based on Piaget.
The general learning process described by Bruner (1966) oecurs in thefollowing manner. First, the child finds in his manipulation ofthe materials regularities that correspond with intuitive regularities he has already come to understand. Notice that what the child does, accordirg to Bruner, is to find some sort of mateh between what he is doing in the outside world and some models or templates that he has already grasped intellectually. For Bruner, it is rarely something outside the learner that is discovered. Instead, the discovery involves an internal reorganization of previously known ideas in order to establish a better fit between those ideas and the regularities of an encounter to which the learner has had to accommodate.
(PAGE 24 - 25)
Bruner almost always begins with a focus on the produetion and manipulation of matcrials. He dcscribcs the child as moving through three levels of represcntation as he learns. The first level is the enactive level., where the child manipulates matcrials directly. He then progresses to the iconic level, where he deals with mental images of objects but does not manipulate them directly. Finally he moves to the symbolic level, where he is strictly tnanipulating symbols and no longer mental images of objects. This sequence is based on Bruner's interpretation of Piaget's developmental theory. The combination of these concepts of manipulation of actual matcrials as part of a developmental model and the Socratic notion of learning as internal reorganization into a learning by discovery approach is the unique contribution of Bruner.
Bruner, in describing the process of mathematics learning, identified three stages in the learning of a new mathcmatical concept: enactive, iconic and symbolic. Optimum learning should pass through these three stages. These stages are identifiable in most of the practical procedures for working in the mathematics laboratory mode.
David Ausubel (1968) has bcen a powcrful (thougli perhaps now a waning) influence on instructional thir.kini];. He stands in opposition to the discovery movement, claiming that much of the apparent superiority of discovery over exposition is due to research generally comparing discovery techniques wiih rote learning approaches. Ausubel argues that much instruetion, particuiarly at highei levels of education, is (and has always been) successfully performed by the process of exposition leading to meaningful reception learning.
He states (1968):
'In reception learning (rote or meaningful) the entire content of what is to be learned is presented to the learner in final form. The learning task does not involve any independent discovery on his part. He 's required only to internalize or incorporate the material . . .
The essential feature of discovery learning . . . is that the principal content of what is to be learned is not given but must be discovered by the leamer before he can incorporate it meaningfully into his cognitive structure. The distinetive and prior learning task, in other words, is to discover something ...
It is evident, therefore, that reception and discovery learning are twoquitc diflerent kinds of processes, and . . . that most classroom instruetion is organized along the lines of reception learning. Verbal reception learning is not necessarily rote in charaeter. Much ideational material (concepts, generalizations) can be internalized and retained meaningfully without prior problem-solving experience, and at no stage of development does the learner have to discovc" principles independently in order to be able to understand and use them meaningfully.'


1.2.4 The expanded model of Gagnc and Briggs
We have already seen that the Bloom model has nothing to say about instructional }  methods, but Gilbert's model and Gagne's model had much to contribute. The of best part of Gagne's books was devoted to the 'conditions for learning' for eachof
his learning categories. These conditions were:
(a)      The internal conditions - readiness to learn. What has to be learned prior to the new learning taking place.
(b)     The external conditions - the particular instructional taeties to be employed.

What we notice about Gagne's model is:
a.        It is rather wooliy about the conditions (particuiarly the external
conditions) for learning problem-solving. One thing Gagne does say is that whereas at the lower levels (including concept learning) the 'discovery approach' does not pay ofT particuiarly (except perhaps in terms of motivation), at the higher levels there are benefits in terms of


long-term retention and the ability to transfer the skill to a greater variely of problems.
b.      Many writers have commented that problem-solving is a leaming category (perhaps several categories) worthy of more attcntion. Even Bloom's model identifies three types of mental processes involved just in problem-solving: analysis, synthesis and evaluation. Polya (1945), Landa (1976) and many other writers draw a distinction between algorithmic and heuristic problem-solving (Landa even subdivides these). It is not all that clear to which type of problem-solving Gagne is referring and which are missing from his scheme.
c.       Other writers (bruner, 1966) commented that creativity and such capabilities as 'learning to learn' are overlooked. Others (Merrill and Wood, 1974) suggest essentially very similar categories but with some difTerences in definitions and nomenclature.
Thus in the I970s Gagne drasticaily inodified his thinking of the I960s,or rather, he attempted to modify and enlarge his model to take into consideration the indicated shortcomings, to placate the humanists, the 'mental process' fans, the 'information processing tiieory' fans and cyberneticians. At the same time, he played down the behaviourist roots of many of his earlicr concepts, whilst ateempting to keep most of tuem in his model (See Figure 1.15).

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