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CHAPTER 2 PERSPECTIVE ON CERTAIN EPISTEMOLOGICAL POINTS OF DEPARTURE REGARDING LEARNING MATHEMATICS
INTRODUCTION
Stewart (1991:20) refers to the significanc e of an adequate study orientation in mathematics to do well in this subje ct:
The children who persisted in the face of failure tended to attribute failure to the lack of effort and motivation. These children were described as showing mastery-orientat ed behaviour, since for them failure was not insurmountable, but could be overcome by additional effort. However, for the learned helpless children, who attributed failure to things beyond their control, there was no point in increasing effort, and so they gave up in the face of failure … following failure, learned-helpless children tended to abandon the correc t problem-solving strategies they had previously used successfully, whereas the mastery-orient ated children tended to develop more advanced and sophisticated strategies to try to solve the insoluble problems.
Van Aardt and Van Wyk (1994:223) emphasise the detrimental effect of mathema- tics students’ inadequate study orientation as follows:
There is general agreement that an increasing number of academically underprepared students are reading for university degrees, with the result that many fail to meet the acad emic demands … recent evidence suggests that the use of effective learning and study strategies is an important factor in determining success at (school and) university level.
Pollock and Wilkinson (1988) declare that the use of adequate study skills is probably the most important requirement for effe ctive study.
Chapter 2 focuses on an evaluative synopsis of certain epistemological approa ches to the study of mathematics to ascertain what constitutes the essential elements of an adequate study orienta tion in mathematics. The objective of this synopsis is to obtain perspective on the way learners learn mathematics. The approach to theory structuring followed in this chapter is an e cle ctic one; in other words, an attempt will be made to indicate that no particular learning theory should be given preference at the expense of others. Ea ch learning th eory represents a particular view of know- ledge. In this study the point of view is taken that each theory is valid to a certain extent and of value when the very nature of an adequate study orientation in mathematics is considered.
Scientifically based learning theories reve al a historic development that to a high degree agrees with synchronous scientific and social values that were given prefe- rence during a particular period in a particular country. Although neither the parti- cular values nor the climate in which most of the learning theories developed can be imposed directly on the South African si tuation, the generalised learning theories must necessarily have an effect on th e local development of psychology as a science. These theories do not continue to develop in isolation but in interaction with one another. Finally these learning theories provide psychologists with the theoretic al foundation for establishing a pr actice with the final objective an accep- table, applicable and suitable intervention in the interest of their clients.
The psychologic al foundations of research on the learning of mathematics are dealt with first.
PSYCHOLOGICAL FOUNDATIONS OF RESEA RCH ON THE LEARNING OF MATHE- MATICS
As far back as 1899 Binet identified three basic ways of research, namely the ques- tionnaire method, observation, and experimentation (Kilpatrick, 1992). He currently receives the same degree of rec ognition for his c ontribution to the psychology of thinking as for constructing the first intelligence test.
Research on the nature of thinking
The measurement of intelle ctual ability
Initially Binet attempted to follow the trend of his time, namely to base his research on intelligenc e on the dimensions of skulls (Gould, 1981). Gall was the founder of phrenology (assessing the state of c ognitio n a ccording to the structure of the skull) whereas Bro ca refined this science as the study of skull measur ement. Möbius’s research on the skulls of prominent mathematicians c onvinced him that there was a relation between aptitude for mathemat ics and the shape and circ umference of the skull. Binet, however, c arefully investigated this phenomenon but c oncluded that there was no conne ction between physic al dimensions and c ognition. He had enough insight to realise (and to state) that a researcher had to set certain intel- lectual tasks for clients (for clients to display their c ognitive skill) in order to obtain any indic ation of the state of their c ognition.
Galton in his turn tried to apply Darwin’s theory of evolution to the study of psy- chology. He was especially interested in proving, by using a num ber of physiologic al and psychologic al tests, that his theory of intelligence was (mainly) the result of heredity. Consequently he c an be regarded as the one who initiated the science of mental testing. His succ essors (Burt, Pe arson and Spearman) continued his work, but his tests were limited in the sense that they involved only limited aspe cts of the responses, namely rea ction time, associatio ns, as well as sensory discrimination. Kilpa trick (1992:8) states the following in this regard:
Had Binet’s ideas about intelligence testing – the use of scores for diagnosis rather than ranking; the rejection of an innate, fixed quality known as “intelligence” – been preserved as his tests migrated across the Atlantic, “ we would have been spared a major misuse of science in our century” … Instead, American psychologists such as Henry Goddard, Lewis Terman, and Robert Yerkes developed out of Binet’s tests a hereditarian theory of IQ that not only had some disastrous effects in its consequences for social policy … but also colored the views of a generation of American psychologists in mathematics education about the prospects for improving mathematical abilities.
The study of mental development
Binet developed his intellig ence scale with a particular purpose in mind.He wanted to identify those learners whose achievement provided indications of special intervention strategies. In other words, he was less interested in the label that a particular score provided than in the assistance that could be given to learners with particular scores.Piaget, a student of Binet, was particularly interested in the proce- dures that learners followed in order to arrive at their answ ers in mathematics – particularly the wrong answers (Flavell, 1963).
Hall, who worked with Wundt, was convinced that child development followed the same pattern as human evolution. He believed that there was not much sense in stimulating intelle ctual development in learne rs. Hall particularly emphasised the sig- nific ance of learners’ interests and their need for motivation.
Stimulation of productive thinking
Külpe broke away from the views of Wundt, namely that:
one could study the structure of consciousness through introspection (Kilpa trick, 1992:8) . The Würtzburg School believed that abstract thinking did not frequently a cc ompany ideation and researchers c onsequently sh ould not study thinking a ccording to thinking content but a cc ording to thinking functions. Prompted by this form of thinking Wertheimer founded gestalt psycho logy. Although this line of thinking primarily focused on perception, it also a cknowledged the processes of creativity, productive thinking and problem solving. Selz, whose work on problem solving in- fluenc ed psychologists signific antly, was also involved in this school of thought. The gestalt psychologists’ work on thinking and reasoning influenced mathematicians’ views on these matters, espe cially when be haviouristic theories were re c eiving the most attention.
To summarise, it appeared that early research on thinking usually followed this pattern: observation of individual or group differenc es under “ typic al” circum- stanc es in the hope of improving a chi evement by providing suggestions or guidelines aimed at optimising learners’ a chievement. The fo cus was usually on facilitating changes over a longer period.
Criticism of research based on thought pro c esses is usually that the test pro c edures used, for example c orrela tion analyses, regression analysis and factor analysis are too c omprehensively statistical in nature and that it is also assumed that ratios are linear and effects “additive ”. Apart from this, early research pra ctically ignored cultural influenc es on thinking. The willingness of researchers in this field to investi- gate natural relationships as a potential sour ce of hypotheses that have to be tested is, however, a positive c ontrib ution worth mentioning.
APPROACHES TO THE LEARNING OF MATHEMATICS (ARITHMETIC) IN THE TWENTIETH CENTURY
Cross (1981) points out that the study of learning pro cesses dates back to the eighties of the previous c entury on acc ount of the work of researchers like Ebbing- haus, Dewey, Thorndike, Watson and Levin. The work of Thorndike will be discussed briefly.
Mechanical drillwork versus acquisition of meaning
From 1900 to approximately 1920 the most general method of learning mathematics was the drill and pra ctice method of Thorndike (Grossnickle, Reckzeh, Perry & Ganoe, 1983). The latter’s views in this conn e ction c an be closely linked to those of Pavlov (in connection with conditioning) and Skinner (in connection with radical behaviourism) (Kilpatrick, 1992).An attempt was made to inculc ate arithmetic al skills and abilities in learners by making use of drillwork, arithmetic steps and memorising c ombinations. Learning was regarded as a form of relationship that could be strengthened by means of repeated drillwork. Learning theoreticians such as Brownell were strongly opposed to aspects of this approach stating, among other things, that not enough allowance was made for the acquisition of insight.
The social approach
This period lasted approximately from 1920 to 1935. In essence this approach boils down to trying to find possible applications of mathematics to true-life situations. The severest criticism levelled against this approa ch c a me from academics who seriously objected to the fa ct that the social utility value of the subje ct was regarded as the major criterion of the significanc e of the c ontents of mathematics, that the systematic study of mathematics was not given adequate attention, that insight into the true meaning of arithmetic was under-emphasized, and that the development of arithmetical skills and ability was neglected.
The meaningful approach
In broad outline this approach was followe d from 1935 to early in the sixties. In brief it implied that the mathematical aspect of espe cially arithmetic received just as much rec ognition and attention as its so cial as pect (arithmetic al knowledge and ability were to be used in everyday life situations) as well.
SUBJECT
TABLES
FIGURES
SUMMARY
CHAPTER 1 TITLE AND CLARIFICATION OF CONCEPTS, (PROVISIONAL) PROBLEM STATEMENT, AIM OF THE STUDY AND PROGRAMME ANNOUNCEMENT
1.1 GENERAL INTRODUCTION
1.1.1 The extent of inadequate achievement in mathematics in South Africa
1.1.2 Attempts to explain inadequate achievement in mathematics
1.1.3 Psychological testing in mathematics
1.2 THEORETICAL FOUNDATION FOR CERTAIN APPROACHES TO LEARNING MATHEMATICS
1.3 STATEMENT OF THE PROBLEM
1.3.1 Aspects of learners’ study orientation in mathematics that should be explored with the aid of a study orientation questionnaire
1.3.2 Aim of a study orientation questionnaire in mathematics
1.3.3 The use of a study orientation questionnaire in mathematics
1.4 DEFINITION OF TERMS
1.4.1 Development
1.4.2 Evaluation
1.4.3 Study
1.4.4 Orientation
1.4.5 Study orientation
1.4.6 Mathematics
1.4.7 Learners
1.4.8 Achievement
1.5 RESEARCH PROCEDURE
CHAPTER 2 PERSPECTIVE ON CERTAIN EPISTEMOLOGICAL POINTS OF DEPARTURE REGARDING LEARNING MATHEMATICS
2.1 INTRODUCTION
2.2 PSYCHOLOGICAL FOUNDATIONS OF RESEARCH ON THE LEARNING OF MATHEMATICS
2.2.1 Research on the nature of thinking
2.2.1.1 The measurement of intellectual ability
2.2.1.2 The study of mental development
2.2.1.3 Stimulation of productive thinking
2.3 APPROACHES TO THE LEARNING OF MATHEMATICS (ARITHMETIC) IN THE TWENTIETH CENTURY
2.3.1 Mechanical drillwork versus acquisition of meaning
2.3.2 The social approach
2.3.3 The meaningful approach
2.3.4 Direct versus incidental learning
2.3.5 The “New mathematics” of the sixties; discovery versus exposition
2.3.5.1 Anti-New mathematics forces
2.3.5.2 Research on cognitive development
2.3.5.3 Poor academic achievement of school-leavers
2.3.6 The Back-to-the-basics movement
2.3.7 Aim of mathematics teaching since the eighties
2.4 BEHAVIOURISM: THE LEARNING OF MATHEMATICS AS THE ACQUISITION OF ARITHMETIC AND ARITHMETICAL SKILL
2.4.1 Introduction
2.4.2 Thorndike’s behaviouristic theory of learning
2.4.3 Brownell’s theory of learning
2.4.4 Gagné’s neo-behaviouristic (cumulative) theory of learning
2.4.4.1 Gagné: Synthesis
2.4.5 The radical learning theory of Skinner
2.4.5.1 Operant conditioning
2.4.6 Connectionism
2.4.6.1 Behaviourism: Synthesis
2.5 SOME COGNITIVE DEVELOPMENTAL AND LEARNING THEORIES RELATING TO THE LEARNING OF MATHEMATICS
2.5.1 The gestalt-psychological learning theory of Köhler
2.5.1.1 Köhler: Synthesis
2.5.2 The gestalt-psychological (verbal) learning theory of Ausubel
2.5.2.1 Ausubel: Synthesis
2.5.3 The cognitive learning theory of Bruner
2.5.3.1 Bruner: Synthesis
2.5.4 The field theory of Lewin
2.5.5 Dienes’ cognitive theory of multiple embodiment
2.5.6 Guilford’s model of the cognitive structure of the intellect
2.5.7 The cognitive learning theory of Vygotsky
THE INFORMATION PROCESSING MODEL OF INSIGHT ACQUISITION
2.6.1 Problem-solving by means of information processing
2.6.2 Information processing by means of metalearning
2.7 THE CONSTRUCTIVIST APPROACH TO LEARNING MATHEMATICS
2.7.1 The constructivist or developmental-procedural learning theory of Piaget
2.7.1.1 Piaget’s views on learners’ cognitive developmental stages
2.7.1.2 The learning and cognitive theory of Piaget: Synthesis
2.7.2 Modern constructivism
2.7.2.1 Modern constructivism: Synthesis
2.8 PROBLEM-CENTRING
2.8.1 What is the problem-centred approach to the learning of mathematics?
2.8.2 Problem-solving in mathematics
2.8.3 Discovery learning
2.8.4 Metalearning
2.8.5 Co-operative learning
2.9 RATIONALE FOR THE DEVELOPMENT OF A STUDY ORIENTATION QUESTIONNAIRE IN MATHEMATICS: THREE APPROACHES
2.10 THE SIGNIFICANCE OF A PHENOMENOLOGICAL AND HUMANISTICEXISTENTIAL APPROACH TO THE LEARNING PHENOMENON
2.11 CONTEXTUALISATION
CHAPTER 3 DEFINITION OF THE CONCEPT “STUDY ORIENTATION” AND A REVIEW OF SOME FACTORS THAT CAN INFLUENCE LEARNERS’ STUDY ORIENTATION IN MATHEMATICS
3.1 INTRODUCTION
3.2 STUDY ORIENTATION IN MATHEMATICS
3.3 HOW DO STUDY ORIENTATION PROBLEMS IN MATHEMATICS ORIGINATE
3.4 REVIEW AND CLASSIFICATION OF CERTAIN FACTORS THAT CAN INFLUENCE LEARNERS’ STUDY ORIENTATION IN MATHEMATICS
3.5 PSYCHOLOGICAL MODELS EXPLAINING STUDY ORIENTATION AND ACHIEVEMENT PROBLEMS IN MATHEMATICS
SYNTHESIS
CHAPTER 4 A CROSS-CULTURAL PERSPECTIVE ON ACHIEVEMENT PROBLEMS IN MATHEMATICS WITH REFERENCE TO THE MEASUREMENT OF A STUDY ORIENTATION IN MATHEMATICS
4.1 BACKGROUND
4.2 CULTURE AND PROBLEMS WITH MATHEMATICS ACHIEVEMENT
4.3 CULTURE-RELATED LINGUISTIC PROBLEMS
4.4 CULTURE-RELATED NON-LINGUISTIC PROBLEMS
4.5 ENCULTURATION AND ACCULTURATION
4.6 A POSSIBLE MODEL INDICATING THE MANNER IN WHICH LINGUISTIC AS WELL AS NON-LINGUISTIC FACTORS AFFECT STUDY ORIENTATION IN SOUTH AFRICA
4.7 CULTURE, MEASUREMENT AND BIAS IN PSYCHOLOGY
4.8 COMPARABILITY AND EQUIVALENCE
4.9 STRATEGIES FOR CROSS-CULTURAL MEASUREMENT
4.10 A MODEL FOR CROSS-CULTURAL MEASUREMENT
4.11 CONCLUSION
CHAPTER 5 METHOD OF INVESTIGATION
5.1 PROBLEM STATEMENT AND MOTIVATION FOR THE INVESTIGATION
5.2 AIM OF THE INVESTIGATION
5.3 RESEARCH DESIGN AND PROCEDURE: THE DRAWING UP AND STANDARDISATION OF THE STUDY ORIENTATION QUESTIONNAIRE IN MATHEMATICS (SOM)
5.4 DATA PROCESSING PROCEDURES
5.5 SUMMARY
CHAPTER 6 RESULTS AND DISCUSSION
6.1 INTRODUCTION
6.2 DATA PROCESSING: STANDARDISATION OF THE SOM
6.3 DATA PROCESSING: COMPARATIVE STUDIES TO DETERMINE THE APPLICABILITY OF THE SOM
6.4 SUMMARY
CHAPTER 7 SUMMARY AND RECOMMENDATIONS
7.1 INTRODUCTION
7.2 FINDINGS AND IMPLICATIONS OF THIS INVESTIGATION
7.3 RECOMMENDATIONS
7.4 SUMMARY
APPENDIX
LIST OF REFERENCES
