Anne D. Cockburn
School of Education and Lifelong Learning
University of East Anglia
"To generalise, or not to generalise, that is the question"
(with apologies to Hamlet and William Shakespeare)
From a very early age an ability to generalise makes our lives easier in many respects. Indeed, developing an awareness of pattern is an important step in becoming a proficient mathematician. Over the years as a researcher and teacher educator, however, Anne Cockburn has observed many cases of 2 – 60 year-olds generalising when it is inappropriate to do so and vice versa. Sometimes such examples are easy to predict. Other times they are not. Using a range of examples spanning early years’ education, mathematics and research on teachers’ professional practice, Anne will highlight some of the issues surrounding generalisation and explore possible implications for the academic community.
Hebrew University of Jerusalem
Ben Gurion University of the Negev and Achva College of Education
Thought processes which lead to generalizations in every day thought processes and in mathematical contexts
Generalizations are the engine which forms concepts in all domains and claims about almost any subject. It seems that it is possible to claim that generalizations are kind of a cognitive drive (if we use Freudian terminology) or cognitive need (if we prefer the terminology of Maslow). If we like to use evolutionary psychology it will be easy to point at the evolutionary advantage of generalizations. Namely, when we were still hunters in the wilderness, generalizations helped us to survive. The talk will point at the thought processes which lead to generalizations. All that is true about non-technical situations. Things are different in mathematical thinking. Here the ultimate goal is that the student will acquire the desirable mathematical behavior. Namely, in mathematical contexts we are supposed to train our mind to form concepts by relying on formal definitions and to establish claims by relying on proofs. This contradicts the spontaneous nature of thinking. Thus, some mathematics educators, in order to facilitate the learning of mathematics, offer to the students strategies which are supposed to imitate the assumed spontaneous way of forming generalizations. They do it by presenting to the students examples which will lead them to the correct generalizations. The talk will focus on the role of examples in everyday thought processes and in mathematical contexts.
Pedagogical University of Cracow
Generalization in the process of defining a concept and exploring it by students
Generalization is one of the most important processes that occurs in the construction of mathematical concepts, discovering theorems, and solving math problems. This process can be analyzed from two different view points: 1) the cognitive theory, 2) the mathematical activity of individuals. In my talk I’ll take into account these two aspects.
The first part will present the problem of creating conceptual classes and types of generalizations of theorems, as presented by the eminent Polish educator, A. Z. Krygowska in her "Outline of mathematics didactic". References to the Dörfler’s theory of generalization will be made.
The second part will include an analysis of examples of generalization activity, disclosed in my research on solving math problems by students at different levels of mathematical knowledge and experience.
Nicolina A. Malara
University of Modena and Reggio Emilia,
Generalization processes in teaching/learning of algebra:
students difficulties and teacher role
After a brief overview of the main problems to be faced in algebra teaching to promote students approach to generalization processes, we recall some research results which give meaningful indications for the teachers practice. Then we focus our attention on two algebra teaching key activities: the study of sequences and the discovery and proof of numerical regularities.
In this frame we shall consider the question of the teacher role in leading the students to engage in this kind of activities. We shall present some short excerpts of classroom work which show the sharp relationship between the teacher’s actions and the students’ behaviors. We shall highlight the importance of the teacher’s awareness at different levels to gain consciousness and control about the effective ways of posing his(her)self in the class when (s)he addresses with the students this kind of questions. We shall conclude underlining the need of a refined teacher’s education on this delicate aspect of teaching which requires a deep study of classroom episodes and above all a systematic careful self-analysis of the teacher’s own practice.