The aims of this qualitative-descriptive study conducted in the context of a partnership between an Elementary School and a College of Education, are to describe the formation of mathematical–scientific concepts by student-teachers, teachers, the methods-supervisor and pupils, and to identify the factors which obstruct this process.

The aims of this study are to describe how scientific concepts in mathematics are formed by student-teachers, by their methods-supervisor, by teachers and by their pupils, and to identify the factors which obstructed this process. The research questions are:

 

This study was conducted in the context of partnership between a College and an Elementary School. The researcher, who was the methods-supervisor, hade noticed that when the student-teachers, teachers and pupils were asked to explain a mathematical idea, they do it by using spontaneous-concepts. She understood that there is a problem with the formation of scientific-concepts, and in order to improve her instruction, she decided to explore how mathematical concepts are formed by the participants in the context of the elementary  school.

The examples described in this study deal with the concept of the “sequential number”. Its lexical definition is: “A number which immediately follows another number”. The acceptable definition in the mathematical community is: “A number that is greater than it’s predecessor by one whole unit in the set of integers numbers”.

 

When entering school, the child is equipped with a “kit of concepts” acquired during his formative years. These daily concepts are characterized by an unconscious understanding and spontaneous use. This period, called "the age of concrete operations", is characterized by the child’s ability to develop scientific-concepts via manipulations with concrete objects (Piaget, 1976). In contrast to this point of view, Vigotzky claims that the development of mathematical concepts does not occur spontaneously as a result of engagement with concrete objects, but during the relationships between the scientific-concepts presented to the child by adults and the child’s spontaneous-concepts which are related to these phenomena (Krapov, 2003). The development of spontaneous-daily concepts and the non-spontaneous-scientific concepts are interrelated. In this process, the role of mediators is crucial. Mediation includes material and symbolic tools, and enhance the role of the teacher as a human mediator (Kozulin, 1998).

Tall and Vinner (1981), emphasize the strong essence of spontaneous concepts, which create a personal image comprised of the sum of all the cognitive constructions, mental pictures, properties and processes related to the pupil’s experiences. This personal image can contradict the scientific-concept. In a state of conflict between the scientific-mathematical concept and the intuitive-spontaneous feeling, the intuitive feeling "wins". This situation makes the teaching and learning of the concept more difficult. Moreover; very abstract concepts cannot be acquired after directed teaching. In order to facilitate the learning process Klausmeier (Marzano,1988) suggests to promote activities such as comparison to similar concepts and to introduce examples and counter examples. From Vigotzkyan point of view, the scientific-concepts are not acquired via simple rote and committed to memory learning, but rather are formed by the learner’s intensive mental activity, and as a result of a dual dialectic process: top-down (from theoretical to empirical definitions) and bottom-up (from empirical to theoretical definitions). The formal learning in school is designed to promote this dialectic process and encourage the passage from a mastery of spontaneous-daily concepts to a mastery of academic-scientific concepts (Krapov, 2003)

 

This study will describe the process of forming scientific-concepts in mathematics, and will focus on the role of the teacher as a mediator.

 

This study is qualitative-descriptive.

The participants:

The data was collected during one academic year and included:

The data was analyzed in a dialectical process between the empirical data, the lexical concept and the mathematical-scientific concept of “sequential number”.

 

The participants' responses to the question "what number is sequential to a quarter" and their explanations, were classified into three groups:

1. Responses characterized by a use of the intuitive-daily concept such as: “the following number” or “the number that immediately follows”. The spontaneous use of the concept “sequential number” was lacking awareness and the capacity for generalization and differentiation. This group included mainly grade 3 and grade 4 pupils, who had never met the concept “sequential number” during any learning activity mediated by an adult. They used the concept spontaneously without feeling the need for re-examination of the terms.

3. Responses which testify to an understanding of the scientific concept “sequential number” in the context of the world of rational numbers.

From the second group participants’ explanation of their own answers, it appears that most of them know that “a sequential number is the number plus one”, even though they will not give the accurate formal definition and will not say “and plus a whole unit.” They also know that between any two fractions there is an infinite number of fractions and can also explain this claim. However, they still made an analogy to a more familiar system of numbers and said that “the sequential number to a quarter is one and a quarter”, “two fifths”, or “0.26”, which demonstrate a limited and erroneous perception of the scientific-mathematical concept “sequential number”. According to the finding that an erroneous analogy is a main factor which obstruct the process of forming scientific-concepts, learning strategies and mediated tools were elaborated.

 

The teacher’s mediation has a crucial role in the reciprocal relations between the mathematical concepts presented to the child and the child’s daily concepts, created in the context of the child’s spontaneous mental activity. The contribution of this study is in offering strategies for dealing with the erroneous analogies and in nurturing analogies that contribute to the development of scientific concepts.