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The Rise and Fall of Behaviourism in Mathematics Education

The Move Towards Mastery Learning Approaches

In fact, much of the post-New Math(s) approach retained some of the New Math(s) emphases. Probability and statistics remained, as did an acceptance of the importance of the function concept. Although the transformation approach to geometry introduced in the New Math(s) was largely dropped, the old Euclidean axiomatic approach was not re-introduced, much to the concern of many mathematicians. Post-New Math(s) courses emphasised the hierarchical development of basic knowledge, skills, concepts, and principles; they also stressed the importance of repeated practice gained by doing large numbers of examples from textbooks or worksheets.

Curriculum theorists justified this back-to-the-basics movement by referring to the work of leading psychologists. The notion of behavioural objectives - that is to say, succinctly stated expected student outcomes - became particularly important, with the writings of Skinner (1953) being called upon. Skinner's theory that cause and effect behaviour could be manipulated by conditioning, gave rise to what became known as programmed learning. Other education psychologists also had major influences on curriculum theory at the time. Benjamin Bloom's (1956) Taxonomy of Educational Objectives became an educational classic, and Robert Gagné's (1967) ideas on learning hierarchies became standard fare in tertiary teacher education courses.

The catchcries in education became "programmed learning," "behavioural objectives," and "individualisation," and New Math(s) textbooks were quickly discarded in favour of new textbooks in which behavioural objectives were clearly defined. In many cases, a chapter of a school mathematics textbook would begin with a statement like "By the end of this chapter, you will be able to . . .," and a set of twenty or more expected behaviours would be listed. Consistent with Gagné's learning hierarchy theory, these objectives were sequenced by the curriculum developer and textbook writer (though not always by the teacher) according to some perceived, pre-determined logical structure. The notion of programmed learning was developed and systematised into what became known as "Mastery Learning," which was popularised in the late 1960s and throughout the 1970s by the writings of the North American education psychologists, J. B. Carroll, Benjamin Bloom, and James Block.

With mastery learning, behavioural objectives for each unit had to be stated unambiguously, and the level of mastery expected for each topic, defined. Criterion-referenced pre- and post-tests (often of the multiple-choice variety), were carefully constructed so that items corresponded to the stated behavioural objectives, and teaching approaches were intended to be designed to enable students to achieve the stated behavioural objectives. Time, and not ability, was taken to be the major education variable, and it was argued that almost any child could achieve mastery of a unit if he or she was given enough time to do so. Students were given pre-tests before they were taught a new topic, and those who reached mastery level on the pre-test proceeded to the next topic immediately. After a teaching treatment, students were given a post-test and if they reached mastery on this test, they then proceeded to the next topic; if they did not demonstrate mastery, they were then given another teaching treatment followed by another post-test, and so on, until they demonstrated mastery.

In line with the emphasis on mastery learning, considerable attention was paid, in the provision of professional development programs for practising teachers, to the writing of succinct behavioural objectives using Bloom's Taxonomy, to the development of criterion-referenced multiple-choice tests, to the organisation of classrooms suitable for individualised learning approaches, and to diagnosis and remediation. However, despite the huge investment by textbook publishers and ministries of education, aimed at providing support for mastery learning approaches in mathematics, during the late 1960s and throughout the 1970s education research gradually began to generate data which cast doubt on the validity of behaviourist approaches to school mathematics.

The Retreat from Mastery Learning Approaches

Ausubel and Sullivan (1970) questioned whether a logically structured hierarchical analysis of the skills and behaviours involved in a topic in mathematics necessarily provided the most appropriate pathway for teaching that topic. Ausubel and other researchers were able to demonstrate, for example, that the provision of so-called "advance organisers" could enable learners to study of a topic productively even though, from a learning hierarchy perspective, they had not acquired some of the specified "essential prerequisite skills." Skemp (1976), in his classic paper on instrumental and relational learning in mathematics, attacked the very foundations of behaviourism by pointing out that, if A, B, C, and D are four steps that appear in a learning hierarchy, teaching students to go from A to B, and then from B to C, and then from C to D, did not necessarily assist them to acquire a holistic understanding of how A, B, C and D are related, and indeed there was no guarantee that someone who had learned each individual step could then return from D to A.

Perhaps the most devastating attack on the application of behaviourism and mastery learning to mathematics education, came from Stanley Erlwanger (1975), whose study of the strict application of mastery principles in an elementary school in the United States revealed that children who had succeeded in passing mastery tests lacked any real understanding of what they were doing. They had no idea of links between skills, were unable to apply the mathematics they had "mastered" and, worst of all, had developed a totally inadequate mechanical view of the nature of mathematics. Teachers in the program had been unable to cope with the different demands placed on them by 25 or more children who were working individually in the same classroom, and often they had not realised that their students had acquired faulty or at best very limited conceptions of the mathematics they had been studying. Yet, often, these same students had been able to demonstrate "mastery" on multiple-choice, criterion-referenced tests.

The eminent Dutch mathematician and mathematics educator, Hans Freudenthal, went one step further by questioning the validity of the research carried out by Bloom and Block and other mastery learning proponents. In a scathing attack, Freudenthal (1979) not only criticised the concept of mastery learning, but also the ways in which mastery learning researchers had applied dubious statistical techniques to bolster their results. Freudenthal was not the first leading educator to question the inferential statistical techniques which had dominated much of Western education research reporting; among others to do so was the Soviet psychologist and mathematics educator, Valdim Krutetskii (1976).

Interestingly, despite a strong move by mathematics education researchers in most Western countries in the 1980s away from behaviourism, the intuitive appeal of mastery learning ideas has continued to capture the minds of educationists in other countries (and this is particularly the case in many Asian countries). Even in some of the Western countries where mathematics educators have discarded behaviourism, ministry of education officials and others, who do not know the pertinent mathematics education literatures, have sought to impose behaviourist approaches on school mathematics programs. The rhetoric associated with the recent national curriculum movements in the United Kingdom and in Australia, for example, have replaced expressions like "behavioural objectives" and "criterion-referenced tests" with "competency-based learning outcomes," but the underlying theory is essentially the same as that of the failed mastery learning movement. However, with millions of dollars invested by governments in supporting such approaches, this recent revisiting of behaviourism in education, embellished as it is in new terminology, continues to gather momentum.

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