# Functions Learning and Teaching

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_96-2

## Keywords

Variable Correspondence Graph Representations Process-object duality Concept image## Definition and Brief History

The notion of function has three different, yet interrelated, aspects. Firstly, *a function* is a purely mathematical entity in its own right. Depending on the level of abstraction, that entity can be introduced – for example – as either a correspondence that links every element in a given domain to one and only one element in another domain, called the co-domain; or as a certain kind of relation, i.e., a class of ordered pairs (in a Cartesian product of two classes), which may be represented as a graph; or as a process – sometimes expressed by way of an explicit formula – that specifies how the dependent (output) variable is determined, given an independent (input) variable; or as defined implicitly as a parametrized solution to some equation (algebraic, transcendental, differential). Secondly, functions have crucial roles as lenses through which *other mathematical objects or theories* can be viewed or connected, for instance when perceiving arithmetic operations as functions of two variables; when a sequence can be viewed as a function whose domain is the set of natural numbers; when maximizing the area of a rectangle given a constant perimeter; or when perceiving reflections, rotations, and similarities of plane geometrical figures as resulting from transformations of the plane; or when Euler’s φ-function (for a natural number n, φ(n) is the number of natural numbers 1,2,…, n that are co-prime with n) allows us to capture and state fundamental results in number theory and cryptography, etc. Thirdly, functions play crucial parts in the application of mathematics to and modelling of *extra-mathematical situations and contexts* (Michelsen 2006), e.g., when the development of a biological population is phrased in terms of a nonnegative function of time; when competing coach company tariff schemes are compared by way of their functional representations; or when the best straight line approximating a set of experimental data points is determined by minimizing the sum-of-squares function, and so on and so forth.

These aspects of the notion of function make this notion one of the most fundamental and significant ones in mathematics, and hence in mathematics education. This is reflected both in the history of mathematics (the term “function” going back at least to Leibniz (Boyer 1985 (1968), p. 444) and in the history of mathematics education, where the notion of function as a unifying concept in mathematics was introduced in the curricula of many countries from the late nineteenth century onwards, following the reform program proposed by Felix Klein (NCTM 1970 (2002), p. 41; Schubring 1989, p. 188). Today, versions of the notion of function permeate mathematics curricula in most countries. However, the different aspects of the notion of function also make it highly diverse, multifaceted, and complex, which introduces challenges to the conceptualization as well as to the teaching and learning of functions.

Against this background, the concept of function in mathematics education has given rise to a huge body of research. The origins of this research seem to date back to debates in the 1960s about the right (or wrong) way to define a function. Thus Nicholas (1966, p. 763) compares and contrasts three definitions (which he labels “variable,” “set,” and “rule”), which, in his view, generate a dilemma, because they are not logically equivalent. The first empirical studies also seem to stem from the late 1960s. Empirical studies focused on the formation of the concept of function, which has also preoccupied the far majority of subsequent research, as is reflected in the seminal volume on this topic edited by Dubinsky and Harel (1992a) and in the relatively recent overview of significant research offered by Carlson and Oehrtman (2005).

## Challenges to the Teaching and Learning of Function

The reason why the concept of function itself has attracted massive attention from researchers is that students (and many pre- or in-service teachers as well, see Even 1993) have experienced, and continue to experience, severe difficulties at coming to grips with the most significant aspect of this concept in both intra- and extra-mathematical contexts (Sajka 2003). More specifically, researchers have focused on identifying and analyzing the learning difficulties encountered with the concept of function, on explaining these difficulties in historical, philosophical, and cognitive terms, and on proposing effective means to counteract them in teaching. In so doing, researchers have introduced a number of terms and distinctions (e.g., between “action” and “process” (Dubinsky and Harel 1992b)).

One important issue that arises in this context is the fact that functions can be given several different *representations* (e.g., verbal, formal, symbolic (including algebraic), diagrammatic, graphic, tabular), each of which captures certain, but usually not all, aspects of the concept. This may obscure the underlying commonality – the core – of the concept across its different representations, especially as translating from one representation to another may imply loss of information. If, as often happens in teaching, learners equate the concept of function with just one or two of its representations (e.g., a graph or a formula), they miss fundamental features of the concept itself. This is also true of the many different equivalent *symbolic notations* for the very same function (e.g., y = x^{2}–1/x, f: x → x^{2}–1/x, in both cases provided x ≠ 0; f: ℜ \{0} → ℜ defined by f(x) = x^{2}–1/x; f(x) = (x–1)(x^{2} + x + 1)/x, x ∈ℜ \ {0}; f = {(x, x^{2}–1/x)| x ∈ℜ \ {0}}; (x,y) ∈ f ⇔ y = x^{2}–1/x ∧ x ∈ℜ ∧ x ≠ 0; x = y^{2}–1/y, y ∈ℜ \ {0}, just to indicate a few). Interpreting and translating between function representations in intra- or extra-mathematical settings proves to be demanding for learners. Of particular significance here is the translation between visual and formal representations of the same function, which for some learners are difficult to reconcile (Kaldrimidou and Moroglou 2009).

Functions come in a huge variety of sorts, types, and cases, ranging from familiar ones (such as linear or quadratic functions of one variable) to abstract and complex ones (such as the definite integral as a real-valued functional operating on the space of Riemann-integrable functions of *n* real variables). The plethora of functions of very different kinds means that students’ concept of function is also delineated by the set of function *specimens and examples* of which the students have gained experience. This is an instance of the well-known distinction between concept definition and *concept image* playing out in a very manifest manner in the context of functions (Vinner 1983), in particular in teaching and learning that focuses on abstract functions. This distinction also proves important when zooming in on special classes of functions (such as linear or affine functions, exponential functions, recursively defined functions, and above all the real and complex functions that appear in calculus and analysis), which have been the subject of study in an immense body of research.

Another demanding facet of the concept of function is the *process-object duality* (cf., e.g., several chapters in Dubinsky and Harel (1992a)) that is characteristic of many functions, especially the ones that students encounter in secondary and undergraduate mathematics teaching. In its process aspect, a function is a device that yields outputs as a result of inputs. In its object aspect, a function is just a mathematical entity which may engage in relationships with other objects, or be subjected to various sorts of treatment (e.g., differentiation at a point, or integration over an interval). Oftentimes the transition from a process view to an object view of function is a severe challenge to students (Eisenberg and Dreyfus 1994).

## Overcoming Learning Difficulties

In response to the observed learning difficulties attached to functions, and analyses of these difficulties, mathematics educators have invested efforts in proposing, designing, and implementing intervention measures so as to address and counteract these difficulties specifically. The overarching result is that it is possible to counteract the learning difficulties at issue, but this requires intentional and focused work on designing rich and multifaceted learning environments and teaching-learning activities that are typically extensive and time-consuming. In other words, the desired outcomes are not likely to occur by default with most students, the outcomes have to be aimed at, and they come at a price: time and effort.

A few examples: One focal point has been to help students develop a process conception of function (in contrast to an action conception), by way of technology (Breidenbach et al. 1992; Goldenberg et al. 1992). Technology has also been used to consolidate students’ concept images so as not to “overgeneralize” the prototypical function examples that initially underpinned their conception. Helping students to develop an object conception of function (by way of reification) has preoccupied many researchers, e.g., Sfard (1992). Another approach has been to focus on unpacking the multitude of complex notation conventions that are at play in dealing with functions (e.g., Sajka 2003)

## Future Research

While research in this area in the past has focused on the learning (and teaching) of the concept of function in contexts when functions are already meant to be present, or presented to students, very little – if any – research has dealt with situations in which students are requested or encouraged to uncover or introduce, themselves, functions or functional thinking into an intra- or extra-mathematical context (for an exception, see Breidenbach et al. 1992). Furthermore, there is a need for future research that focuses on designing teaching-learning environments that help to generate transfer of the notion of function from one setting (e.g., real functions of one variable) to another (e.g., functions defined on sets of functions).

## Cross-References

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