Students’ Understandings of the Distinction between Function and Relation

The term 'function' means the mutual connection and dependence of things, in any way, from a dynamic/operational perspective. Therefore, function must be viewed both from the aspect of things that are in mutual interdependence, that is structurally, spatially, statically, geometrically, and from the aspect of procedures, that is in operation, dynamically and algebraically. The concept of the function is one of the two basic notions in modern mathematics, the other being that of Relation. The first has had an interesting evolution and gives us an example of the trend in mathematics to extend and generalize its concepts. Function is greatly involved in mathematics, which is why famous mathematicians (e.g. Mac Lane, 1986) support its use as a unifying and centralizing principle in organizing and teaching of the respective lesson. This concept seems to be the natural and main guide in selecting, organizing and developing mathematical subjects in general and particularly in mathematical texts. The realization of those difficulties out of the entire mathematical community has as a consequence the tremendous increase of the related research and publications, during years. Researchers such as Sierpinska (1991), Sfard (1989, 1991), Harel and Dubinsky (1992), Vinner (1983), Tall (1991) have expressed their views about the subject This research is on mathematics students‘ understandings of the distinction between the concepts of function and relation. The main research question investigates the students‘ awareness of the ‗‗discrimination between the concept of function and relation‘‘. A secondary research question linked to the main question is concerned with the ‗‗discrimination between the dependent and the independent variables‘‘. The survey was designed, in two phases; Phase 1 included the completion of a questionnaire by the students .The questionnaires were distributed to 15 students of class XIIth of Infant Jesus School , Patna city . The 1st, the 2nd and partially the 3rd questions of the questionnaire are linked to the main research question. The 3rd and the 4th questions are connected to the secondary question. The Questionnaire: 1. Which of the following relations are function relations? Make the necessary corrections to the rest of them, in order to transform them into functions. Answer only in YES or NO.(Horizontal line as x-axis and vertical line as y-axis.)

(A)

(C)

(D)

(E)

2. Give two examples of arbitrary relations that are not functions (one described graphically and one analytically (with an algebraic formula). 3. What happens to the graphical representations of the following functions if the line on the graph is rotated by 900? Are they still functions? Give a short justification of your answer.(Horizontal line as x-axis and vertical line as y-axis.) 4. In which situation(s) does a 900turn of a function‘s graphical representation represent a function? Which general rule would you use? In Phase 2, students were interviewed and asked about their thoughts while completing the questionnaire. Summarizing the main findings of our research, we observed the following: • All students interviewed, except one, gave only examples of one - one functions. When they recalled the definition they stereotypically said ‗‗when one element of a set corresponds to one element of the other set‘‘, neglecting to add: ‗‗and only one‘‘. They also had difficulties in giving a good

‗‗many- to-one‘‘ condition in the definition. • The students used mnemonic rules: 6 students out of 15 gave the same example the circle example as a graphical representation of an arbitrary relation that is not a function. • 4 of the 15 students gave wrong answers to the 1st as well as the 2nd question of the questionnaire, where they were asked to recognize the difference between the concepts of function and arbitrary relation. • 8 of the 15 students gave wrong answers to the 3rd question, and 11 of the 15 gave wrong answers to the 4th question of the questionnaire. Moreover, only 2 students in the interviews gave examples of ‗‗a single-valued but not uniquely invertible function‘‘.

It is evident from the results of the questionnaires and the interviews that students experienced difficulties in answering all four questions. Both the students‘ examples and their application of the definition revealed a constrained perception of the function concept. Reflecting on the definition of function by Dirichlet, we observe the priority that it attributes to the dependent variable, stating the necessity of the dependent variable being uniquely determined, while the inverse is not always uniquely determined. Therefore, in order to overcome the obstacle of confusing the function with a relation, we suggest a teaching of function oriented towards activities which demand the measure of a magnitude, to which we do not have immediate access. Such an example might be measuring heights, when there is no alternative way but the use of the tangent function for acute angles.

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PG Department of Mathematics, Patna University, Patna, Bihar