An image function, f(x,y), is a function with both x and y being positive integers (Gonzalez & Woods, 2017). Any combination of such an x and y will form an ordered pair that will denote the position of a particular pixel in the image. Corresponding to each ordered pair is a unique colour. Typically, the different colours are represented using the RGB (red, green, blue) colour space. Any specific output of an image function is then an ordered triple providing the specific combination of red, green and blue. Typically, a scale of 256 different shades of red are used and the same for green and blue (Gonzalez & Woods, 2017). If we then let the first shade be represented by 0, the last shade would then be represented by 255. Using these typical values, 256³ combinations of red green, and blue are possible. For example, the triple (255,0,0) will be bright red as it contains the full complement of red and zero contributions of green and blue. (255,255,0) is bright yellow, (0,255,0) is bright green and (57,229,212) would be called turquoise by some.

If we only consider the possible outputs where the three components of each triple are equal, we end up with what is commonly referred to as a greyscale image, where outputs are shades of grey. For example, (0,0,0) is black, (255,255,255) would be white and (30,30,30) would be a dark grey. The image in Figure 1(b) is an example of a greyscale image. As the three values in each triple will be equal, the outputs for greyscale images each consist of a single number that represents the light intensity at a particular pixel.

An example: Consider Figure 1b, the greyscale image of the horse’s eye.

Domain: This image has exactly 51 rows and 91 columns. The domain of this image function, is the set of ordered pairs: {(x,y)|1≤x≤51,1≤y≤91,x∈Z+,y∈Z+}.Z+ is the set of positive integers.

Range: The word range can refer to two different concepts, namely the codomain and the image of the function, so care should be taken in using it. The codomain for a greyscale image is easily specified as the set {s∈Z+|0≤s≤255}. This is then the set of shades of grey from which any greyscale image could be ‘choosing’. When the term range is referring to the image of the function, it will consist of all shades of grey actually present in the particular ’picture’. Here then the image of the function and the picture-image of the function are the same set. The picture set would normally have repeated values or colours and would thus be a different multiset from the function image.

Consider more aspects of image functions.

An image function would seldom be surjective. With colour images using the RGB colour space, we have a total of 256³ = 16 777 216 unique elements in the codomain and most often many of these colours would not be present in the image. Being closer to surjective is normally desirable when it comes to images, as this would generally mean the image has higher contrast. Greyscale images typically have (only) 256 unique elements in the codomain; thus, being surjective has a much higher probability than in the case of colour images. It is clear that most images would not be injective either because it is highly probable that different pixels have exactly the same colour or shade of grey.

As for all functions, the inverse will exist if the function is injective. In the previous paragraph, we saw that it is highly improbable for an image function to be injective and consequently it is highly improbable for the inverse to exist. With the high resolution of modern cameras, it is quite common for digital images to consist of millions of pixels. For greyscale images of such high resolution, it would then be impossible to have an inverse, as greyscale images only have 256 output options available. Even for colour images with 16 777 216 possible output options, it will still happen often that at least two pixels will have the same colour. Therefore, the probability of the inverse existing is small.

Consider any point (x₀,y₀) in the domain of our image f(x,y). Then we can show that:

lim(x,y)→(x0,y0)f(x,y)=f(x0,y0)

Therefore, the image is continuous on its domain.

Proof: Let ∈ > 0. Let 0 < δ < 1. If |(x−x0)2+(y−y0)2|<δ<1 then f(x,y) = f(x₀,y₀) because the domain of f is a subset of Z × Z. Therefore, |f(x,y)−f(x0,y0)|=0<∈ As the limit exists at any point in the domain and the limit is equal to the function value at that point, the function is continuous at any point in its domain.

A digital image is not differentiable at any point, yet a discrete derivative in the form of a difference quotient plays an important role in image processing. Applications where sudden changes such as steps, ramps, edges, lines or isolated dots need to be identified or accentuated often rely in part on some discrete implementation of a derivative. From calculus, we know that the derivative of a constant is 0, which translates to the important requirement of derivative-based filters to give back a small or even zero response in a homogeneous region of an image. See for example Gonzalez and Woods (2017) for more on the implementation of derivative filters and, for example, Shrivakshan and Chandrasekar (2012) for more on edge detection techniques through the use of derivative filters.

The concept or notion of a function is in its essence quite abstract but is often understood at a level where much of the abstract nature is not truly comprehended or might even be entirely lost. A student might for example directly equate the function concept to the existence of a formula (Dubinsky & Wilson, 2013; Sierpinska, 1992; Vinner & Dreyfus, 1989). One of the prominent indications of a lack of depth in the understanding of the function concept is the restrictiveness applied to what constitutes a function. If a student starts to fixate on particular types or certain representations, they lose much of the richness of the function concept.

Being able to recognise a certain formula or graph as (representing) a function is of course a necessary skill, but not sufficient in providing the student with the correct concept aspects and cognitive reasoning to be able to grasp and utilise higher mathematical concepts. For example, something as immediate as the inverse of a function, concepts such as limits, derivatives and not forgetting ideas that are even more abstract, such as topological homeomorphism and category theory, remain out of reach. Thompson (1994, p. 39) argues that a fundamental difficulty is students’ lack of connections between the various representations of the same function. What is it that is being represented? Thompson names this ‘something’ the ‘core concept of function’, that which is left unchanged when moving between the different representations.

What it boils down to is that students sit with an inadequate or erroneous function concept image. According to Tall and Vinner (1981, p. 151), the concept image constitutes the ‘total cognitive structure that is associated with the concept’. This entails all definitions, properties, ideas, theorems and examples that a student has grouped under the heading of function over their mathematical career so to speak. Although a student may know the formal definition of a function, when exposed to a problem, the full concept image will be utilised to solve the problem. Doorman et al. (2012, p. 1245) also consider the concept image important and had as one of their specific goals the overcoming of a ‘too-limited’ function concept image. They investigated a new learning arrangement incorporating a computer tool to foster the transition from an operational to a structural view of functions. Within the particular setting, they report some success in overcoming difficulties with integrating operational and structural aspects and providing an explorative environment with respect to the aspects of covariation (Doorman et al., 2012, p. 1262).

As the function concept is fundamental, yet misunderstood, the suggestion is that students should be introduced to the idea in such a manner that the resulting concept image will be as rich and accurate as possible. It is in these respects that the exploration of digital images considered as functions, or image functions for short, could be particularly useful.

Much research has been done in confirming the difficulty with the understanding of the function concept and addressing this difficulty. Recent research includes that of Chimhande, Naidoo and Stols (2017), which confirms that the difficulty is prevalent at school level. They showed that the mental constructions were typically at the action level of understanding, which is the lowest level according to the APOS theory (Arnon et al., 2014). Doorman et al. (2012) explored the use of computer tools in aiding the transition to a structural view of function, that is, the object level of understanding. Makonye (2014) also provides a theoretical analysis focusing on the use of multiple representations to foster a nuanced concept image through approaches where the function concept is kept embedded in students’ reality as far as possible. Other research focuses on specific aspects of the function concept and not on functions in general. Bansilal, Brijlall and Trigueros (2017) explored pre-service teachers’ understanding of injections and surjections through an APOS study. They found that most participants were at the action level. Maharaj, Brijlall and Govender (2008) explored the use of instructional design worksheets in advancing pre-service mathematics students’ understanding of the concept of continuity of single-valued functions. They found most participants were able to construct internal processes to make sense of continuity.

This article discusses a new pedagogy. This refers to the new approach reported on in this article, which used the field of digital image processing to gather ‘mathematics for teaching’ (Hoover, Mosvold, Ball & Lai, 2016, p. 4). The specialised mathematical knowledge concerning image functions was used in a novel approach to the problematic teaching and learning of the function concept.

Working with constructivist ideas, essentially that learning is built upon previous learning, Dubinsky and McDonald (2001), as well as others before them such as Breidenbach et al. (1992), formulated the APOS framework (Arnon et al., 2014) for modelling the learning of mathematical concepts. Using the APOS framework, the development of the function concept can be modelled where the conceptualisation passes through stages in a non-linear way, generally starting with actions (A), then processes (P), objects (O) and finally mental schemas (S). The non-linear here refers to the notion that the learning does not exclude the possibility of moving along different paths between the stages. For example, when busy conceptualising the function concept at the process stage, it might be useful or necessary to rethink and expand on one’s conceptualisation at the action level.

At the action stage of understanding, an external stimulus such as an expression or an equation is needed to proceed, and the student cannot yet work with the concept entirely in the mind. In terms of image functions, we shall see in activity 1 of the IFI that explicit instructions are given to create a new image, one pixel at a time, by shading cells in a grid. At the process stage, the student can manage to construct the concept in the mind and also think about the underlying actions that make up the process, without actually performing any of these actions. In terms of image functions, the function as a process is realised when one can imagine how the possibly millions of pixels all get their respective colours (values) at the same moment when the photograph is taken. The number of pixels determines the domain, and which colours they potentially can get determines the codomain. When the student is at the object stage, the process has been encapsulated. The student is then able to think about other actions and processes that might be performed on this object. The student has then gone from an operational view to a structural view. In terms of image functions, the object level makes it possible to think about transformations, like contrast stretching, that can be applied to images. Images can be mentally grouped according to certain criteria and as such form sets of functions. At the schema stage, a student will be able to move freely between considering and using the concept as an action, process or object (Arnon et al., 2014, p. 30; Asiala et al., 1996, pp. 7–8). When considering which transformations would potentially enhance an image, the thinking is mainly at the object level, but when it comes to planning and performing the transformation, the thinking must be at the process and action levels.

The APOS theory can help us understand how the learning takes place by explaining what we see when participants are trying to ‘construct their understanding of a mathematical concept’ (Dubinsky & McDonald, 2001, p. 1). The proposed mental structures needed for learning a mathematical concept are captured in what the APOS theory refers to as the GD. This GD is the theoretical blueprint against which the intervention of this article, the IFI, was measured.

Description: The student is asked to draw an 8 × 8 grid on paper as in Figure 2a. The student must then take a pencil and shade the blocks at positions B2, B3, C1, D1, D5, D6, D8, E1, E4, F2 and F3. Accurate shading leads to the result given in Figure 2b.

Students are asked to reflect on the activity by letting them provide answers to questions pertaining to uniqueness aspects of functions and asking questions to let them think about the choices that can be made with respect to input and output.

Questions asked as a part of this activity:

To keep the activity and the IFI interactive, students are required by the software to first answer the questions before being able to proceed. The question formats vary between multiple choice and typing an answer. From this activity, there is a natural flow in letting the student discover that a photograph or digital image can be interpreted as a mathematical function.

Connecting to the GD and the conceptualisation indicators (CIs): This activity aims to connect the repeated action of assigning a shade of grey to a specific position in the grid to a function value that is assigned to specific input. This activity is therefore aimed at helping students to construct the function concept as an action, which is the first stage of understanding according to the APOS theory. By repeating the shading action, an attempt is made at facilitating the interiorisation mechanism as was described in the GD. The instructions of this activity connect with the external prompts (see the CIs) or explicit steps needed by a student whose understanding is at the action level. The reflective questions of this activity connect with the CI criteria that the student will know the definition of a function and will as such be led to discover that an image can be interpreted as representing a function.

Description: Students visualise some function outputs by using the colour editor in the commonly available software Paint. Students are prompted to try different combinations of red, green and blue and they can then see what those combinations look like. This is an easy-to-use experiential playground where students are asked to try out for example negative values or non-integer values or values larger than the maximum used in the colour scheme and then observe what the software does to these inputs. The software automatically adjusts an inappropriate input in a default manner. If a negative value is typed, the value is changed to the absolute value. If a number larger than 255 is typed, it is automatically changed to 255. If a non-integer is typed, it is changed to the last digit that was typed.

Connecting to the GD and the CIs: This activity allows students to engage with the idea of the range of an image function practically. At a later stage, students must also think about the domain by realising that the image only has a finite number of rows and columns. The concept of the domain of a function is thus a quite practical ‘thing’ with respect to image functions. This activity connects to the GD mechanism of interiorisation to assist in going from an action understanding to a process understanding. Furthermore, it connects to the CIs by attempting to broaden the student’s understanding of the definition of a function, by emphasising the necessity of the domain and range. This is necessary for a process understanding of the function concept.

Description: Students are brought back to the image function formed in Activity 1, but now both the rows and columns are depicted by positive integers as in Figure 3.

Reflective questions are used in this activity to assist students in constructing the function concept as a process:

After a student completes these questions, a description is given comparing the point-by-point, successive way in which they shaded each individual block to what will happen inside a camera. Inside the camera all the ‘blocks’ get shaded simultaneously. The function is then a process of taking the entire domain at once and filling it with the range. Students are again led to think about the specifics of the domain and the range:

They are also led to think about the uniqueness property and the aspects of being injective and surjective:

Finally, thinking about the inverse process is also introduced here, for example questions such as:

Connecting to the GD and the CIs: As described in the CIs, at the process level a student should be able to mentally construct the function as the complete transformation of the domain to the range. This idea is captured in how the camera captures all of the pixels’ values simultaneously. This activity also explicitly deals with the aspects of the reversal of the function process and the existence of an inverse and confronts students with a fuller grasp of the definition of a function while at the same time having no formula. Having a function without a formula was also addressed in Activity 1, but as it is one of the common conceptual difficulties associated with the function concept (Dubinsky & Wilson, 2013, p. 85), it is valuable to address it again.

Description: With this activity, the aim is to assist students in constructing the function concept as an object. The story in which the intervention is set is brought to a peak here. The student sees how the knowledge of image functions, together with simple contrast stretching, is used to enhance the photograph discussed at the start of the intervention. It is enhanced to such a degree that sufficient information can be gathered from the image to assist in identifying the place where the photograph was taken. The contrast stretching seen in this activity is achieved through function composition.

As an introduction, the activity lets the student explore contrast stretching with pen and paper. The student is asked to draw the 5 × 4 grid as pictured in Figure 5a.

The number of vertical lines, v, in each cell (pixel) can be considered the colour of that cell. Figure 5a is then an image function, say f(x,y). The student is then asked to draw a second 5 × 4 empty grid and then fill in its values by applying the function g(v) = 4v – 4 on the original image of Figure 5a. A new image function h(x,y) is thus created through function composition. We obtain Figure 5b through the function composition h(x,y)=g(f(x,y)). For example, h(1,2)=g(f(1,2))=g(2)=4(2)−4=4. This composition stretches the contrast to such an extent that we can now see the number 5 or the letter S present in the image. The 5 (or S) was of course already present in the original image, but it was difficult to distinguish it from its background. It was difficult to distinguish due to the low contrast of the original image.

Once the introductory contrast stretching is completed, the student is brought back to the photograph associated with the missing person’s case. This photograph is seen here in Figure 6a.

Students are initially asked to describe in their own words what is wrong with the photograph, or consequently the (image) function in Figure 6a. How can a function be ‘wrong’, or for that matter be described? As an object, the function acquires global properties such as having low contrast, therefore a small average difference between adjacent pixels across its domain.

This activity lets a student realise that a low-contrast image can be improved by regarding it as one single thing – an object – that can be transformed by another function. Function composition is used to transform the original image function, f(x,y), into a new and improved image function g(f(x,y)).

newobject=g(oldobject)

Connecting to the GD and the CIs: If we carefully design the transformation function, g, we can obtain the desired results and again describe the new function as a whole. A function then becomes a noun and a noun can be described by adjectives. We might say that the new function is beautiful, it is clear, it has high dynamic range, it is smooth or, as was planned for the image of the intervention, it has improved contrast as can be seen in Figure 6b. This ability of seeing the function as a whole and not as something that you do, but rather something that can be acted on, is described in the GD at the object level. The function composition used in this activity fulfils the role of a process acting on the function process. This is described as encapsulation in the GD and is also a requirement in the CIs. Through acting on the function as object and reflecting on properties of the function such as contrast or brightness, a student could start to realise that image functions are a type of function, similar to how exponential or trigonometric or linear functions are different types of functions. This connects to the CI of being able to think about and convey the global properties of a type of function. Furthermore, a student can be led to grasp that different photographs of for example the same scene or person form a set of image functions. Mentally constructing a set with functions as its elements is a further indicator from Table 2 of being at the object level of conceptualisation with respect to the function concept.

The IFI follows the APOS general trajectory in the sense that information and in particular the activities are ordered to first let students retouch on the function concept as an action, then move on to a process and then finally to the function as an object. However, this trajectory is not the only possible path through the levels. As was discussed in the theoretical framework, in general the learning can pass through the APOS stages in a non-linear way (Arnon et al., 2014). In the discussion of the four activities before, it was already indicated how the activities aim to guide the student to construct the desired mental constructs that emerged in the GD and link to the CIs indicators of Table 2. To expand on the previous evaluation, we now track the GD while taking into account the IFI’s underlying path through the APOS stages: action, process and finally object.

In keeping to the second and third design principles, the IFI has to form a basin wherein the conceptual difficulties and other required concept aspects associated with the function concept can be explored in a familiar context. In the IFI, that context is created using photographs. To see if the IFI is compliant, we will look in turn at the most commonly occurring function concept difficulties (Dubinsky & Wilson, 2013, pp. 85–86) and other function aspects:

The IFI was implemented in a classroom setting with a group of 27 students in a first-year Calculus course. This is not the intended method by which the intervention will be implemented, as it was designed to be a self-directed mini module where a participant follows their own pace and can actively engage with the various activities of the intervention. However, to gauge the initial reaction of participants to the material and activities, it was decided that a classroom setting, together with a questionnaire at the end, would be sufficient and still enable us to achieve the objective of a proof of principle.

This group was chosen for convenience but fulfilled the minimum criteria of having prior knowledge of the function concept. From casual observation, the group had male and female members and these members were from at least three different ethnic backgrounds.

The questionnaire was handed out to all 27 participants and it was made clear that participation was voluntary and would be anonymous. No personally identifying information was asked, as this was not deemed necessary for a proof of principle endeavour, thus simply testing if the IFI could be viable. The questionnaire consisted of three grammatically closed, but conceptually open, questions (Worley, 2015, p. 19) as this still allowed any elaboration the participant might wish to provide.

The method of content analysis with emergent coding was used (Maree, 2016, p. 111). The responses to the three questions were searched for any indications relating to the objective of the questionnaire. After reading the responses a few times, four themes were identified:

It was not the intention of the questionnaire to deliver a generalisable result. The questionnaire was part of the effort to establish a proof of principle. Proof of principle can be interpreted as looking for proof that the IFI can have value, at least in some settings with some participants. To increase the credibility of the qualitative data analysis, colleagues were asked to independently verify themes, occurrences and results.

In the data analysis section, four themes were discussed that emerged from repeated reading of the participants’ responses. The responses were then analysed individually within the structure provided by the four themes.

This was done while keeping in mind that we were looking for indications that the participant experienced a broadened understanding of the function concept. In the analysis to follow, these three questions (Q) were asked: