Mathematics Teaching, Mathematics Teachers and Mathematics

By McQualter, J. W. | Australian Mathematics Teacher, Fall 2016 | Go to article overview

Mathematics Teaching, Mathematics Teachers and Mathematics


McQualter, J. W., Australian Mathematics Teacher


On teaching mathematics

Every mathematics teacher has differing views about how mathematics should be taught. These views overlap to form a body of accepted professional practice. Each mathematics teacher has his own way of selecting, organising and presenting mathematics to pupils. These have been called private theories about teaching mathematics (Bishop, 1971). The development of these private theories is the result of mathematics teachers feeling that public theories on mathematics teaching and learning cannot provide all the answers to the problems faced by individual teachers.

Each teacher has developed his or her own set of beliefs about how to teach mathematics. Such beliefs can be grouped in four categories: beliefs about mathematics, beliefs about child development, beliefs about education psychology and beliefs about the relation between learning and teaching. (Rogers, 1979).

The mathematics teacher is faced with a different set of circumstances in each lesson. The mathematics teacher reacts to these circumstances in different ways. The teacher's reaction depends on beliefs held about how to teach mathematics, what the outcomes to be achieved are and the outcomes actually achieved. Despite the vast amount of research that has been conducted on teaching in general, and mathematics teaching as part of this general research, we still cannot describe with any certainty the major components of mathematics teaching and the particular combination of variables which produce the outcomes the mathematics teacher desires (Begle, 1979; Shumway, 1980).

Teaching is a complex activity. The teacher is a major variable in this activity. It is the teacher who has to plan and present the lesson, the programme, the course. It is the teacher who operates on the mathematics to be learned: selecting, sequencing, presenting and evaluating. The teacher acts as a filter for the official or formal curriculum; that which society expects to be taught. The teacher adapts, re-organises and turns the official curriculum into the taught curriculum.

As Begle has pointed out, research on mathematics teaching has not yet established what relationship exists between the taught curriculum and the learned curriculum: what the pupil actually learns, as revealed by tests, teacher questioning and pupil attitude. Teachers monitor the learned curriculum by a variety of means and use the results to change, and refine the curriculum. In this monitoring process the beliefs held by the teacher exert an influence on the decisions made (Rogers, 1979). The monitoring process goes on inside the classroom during the lesson and outside the classroom, both before and after the lesson; the planning and evaluating stages.

The sources of a teacher's knowledge and beliefs about how to teach mathematics are many. One source is the nature of the subject matter being taught. The subject matter to be taught influences the nature of teaching, the role of the teacher and teacher decision making.

The nature of teaching

Any thinking about teaching must consider two main aspects of teaching: instruction and classroom management.

Classroom management concerns such matters as teaching context and procedural activities: discipline, control, motivation, teacher personality, teacher-pupil interaction and like matters. In such activities the concern being with the group and individual psychology of the classroom irrespective of what subject matter is being taught. Much of the research on classroom management has been carried out using observational instruments producing quantitative data. Where quantitative data is not produced from classroom observation, for example commentaries on classroom behaviour, the analyses have produced 'high inference' relationships between observed actions. This is in contrast to 'low inference' relationships produced from the quantitative data (Cooney, 1980). Yet none of these relationships have lead to the development of accepted principles of mathematics teaching. …

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