Instruction which enables students to develop higher mental processes

Ev#ua//on/,n ~ ,

1979, VoI. 3, pp. 173-220. Pergamon Prim Ltd. IMtntld in Grim Bdudn.

INSTRUCTION W H I C H ENABLES STUDENTS TO DEVELOP HIGHER M E N T A L PROCESSES

Tamar Levin School of Education, University of Tel Aviv, Israel

CONTENTS

Page I.

THE PURPOSEOF THE STUDY

174

2.

A REVIEW OF PREVIOUS RESEARCH

177

3.

THE THEORETICAL MODELAND ITS USE

186

4.

THE RESULTSAND THEIR INTERPRETATION

200

5.

IMPLICATIONS

211

ACKNOWLEDGEMENT

217

REFERENCES

218

17"3

1. The Purpose of the Study

One of the most important questions regarding the effectiveness of school learning is how well students can transfer their learning to new situations. Educators throughout the world desire to develop problem solving s k i l l s and various higher mental processes in the schools. They believe that higher mental processes enable students to use their knowledge more effectively and efficiently even long after the students leave the schools. This means that i f school learning is to be useful to the individual as well as to society, school instruction must emphasise the development of high levels of thinking in a wide variety of higher mental process objectives. One class of educational objectives among the larger class of higher mental process objectives is known as application objectives. Application is defined as the use of abstractions and generalisations in particular and new situations (Bloom et a l , 1956). The a b i l i t y to use learned concepts, rules, principles and other abstractions in new situations is thus referred to as the ' a b i l i t y to apply'. The a b i l i t y to apply is based on the assumption that i f students are provided with the appropriate learning experiences, they become able to use principles and generalisations in order to solve problems that are new to them. The development of students'ability to apply rules and principles in a variety of new problem situations is the main theme of the present study. I t explores the nature of the learning conditions that enable students to develop and use this a b i l i t y to apply rules, and evaluates their effectiveness in experimental situations involving a number of school classrooms. The a b i l i t y to apply rules and principles is tested in a wide variety of new and complex problem situations. Application a b i l i t y has been long considered a highly desired learning outcome. There are three major reasons for i t s importance. First, there is evidence that when this a b i l i t y has been developed i t is a permanent acquisition of learning and is transferable to a great variety of problem situations and circumstances (Bloom et a l , 1971). Second, application a b i l i t y by its nature is one means of helping individuals to relate their learning to day-to-day living. Adults, especially, have to make many decisions, choices and evaluations based on rules and principles that were acquired previously. Since the l i f e situations that an individual may encounter are so diverse and sometimes unique, i t is essential that people be helped to acquire the intellectual tools necessary to cope with many unexpected situations. Third, i t is believed that application enables an individual to cope with the tremendous and rapid changes in society and that i t helps to develop secure and capable human beings in modern society where i t is almost impossible to foresee the particular changes that may occur. 174

Instruction whicheneblesS~den~ m Deveiop High~MentelProceu~

175

The study of school learning is usually considered in relation to three major questions: I. 2. 3.

What educational objectives should the schools seek to attain? What learning experiences can be provided to students that are l i k e l y to bring about the attainment of these objectives? How can the effectiveness of learning be evaluated by the use of appropriate instruments and data gathering procedures? (Tyler, 1949)

Educational objectives refer to educational ends and they are the e x p l i c i t specification of the kinds of changes in the students that are expected to occur as a result of the educative process. That is, they describe and i l l u s t r a t e the kinds of behaviours and processes that students are expected to acquire. Learning e~:pe~emoes refer to the interaction between the student and the external conditions provided to him in the learning environment. They include student a c t i v i t i e s , the teaching procedures and the available learning materials. The evaluation process is essentially the prpcess of determining to what extent the educational objectives have actually been achieved by the educational process. I t includes the gathering of evidence on the extent to which desired and intended objectives have been learned. Over the past two decades major progress has been made by educators in defining, stating and clarifying a wide range of educational goals. These include simple objectives such as knowledge and recall of the facts and principles of a subject field as well as more complex intellectual a b i l i t i e s such as application, inference and other higher mental processes. Individuals and groups of educators have worked out different classifications and definitions of educational goals. In addition, some groups have developed a rationale of why these goals should be pursued by the schools (Bloom et al, 1956; Gagn~, 1965; Mager, 1962; Popham and Baker, 1970; Scriven, 1967). Likewise, major successful efforts have been made by evaluators to develop valid and precise instruments and sophisticated data gathering procedures to evaluate the extent to which students actually have achieved a wide range of goals including application and other higher mental processes. Although some progress has been made in specifying and creating the appropriate learning experiences that are necessary for the attainment of specific knowledge, the learning conditions and the teaching procedures that are necessary to achieve the more complex educational goals remain relatively unclear. Partly as a result of this gap in our understanding of the learning conditions required for higher mental processes, educational practices throughout the world are s t i l l primarily concerned with the acquistion of knowledge (Bloom, 1974). For example, the IEA science survey in nineteen countries (Comber and Keeves, 1973) indicates clearly that in almost every country students perform best on the simple s k i l l s involving knowledge, perform less well on more complex s k i l l s involving some interpretation and perform least well on problems requiring application and other complex i n f l u ences. Moreover, i t is evident that the pattern of decreasing performance from simple to more complex s k i l l s and a b i l i t i e s reflects the opportunities to learn these different objectives in the classroom. That is, although educators profess more complex objectives for education than merely knowledge of specific information, rules and concepts, the actual emphasis in the classroom is largely on knowledge. Furthermore, there is a prevailing view among both teachers and researchers that application a b i l i t y as well as other more complex educational objectives can be developed only in those students who have exceptionally high intelligence and aptitude scores. This idea has been frequently determined by the fact that high intelligence students were found to use prior learning more readily than do low intelligence students. Also, high correlations between measures of students'

176

~ Levin

intelligence or aptitude scores and their a b i l i t y to apply acquired knowledge has often been taken as an indication of a causal link between general capabilities that an individual possesses and his a b i l i t y to learn application as well as other higher mental process objectives. In contrast to these views, is the research literature which demonstrates that when the instructional factors in a learning situation are weak or inappropriate, the personal characteristics of an individual become strong influences on the learning and achievement. Whereas, when instruction is appropriate to the objectives and the learners i t markedly reduces the role of such personal factors as intelligence and aptitudes in learning relatively complex s k i l l s and a b i l i t i e s . One of the problems in this study is to identify the learning conditions which reduce the role that intelligence and aptitude play in determining students a b i l i t y to apply rules and principles in new situations. Given the existing gap between what is expected of school learning, what is known, and what is being accomplished, the need for establishing links between educational objectives, educational experiences and evaluation procedures for application objectives becomes apparent. This research was addressed to this need. I t chose to concentrate on the study of a specific educational objective - applications of rules to new situations. The research was further limited to the subject field of mathematics. A set of probability rules were selected as the rules to be learned and applied by ninth grade students in their real classroom setting. The study was concerned with the three major questions: I. 2. 3.

What is the relationship between knowledge of learned rules and the a b i l i t y to apply them in a variety of problem situations? What are the relative effects of different learning experiences on the students' a b i l i t y to apply the rules in new situations? and Can the relationship between intelligence or aptitude scores and the a b i l i t y to apply learned rules be markedly altered by instructional practices?

2. A Review of Previous Research

APPLICATION OF PRINCIPLES TO NEW SITUATIONS Application is defined as the use of abstractions in particular and concrete problem situations. An abstraction may be a general idea, rule, procedure or organised method. I t could also be a technical principle or theory which must be remembered and used. According to Bloom et al (1971) the ' a b i l i t y to apply' implies that there is a general a b i l i t y that students develop such that when faced with new problems and situations, they can make use of previously learned relevant abstractions. A student is able to apply principles and other general ideas i f , when faced with new problem situations, he is able to deal with them without having to be prompted as to which abstraction he should use or how he should attack the problem. The study of the a b i l i t y to apply rules or principles requires that one c l a r i f y the nature of the specific behaviours involved in the application process. I t also requires that the researcher specify the kinds of problems that constitute appropriate problem situations. Specification of the stages involved in an application process is considered in the 2c~o~mj of Edu~atY~l Objectives (Bloom et al, 1956). I t includes specific suba b i l i t i e s or s k i l l s such as the identification of or the search for familiar elements in the problem, the classification of the problem to a familiar type, and the selection of abstractions suitable to the particular type of problem and their use.

More specific behaviours underlying the a b i l i t y to apply rules or principles are also recognised by Bloom et al (1971). These include the students s k i l l s in restating the problem, specifying the limits within which a particular abstraction is relevant or true, and explaining new phenomena in terms of known principles. Also included are s k i l l s in predicting the consequences of new circumstances by the use of an appropriate principle and justifying the selection and the use of a relevant and appropriate principle. Embedded in the definition of application of principles and generalisations is the term 'new situation' This reflects the idea that the appropriateness of the problem situation must be determined with respect to the previous experiences that the student has had. I t means that problem situations must be either new to the student or need to contain new elements compared with the learning situations in which the principles or other abstractions were learned. I t is not a new problem or situation 177

178

~ Levin

i f i t is exactly like others solved in class except that new quantities or symbols are used. I t is also not a new problem i f i t is the same as one solved in class with only some new names or slight changes altering the original form. This distinction is essential to the construction of appropriate application problems and i t is crucial in differentiating application of principles from the simpler a b i l i ties such as knowledge and comprehension. A problem is new i f the student has not been given instruction or help on a given problem, and he must either modify the statement of the problem in some way before attacking i t , put i t in a form or some model before he can bring the previously learned principles to bear on i t , or search through his memory for relevant principles. I t is also a new problem i f the student must use learned principles somewhat differently from the way he has used them previously. The design of new problem situations is a d i f f i c u l t task for the evaluator. I t requires that the evaluator be able to determine the type of problem situations that the student was exposed to, in order to avoid them. I t may require that he creates new situations based on real l i f e problems or f i c t i t i o u s ones. The specification of the behaviours and the characteristics of problem situations can be of great value for both evaluators and teachers. A teacher who is acquainted with these specifications should be in a better position to plan educational experiences that will help students to develop the a b i l i t y to apply rules or principles. This need to practice each specific behaviour required by an application problem has been emphasised by Tyler (1949) as an appropriate and a desirable way to develop the application of rules or principles in students.

EVALUATION PROCEDURESFOR APPLICATION OF PRINCIPLES The process of evaluation is essentially the process of determining to what extent the educational objectives are actually being realised by learning and instruction. I t also becomes aprocess of finding out the strengths and weaknesses of the specific instructional procedure and learning experiences as well as testing the validity of the rationale upon which these experiences were developed. I t is possible to determine the requirements for evaluating the application of rules from the foregoing discussion of the behaviours underlying the application process and the kinds of possible problems that constitute problem characteristics. Some of the general requirements are: I. 2. 3.

One or more of the behaviours required in an application process should be sampled by the problem; The problem should be new or in some way different from those used in instruction; and The problem should be solvable in part by the use of appropriate rules or principles.

One should realise that since several specific behaviours are required for a solution of an application process, i t is possible to assess some of these specific behaviours without actually asking the student to carry out the entire application process. Thus, for example, i f the student is asked to restate the problem and determine the necessary rule for its solution, this may provide evidence on the student's grasp of what is required in the problem. One cannot be sure from this that the student can actually solve the total problem. I t is also possible to construct problems that require the student to carry out the entire application process. This will enable the teacher and the evaluator to get a more complete indication of the student's general a b i l i t y to apply learned rules and principles.

Instruction whichm'mblesSmden~ ~L~mlop Higher Mentall~oceues

179

Application problems by definition need to be new to the student. Since a great variety of new problems can be designed, i t is important to know whether the nature of the problem situation influences the appraisal of students a b i l i t y to apply rules and principles. Will a student who is able to apply a rule in a problem taken from real l i f e situations be able also to apply the same rule in an a r t i f i c i a l problem situation? There are indications in the literature which suggest that this is not necessarily true. That is, a student who is capable of solving one application problem may not be able to apply the same rule in somewhat different problem situations. Horrocks (1946), for example, illustrated that the inter-relationships between performance on three case studies each of which measure a b i l i t y to apply rules of adolescent developmentare positive but relatively low. The fact that each case study was designed to deal with a different aspect of humandevelopment such as emotional, social and physical aspects suggests that a b i l i t y to apply learned rules could be considered not as one general a b i l i t y but rather as specific a b i l i ties which depend on the context of problem and on the nature of the rules. A similar idea is reflected in more recent work by Scandura (1967). Scandura explored the relationship between knowledge of rules, the nature of the rules and performance on application tasks. In the context of a number game, he presented to his subjects statements of three rules by which the person making the f i r s t move can always win the game. Someof these principles were particular to specific numerical conditions within the game, whereas others were more general. Scandura tested his subjects on problem situations which were defined as 'within the scope of the rules' Theseproblems matched the more specific rules required for winning the game. He also tested his subjects on problem situation defined as 'beyond the scope of the rules', which essentially required the knowledge of a more general rule. The findings indicated that students were not equally capable of applying the learned rules in these two different situations. Successful performance on application of rules was more noticeable on problem situations defined within the scope of the rules. On the basis of related research Scandura (1968) suggests that i f students are able to apply learned rules in new problems which differ from previous learned problems along one or more dimensions, i t is highly l i k e l y that they w i l l be able to apply the rules to a great variety of problems which include these same dimensions. Scanduramakes i t evident that each of the major dimensions must be sampled i f we are to generalise about the students a b i l i t y to apply rules. The evidence in the literature makes i t clear that application of rules to new s i t uations should be viewed as a multi-dimensional t r a i t . The specific t r a i t s may very well reflect different qualities of the a b i l i t y to apply. The literature also makes i t clear that i t is essential to explore the nature and dimensions of possible problem situations i f we are to include them in a set of evaluation measures of the a b i l i t y to apply rules. Few theoretical speculations and analytical studies exist which may help to identify elements and dimensions of problem situations to guide the development of a wide variety of problems. Three general types of problem situations are listed in the Taxonomy: fictional situations, situations that are simplified versions of complex material which the student is not l i k e l y to have had contact with, or known situations with a new slant which students are unlikely to have thought of previously. More specific factors affecting the nature and structure of problem situations are offered in some experimental studies within the area of mathematics education. For example, in an attempt to study the d i f f i c u l t i e s that students have in solving arithmetic problems, Loftus and Suppes (1972) and Jerman and Reese(1972) focussed on the variables that characterise the problems as well as their structural relationships. Their results showed that variables such as the length of the problems, the minimum number of different operations required to reach a correct solution, and the nature of the sequence of the problems are important in determining word problem d i f f i c u l t y . Similarly, Wilson (1967) explored the potential dimensions invol-

180

T. Levin

ved in problem situations in arithmetic verbal problems and indicates that the objects dealt with in a problem, the events they create and the required activities constitute potentially different domains of problem situations. A somewhat different approach to the study of problem situation characteristics is exemplified by the work of Kruteskii (1969). In a series of experiments, Kruteskii observed and interviewed children while they were solving mathematical problems. He attempted to classify problem situations on the basis of their anticipated relation to specific mathematical aptitutes. He established several classes of problems such as problem situations whose nature suggests the use of visual aids in the solution process, unrealistic problems in terms of inconsistent data, or changeable content problems where changing of the content may or may not reflect changes in the basic mathematical operations to be performed for a correct solution. The examples cited above makes i t clear that although l i t t l e guidance is offered by the literature as to what are the important dimensions of problem situations, such dimensions can be defined and used. Based on the view that application of rules and principles is a multi-dimensional t r a i t and that its manifestation is also a function of problem situations, i t seems apparent that for a valid evaluation of the ability to apply rules and principles we need to use some different problems. These could make use of a wide variety of new problem situations that vary in terms of defined dimensions, or the problems should cover the range of behaviours required by an application process.

THE CONDITIONS UNDERWHICH STUDENTSUSE THEIR LEARNING IN NEW SITUATIONS In the past, educators and researchers attempted to describe and explain the ability to apply rules and principles from the transfer-of-training point of view. This paradigm related prior learning to subsequent learning and performance (Klausmeier and Davis, 1969; Schultz,1960). I t implied that the ability to apply learned rules largely depends on the previous learning experiences that the student has. In particular i t emphasised that the student who has learned and mastered a set of rules will be able to better apply them than a student who did not master the same rules. Several of the earliest experimental studies of transfer became standard references for educators even though the studies did not really deal with school learning. Probably one of the best known view of transfer was offered by Judd (Ig08). Judd's conception of transfer was f i r s t presented along with his report of the famous study of throwing darts at targets under water. In this study one group of boys was given an explanation of the principle of refraction whereas the second was not. Both groups then threw darts under water. After this task was mastered, the subjects were asked to throw darts at a target under water when the conditions of the amount of refraction and the apparent position of the target were changed. The group that had learned the general principle adapted more rapidily to the second task than did the other group. Similar results were demonstrated by Hendrickson and Schroeder (1941) who repeated Judd's experiment in a modified form. Thorndike and Woodworth (1901) found that practice in judging the size of rectangles led to the most improvement on another set of rectangles but less to a set of nonrectangular shapes. Dealing with more school-related subjects, Horrocks (1946) examined the relationship between knowledge of rules in human development and the ability to apply them (see page 7 ). His results indicate that the students' knowledge of rules is positively but not highly related to their ability to apply the learned rules in new situations. More direct and specific information of the relation between

~structionwhichenablesS~o~n~ m Develop H~herMenta/Proc~s~

181

knowledge and application of rules is also demonstrated in Block's study (1970). Block has shown that students who mastered s k i l l s and principles in matrix a r i t h metic were better able to apply these s k i l l s and principles to new and more complex problems than were students who had not mastered these same abstractions. Furthermore, Block demonstrated that the level of mastery on the pre-requfsite principles greatly influenced the a b i l i t y to apply them. For example, the attainment of the 95 per cent level of mastery of the principles enabled a higher proportion of students to apply them to a set of new problems compared with the students who attained only an 85 per cent (or lower) level of mastery on the same principles. While there are differences among researchers in viewing the mechanism and the nature of transfer, there is agreement in the literature that knowledge of rules and principles is a necessary pre-requisite for the attainment of the a b i l i t y to apply these rules and principles. This means that application of rules to new situations is dependent on knowledge of the rules, but i t does not imply that knowledge of rules alone guarantees that an individual w i l l be able to apply the knowledge at a high performance level and to a broad spectrum of problem situations. The contention that knowledge of concepts, s k i l l s and rules is a necese~ b u t u t 8 u ~ f i e i ~ t c o t i l l o n for the a b i l i t y to apply them is a fundamental and challenging principle in the work of Gagn((1965). During the last decade Gagn~ has formulated a useful way of viewing human learning. Having examined existing theories of learning, he believes that learning of different facts, concepts and rules is a cumulative and hierarchically organised process. This suggests that one can derive, on the basis of logical and pedagogical analyses, hierarchies of s k i l l s and a b i l i t i e s for each desired learning outcome. These hierarchies consist of super-ordinate and sub-ordinate relevant s k i l l s , facts and concepts. According to Gagn~, the acquisition of sub-ordinate facts, s k i l l s and concepts facilitates the transfer of learning to the more complex learning types such as rules, application of rules and other kinds of problem solving. Yet, although Gagn~ has emphasised the potential i n t e l lectual power of this property of transfer, he also realised its limitations. He thus posited that learning occurs not only as a result of a set of a b i l i t i e s an individual has acquired, but that learning is also affected by the external instructional conditions. According to him what is appropriate instruction depends on the kinds of s k i l l s and a b i l i t i e s that are being learned. Each defined set of desired learning outcomes requires somewhat different instructional processes. These should be clearly defined and implemented i f teachers desire their students to have a large repertoire of s k i l l s and a b i l i t i e s . The recognition of the virtues as well as of the limitations embedded in the transfer paradigm implies that further evidence is necessary to determine what else beyond pre-requisite knowled)e students should learn in order to be able to supply their knowledge. The view of Gagn~ is useful in suggesting some of the specific learning and instructional experiences needed to develop this a b i l i t y in students.

INSTRUCTIONAL METHODSWHICH ENABLESTUDENTSTO APPLY PRINCIPLES IN NEW SITUATIONS The literature indicates that i t is possible to design different kinds of learning experiences that affect students' a b i l i t y to apply rules and principles to new s i t uations. One general type has been suggested by Harlow (1949, 1959). This type of experience refers essentially to general practice on defined sets of problems. Harlow carried out a range of experiments, using children as subjects in some and monkeys in others. A typical study was one in which monkeys were presented with three objects, two of which were similar in some characteristic such as the size,

182

7". Levin

shape, or position and the third different. The task was to respond consistently to the different or the 'odd' objects. After the f i r s t task, a different set of objects would be presented, but again with two similar and one different stimuli. I t was found that after a few 'oddity' problems, the monkeys could solve many new other problems of the same form very quickly. On the basis of his results, Harlow claimed that the monkeys had learned a s k i l l which is applicable to a whole class of problems. Children ranging in age from two to five years learned more rapidly than the monkeys, but the results were otherwise identical. In his studies Harlow has demonstrated that the a b i l i t y to apply acquired principles could be developed when the i n i t i a l learning of the principle is followed by practice on a large number of problems that are organised in an increasing order of d i f f i c u l t y . The problems must also have a common factor, and share the same general solution but may differ on specific stimuli such as colour, shape cr size,~ either from problem to problem or from problems which were used during the i n i t i a l learning procedures. Based on his findings, Harlow has claimed that these learning practices transform an organism from a creature that adapts to the environment by the method of t r i a l and error to one that adapts by using thinking s k i l l s and hypotheses searching. Influenced by Harlow's work, Adams (1954) has studied whether similar effects could be produced by giving students practice on the same type of problems rather than on a variety of new types of problems. Adams carried out his study with airmen trainees. He used a series of discrimination problems. Each problem was built of a pair of stimuli such as circles, triangles and other more abstract figures. Two groups of subjects were trained to solve discrimination tasks. One group practiced on a set of problems which share the same general solution for all problems but the stimuli characteristics varied from problem to problem. The second group practiced on the same problem. Following training, subjects in the two groups were tested on their discrimination a b i l i t y in similar and in new problems. Adam's study demonstrated that subjects who practiced on one type of problem performed better on the transfer tasks than subjects who practiced on a variety of problems. The discrepancy between the findings led to a series of studies investigating the effects of these two practice experiences on the development of students'ability to apply learned rules and principles. The discrepancy was partly resolved by Duncan (1958). Duncan studied transfer of perceptual-motor paired associates tasks as a function of two variables: degree of variation in training which was defined in terms of the number of different sets of training stimuli, and the amount of training. The second variable was considered necessary since i f total practice is equal for all degrees of varied training, increasing the number of tasks means decreasing the amount of practice on each. Different groups were trained with varying numbers of tasks using different sets of stimuli for varying number of days. Following training, all the subjects were tested for transfer to new sets of stimuli. The results showed that among groups trained with different sets of stimuli, transfer increased as a direct function of degree of variation. Ingeneral, when the total number of problems was equal all degrees of varied training produced better transfer than constant training. Duncan defined varied training in terms of the number of tasks (variations) introduced during practice. On the basis of his results, Duncan has claimed that since certain degrees of varied training may be more advantageous than others, training could be treated as a continuum with constant training as one extreme end. For a given number of training problems, maximum v a r i a b i l i t y will be reached i f each single problem differs from the other problems on some well defined characteristic. Based on Duncan's results, the findings of Adams could be explained by the lack of sufficient variety in his varied practice experiences.

Instruction whichenabtesS~den~ W Develop H~herMentaiProcess~

183

Variety in task examples involving application of rules was studied by Gagn~ and others (1965) and Gagn~ and Bassler (1963) with sixth grade students learning nonmetric geometry. Three different degrees of variety were employed in the examples which followed the presentation of each learned rule. In terms of scores on a final test covering these rules' application, significant effects of variety of examples were not found. I t may be noted, however, that the level of mastery of the pre-requisite s k i l l s and knowledge was not high and this fact may have affected the results. Whenthe same students were tested on an equivalent test two months later, there is indication that beyond a certain point v a r i a b i l i t y in examples has no significant advantage. This finding may suggest that difference in variety of practice problems could be placed below or above certain c r i t i c a l thresholds, leading to large differences in the a b i l i t y to apply learned rules and principles. This idea is consistent with Duncan's view of training as a continuum with constant train. ing as the one extreme end. Traub (1966) defined variety in terms of problem heterogeneity. To make his arithmetic problems heterogenous, he varied the problem context and constructed his examples so that the size of the numbers, the signs of the numbers and the portion of the number line employed was varied from one problem to the next. This in fact resulted in a variation in the answers on successive problems. Traub found that the group which was instructed with heterogenous problems was superior to the group practicing with homogenousproblems on a test administered after the practice session. Although previous research has shownthat constant practice experiences have some, though limited, impact on the a b i l i t y to apply learned s k i l l s and principles, there are studies which focus on the potential negative consequences of this type of practice. I t has been demonstrated, for example, by Luchins (1942); Luchins and Luchins (1950); Guetzkow (1951); and Logan and Wodtke (1968) that an over emphasis on specif i c behaviours acquired through constant practice may develop fixation in the thinking behaviour of students. They furthermore recognised that over-learning through constant practice may cause an inhibitory effect which could ultimately prevent students from future learning and restrain their a b i l i t y to perform on similar and related tasks. This phenomenonis sometimes called 'functional fixedness' or ' r i g i d i t y ' . I t emphasises that over practice with problems or solution of problems that are very similar can have a negative effect on the students' a b i l i t y to use the learned s k i l l s and knowledge in new and different situations. Although a definitive conclusion about the relative effectiveness of variety versus constant practice experiences cannot be drawn from the studies cited above, these studies indicate that under relatively controlled and laboratory experimental conditions the amount of variety of practice problems can have an effect on the development of the a b i l i t y to apply rules and principles to new situations. I t s t i l l remains for future research to determine the conditions under which variety in practice problems is important and what their precise learning outcomes are. This is particularly necessary for rules and principles to be learned by students under classroom conditions.

THE ROLEOF INTELLIGENCE IN LEARNINGTO APPLY PRINCIPLES Educators and researchers have for a long time recognised that differences in i n t e l ligence among students are closely related to all kinds of school performances. Virtually all measures of academic achievement show strong relationships with IQ scores and other measures of academic aptitude. In most heterogenous classrooms from one fourth to one half of the variation in school achievement appears to be related to scores on an intelligence test. Another way of saying i t is that the

184

T. Levin

correlation between these two scores ranges from about +0.50 to +0.70. Nevertheless, finding a relationship between two variables does not demonstrate the reasons for i t . Since schools seek to change a wide range of cognitive abilities of students, i t is essential to explore the role of intelligence in the development of the a b i l i t y to apply rules and principles to new situations. Symonds (1964) has found that highly intelligent students are best able to make transfer of learned material to situations that are remote from those in which the learned material was f i r s t introduced. He also found that they are best able to transfer principles that are more abstract. Similarly, Werner (1930) and Craig (1953) indicated that individuals with higher intelligence scores show greater transfer a b i l i t y , seek out new relationships, and generalise to a larger degree than do individuals with lower intelligence scores. Green (1964) has also claimed that students whose scores are in the top one third of an intelligence test use prior knowledge more readily than do students with lower scores. Consequently, according to Green, in complex and sequential learning differences in performance are clearly related to differences in intelligence scores. Green, however, argues that when learning tasks do not require much transfer of prior knowledge, the differences between bright and dull students are reduced or disappear. This means that i f appropriate instruction is provided to students to learn the necessary skills for effective transfer, the role that intelligence will play in determining successful learning can decrease. Or to put i t the other way around, when instruction is not appropriate, a new learning task which requires the use of previously learned knowledge "becomes a puzzle and i t takes more intelligence to solve that puzzle" (Green, p.93). There is evidence that many of the behaviours necessary for application of rules and principles can be learned by low IQ students whose scores are at the bottom one-third or one-fourth of the distribution on an intelligence test. Thus for example, Duncan (1958) who used different practice experiences (see page io ) has found that the best one-fourth of the students did not improve as much after the practice as the poorest one-fourth; partly because the best one-fourth was already near the upper limit of possible performance. Similarly, Lundsteen (1970) who manipulated different learning experiences in an attempt to produce problem solving performance, found the greatest improvement in problem solving performance among the students who are at the bottom one-third of an intelligence test score d i s t r i bution. A different and a more general manifestation of the role played by intelligence in the explanation of school learning has been suggested by Bloom (1976). Bloom has demonstrated, in the framework of sequential learning, in both experimental and observational studies that general intelligence scores are predictive of school learning outcomes only to the extent to which an intelligence test overlaps with some of the measures of the more specific cognitive characteristics of students, namely previous achievement on relevant tasks. This implies that the students' intelligence plays a major role in explaining and predicting outcomes only when school experiences do not provide the students with the appropriate pre-requisite skills required by the learning outcome measures. The notion that intelligence determines learning is recognised as limiting what school and teachers in their classes can do. Evidence on the feasibility of improving the a b i l i t y of low IQ students to solve new problems is therefore of great value to educators since i t reveals the potential power of schools under more ideal conditions of instruction. I t also shows that schools could accomplish their goals with a larger proportion of students than they previously did. Specification of the learning conditions that can reduce the role of intelligence in predicting students performance on the knowledge of facts, rules and principles is already

Instruction which ermble= Students to Develop Higher Mental Processes

185

provide in the literature. It remains for further research to identify the type of learning experiences which can reduce the role played by intelligence in determining students ability to apply learned rules and principles in new situations.

3. The Theoretical Model and its Use

The previous review of the literature sets the basis for the development of a conceptual model to guide further study of the learning and evaluation of the a b i l i t y to apply rules in new problem situations.

THE ABILITY TO APPLY RULES CAN BE MEASUREDIN A WIDE VARIETY OF PROBLEMSITUATIONS The literature suggests that a b i l i t y to apply abstractions, such as rules and principles, should be regarded not as a single general t r a i t but rather as a set of specific a b i l i t i e s (Horrocks, 1946; Scandura, 1967). Other researchers are generally agreed that these more specific t r a i t s are a function of the nature of the problem situations. This means that a student may be able to apply a rule to a new problem situation taken from real l i f e situations, whereas, the same student may find i t d i f f i c u l t or impossible to carry out a similar solution process in an a r t i f i c a l problem situation. Unfortunately, however, the literature does not provide sufficient evidence for a set of generally accepted and well defined dimensions within which the a b i l i t y to apply rules and principles in new situations may be assessed. The characteristics of problem situations which can influence the a b i l i t y of students to solve problems were discussed earlier (Kruteskii, 1967; Loftus and Suppes, 1972). This research indicates that altering particular characteristics of a problem situation determines, to some extent, the level of d i f f i c u l t y students will have in solving i t . But even more important, both teachers and evaluators can make use of these dimensions of problems as one basis for developing a great variety of problem situations. In view of a large range of possible problem situations, this research chose to study the a b i l i t y to apply rules as a function of two structural characteristics of problem situations; familiarity and complexity. At one end of the familiarit~ dimension are situations which are relatively familiar to the students in terms of the general context and structure of the problems. The familiar problems are similar to those problem situations used in the learning of the rules or their practice. At the other extreme end of the familiarity dimension are problem situations which are very different in context and structure from situations used as illustrations in previous learning. The complexity dimension is based on the number of rules or 186

~structionwhichenabtesS~denB ~Devw~p H~herMentalPmce~es

187

principles required for solutlon of applicatlon problems. At one end of this dimension only a single rule is required for the solution, while at the other extreme, many rules are required. These two dimensions ( f ~ Z ~ ' ~ t l / and c o ~ - ~ t ~ j ) serve as major c r i t e r i a for the construction of measures of performance on application of rules in this study. The model developed here conceptualises application achievement as a multi-dimensional t r a i t defined on two qualitatively different dimensions. These dimensions are characterised by the degree of familiarity and complexity of a problem situation. I t is assumed, on the basis of the literature, that i t is easier to apply learned rules in application problems that are more familiar to the students than to apply the same set of rules in less familiar problem situations. I t is also assumed that 'simple' problems which require only one rule for a correct solution are easier than the more 'complex' problems which require two or more rules for a correct solution. THE CHARACTERISTICS OF LEARNINGEXPERIENCES DETERMINEHOWWELL STUDENTSCAN APPLY RULES There is a body of evidence to indicate that students are not able to apply rules and principles to new situations unless they have learned the rules and principles to a satisfactory level of mastery (Block, Ig7O; Gagn6, 1965). This means that /¢no~Zed~/e of rules is a necessary pre-requisite for the ~ i Z i ~ j to ~pply these same rules in a variety of new situations. I t is l i k e l y that students who have learned a set of rules to a mastery level w i l l be better able to supply them in new situations than w i l l students who have learned these rules to a lesser degree. The mastery learning strategy (Bloom, 1968) specifies the necessary learning and instructional conditions under which students are able to acquire knowledge or rules and principles to a pre-determined level of mastery. This stratei~y provides each student with the amount of time and help he needs in order to reach the pre-set mastery criterion. I t uses frequent testing to provide immediate and specific information feedback to the students on what they have learned well and what they s t i l l need to learn, and corrective procedures by which the learning d i f f i c u l t i e s can be resolved. The mastery learning strategy enables most of the students reach a preset level of mastery and to become very slmilar to each other in terms of their learning outcomes. Knowledge of rules was recognised to be a necessary l ~ t not s u f f i c i e n t condition for the attainment of the a b i l i t y to apply rules and principles. For this study, this means that in addition to the knowledge of the relevant rules, students must be provided with appropriate learning experiences and practice to improve their a b i l i t y to apply learned rules. The present model conceptualises these experiences as process pre-requisites; that is, learnlng experiences that are intended to develop the s k i l l s involved in the application process. This model includes two distinct and contrasting types of learning experiences. One type of learning experience includes mainly practice problems which provide students with opportunities to practice each s k i l l required by the application process. Tnis type of learning experience also provides a variety of problem s i t uations in which each of the s k i l l s can be practiced. The previous review of the literature suggests that varied learning experiences of this type have a good potential for helping students to learn how to apply the rules to a wide range of new problems.

In contrast, the second type of learning experience includes only a restricted set of problems. I t also provides students with opportunities to practice only a ltmt-

188

T. Levin

ted set of the s k i l l s underlying the application process. There is some evidence in the literature to indicate that such a restricted type of learning experience results in their being able to apply the rules in only a very narrow range of possible situations. (A more detailed description of these two types of learning experience is presented later in this chapter.) At this point, the model developed here includes two components which are anticipated to be causally related; Instructional and Learning Conditions and Achievement in Application of rules. One component, the instructional and learning conditions, specifies students knowledge of the rules and their practices in s k i l l s underlying the application process. In other words, i t specifies the knowledge and the process pre-requisites. The second component, achievement in application of rules, refers to students a b i l i t y to apply rules to problem situations defined on the familiarity and the complexity dimensions. In particular, the model indicates that there are two levels of knowledge of rules that students have; a high mastery level (85 per cent) and a lower level of mastery (50 per cent). These two levels of mastery of the knowledge of the rules are anticipated to determine students' a b i l i t y to apply rules to a wide variety of application problems. Therefore, the f i r s t hypothesis of this study is: groups of students who possess more of the necessary ~,e-requisites (85 per cent mastery level of rules' knowledge) will score higher on measures of the ability to apply their knowledge than groups of students who have fewer of the neeessary pre-requisitee (50 per cent mastery ~evel of ruses 'knowledge) (see hypothe-

sis l in Fig. 3.1). The model also indicates that in addition to knowledge of rules, students practice in application s k i l l s contributes greatly to their a b i l i t y to apply learned rules to new situations. Two types of application learning experiences are included and both are expected to have some additional effects on students' application achievement over that of knowledge of rules only. This reasoning leads to the second hypothesis of the study: following the knowledge of the rules to a maatery level (85 per cent), students with experiences in application skills will score higher on the measures of applicatlon achievement than students who do not have learning experiences in application skills) (see hypothesis 2 in Fig. 3.1).

The model indicates that there are two different kinds of learning experiences in s k i l l s required by the application process. These distinct experiences are assumed not only to have different effects on the general a b i l i t y of students to apply a set of rules, but also to have differential effects of students' a b i l i t y to apply the rules on the familiarity and the complexity dimensions of application. I t is anticipated that the more varied and specific type of practice experiences w i l l enhance students a b i l i t y to apply learned rules to a wide range of new problem situations such as the unfamiliar and the more complex application problems. In contrast, the more constant and general type of experiences in application s k i l l s are more l i k e l y to develop the a b i l i t y to apply learned rules in only these problem situations which are familiar and simple. These speculations lead to the third hypothesis of the study: following the knowledge of the rules to a mastery level, the quality of students' experiences in application skills determines their ability to apply the ruses in a range of application problems (see hypothesis 3 in Fig. 3.1).

THE LINK BETWEEN INTELLIGENCE AND APPLICATION OF RULES DEPENDSON THE LEARNING EXPERIENCES PROVIDEDTO THE STUDENTS Intelligence is often regarded as a personal attribute of a student that is relatively stable and highly general. The literature indicates relatively strong relationships between measures of students' intelligence scores and various measure of school performance, including the a b i l i t y to apply learned rules and principles

Instruction which enables Students to Develop Higher Merit# ~ I

Instructional and Learning Condttt oes A.

Knowledge of Rules (50 per cent level of of mastery)

B.

Knowledge of Rules

(85 per cent level of mastery)

C. Knowledge of Rules (85 per cent level of mastery) + Application Skills (Constant and General Practice) D. Knowledgeof Rules (85 per cent level of mastery) + Application Skills (Varied and Specific Practice)

Performance on

Application of Rules I.

Familiar and Less Familiar Problems Simple and Complex Problems

>

'I (conditions A L /

/

,

; l~°srdu s~ it°nSndBDi')/ Hypothesis 4

~othesis3 Students' Characteristics i

General Intelligence IF/ 11 2. Specific Aptitude ~

relates students characteristics and application achievement (conditions A, B, C, D)

~

I.

I

Fig. $.1 The conceptual model of the 8tud~

189

190

T. Levin

(Green, 1964). These relationships have been frequently taken as indication teat a student's i n t e l l i g e n c e scores determine not only his a b i l i t y to learn but also his a b i l i t y to apply his knowledge in new problem s i t u a t i o n s . In contrast, there is evidence that the a b i l i t y to apply acquired knowledge, in p a r t i c u l a r for low i n t e l ligence students, depends on the previous learning experiences that these students have had. This implies that i f students are provided with appropriate i n s t r u c t i o n to learn the s k i l l s required for an e f f e c t i v e t r a n s f e r of knowledge, the link between application achievement and i n t e l l i g e n c e can be reduced. Under ideal learning and i n s t r u c t i o n a l conditions the correlations could decrease to something approximating zero. At this point, the model underlying this study in i t s f i n a l form is established. It includes three components, student's c h a r a c t i s t i c s , i n s t r u c t i o n a l and learning conditions and application achievement, which f i t together in the schematical presentation displayed in Fig. 3.1. The relationships among the three parts of the model ~re based on the following r a t i o n a l e : i f learning conditions are provided to a group of students in order to learn a set of rules to a low level of mastery, and to attain a large v a r i a b i l i t y in students' scores then, the high i n t e l l i g e n c e students w i l l know most rules and the low i n t e l l i g e n c e students w i l l know fewest rules. Consequently, students i n t e l l i g e n c e measure w i l l determine students a b i l i t y to apply the rules in new situations (Condition A in Fig. 3.1). Also, i f learning experiences provided to students focus on the knowledge of the rules only, some of the high i n t e l l i g e n c e students w i l l be able to apply the rules in new problem situations while most students with low i n t e l l i g e n c e scores w i l l not be able to apply the same rules to a v a r i e t y of new problems. Therefore, students i n t e l l i g e n c e measures w i l l predict t h e i r a b i l i t y to apply the rules in new application s i t u a t i o n s . (Condition B in Fig. 3.1) But, i f both adequate level of mastery of the rules is attained by the students and they are provided with opportunities to practice in application s k i l l s , then students i n t e l l i g e n c e scores w i l l have less e f f e c t on t h e i r a b i l i t y to apply the learned rules. Moreover, the more varied and s p e c i f i c these practices in application skills are, the less students i n t e l l i g e n c e score w i l l predict t h e i r a b i l i t y to apply the rules (Condition D in Fig. 3.1) compared with i t s p r e d i c t i v e strength under more constant and general type of practices (Condition C in Fig. 3.1). This reasoning underlies the fourth hypothesis of the study: the c o r r e l a t i o n s between students'intelligence measures and their ability to apply rules in new problems depend on the quality of the learning experiences provided to the students. (See hypothesis 4 in Fig. 3.1) The conceptual model in i t s f i n a l form indicates the profound importance of learning and i n s t r u c t i o n a l experiences in e i t h e r improving or depriving students a b i l i t y to apply rules to a variety of application s i t u a t i o n s . In other words, i t anticipates the existence of causal r e l a t i o n s h i p s between the nature of the i n s t r u c t i o n a l and the learning experiences provided to the students and t h e i r a b i l i t y to apply rules in a v a r i e t y of new problems. Also, these experiences are assumed not only to a f f e c t s t u d e n t s ~ a b i l i t y to apply learned rules but also to a l t e r the r e l a t i o n s h i p between students'stable c h a r a c t e r i s t i c s such as general i n t e l l i g e n c e or s p e c i f i c aptitude and students' performance level on application of rules. Since causal relationships underlie this model, i t s examination is required to design an experimental study. Central to this study is the development and implementation of i n s t r u c t i o n a l and learning experiences which adequately represent the four experimental conditions specified by the model. These are discussed in the following section.

Instruction which enables S~den~ m Develop High~MentalProcesses

191

THE USE OF A MASTERY LEARNING STRATEGY FOR THE LEARNING OF RULES One purpose of the present study was to determine how and to what extent students' knowledge level of a set of rules determines their a b i l i t y to apply the rules to a variety of problem situations. To control and alter the level of mastery of the knowledge of the rules a mastery learning strategy (Bloom, 1968) was used. The mastery learning strategy provides students with instructional and learning opportunities so that each student gets the time and help he needs to reach the prescribed mastery criterion. At the heart of this strategy are the diagnostic assessment procedure of students' learning process accompanied by feedback and corrective procedures to guide students furthcr learning. This means that i f a student does not reach the pre-set criterion level in his f i r s t t r i a l of learning a given task, he receives specific information as to what he had learned well and what he had not learned well, and is given additional time and instruction about how his mistakes can be corrected. To re-learn or correct his mistakes the student is provided with additional instructional materials and learning aids. For each learning task, unit or objective to be attained, a student can repeat feedback-corrective procedures one or more times until he reaches the pre-set level of mastery. Students who attain or surpass the required performance level continue their learning of the subsequent unit, task or objective. In order to u t i l i s e the mastery learning strategy a set of learning materials was developed. I t includes three sequentially organised learning units (semi programmed) which covered the terms, concepts and the rules to be learned; three additional learning units with different types of explanations to the same term, concepts and rules (review units); a double set of achievement tests for each learning unit which were to be used immediately after the learning of a certain unit is completed, and f i n a l l y reference sheets to describe learning and instructional sources that the students could use to correct t h e i r mistakes in addition to the review units.

THE DESIGN OF LEARNING EXPERIENCESFOR THE DEVELOPMENTOF THE ABILITY TO APPLY RULES In order to determine to what extent we can develop student's a b i l i t y to apply learned rules to a variety of new problems, specific instructional and learning experiences were developed. Four types of learning experiences have been recognised and suggested by the l i t e r a ture to influence the development of application a b i l i t y . One is the varied practice advocated by Harlow (1949) and Duncan (1958). This type of learning emphasises students' practice on a large number of problems that share the same general solution but d i f f e r on specific stimulus either from problem to problem, or from problems that were used in previous learning. The second is the constant practice which was suggested mainly by Adams (1953). In this type of learning experiences the repeated presentation of the same type of problems are used as practice problems. The third type is the over learning practice (Luchins, 1942) which is characterised by an over emphasis on a specific behaviour or a problem situation acquired through constant practice. The fourth type of learning experiences was suggested by Tyler (1949). This is the learning of each specific behaviours underlying the application process as inferred from a pedagogical or logical task analysis of the process. In view of a much larger range of possible learning experiences that can be designed on the basis of the literature review, this research chose to concentrate on only two different types of experiences. These were largely based on a synthesis of the

192

T. Levin

characteristics involved in the four approaches cited above. One is based on a synthesis of Adams' approach to constant practice and Luchins' view of over-learning through constant practice. This type is labelled oonstant and general. The second integrates both Duncans' and Harlows' approach to varied practice, and Tylers' approach to practice specific behaviours underlying the process of application of rules. This type of learning is labelled varied and 8peoific. The two types of learning experiences take the form of learning of distinct sets of problems. Further, these two learning experiences are considered to be distinct in terms of their structural characteristics as well as their anticipated effects on the development of the a b i l i t y to apply rules to a variety of new situations. Constcmt and general learning experiences - In this type of learning experience the problems are classified according to the rules required for their solution. For each single rule, a student needs to solve a set of problems that require the use of this rule only. Within each set of problems, the practice problems are randomly organised. Yet, since a number of rules are to be learned by a student, the sets of problems are organised according to the sequence in which the rules were learned. This is very typical in mathematics and the sciences but i t may also happen often in other subject areas studied in the school curriculum.

Problems of this type of learning experience are similar to the problem situations which were used in the i n i t i a l learning of the rules. Further, the problem situations are very similar in many ways. That is, problem situations are of very limited nature in terms of their context, wording and presentation form. The problems used in this study, for example, focus mainly on events that describe simple situations in which chance occurrences are i l l u s t r a t e d such as throwing dice or coins. The problem situations are concerned with a small number of objects such as one or two dice, one or two coins, or ten to twelve pieces of paper. These constitute a relatively small number of elements to be considered in solving an application problem. Also, the situations emphasise mainly the numerical properties of these objects with the use of a limited set of integers. No additional aids in the form of a diagram, a picture or other illustrations are involved. The presentation modes of the problems are verbal only and do not include the use of any manipulative materials. In order to solve correctly a problem in this type of learning experience, a student needs to use only one rule at a time. This means that a student has to practice a relatively simple application process. Under these learning conditions a student has to practice only a limited number of s k i l l s required by an application problem. For example, only rarely does a student need to determine which previously learned rules are needed for a solution of a problem, or to j u s t i f y their use. Under these conditions a student frequently needs to identify the similar elements in a problem and to use the rule appropriately. As specified here, these learning experiences emphasise the homogeneity of the problem situations and the similarity in the solution process required by each problem. Due to the restricted type of experiences and the opportunities provided to students to practice only a limited set of behaviours underlying an application process, these learning conditions are believed to be effective in developing application a b i l i t y in students. But, i t is also l i k e l y that under these conditions, the student's a b i l i t y to apply rules will be limited to a narrow range of possible problem situations, and mainly to those problems that resemble the practice experiences. In order to provide students with the constant and general type of experiences learning materials were developed to f i t the specific characteristics of these experiences. These materials consist of a set of practice problems that students are to solve in the order prescribed for them. The problems are classified according to the rules required for their solution, and the statement of each rule preceeds the set of its practice problems. The sequence in vA~ich these classes of problems are presented to the students f i t s the order in which the rules were learned

InsfructionwhichermblesSWden~m DevelopH~herMentelProcezses

193

in the classes; from the easier rules to the more d i f f i c u l t ones. To further meet the requirements of this type of learning experience, the practice problems deal only with limited types of problem situations that closely resemble the problem situations in which the rules were taught. In their final form, these learning materials include 27 problem situations. Since more than one question is posed for each problem situation, the materials include 79 specific questions to solve.

Varied and specific learning experiences - A second type of learning experience is largely based on Duncans', Harlows' and Tylers' approaches to the development of application a b i l i t y . Their approach emphasises the practice of the specific s k i l l s underlying an application process with a relatively heterogeneous sample of problem situations. On the basis of these two requirements, practice problems in this type of learning experience were classified into three sets of practice problems. The nature of this classification was guided by research evidence concerning three d i f ferent dimensions of problem situations that affect the d i f f i c u l t y level of a problem. These dimensions include the number of objects involved in a problem, the types of objects that the problems are concerned with, and the presentation mode of the problems such as verbal, figural or pictorial modes. The problems used in this study, for example, include a small number of objects such as one or two coins or dice and a larger number of objects such as four or five dice and 70 people or many slips of paper. Also, problems in this type of learning experience differ in terms of their context. I t includes problems that are relatively concrete and familiar to the students such as children's games or clothes, and problems that are more abstract such as complex geometrical shapes. Finally, the problems differ in their mode of presentation. There are problems which are accompanied by pictorial figures of the objects involved, such as cards or regular tetrahedrons. There are also problems that include tables of numerical data, and verbal problems without additional aids such as diagrams or pictures. Under these learning conditions, the practice problems within each set are either organised in an increasing order of d i f f i c u l t y , or classified according to similar properties. Thus, in this study the more concrete problems are presented separately from those that are more abstract. Partly because of the way in which the problems are organised, and partly because of the data and the questions posed, a student needs to carry out the entire application process to solve each problem. This means that the student has practice in each of the specific s k i l l s required by the application process. In many of the problems a student needs to search for familiar and unfamiliar elements, to restate the problem, to predict the consequences of using a particular rule, to j u s t i f y its use and to specify the limits within which the particular selected rule is relevant. I t is also necessary that a student makes use of most of these processes in those problems that require more than one rule for a correct solution. Since the varied and specific type of learning experiences provide students with opportunities to practice most of the specific behaviours required by the process of application of knowledge to new students, i t is likely that under these learning conditions students will develop an a b i l i t y to apply rules to a wide variety of new problem situations. Special materials were developed to actually employ the varied and specific type of experiences. These include two major components. The f i r s t consists of a set of practice problems that students are to solve in the order specified for them. These problems are classified according to the number of objects involved (few and many), the type of objects that the problems are concerned with (concrete and abstract), and the presentation mode of the problems (figural and verbal). A statement describing the specific dimension being manipulated in a particular class of problems preceeds each set of problems.

194

T. Levin

As required by the characteristics of the varied and specific type of experiences, each problem requires at least one rule to arrive at a correct solution. No indications are provided to the students regarding the appropriate rules needed for each problem. The problem situations d i f f e r markedly from each other in their context as well as structure. The second component of the learning materials includes a modified version of the application process diagram (To~ono~ of E~oationa~ Objectives, Bloom, et a~, 1956). Supplemented to this diagram are guided questions and instructional cues pertaining to each behaviour, s k i l l or process presented in the diagram. To use these materials appropriately, stude~ts are required to follow each stage displayed in the diagram and to make use of the questions and suggestions available to them. On average each application s k i l l or behaviour of the diagram is followed by four directive questions or suggestions. In their final form these learning materials include 15 different problem situations with four specific questions for each problem. Table 3.1 describes the general nature of these two types of learning experiences and juxtaposes them with respect to three general dimensions: the organisational nature of the practice problems, the structural characteristics of the problems, and the type of application process required to solve the problems. I t is believed that these two types of learning experiences are generalisable to a range of application objectives set by schools. The Experimental Design In order to investigate the hypotheses of the study, a particular form of an experimental design was required. The following section describes the experimental conditions and procedures necessary to test the hypothesised relationships among the components of the model. The f i r s t hypothesis of the study anticipated that a group of students with greater knowledge of a set of rules will be better able to apply these rules than a group of students with less knowledge of the same rules. In order to test this hypothesis, two study groups were formed. One group of students were to learn a set of rules to a pre-set standard by the use of the mastery learning strategy. The second group of students were to learn the same rules to a lower standard by a more conventional learning strategy. These two groups are l i k e l y to d i f f e r in their achievement level of knowledge of the rules. The mastery learning group is required to attain a pre-set criterion level (85 per cent mastery) on the knowledge of each rule. In contrast, the more conventional study group learns each single rule under conditions in which a standard criterion is not specified and feedback and corrective procedures are not provided. Thus the non-mastery group is l i k e l y to have a lower performance level (about 50 per cent mastery) on the knowledge of the rules. Furthermore, while students in the mastery group as a whole will become more similar to each other in terms of their achievement, the non-mastery group members will become more dissimilar with regards to t h e i r achievement level. By comparing the means of these two groups in their a b i l i t y to apply the learned rules, the causal contribution of rules' knowledge to rules' application can be inferred. The second hypothesis of this study suggests that beyond a certain mastery level of the knowledge of the rules, students who are provided with some form cf practice s k i l l s underlying the application process w i l l be better able to apply the learned rules than will students who received no additional practice in application s k i l l s . For the purpose of testing this hypothesis two additional study groups were formed.

Instruction which enables Students to Develop H~gher Mental Processes

195

TABLE 3. l A description of the learning experiences for application of rules

Learning Experiences

Constant and General

Varied and Specific

I. Classification

Problems are classified according to stated rules required for solution

Problems are classified with respect to specific problem characteristics

2. Sequence

Sequentially ordered with respect to rules' d i f f i c u l t y

Semi-sequentially ordered with respect to specific characteristics

I. Number of objects*

One or two

One, two, three, five, seventy and more

2. Nature of objects

Mostly quantitative (countable objects)

Quantitative and Qualitative

3. Presentation modes of objects

Verbal only

Verbal, figural and pictorial

4. Number of rules required for a solution

Always one for each problem

Always more than one for each problem

Type of application Process Practiced

Limited Restricted to a narrow range of behaviours underlying the application process

Broad Emphasis on specific behaviours of the application process in view of the general process

Problem Or~anisation

The Structure of The Problems

* The objects determine the context of the problem (for example, dice, coins, spinners) In both groups students were to learn the rules to a similar level of mastery, but in addition were provided with the constant and general type of experience while the varied and specific type of experience was provided to the second group of students. By comparing the mean scores on the test of a b i l i t y to apply rules between the group of students who mastered the rules only and the groups which had mastered the rules as well as had experiences in the application, the differential effects of learning experiences in application skills can be determined. To set a solid ground for combining the two groups of students who were provided with different types of application experience, the actual learning time required for these two experiences was controlled. That is, students provided with the constant and general type of experience were to spend a similar amount of practice time as were the students who experienced the varied and specific type of experience.

196

T. Levin

The inclusion of these two study groups in the design of the study enables us to test the third hypothesis of the study. This hypothesis states that students who are provided with the varied type of practice experience (varied and specific in addition to mastery of the rules) w i l l apply the learned rules to a higher performance level than students who are provided with the more restricted type of practice experience in application (constant and general) in addition to mastery of the rules. Further, i t is believed that the d i f f e r e n t i a l effects of the varied and specific type of experiences over the constant and general type on students a b i l i t y to apply rules w i l l be strongest for the less familiar and the more complex application problems. I t is also believed that application experiences of the constant and general type w i l l be most effective in application of rules to the simple and familiar problems. Since students in these two groups spend the same amount of practice time, an estimation of their di f f erent i al effects on students' a b i l i t y to apply rules can be inferred by comparing the mean scores between the groups. The design this far includes four study groups which actually represent four d i f ferent types of learning conditions; non mastery learning conditions directed to the knowledge of the rules (group l ) , mastery learning strategy directed to the knowledge of the rules (groupg, mastery learning conditions towards the knowledge of the rules supplemented by the constant and general type of application practice (group 3), and f i n a l l y mastery learning conditions towards the knowledge of the rules followed by the varied and specific type of application experience (group 4) (see Fig. 3.2).

Group l

Learning rules under Testing on the Non-Mastery Strategy ..~p. Application and Testing on Knowledge of rules of rules

Group 2

Learning rules under the Testing on Mastery Strategy and -~D-Application Testing on Knowledge of of rules rules

Group 3

Learning rules under the Mastery Strategy and Testing on Knowledge of rules

Experiences in Testing on Application Skills --D~Application ("Constant and of rules General" type)

Group 4

Learning rules under the Mastery Strategy and Testing on Knowledge of rules

Experiences in Testing on Application Skills ---D-Application ("Varied and of rules Specific" type)

Fig. 3. 2 The experimental design

Instruction which enebtesStudents to DevelopHigher Mental ~

197

This experimental design enables us also to investigate the fourth hypothesis of the study. According to this hypothesis, the relationship between students general intelligence ~easures and their a b i l i t y to apply rules depends on the learning conditions provided to the students. In particular i t anticipated that the more speci f i c and relevant the learning conditions tend to be, the less will the intelligence measure be able to predict students' a b i l i t y to apply. For the purpose of testing this hypothesis, the correlation coefficients between intelligence measures and application achievement measures will be computed for each study group. Due to the experimental nature of the study, students were randomly assigned to the four study groups in order to form comparable and heterogeneous groups on such relevant measures as students general intelligence scores and mathematics aptitude. Also, the four defined experimental conditions were randomly assigned to the four study groups. I f differences between the groups, on the relevant variables such as intelligence measures, are detected despite randomisation procedures, a statis-, tical technique which can allow for these differences will be used (analysis of covari ance). The design of the study was carried out with ninth grade students who learned a set of four probability rules in their regular school setting. This particular grade level was chosen to f i t the requirements set by the rules to be learned.

INSTRUMENTS AND MEASURESUSED IN THE STUDY This section reports on the instruments used to measure each of the three major variables involved in this study; application achievement, knowledge pre-requisite and measures of student~ intelligence.

Achiev~nent in application of z ~ s - in order to measure student~ a b i l i t y to apply learned rules, 35 test items were constructed. These items sample the two dimensions of problem situations conceived in this study, the f c ~ n i l i ~ and the oomp l e ~ dimensions. Fc~niZio.r problems are those which resemble previous learned problem situations in terms of their context and structure, whereas less familiar problems differ in context and structure from the ones used in previous learning. Familiar items are concerned with both objects that are well defined and are similar to events that students actually face in a day-to-day l i f e . For example, the problems deal with the days and months of the school year, school examination, a v i s i t to a dentist etc. In contrast, problems of the less fc~n~IY~zr type are concerned with genetics and physical t r a i t s , rocket mechanism, molecular movements etc. Someof the less familiar items were taken from advanced s t a t i s t i c and probability books (Probc~bilit~ wi~h Statistical Application by Mosteller, Rourke, Thomas (1970); Modern Elementar~ Statistics by Freund(1970) and were adapted for the present study. The oomple~t~ dimension of application problem situations has to do with the number of rules required to reach a solution to the problem. Sidle problems require the use of only a single rule for a correct solution, whereas, many rules are needed for the correct solution to co~lex problems. The simple items were constructed so that a student needs to use only one rule at a time in order to reach a correct solution. The complex items were constructed so that a student needs to use 2, 3, or 4 rules, to arrive at a correct solution.

198

T. Levin

In order to ensure content v a l i d i t y of the application test, two independent subject matter specialists lwho were acquainted with the learning materials) judged the degree to which each test item adequately matched the required characteristics. A relatively high degree of agreement (0.90) between the judges was established. The construct v a l i d i t y of the test ~as also examined by using a group of 12 students. Following their learning of the rules, these students were tested on the test of application of rules. The achievement pattern indicated that less familiar items are indeed more d i f f i c u l t to solve than familiar items. Also, as anticipated, complex problems were found to be more d i f f i c u l t than simple ones. o f r u l e s - in order to measure the level of mastery reached by students on knowledge of the r u l e s , a 30 item t e s t was developed. The items in this t e s t are based on the learning objectives and materials that were used during the course of the study. The content v a l i d i t y of t h i s t e s t was examined by two independent subject matter s p e c i a l i s t s . A 95 per cent level of agreement between the judges was established regarding the extent to which the t e s t items adequately cover the knowledge of rules included in the learning materials.

Knowledge

measures - to measure the students' i n t e l l i g e n c e l e v e l , the National Educational Development Test (NEDT) was used. The NEDT battery is known to measure a broad aspect of general educational development of students. A composite measure cf performance on f i v e educational tests such as verbal a b i l i t y , mathematical usage and reading of social studies and natural sciences was used in t h i s study as an index of students' general i n t e l l i g e n c e score. In a d d i t i o n , students' scores on the NEDT Mathematical Usage subtestwereused as an index of students' aptitude for mathematics. This t e s t measures students' a b i l i t y to use mathematical p r i n c i p l e s in formal exercises and in practical q u a n t i t a t i v e problems. Intelligenoe

The r e l i a b i l i t y of the composite score, used as a measure of i n t e l l i g e n c e , in this study is estimated by the use of the Kuder Richardson to be about 0.97. I t s pred i c t i v e v a l i d i t y in terms of predicting grade point average was estimated to be between 0.69 and 0.77. These estimations ~ere obtained by a study of the t e s t results and teacher's grades with ninth grade students. The r e l i a b i l i t y of the Mathematics Usage t e s t was estimated to be about 0.83; while i t s p r e d i c t i v e v a l i d i t y of students grade point average was estimated to be about 0.56.

THE RESEARCH PROCEDURES The experiment was conducted in a comprehensive suburban high school with I00 ninth grade students from four d i f f e r e n t classrooms. Two teachers aided in the study. The study used approximately 15 hours of students time over a period of one month. To f i t the school schedule, the experiment was conducted during the students' regul a r mathematics sessions. Prior to the experiment, students were assigned to four study groups as required by the design of the study. Since cut of the four o r i g i n a l i n t a c t classes two had t h e i r mathematics sessions in the same period as did the remaining two classes, the procedure of random assignment of students to study groups was s l i g h t l y altered. That i s , random assignment of students to study groups was carried out only w i t h i n classes that took place in the same period. Within these r e s t r i c t i o n s , h a l f of the students in each class were randomly assigned to each of the two study groups. The degree to which t h i s procedure formed comparable study groups was examined using the i n t e l l i g e n c e scores and a p a r t i a l knowledge of the rules. In a d d i t i o n , as required by the design of the study, the four d i f f e r e n t treatn~nts (learning cond i t i o n s ) were randomly assigned to the four study groups.

Instruction which enables Students to Develop Higher Mental Processes

199

Once the four study groups were formed, the experiment proceded as follows (see Fig. 3.1): students in three out of the four study groups (2, 3, 4 in the design) learned probability rules to a pre-set criterion level of 85 per cent on the knowledge of each rule through the mastery learning strategy. By the use of frequent testing, feedback and correcting students' mistakes, in approximately 8-9 hours most of the students in these groups completed their learning of the rules. Students in the remaining group (group l) learned the same rules by more conventional, non mastery procedures. They used the same learning materials but they were neither required to reach a pre-set standard nor were they provided with systematic feedback information and correctives guidance. This group of students completed its learning of the rules in about 6-7 hours. Following the learning of the rules, students in the non mastery group (1) and students from one of the mastery groups (2) were tested on the application test. The remaining students in the two of the mastery groups (3 and 4), depending on their practice experiences, started their practice in application skills following an introduction to the specific learning activities required of them. At the end of each practice session, students' work sheets were checked and returned to the students with written comments regarding specific mistakes and possibilities of improved learning. The working sheets were not graded. Both study groups finished their practices in about 23 - 3 hours. Following these practice experiences, these students were tested on the application test. Out of the I00 i n i t i a l participants, complete data on all measures was available for 86 students, classified as 20, 22, 22, and 22 for groups l , 2, 3 and 4 respectively.

4. The Results and their Interpretation

On the basis of the conceptual model developed previously, four hypotheses were derived. The result of testing these hypotheses is the main theme of this chapter. I t is concerned with the analysis and the interpretation of the collected data and reports the findings in relation to each hypothesis of the study.

NECESSARY BUT NOT SUFFICIENT LEARNING CONDITIONS FOR THE ABILITY TO APPLY RULES The main purpose behind the f i r s t hypothesis of the study was to establish the relationship between performance on application of rules and the knowledge prerequisites, as a necessary but not s u f f i c i e n t condition. On the basis of the conceptual model i t was hypothesised that groups of students who possess more of the necessa~j prerequisites (86 per cent mastery level of rules ' knowledge) will score higher on measures of the ability to apply their knowledge tha~ students who have fewer of the necessary prerequisites (50 per cent m~stery level of rules' knowledge). In order to test this hypothesis, two groups were used and examined. One group learned the rules under the mastery learning conditions. In this group, students were provided with time and help in the form of feedback and corrective procedures were provided to the students beyond the original instruction on the rules. These groups are referred to as groups 2 and l , respectively (see Fig. 3.2). Table 4.1 presents the mean achievement scores and the standard deviations of the measure on rules' knowledge for the two groups. I t is apparent from Table 4.1 that the mastery group scored significantly higher on the measure of rules' knowledge than did the groups of students who were not provided with the same learning conditions. Although during the learning process students in the mastery group achieved 85 per cent mastery level on each rule separately, the mean score of this group on the measure of rules' knowledge at the end of their learning reached a lower level of mastery (75 per cent). Table 4.1 reports also the v a r i a b i l i t y of students with the mastery learning conditions, students become more similar in their achievement on the knowledge of the rules than do the students in the non mastery group.

200

Instruction wtdch ermbles Students to Develop Higher Mental ~

deviation of students aohievement on the TABLE 4.1 Means and 8 t a ~ knowledge of rules and on application of rules for the nonmastery and the mastery groups.

Mastery Group

Knowledge of Rules

Application of Rules

Mean No. of correct responses

Percent

SD

Percent

SD

Mean No. of correct responses

22.5

75

4.2

I0.9

31

3.7

15.7

52

4.7

8.1

23

5.0

Non-

Mastery Group

F=23.5 p
F=4.46 p
TABLE 4.2 ~eans and standard deviations of k n ~ e d g e of rules and performance on application for the master~ groups with and without additional experiences.

Knowledge of Rules

Application of Rules

Mean No. of correct responses

Mean No. of correct responses

Percent

SD

Percent

SD

Groups 3 and 4 Mastery + Practices

22.0

73

3.9

15.9

45

4.4

Group 2 Mastery Only

22.5

75

4.2

I0.9

31

3.7

F=O.18

F=20.13 p
201

202

T. Levin

Having established two groups of students whose performance level on the knowledge of the rules d i f f e r significantly, i t is now possible to examine the relationship between knowledge of rules and the a b i l i t y to apply the learned rules. This relationship can be studied in two different ways; associationally and experimentally. The associational relationship between rules' knowledge and the application of rules was investigated by computing the correlation coefficients of these two measures in each of the two groups, separately and combined. In the mastery learning group the correlation between knowledge of rules and application of rules was positive and highly significant (+0.69; sig.p<.Ol). The corresponding correlation for the non mastery group was somewhat lower but s t i l l significant (+0.56; sig.p<.05). Also the correlation for the combined groups was found to be positive and significant (+0.62; sig.p<.05). These results mean that there is a relatively strong relationship between students knowledge of the rules and their a b i l i t y to apply the rules in new problem situations. The higher the level of knowledge of rules, the higher is l i k e l y to be the students scores on a measure of rules' application. The results this far demonstrate that not only are we able to alter students performance level on the knowledge of the rules with the use of instructional strategies, but also that knowledge of rules is strongly related to students' a b i l i t y to apply the same rules in new situations. That is, knowledge of rules highly pred~ots students a b i l i t y to use the rules in a variety of new situations. These results set the basis for further examination of the possibility to determine students performance level on their a b i l i t y to apply rules by selectively manipulating their performance level on the knowledge of the rules. That is, we can now examine whether rules' knowledge and the a b i l i t y to apply the rules are eau~aZly linked. In particular, we can now compare the performance level on the measure of the a b i l i t y to apply rules of the two groups of students who reached different levels of mastery on the knowledge of the rules. Table 4.1 presents the mean achievement scores and the standard deviations of the two groups (mastery and non mastery groups) on the measures of the a b i l i t y to apply rules. The findings indicate that the mastery learning group scored significantly higher on the measure of rules' application than did the non mastery group. I t is also evident that students' scores on application of rules in the mastery group are less varied than students' scores in the non mastery group. The results make i t clear that the group of students who mastered the rules to only 52 per cent mastery level reached only a chance mean score on the measure of the a b i l i t y to apply the rules. In contrast, a higher mastery level of knowledge of rules (75 per cent) brings the application achievement score significantly above the chance level. Similar analysis was carried out by taking into account the i n i t i a l differences that were found between the two groups in their mean scores on the measure of mathematics aptitude. The technique used to make this adjustment was Analysis of Covariance. The results indicated again that the mastery group scored significantly higher on the measure of rules' knowledge compared to the non mastery group, and the mean score on rule application of the mastery group was s l i g h t l y above the chance level reached by the non mastery group. The findings of the f i r s t hypothesis of the study established significant associational as well as causal links between students performance level on the knowledge of the rules and their a b i l i t y to apply the rules in new problem situations. The findings make i t clear that within a group of students, those who possess more knowledge of the rules are better able to apply these rules in new situations than students who possess less of the rules (associational links). But, the findings also make i t clear that by selectively providing or depriving students help in the form of feedback and correctives, i t is possible to determine students' level of learning of a set of rules. The f e a s i b i l i t y of manipulating students knowledge of

Instruction which enables Students to Develop Higher Mental Processes

203

a set of rules enable us also to determine their performance level on the a b i l i t y to use the same rules in new problem situations (causal links). This means, that the higher the mastery level on rules' knowledge attained by groups of students, the higher is their performance on a measure of their a b i l i t y to apply the learned rules. These results support the contention found in the literature that knowledge of rules is a ~cessaI~j but not ~ffic~ent condition for the a b i l i t y to apply rules in new problem situations.

NECESSARY AND SUFFICIENT LEARNING CONDITIONS FOR THE ABILITY TO APPLY RULES Since knowledge prerequisites are believed to serve as a necessary but not s u f f i cient condition for performance on application of rules and principles, this study attempted to determine the supplementary conditions which are sufficient for the development of the a b i l i t y to apply rules in new situations. These are believed to be instruction and practice in the s k i l l s involved in the application of principles or rules to new problems. The second hypothesis of the study states that f o l / ~ n g the knowledge of the rules to a mastery level (85 per cent), students with learning of application skills will score higher on the measure of application achievement than students who do not have learning experiences in application skills. In order to test this hypothesis, two groups were formed and examined. One group of students learned the rules under the mastery learning strategy to a pre set (85 per cent) mastery level (group 2). In the second group, students learned the rules to the same mastery level and in addition received learning experiences in application s k i l l s (groups 3 and 4). Thus, for the purpose of testing this hypothesis i t was necessary to combine groups 3 and 4 (see the design of the study in Fig.3.2) since students in these two groups have received instruction and learning experiences in the application process. Table 4.2 reports the mean achievement scores and the standard deviations of the rules' knowledge and the total measure of rules' total application for the two groups under study. The findings in Table 4.2 indicate that the two groups are comparable in their mean achievement scores on the measure of the knowledge of rules. This comparability makes for clearer estimations of the effects of learning experiences on the a b i l i t y to apply the rules to new problems. The findings clearly demonstrate that students who had learning experiences in solving application problems (following their i n i t i a l learning of the rules) scored higher on the total measure of the a b i l i t y to apply rules than students who did not receive these experiences. (Similar results emerged when the examination of this relationship was repeated, adjusting the i n i t i a l mean differences in mathematics aptitude scores between the two groups.) The results suggest that once students reach a certain level of mastery of rules' knowledge, students with learning experiences in application of rules are more capable of applying the rules than are students not provided with these experiences. I t is, therefore, clear that the study established a differential effect of the presence versus the absence of instructional and learning experiences in application s k i l l s on the a b i l i t y to apply the rules.

204

~ Levin

LEARNING EXPERIENCES WHICH ENABLE STUDENTS TO APPLY RULES TO A WIDE VARIETY OF PROBLEM SITUATIONS

While the preceeding hypothesis investigated the effect of the presence versus the absence of learning experiences directed to s k i l l s underlying an application process, the third hypothesis examines the effects of the quality of the experiences on students' performance on application of rules to a variety of new situations. Based on the literature, this study hypothesised that following the kno~lec~e of the rules to a mastery level, the quality of students ' experiences ~n application skills determines their ability to apply the rules ~n a range of application problems.

The present study chose to concentrate on two distinct types of learning experiences directed towards selected behaviours underlying the application process. One is the constant and general type which emphasises practice on application problems limited in their structure, wording and form and in which the solution to each problem requires the use of only one rule. In contrast, the second type, vc~ed and specific includes practice on application problems which are of varied nature in terms of wording, form and context as well as problems which require the use of more than one rule for a correct solution. Generally, i t was anticipated that following an i n i t i a l mastery level of the rules, students who were provided with experiences that emphasise wider aspects of the application process (the varied and specific type) would score higher on the less familiar and the complex measures of rules' application than students who were provided with the more restricted type of experiences, the constant and general type. I t was also expected that students who were provided with restricted type of experiences would score higher on the familiar and the simple measures of application of rules than students who were provided with broader practice experiences. In order to test this hypothesis, students' scores on application of rules were compared between two groups. In one (group 3), following the learning of the rules to a mastery level, the students were provided with the constant and general type of experiences. In the second group (group 4), following the rules' knowledge learning to the same criterion level, students were provided with the varied and specific type of experience. Both groups spent the same amount of time in the instruction and practice of these application processes. Table 4.3 presents the mean and standard deviation of students knowledge and application achievement scores of the two groups. I t is apparent from Table 4.3 that the two groups were comparable Mth respect to their mean performance level on the knowledge of the rules. I t is also evident that students who practiced with the type of experiences that foster the specific behaviours of the entire application process (varied and specific) scored significantly hlgber on the total measure of application of rules than students who were provided with the more restricted type of learning experience (constant and general). These results suggest that the a b i l i t y to apply rules is best learned when experiences provided to students emphasise practice on a wide range of behaviours underlying the process of application within a context of a varied and cowlex practice problem. In contrast, by narrowing the range of s k i l l s and behaviours required by an application process and by limiting the characteristics of the practice problems to highly similar and simple ones, we may produce lower effects on students a b i l i t y to apply learned rules.

InstruclJon which enables Students to Develop Higher Mental ~

2~

TABLE 4. $ Means and standard deviations of knowledge of rules and performance on application of rules for the two mastery groups with additional experiences in application skills. Knowledge of Rules

Application of Rules

Mean No. of correct responses

Mean No. of correct r e s p o n s e s Percent

Percent

SD

,,

Group 4 Mastery & Varied and Specific

22,1

Group 3 22.0 Mastery & Constant and General F=O.O05

,

SD

n

m

73

4.0

17.9

51

4.3

73

3.9

13.9

39

3.6

F=I0.87 p
To demonstrate further the differential effects of practice experiences in application s k i l l s on students a b i l i t y to apply learned rules on a roncje of problem situations, the two groups (groups 3 and 4) were compared also on the specific measures of rules' application. The results (see Fig. 4.1) demonstrate that both groups which had mastered the rules and in addition received different types of practice in application s k i l l s reached approximately similar high scores on the measure of their a b i l i t y to apply the rules in familiar and simple problems (80 per cent and 77 per cent for groups 3 and 4 respectively). In addition, both groups approached only a chance level score on the measure of rules' application in less familiar and complex situations (18 per cent and 30 per cent for groups 3 and 4 respectively). The major differences in the performance level en application of rules between these two groups were demonstrated on only two measures, the f~niliomond complex measure of the a b i l i t y to apply rules and the less f ~ i l i o m and sgnpZ~ one. The group which received practice in the varied and specific type experiences scored s i g n i f i cantly higher on these two measures (54 per cent and 50 per cent respectively) compared with the group of students who were provided with the constant and general type of practices and who performed only slightly above the chance level on these same measures (35 per cent and 38 per cent respectively). Considering the entire range of the application measures and their different structural properties, these results suggest that i t is possible to design distinct types of learning experiences which produce quantitative and qualitative effects on students a b i l i t y to apply rules and principles to novel application situations. One type of experience emphasises practice and instruction on a wide range of application skills within the use of an heterogeneous set of practice problems. In contrast, the second type includes homogeneous sets of practice problems that enable students to learn and practice only a limited nu~er of application s k i l l s . These two types of experiences were found to be equally good in developing students a b i l i t y to apply rules to simple and familiar problems. Hov:ever, practice with ~ r e hetergenous and complex sets of problems facilitated students' a b i l i t y to apply rules on a ~'~d~r range of application problems than did practice on a set of more homogeneous and simple problems. In particular, the results suggest that by varying the context of the practice problems as well as their complexity, students acquired the a b i l i t y to

206

T. Levin

O Group 4 - Mastery of rules + Varied & Specific A Group 3 - Mastery of rules + Constant & General •

Group 2 - Mastery of rules only Group 1 - Non mastery of rules

100

90

8O

70

l

4, l

•,

\

\\

\

&

\

60

\

\ \

8 u

50

\ ,

\

A

"o.

\

- ox

\ \ '

\

\\

,

8

,

\

\

4O •

30

\

~- -. \

chance score

\

"\ -.

\

20

\\

\ "~,

,

\

\ \

\

"•

\

\ \

\

"\

10

0 Familiar & Simple

Fi~. 4.1

Familiar & Complex

Less Familiar & Simple

Less Familiar & Complex

Mean scores on the specific measures of application achievement for ths four study groups

Instruction which ermbles Students to Develop Higher Mental Pmc~J~____8 207

apply rules problems of application ed rules or situations.

and principles in either complex problems of familiar types or in simple less familiar type. In contrast, by limiting students' experiences in s k i l l s to a narrow set of practice problems, their a b i l i t y to use learnprinciples is highly restricted to a set of familiar and simple problem

That is, by implementing qualitative different types of practices in application s k i l l s , we are capable of determining not only the different performance levels of students' a b i l i t y to use acquired knowledge, but also to produce different qualities of this a b i l i t y . These results support the third hypothesis of the study and indicate that beyond the mastery of the knowledgeof rules, different experiences in the process underlying the application of rules can produce different qualities of students' a b i l i t y to apply rules to a variety of application problems. To e ~ up the effects of the different learning conditions of this study on students a b i l i t y to apply rules, i t is important to note the achievement pattern of students on the various measures of this a b i l i t y for each study group. These patterns are displayed in Fig.4.1. I t is apparent from Fig. 4.1 that i f a group of students learn a set of rules under learning conditions which deprive them of feedback and corrective procedures in order to master the rules, their a b i l i t y to apply these rules is very limited and approaches the chance level score on each of the measures of application achievement. However, i f additional time and help in the form of feedback and corrective procedures are provided to a group of students and consequently they reach a satisfactory level of rules' knowledge, they are more capable of applying the learned rules to a performance level which exceeds the chance level score. In particular, mastery of rules' knowledge, compared with non mastery of the rules, was more effective in developing the a b i l i t y to apply rules to familiar and simple application problems, and to a lesser degree to the familiar and the more complex problems. However, knowledge of rules only does not, in this study, enable either groups to do better than a chance level on the less familiar and complex application problem situations. I t is apparent from Fig. 4.1 that when in addition to the mastery of the knowledge of the rules, students are provided with experiences in application s k i l l s , their a b i l i t y to make use of these rules in the solution of application problems developed substantially above the chance level scores. I t is clear that mastery of rules' knowledge supplemented by the more restricted type of practice in application s k i l l s (constant and general) is superior to the mastery of rules' knowledge only, mainly in simple application problems. I t is also evident that the more varied and specific type of practice experle~ce is superior to the mastery of the rules only, on all measures of application except the less familiar and complex one. That is, the results suggest that in order to develop students' a b i l i t y to apply rules in only familiar and simple problem situations, mastery learning conditions supplemented by either the varied and specific type of practice or the constant and general one would be equally appropriate. However, in order to develop students' a b i l i t y to apply learned rules and principles in complex problem situations or in less familiar and simple application problems, the varied and specific type of practice w i l l be more effective than the constant and general type. The results of this study demonstrate that practice experiences in application s k i l l s have the potential to either develop or to l i m i t the a b i l i t y of students to apply learned rules in a certain direction. Practice on wide aspects of the application process coupled with practice on an hetergeneous set of complex problems develop students' a b i l i t y to apply their knowledge to a broader range of new application situations. In contrast, practice on a narrow set of homogeneous and simple problems combined with the use of only a small number of application s k i l l s , limits

208

T. Levin

students' a b i l i t y to apply their knowledge on a range of application problems, mainly to familiar and simple ones. Generally, therefore, the study clearly suggests that the structure and the nature of previous learning experiences provided to students play a major role in determining the level and the kinds of students' a b i l i t y to apply rules in new situations. While knowledge of the rules serves only as a necessary but not sufficient condition for the a b i l i t y to apply rules in new situations, knowledge of rules supplemented by practices in application s k i l l s serve as necessary and sufficient conditions for the a b i l i t y to apply rules to a great variety of problem situations. I t s t i l l remains however for future research to explore what are the sufficient learning conditions that will enable us to develop student~ a b i l i t y to apply learned rules to a wider range of problem situations, including novel and con~)lex application problems.

THE EFFECT OF THE LEARNINGCONDITIONS ON THE RELATIONBETWEEN INTELLIGENCE AND STUDENTS'ABILITY TO APPLY RULES. The f i r s t three hypotheses of the stud),, which grew out of the theoretical model, were intended to determine to what extent and how students' learning experiences are capable of improving their a b i l i t y to apply rules in new situations. The main purpose behind the fourth hypothesis of the study was to examine the role played by these learning experiences in altering the relationship between measures of students intelligence and application achievement. The study hypothesied that the correlations b e ~ e e n students~ intelligence measures and their ability to apply rules in new situations depend on the quality of the learning experiences provided to the s tudents.

Since the study made use of two measures of students general cognitive a b i l i t i e s , the examination of the relationship between these two measures and students' performance on the application of rules was carried out separately for each one of the measures. One is a subject nwitter related measure, students mathematic aptitude, which measures students' a b i l i t y to use mathematical principles in formal exercises and in practical quantitative problems. The second, general intelligence measure, is viewed as an index of total educational development and as a good predictor of overall academic success. In order to test the hypothesis stated above, the correlation coefficients between the measures of students' mathematical aptitude, intelligence scores and their performance level on application of rules were calculated for each study group in this research. These groups reflect, unequivocally, well defined and different instructional and learning conditions. Fig. 4.2 shows the correlations between measures of students' a b i l i t y to apply rules and each of the two measures of students' cognitive characteristics. In the analysis of these results i t is important to notice the trend of the correlation coefficients with respect to the specific learning and instructional conditions experienced by each of the study groups. For example, when we examine the correlation coefficients between students' mathematics aptitude and their a b i l i t y to apply rules in new problem situations, i t is evidence that the correlation approached the highest value for the minimal learning conditions of this study, in which students learned the rules to a knowledge level only under a more conventional non mastery strategy (groups). The coefficient is positive and highly significant. It is also apparent that the corresponding correlation approached a minimum positive value for the maximal learning conditions of this study (group 4); these combine the mastery of rules' knowledge with learning experiences in skills underlying the

Instruction which enables Student8 to Develop Higher Mental PmcNses

209

0.70.

0.60, a~ (-

0.50 o u

,. ..

/o, .

\,

\ ..

\,

/

"

\

0.40

co

"5 L S,.. 0 U

\-.

0.30

,

eMathematics Aptitude

\

0.20

"oGeneral Intelligence

O.lO

Fig.

4.2

Non Mastery of rules

Masteryof rules

Masteryof rules + Constant & General

Masteryof rules + Varied & Specific

(group l)

(group 2)

(group 3)

(group 4)

Simple correlation coefficients between measures of students' general intelligence, mathematics aptitude and application achievement a8 a function of the learning conditions

application process in a variety of problem situations. The correlation coefficient in these conditions is much lower and does not differ significantly from chance. In between these two extremes, the trend of decreasing correlations among the four study groups is apparent, with only one exception for the general intelligence measure. The differences between the correlation coefficients and in particular the trend of decreasing correlations between students aptitude scores and their a b i l i t y to apply their knowledge in new situations suggests that the role of students' characteristics in determining students' learning outcomes depends on the quality of experiences provided to them. Under the most favourable learning conditions, students' aptitude or intelligence scores do not determine students' learning. Yet, when less favourable conditions are provided to the students, aptitude and intelligence scores strongly influence students' learning outcomes. Under these learning conditions, students' aptitude and intelligence scores highly predict students' a b i l i t y to apply rules in new situations. That is, the higher the level of intelligence or aptitude scores, the higher is likely to be the students' scores on a measure of rules' application.

210

T. Levin

Of importance is to emphasise that, in this study, the most favourable learning conditions are implemented under conditions of group instruction rather than under an individualised learning strategy. These conditions include the mastery of the necessary learning components (knowledge of rules) as well as the learning of the sufficient conditions for the desired learning outcomes ( s k i l l s on application of rules). The results suggest that as students are provided with instructional and learning conditions that are necessary and s u f f i c i e n t ; and which are relevant and specific to the desired learning outcomes, students' characteristics such as their specific aptitudes to a subject area or their general intelligence do not serve as a barrier to their successful learning and performance. In contrast, under less appropriate conditions, students' learning largely depends on their aptitude or intelligence scores. Under these conditions, those students who score higher on intelligence or aptitude measures are better able to apply rules in new situations than students with lower scores on these measures. The results of this study demonstrate that learning experiences could be designed ~vhich have the potential to either increase or to decrease the degree to which students'characteristics determine students' learning and performance.

5. Implications for Education and for Educational Research

Having intepreted the findings of the study, i t is now appropriate to discuss what those findings mean for educational practices, educational research and evaluation, and for education in general.

HIGHER MENTAL PROCESSESOBJECTIVES CAN BE DEVELOPED IN THE SCHOOLS The f i r s t major implication of this study is concerned with the f e a s i b i l i t y of developing higher mental processes objectives in the schools. Many educators throughout the world desire to develop problem solving s k i l l s , inferential thinking and various higher mental processes in the schools. They believe that i f learning and instruction are to be useful and relevant to the individual as well as to society, schools need to improve students' intellectual skills rather than place emphasis on the more limited objectives dealing with specific information, facts, principles or methods. Educators as well as researchers believe that progress in developing higher mental processes objectives can be expected i f we can identify the ways of teaching intellectual processes to the students in their classrooms. Great strides have been made by educators in defining and clarifying the nature of higher mental processes objectives, and evaluators have done much to develop valid and precise instruments to measure these objectives. In contrast, very l i t t l e has been done to specify the instructional processes and the kinds of learning experiences needed by the students to acquire these objectives. Partly as a result of this gap in our understanding of the learning and instructional conditions needed for higher mental processes, some educators have become pessimistic about what can be done in the schools to improve higher levels of thinking of students. The present study provides strong empirical evidence that higher mental processes objectives can be developed in the schools. In particular, i t demonstrated that by planning and implementing specific educational practices, i t is possible to develop students'ability to apply learned rules in a great variety of new problem situations It furthermore demonstrated that different kinds of learning and instructional procedures can lead to different learning outcomes within this larger domain. Under one set of learning conditions, students acquired the a b i l i t y to apply learned rules on a wide range of new problem situations. In contrast, another set of learning con. ditions made i t possible for the students to apply learned rules to only a very limited set of new problem situations. 211

212

T Levin

The implications of these findings for schools and schooling are important. It makes i t clear that i f the school objectives emphasise higher levels of thinking such as the student# a b i l i t y to use rules and principles, schools and teachers are capable of achieving these goals. School learning goals need not be limited only to specific and unusually s k i l f u l teachers, or to students with unusually high levels of intelligence or aptitude. Instructional procedures, learning experiences and learning materials carefully developed can enhance the students'ability to achieve higher mental processes objectives. The research also makes i t clear that i f we do not develop such procedures, experiences and learning materials, we limit the students'ability to learn to solve new problems. In summary, this study suggests that i t is within the capacity of educators and education to produce changes in students' learning outcomes for a wide range of educational objectives. When learning opportunities are provided to the students that are not appropriate and relevant to the desired learning goal, we can expect to have l i t t l e impact on such learning outcomes for most of the students. But, when learning opportunities are provided to the students which are specific and relevant to the desired goal, much can be achieved in improving these learning outcomes for most of the students. I t is the a v a i l a b i l i t y as well as the structure of educational practices that either f a c i l i t a t e or deprive students of the opportuni t y to acquire higher levels of thinking.

MOST STUDENTSCAN LEARNHIGHER MENTAL PROCESSESOBJECTIVES The second implication of this study is concerned with our conception of education and educational practices in view of the question of individual differences in school learning. There is a prevailing view among many teachers and researchers that complex cognitive educational goals can be developed only in those students who have exceptionally high intellience or aptitude scores. This view has been an educational restraint for a long time. This idea has led many teachers to believe that such stable and relatively general characteristics of students as intelligence or aptitude limit the influence of school and instruction on students'learning. Moreover, this view of schools and human learning has resulted in a weakening of the search for effective instructional procedures directed to these higher mental processes objectives. In contrast to these views, this research provides empirical support for the con= tention that the relationship between intelligence or aptitude measures and measures of school achievement is a function of the learning conditions provided to the students. I t demonstrated that the nature and the quality of the educational practices provided to students determine which students will attain high levels of learning and achievement. The study clearly demonstrated that when educational practices are weak and inappropriate to the desired goal, students characteristics such as intelligence or specific aptitudes are l i k e l y to determine their a b i l i t y to learn and perform these complex goals. In contrast, when educational practices are appropriate and effective for the desired goals, these characteristics of the students do not provide barriers to the development of students' a b i l i t y to learn and use higher mental processes. That is, the teachability of particular mental processes is not restricted to a small number of the most able students. In particular, the study demonstrated that under the least favourable set of learning conditions, students' mathematics aptitude scores highly predict their a b i l i t y to apply rules in new situations. Whereas, when students were provided with additional time and helped to master the rules, students' mathematics aptitude becomes less powerful in determining t h e i r a b i l i t y to apply learned rules. Furthermore, the study demonstrated that even more favourable aptitude scores on their a b i l i t y

Instruction whichermblesS~den~ W Develop H ~ h e r M e n t a l P r o ~

213

to use learned rules in a great variety of novel s i t u a t i o n s . When students not only mastered the rules but in addition were provided with practices in application s k i l l s on a range of practice problems, student~ mathematics aptitude scores accounted only for four per cent of their achievement on application of rules. That is, the relation between students mathematical aptitude and their a b i l i t y to apply rules becomes very small and does not differ from chance.

The study therefore suggests that beyond this specific objective, as learning conditions become more effective, relevant and specific to the desired educational outcomes, the greater is the power of educational practices in enabling most students to attain these goals. I t furthermore suggests that when very effective educational experiences are provided to the students, the effects of characteristics such as intelligence or aptitude on student learning are likely to reduce to something approximating zero.

THE ESSENTIAL COMPONENTSOF EDUCATIONEXPERINECESFOR HIGHERMENTAL PROCESSES The search for the design of effective educational practices for higher mental processes objectives, sets the basis for the third major implication of the research. This study differentiated between two essential sets of instructional and learning conditions; necessary but not sufficient, and necessary ~ d sufficient conditions for learning. The study provided further empirical evidence for the view that students are unable to apply rules in new situations i f the knowledge of the rules was not i n i t i a l l y mastered. That is, knowledge or rules is a neeessaz~ but not sufficient condition for the a b i l i t y to apply rules. I t demonstrated that with the use of appropriate feedback and corrective procedures, i t is possible to bring almost every student to a pre-determined level of mastery on the knowledge of the rules. When students were provided with these more effective learning conditions for learning the rules, students developed the a b i l i t y to apply these rules in new situations. However, the study also suggests that the notion of necessary but not sufficient learning components, can no longer be conceived of without understanding its main effects of a range of possible learning outcomes. I t demonstrated that mastery of rules as a necessary but not sufficient learning condition, has the greatest differential effect on students' a b i l i t y to use the learned rules in only a limited set of problem situations; mainly those that are similar to problems used in the previous instruction of the rules. That is, the necessary but not sufficient learning components contribute to learning outcomes but also limit these outcomes unless educators go beyond them. The educators must be able to ascertain in what ways the necessary components operate and what their limits are. This study suggests therefore that beyond the mastery of knowledge, there are some other learning components required to develop students' a b i l i t y to use acquired knowledge in a variety of new problem situations. In the search for additional components to the necessary ones, the findings of this study clearly demonstrated that when opportunities are provided to students to practice some of the skills implied by the learning goal, students' learning of complex educational goals improvessubstantially on a range of learning outcomes within the goal. Therefore, i f i t is desired to promote student learning on a chosen dimension and to a higher level of performance than allowed for by the necessary and not sufficient components of learning, additional practices on the behaviours underlying the general process demanded by the learning goals are required.

214

~ Levin

More generally, the study suggests that planning of educational practices for a defined class of educational goals should include the specification of the essential components directly oriented toward the underlying s k i l l s of the desired goal. In other words, planning that preceeds effective design for the learning of a class of objectives should involve the specification of the necessary and sufficient conditions for learning. This includes what may be called the instructional and learning structure for any class of educational goals and for the different dimensions within the goal. I t needs to include the learning conditions that f a c i l i t a t e the mastery of the necessary but not sufficient components of learning as well as the conditions that further develop the behaviours required by the desired educational goals. The role of planning education is suggested to be the development of the variety of learning and instructional conditions which are demonstrated to be essent i a l for a range of desired learning outcomes. Great progress has been made by educators in identifying and implementing teaching methods and classroom procedures which have a necessary but not sufficient relation to important educational objectives. Such methods and procedures work reasonably well for a relatively small proportion of the students. What has not been done on a large scale is to identify and implement methods and classroom processes which produce the necessary and sufficient conditions for most of the students to learn the same objectives to a high level. This study provides further empirical evidence on the f e a s i b i l i t y to alter students' learning outcomes for the necessary components of learning. I t demonstrated that i f we desire to achieve a pre-determined level of performance of the necessary component of learning, either providing or depriving students of the time and help they need can produce learning outcomes in the expected direction and magnitude. Some c r i t e r i a for the development of necessary and sufficient learning conditions for a defined educational goal were also suggested. Generally, this study made use of selected instructional and learning principles in order to enable students to learn the behaviours underlying the process of application of rules. In particular, the study used learning principles regarding the sequence of the learning materials and the practice problems, their mode of presentation, the extent of v a r i a b i l i t y of the experiences, the cues embedded in these learning experiences and the specification of the s k i l l s required by the desired goal. The findings of this study demonstrated that when equal learning time was available to students, the specific characteristics of the practiced problems as well as the nature of their organisation produced different effects on students'ability to apply rules on a range of new problem situations. Practice on a narrow and homogeneous set of problems that require the use of only some s k i l l s in application yielded only a limited type of a b i l i t y to apply rules; mainly on familiar and simple problem situations. Practice on a broader range of application problems which demands most of the s k i l l s underlying an application process, yielded much more effective learning to a wider range of new and complex problem situations. This study suggests that learning and instructional principles can be used to design appropriate learning experiences and educational practices for both the necessary and sufficient components of learning. I t does not consider the principles used in this study to be the only sufficient set of possible principles to be employed in further research. I t regards these as some minimal c r i t e r i a for the development of appropriate experiences by teachers and curriculum developers. Moreover, the study suggests that further research is needed to explore under what learning conditions we w i l l be able to develop students J a b i l i t y to use acquired knowledge in a wider range of complex problem situations. By emphasising the powerful effects of educational practices on learning and achievement, the three major implications of the study for education and for educational research are highly important. On the one hand i t places greater responsibility on

Instruction whichermblesStuden~ m Develop H~herMentalProcuses

215

educators and schools for improving the more complex 9oals of learning for these students. On the other hand i t suggests that much can be done in the schools and classrooms in order to achieve complex and desired educational outcomes.

THE USE OF CAUSAL INFERENCES IN EDUCATIONAL RESEARCH The second set of implications is concerned with the methodology of educational research for school learning. This study established causal links between educational goals, educational practices and evaluation procedures. I t made use of analytical procedures to identify the nature of an educational goal, the necessary evaluation measures and the required conditions for the learning of this goal. Experimental procedures were then used to determine the effects of varying types of specific learning conditions on the specific measures of students'learning outcomes. Much research on school learning takes the form of 'surveys'. This approach enables researchers to establish associational relationships between educational experiences as they are taking place in schools and students'achievement on a desired set of learning outcomes. Associational relationships between educational experiences and student learning provide relatively l i t t l e guidance as to the specific elements in the learning experiences that can directly affect and alter the measured educational outcomes. Strong associational relationships among a set of variables, at the least have some predictive value and at the most are suggestive of possible causal relationships among the variables. In contrast, the e~e2"/mcntal approach, when well used, enables researchers to determine causal links between educational practices and student learning for a defined set of learning outcomes. I t enables the researcher to know when his analysis of the relevant causal conditions of learning has been valid as well as what is s t i l l required for the learning conditions to achieve the desired goal. The f i r s t methodological implication of this study is concerned therefore with the feasibility of determining causal links between educational practices and educational outcomes for a wide range of educational goals. In particular, i t demonstrated that by applying analytical procedures to existing knowledge of learning and instructional principles i t is possible to design educational experiences that influence student learning. This approach enables us to determine the learning conditions that facilitate student learning in the desired direction. I t also enables us to determine the conditions which hinder student learning and performance More specifically, this research demonstrated that with additional learning time and with the use of feedback and corrective procedures, i t is possible to determine students'level of learning a set of rules. These conditions enabled us also to improve students' ability to apply the rules to familiar and simple problems. The use of feedback and corrective procedures supplemented by different types of experiences in application skills, made i t possible to produce learning of application of rules in a variety of new problems. This study suggests that i f education is to produce changes in students'learning, and i f learning outcomes are to be evaluated by valid and precise measures, educational research must make better use of the experimental approach to establish causal links between educational practices and educational outcomes for a variety of desired educational goals.

216

~ Levin

RESEARCH ON THE SIGNIFICANT VARIABLES DETERMINING SCHOOLLEARNING The search for the necessary variables to be used in further research sets the basis for the second methodological implication of this research. While this study made use of only some minimal c r i t e r i a for the development of appropriate educational experiences, i t generally suggests that the exploration of potential explanatory conditions of school learning needs to concentrate on variables that satisfy three sets of conditions. First, the variables need to be related to learning outcomes and to have defined effects on these outcomes. Second, they must be alterable and t h e i r effects need to be assessed within a relatively short time span. Third, the variables need to be conducive to effective implementation by most of the teachers regardless of their specific personality characteristics or other stable qualities. The exploration of possible causal relationships between educational practices and educational outcomes should emphasise specific sets of alterable learning and instructional conditions. The second methodological implication of this study is concerned therefore with the Traditiona l l y , measures of relatively stable characteristics of students such as intelligence, aptitude scores or other unalterable variables of students'background have been frequently used as explanatory variables of students'learning. Also, much research has focused on the relationships between the characteristics of the teachers and the learning outcomes of the students. Relatively stable indicators of students' learning and performance do not provide insights into those attributes of educational experiences which, i f modified, could lead to improved learning and instruction. Consequently, such an approach to research can only lead educators and researchers to a despair about the potential effects of schools in developing students" learning, particularly with regard to the development of higher level of thinking in students.

nature of the variables to be explored in research on school learning.

In contrast, i f educational research is to be useful to teachers, principals, curriculum developers or policy makers, i t must focus on those aspects of educational practices that can be utilised in the schools. Moreover, i t needs to strive to identify not only instructional and learning conditions appropriate for a range of educational goals, but also those conditions that are effective for most of the students. This research demononstrated that with such an approach i t is possible to design and implement, in a school setting, a set of educational experiences that produce different educational outcomes. This research focussed on a particular educational objective within a general category of educational goals. The application of mathematical rules to new situations is a specific objective within the larger class of application of principles to a new problem situation. I t is believed, however, that the results of this study have many further implications on a range of more complex educational goals and to d i f ferent subject areas learned at school; such as the different sciences and language development. Further research is certainly needed to identify the specific learning conditions and classroom processes under which students can develop high levels of thinking in a great variety of complex cognitive goals. The methodology underlying this study is believed to be useful for research of that nature. The results of such research can entail the development of educational experiences for a wide range of complex educational goals within the cognitive domain. Eventually, such an approach to educational research in this area could serve as a basis for developing a taxonomy of learning and instructional experiences for a wide range of cognitive educational goals. This would be of great value in providing teachers and curriculum developers the means to accomplish their major goals of education.

Acknowledgement

The author wishes to acknowledge with profound gratitude the encouragement, help and guiding s p i r i t of Professor Benjamin Bloom, which were generously granted to her during the research for and writing of this monograph. Deepthanks are expres. sed for the unflinching support and warm comradeship of Nissim, which were highly essential to the formation and final production of this work.

217

References

Adams, J.A., 'Multiple versus single problem training in human solving', Journal of Experimental Psycholo~,v, 48, p.8015, 1954. Block, J.H., 'The effects of various levels of performance on selected cognitive, affective and time variables', unpublished Doctoral dissertation, University of Chicago, 1971. Bloom, B.S. et al, (Eds.), Taxonomy of Educational Objectives: Handbook l Cognitive Domain, New York: David McKay Co. Inc., 1956. Bloom, B.S., 'Learning for mastery', Evaluative Comment, I , ( 2 ) , 1969. Bloom, B.S., Hastings, J.T. & Madaus, G.Fo, Handbook on Formative and Summative Evaluation of Student Learning, New York: McGraw-Hill Inc., 1971. Bloom, B.S., 'Implications of the IEA studies for curriculum and instruction', School Review, 82, p.413-434, 1974. Bloom, B.S., Human Characteristics_ __ and School Learning, New York: McGraw-Hill, 1976. Comber, L.C. & Keeves, J.P., Science Education in Nineteen Countries: International Studies in Evaluation l , New York: John Wiley, 1973. Craig, R.C., The Transfer Value of Guided Learning, New York: Teachers College, Columbia University, 1953. Duncan, C.P., 'Transfer after training with single versus multiple tasks', Journal

of Experimental Ps~ch91o~, 55, pp.63-72, 1958. Frund, J.E., Modern Elementary S t a t i s t i c s , Englewood C l i f f s , N.J.: Prentice-Hall Inc., 1970. Gagn~, R.H. & Bassler, O.C., 'Study of retention of some topics of elementary nonmetric geometry', Journal of Educational Psychology, 54, pp.123-131, 1963. Gagn~, R.M. et al, 'Some factors in learning non-metric geometry', Monograph of Social Research and Child Development, 30, pp.42-49, 1965.

218

MstructionwhichermblesS~den~ W Devetop H~herMentalPtocesses

219

Gagn~, R.M., The Conditions of Learning, New York: Holt, Rinehart and Winston Inc., 1965. Gagn6, R.M., 'The reasons of specifying objectives', in Glaser, R., (Ed.), Teaching Machines, 14, Washington: NEA, 1965. Green, D.R., Educational Psychology, Englewood Cliffs, N.J.: Prentice-Hall Inc., 1964. Guetzkow, H., 'An analysis of the operation of set in problem solving behaviour', Journal of General Psychology, 45, pp.219-244, 1951. Harlow, H.F., 'The formation of learning sets', Ps~c.hological Review, 56, pp.51-65, 1949. Harlow, H.F., 'Learning sets and error factory theory', in Kock, S., (Ed.), Ps~ychology: A Stud~ of a Science, Stu~ l: Conceptual and Systematic Role: General Systematic Formulations, Learning and Special Processess, New York: l~F~Graw-Hi~l, pp.492-537, 1959. '" Hendrickson, G. & Schroeder, W.H., 'Transfer of training in learning to hit a submergedtarget ', Journal of Educational Psychology, 32, pp.205-213, 1941. Horrocks, J.E., 'The relationship between knowledge of human development and a b i l i t y to use such knowledge', Journal of Applied Psychology, 30, 1946. Jennan, M. & Reese, R., 'Predicting the relative d i f f i c u l t y of verbal arithmetic problems', Educational Studies in Mathematics, 4, pp.306-323, 1972. Judd, C.H., 'The relation of special training to general intelligence', Educational Review, 36, pp.28-42, 1908. Klausmeier, H.J. & Davis, J.K., 'Transfer of learning', in Ebel, R.L. (Ed.), Encyclopedia of Educational Research, pp.1483-1493, 1969. Krutetskii, V.A., 'An analysis of the individual structure of mathematical abilities in school children', in Kilpatrick, J. & Wirszup, I. (Eds.), Soviet Studies in the. Psychology of Learning and Teaching Mathematics, Calfornia: Stanford University, I I , pp.59-I04, 1969. Loftus, E.F. & Suppes, P., 'Structural variables that determine problem solving d i f f i c u l t y in computer-assisted instruction', Journal of Educational Psycholo~y, 63, pp.531-542, 1972. .L .. Logan, T.H. & Wodtke, K., 'Effects of rules of thumb on transfer of training', Journal of Educational Psychology, 5, pp.147-153, 1968. Luchins, A.S., 'Mechanisation in problem solving', Psychological Monograph, 54, p.54, 1942. Luchins, A.S. & Luchins, E.H., 'New experimental attempts at preventing mechanisation in problem solving', Journal of Genetic Psychology, 42, pp.279-297, 1950. Lundsteen, S.W., 'Manipulating abstract thinking as a sub ability to problem solving in the context of English curriculum', American Educational Research Journal, 7, pp.373-396, 1970. Mager, R.F., Pr~aring Instructional Objectives, Palo Alto: Fearon, 1962.

220

r. Levin

Mosteller, F, Rourke, R.E.L., & Thomas, G.B. Probaility, Reading, Mass.: Addison Wesley Publishing Company Co. Inc., 1970. Popham, W.J. & Baker, E.L., Esta.blishin.9 Instru.ctional Goals, Englewood C l i f f s , N.J.: Prentice-Hall, ]970. Scandura, J.M., Woodward, E. & Lee, F., 'Rules generality and constancy in mathematics learning', American Educational Research Journal, 4, pp.303-319, 1967. Scandura, J.M. & Durnin, J.H., 'Extra-scope transfer in learning mathematical strategies', Journal of Educational Ps~cholo~y, 59, pp.350-354, 1968. Schultz, R.W., 'Problem solving behaviour and transfer', Harvard Educational Review, 30, pp.61-77, 1960. Scriven, M., 'The methodologyof evaluation', in Tyler, R., Gagn6, R.M. & Scriven, M., Perspectives of Curriculum Evaluation, AERA, Monograph series in Curriculum Evaluation l , Chicago: Rand McNally, 1967. Symond, R.M., What Education Has to Learn from Psychology, New York: Bureau of Publications, Teachers College, Columbia University, 1964. Thornike, E.L. & Woodworth, R.S, 'The influence of improvement in one menta] function upon the efficiency of other functions', P~cholo~icaI Review, 8, pp.247-261, 1901. Traub, R.E., 'Importance of problem heterogeneity to programmed instruction' Journal of Educational PsychoIogy, 1966. Tyler, R.W., Basic Principles of Curriculum and Instruction, Chicago: University of Chicago Press, 1949. Van Engen, H. et al, Charting the Course for Arithmetic, Chicago: Scott, Foresman and Co., 1960. Werner, O.H., 'The influence of the study of modern foreign language on the development of desirable a b i l i t i e s in English', S_tudies in Modern Language Teachin9, 17, 1930. Wilson, J.W., 'What s k i l l s build problem so]ving power?', The Instructor, 76, pp.79-80, 1967.