definition of function Archives - Mathematics for Teaching

Strengths and limitations of each representation of function

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Function is defined in many textbooks as a correspondence between two sets x and y such that for every x there corresponds a unique y. Of course there are other definition. You can check my post on the evolution of the definition of function. Knowing the definition of a concept however does not guarantee understanding the concept. As Kaput argued, “There are no absolute meanings for the mathematical word function, but rather a whole web of meanings woven out of the many physical and mental representations of functions and correspondences among representations” (Kaput 1989, p. 168). Understanding of function therefore may be done in terms of understanding of its representations. Of course it doesn’t follow that facility with the representation implies an understanding of the concept it represents. There are at least three representational systems used to study function in secondary schools. Kaput described the strengths and limitations of each of these representational systems. This is summarised below:

Tables: displays discrete, finite samples; displays information in more specific quantitative terms; changes in the values of variables are relatively explicitly available by reading horizontally or vertically when terms are arranged in order (this is not easily inferred from graph and formula).

Graphs: can display both discrete, finite samples as well as continuous infinite samples; quantities involved are automatically ordered compared to tables; condenses pairs of numbers into single points; consolidates a functional relationship into a single visual entity (while the formula also expresses the relationship into a single set of symbols, individual pair of values are not easily available for considerations unlike in the graph).

Formulas/ Equations: a shorthand rule, which can generate pairs of values (this is not easily inferred from tables and graphs); has a feature (the coefficient of x) that conveys conceptual knowledge about the constancy of the relationship across allowable values of x and y — a constancy inferable from table only if the terms are ordered and includes a full interval of integers in the x column; parameters in equation aid the modelling process since it provides explicit conceptual entities to reason with (e.g. in y = mx, m represents rate).

It is obvious that the strength of one representation is the limitation of another. A sound understanding of function therefore should include the ability to work with the different representations confidently. Furthermore, because these representations can signify the same concept, understanding of function requires being able to see the connections between the different representations since “the cognitive linking of representations creates a whole that is more than the sum of its parts” (Kaput, 1989, p. 179). Below is a sample task for assessing understanding of the link between graphs and tables. Click solutions to view sample students responses.

How do you teach function? Which representation do you present first and why?

Reference

Ronda, E. (2005). A Framework of Growth Points in Students Developing Understanding of Function. Unpublished doctoral dissertation. Australian Catholic University, Melbourne, Australia.

Evolution of the definition of function

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How do you define function? Do you teach relation first before teaching function? Does knowing about relation a pre-requisite to function understanding?

The concept of function “was born as a result of a long search after a mathematical model for physical phenomena involving variable quantities” (Sfard, 1991, p. 14). In 1755, Euler (1707-1783) elaborated on this conception of function as a dependence relation. He proposed that, “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (p. 15). Seventy-five years later, Dirichlet (1805-1859) introduced the notion of function as an arbitrary correspondence between real numbers. About a hundred years later in 1932, with the rise of abstract algebra, the Bourbaki generalised Dirichlet’s definition. Thus, function came to be defined as a correspondence between two sets (Kieran, 1992). This formal set-theoretic definition is very different from its original definition. Function is no longer associated with numbers only and the notion of dependence between two varying quantities is now only implied (Markovits, Eylon, & Bruckheimer, 1986). The Direchlet-Bourbaki definition allows function to be conceived as a mathematical object, which is the weakness of the early definition. However, the set-theoretic definition is too abstract for an initial introduction to students and is inconsistent with their experiences in the real world (Freudenthal, 1973; Leinhardt, Zaslavsky, & Stein, 1990; Sfard, 1992).

Textbooks, which often define function as a set of ordered pairs usually start the discussion with relation and introduce function as a special kind of relation. But relation is more abstract than function. Thus the supposed pedagogical value of having to learn relation first before one understands function is, in the opinion of Thorpe (1989), wrong. Freudenthal (1973) also expressed strongly that “to introduce function, relations can be dismissed” (p. 392). Thorpe went on to say that the use of the set-theoretic definition which defines function as a set of ordered pairs “was certainly one of the errors of the sixties and it is time that it were laid to rest” (p. 13). Amen to that but only until a certain grade level.

My references:

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht-Holland: Reidel.

Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research, 60(1), 1-64.

Markovits, Z., Eylon B. A., & Bruckheimer, M. (1986). Function today and yesterday. For the Learning of Mathematics,6(2) 18-28.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-84). (MAA Notes no. 25) Washington DC: Mathematical Association of America.

Thorpe, J. (1989). Algebra: What should we teach and how should we teach it? In S. Wagner & C. Kieran (Eds.) Research issues in the learning and teaching of algebra (pp. 11-24). Reston, VA: NCTM.

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