The concept of a function appears in many disciplines, especially within mathematics. In the Victorian educational context, students are typically introduced, albeit in a basic abstract manner, to the concept of mathematical functions in Level 3 (Victorian Curriculum and Assessment Authority, 2018b, p. 43), but it is not until Level 7 that the use of pronumerals in functions is introduced (p. 63). Transformations of functions, whilst somewhat introduced in Level 9, is not fully covered until Mathematical Methods Units 3&4, a final year elective subject! In this paper it will be argued, based upon the research literature and the author’s experience as a mathematician and teacher, that a fundamental change is required to the manner in which functions and transformation of functions are taught in the Victorian curriculum.
Arguments about what should and should not be taught to students, as well as arguments about the methods employed in teaching students, have raged since antiquity. From the ancients such as Socrates, Plato and Aristotle through to more modern writers such as Skinner, Chomsky, Piaget, Vygotsky, Ausubel and Novak, the pedagogical arguments abound. The behaviourist models of learning, pervasive through the 1950s and 1960s, argued that “learning could only occur if one was presented with the right stimulus” (Ovens, 2017), reducing the teacher to nothing more than a “transmitter of information”. Educational practice is still somewhat corrupted by the behaviourist models with the prevalence of didactic classes, especially in the tertiary sector, wherein the students are simply presented with information, often via nefarious PowerPoint slideshows.
Modern Learning Theories
Piaget’s work with young children (e.g. Piaget (1930, 1972)) and similarly the work of Vygotsky (e.g. Vygotsky (1935)) suggested that children “were not simply empty vessels into which a teacher “poured” knowledge, as espoused by the behaviourist models” (Ovens, 2017). Translations of their works enabled the English-speaking world to recognise their contributions and also to stimulate strong questioning of the behaviourist models of learning. Educational practitioners and researchers began talking about how the learner constructs “their own meaning from the information around them, often alongside, and sometimes in conflict with, their teacher-guided instruction, giving rise to the term of constructivism” (Ovens, 2017).
The new constructivist paradigm of learning quickly became popular, to the point where opposing constructivism was tantamount to a form of heresy, with those in opposition often treated as if they had some kind of educational leprosy. Simpson (2002), however, reminds us that constructivism is not “an instructional methodology; constructivism is an epistemology, a philosophical explanation about the nature of knowledge” (p. 347). Although popular, constructivism is not without its critics (Pegues, 2007).
The constructivist idea that a learner creates their own rules for how things work based upon surrounding information led some authors to describe certain teaching practices as “scaffolding” designed to assist the learner to arrive at the correct rules (e.g. Bhattacharjee (2015); Garfield and Ben-Zvi (2007); Jones and Brader-Araje (2002); Karagiorgi and Symeou (2005); Kelly (2008); Taber (2011); Wild and Pfannkuch (1999)). One of the key concepts of this technique of scaffolding is that it should “help students perform just beyond the limits of their ability” (Bhattacharjee, 2015, p. 69), a noble and worthy goal which, it could be said, is the main objective of all educational practice.
Mathematics is an ancient discipline, obsessed with patterns and a cornerstone of the sciences, replete with formulae and rules. Mathematics, therefore, might seem an ideal environment in which to apply a constructivist paradigm to the learning and teaching of the discipline. It could be argued that mathematics has an inherent constructivist nature given that we mathematicians try to create rules from patterns we observe and then update these rules to include new observations that contradict the existing rule. Expecting students, however, to observe and somewhat independently rediscover the established rules of mathematics would be onerous on the poor learner to say the least. This is especially evident if we consider that Euclid’s Thirteen Books of the Elements, which are themselves over 2500 years old, whilst extensive, only cover the basis of three-dimensional geometry. Similarly, if we consider the modern masterpiece Calculus by Spivak (1994) we can see that the prologue discusses twelve properties of rational numbers (p. 9), but covers some significantly complex prerequisite mathematical theories before introducing, in chapter 8, a thirteenth property that gives us the set of Real numbers (p. 133). It would be insane to suggest that learners would need to rediscover all of this knowledge for themselves, even if guided by a teacher. Instead, as is current practice, learners should be presented with the basic rules and guided towards developing their mathematical proficiency.
Mathematical proficiency, to be clear, is not simply the ability of a student to recall formulae or utilise technology to obtain some desired result, rather it is the student’s conceptual understanding of mathematical techniques and their appropriate application; the difference between training and genuine education. “The first serious error we often meet in considering the role of mathematics is the confusion of education with training.” (Gullberg, 1997, p. xvii). Sullivan (2011) discusses, in Section 5, six “key principles for effective teaching of mathematics”, with principle six perhaps being most important in the context of mathematics education. Principle six states:
“Fluency is important, and it can be developed in two ways: by short everyday practice of mental processes; and by practice, reinforcement and prompting transfer of learnt skills.” (p. 29).
Unfortunately, the predominant implementation of this principle, judging by modern textbooks, appears to be, in this author’s opinion, that students are not required to attain proficiency, but rather have their time occupied by repetitive exercises. Furthermore, it appears that, as Gullberg eloquently describes, “Students, and their parents, believe that mathematics education should consist exclusively of the acquisition of a set of skills that will prove useful in their later careers; so the skills must be learned, that is, committed to memory, and no real understanding need occur.” (Gullberg, 1997, p. xvii).
There have been two key themes that recur across many articles about the teaching of mathematics at both primary and secondary schooling levels. First is the theme of engagement, either as a lament of how current practices are not engaging students in their mathematical studies, or, as an exhortation to teachers to modify their practices to engage or reengage their students. Second is the theme of a need for institutional change to address the finding that “Our institutions of formal education do not help most students to learn science with understanding.” (Anderson, 2007, p. 5). Mathematics education faces not only the challenge of trying to develop a learner’s conceptual understanding, but also the battle against the “perception that it is okay to be bad at mathematics [which] often stems first from the parents” (Erwin, 2015, p. 5). Combine that with the fact that “there is now a shortage of qualified science, mathematics and information and communications technology teachers, and the participation rates of Australian school and tertiary students in STEM disciplines remain a matter of concern” (Office of the Chief Scientist, 2014, p. 21) and it is clear that developing a high level of proficiency in mathematics in Victorian schools is an uphill battle. This battle, however, is one that must be fought and won.
What is in the Current Victorian Curricula?
The current Victoria curricula is prescribed across two documents, the first covering the compulsory components, listed as levels from Foundation to Level 10/10A (Victorian Curriculum and Assessment Authority, 2018b), where the level refers to an achievement standard rather than a grade/year level. The second document covers the components of the Victorian Certificate of Education (VCE), called “Study Designs”, that are typically undertaken in the final two years of post-compulsory secondary schooling. These study designs include details on the unit content, including the expected outcomes, divided up as key skills and key knowledge, and the structure and conduct of assessments. This second extensive tome is broken up into the various disciplines and subjects with the mathematical units detailed in a separate ninety page document (Victorian Curriculum and Assessment Authority, 2018a).
The Foundation to Level 10/10A curricula (Victorian Curriculum and Assessment Authority, 2018b) first introduces the idea of a function to students at Level 3 (p. 43), then to the Cartesian coordinate system at Level 6 (p. 60) and then, finally, the concept of using a rule with pronumerals to plot a graph at Level 7 (pp. 65-66). It should be noted, however, that there is a strong emphasis on the use of technology to help students understand the concept of a function and the effect of simple transformations, generally with only linear functions. Level 8 then continues the work on linear functions and simple non-linear functions, again with an emphasis on plotting graphs (p. 72). Level 9 lists a category for “Linear and non-linear relationships” (p. 77), which one might argue is considerably overdue by this stage of the document. Again, there is strong emphasis on linear functions and the use of technology with consideration of only “simple non-linear relations”. Level 10, the final level of compulsory secondary mathematics, marks the first point in which students begin to “explore” the effects of transformations, under item VCMNA339 (p. 83), but again only on simple non-linear functions.
Level 10A, an optional level providing “additional content for students to be extended in their mathematical studies” (p. 87), lists the following additional items (p. 88):
- “Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (VCMNA359)”
- “Solve simple exponential equations (VCMNA360)”
- “Apply understanding of polynomials to sketch a range of curves and describe the features of these curves from their equation (VCMNA361)”
- “Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts (VCMNA362)”
- “Use function notation to describe the relationship between dependent and independent variables in modelling contexts (VCMNA363)”
- “Solve simultaneous equations using systematic guess-check-and-refine with digital technology (VCMNA364)”
It should be noted that the students, up to this point, have not yet investigated much beyond simple non-linear functions and they may have only just been introduced to surds and logarithms.
Turning to the curricula for the elective mathematical units undertaken by VCE students (Victorian Curriculum and Assessment Authority, 2018a), typically during their final two years of post-compulsory secondary education, seven subject pairs of semester-long units are described (p. 7). Of most interest to this paper, naturally, are the units involving functions, namely those of Mathematical Methods and Specialist Mathematics. Although both General Mathematics and Further Mathematics include consideration of linear equations and solutions of simultaneous linear equations, neither of these units, regrettably in this author’s opinion, extend the students to substantive work on non-linear equations. Mathematical Methods and Specialist Mathematics both include detailed consideration of functions and the calculus, with Mathematical Methods Units 3 and 4 being a co-requisite of Specialist Mathematics Units 3 and 4. Mathematical Methods is broken up into four “Areas of Study”, namely “Functions and Graphs”, “Algebra”, “Calculus” and “Probability and Statistics”. In Unit 1, these areas include the following relevant dot points (pp. 31-32):
- “graphs of power functions for and , and transformations of these graphs to the form where and ”
- “graphs of inverse functions”
- “transformations of the plane and application to basic functions and relations by simple combinations of dilations (students should be familiar with both ‘parallel to an axis’ and ‘from an axis’ descriptions), reflections in an axis and translations, including the use of matrices for transformations”
The learning outcomes for this unit include (pp. 33-36):
- “the definition of a function, the concepts of domain, co-domain and range, notation for specification of the domain (including the concept of maximal, natural or implied domain), co-domain and range and rule of a function
- “the effect of transformations of the plane, dilation, reflection in axes, translation and simple combinations of these transformations, on the graphs of linear and power functions”
- “the matrix representation of points and transformations”
- “sketch by hand graphs of linear, quadratic and cubic polynomial functions, and quartic polynomial functions in factored form (approximate location of stationary points only for cubic and quartic functions), including cases where an x-axis intercept is a touch point or a stationary point of inflection”
- “sketch by hand graphs of power functions where and and simple transformations of these, and graphs of circles”
- “describe the effect of transformations on the graphs of relations and functions and apply matrix transformations, by hand in simple cases”
- “the role of parameters in specifying general forms of functions and equations”
- “the relation between numerical, graphical and symbolic forms of information about functions and equations and the corresponding features of those functions and equations”
Mathematical Methods Unit 2 adds circular functions, exponential and logarithmic functions to the list of functions for students to learn (p. 37). Mathematical Methods Units 3&4 then review most of these functions and finally introduce students, albeit with different and arguably confusing symbology, to the expression for the transformation of functions, . Specialist Mathematics assumes that students are already familiar with functions and their transformations through their studies of Mathematical Methods and adds ellipses, non-rectangular hyperbolae and reciprocal circular functions to their repertoire.
As can be seen in the VCE study designs, the students of Mathematical Methods are expected to learn and focus on the areas of visualising and sketching functions of the form where is one of the following:
- Power functions, .
- Circle functions, ,
- Exponential functions, , esp.
- Logarithmic functions, , esp. and
- Circular functions, , and
Along with transformations of the above functions as , where the parameters . Note that the current study design removed the Modulus functions, , which can be similarly transformed.
How do Transformations of Functions Work?
Transformations of functions, regardless of whatever the base function may be, work exactly the same. Transformations are made up of a combination of three operations (translations, dilations and reflections) in one or both of the dimensions (horizontal and vertical) of the Cartesian plane (). Translations shift the position of the graph, dilations stretch or squash the graph, whilst reflections rotate the graph about its centre either horizontally or vertically. These transformations can be applied in any sequence, however, in the absence of an explicit script, operational precedence prevails.
A horizontal translation of units in the positive -direction simply means that we “pick up” the function and move it units to the right; it does not change the type of function. Similarly, a vertical translation of units in the positive -direction simply means that we, again, “pick up” the function and move it units up without changing the type of function. A horizontal dilation of factor stretches or squashes the graph in the -axis, again without changing the type of function. If the value of is negative, then it also represents a horizontal reflection, i.e. it rotates the graph horizontally about its centre without changing the type of function. A vertical dilation of factor stretches or squashes the graph in the -axis, again without changing the type of function. If the value of is negative, then it also represents a vertical reflection, i.e. it rotates the graph vertically about its centre without changing the type of function.
It is important, in this author’s opinion, that students are introduced to transformations using these parameters, with this exact symbology, to allow linkages to be created between quadratics in turning point form () and all other functions.
Criticism of the Current Curricula
The first major criticism one has about the current treatment of the teaching of mathematical functions and transformations is the lack of focus on development of the conceptual understanding without technology. Technology can be useful to help graph a function, but when the curricula places strong emphasis on graphing via technology, it is only a matter of time before the default practice is to train students how to graph via technology instead of focusing on developing the students’ understanding. The focus on a technology first approach robs students of the opportunity to develop their skills in visualising functions from their equations.
Similarly, the focus in the learning outcomes on “the matrix representation of points and transformations” without an outcome that requires the same without matrices encourages students to erroneously believe that the order of transformations is always dilations, followed by reflections and then translations. Taking the transformation of described by the matrix equation , operational precedence tells us that the matrix multiplication () occurs first followed by the addition, where matrix represents the dilations and reflections and matrix represents the translations. Alternatively, if the transformation of the function, , is given as , then operational precedence tells us that first one must find , then this result is multiplied by before the function is evaluated at this point, then multiplied by and then finally is added. In terms of what these parameters represent, this means that first the horizontal translation is applied, then the horizontal dilation/reflection, then the vertical dilation/reflection and then finally the vertical translation. As such, it is important that students are aware of the differing methods of transformation and the differing orders of transformations.
Thirdly, this author would argue that a vast amount of content is unnecessarily delayed from being introduced to students. The concepts of pronumerals, Pythagora’s Theorem, Logarithms and Indices and Circular functions were all a part of this author’s primary school education and yet the present curricula, for example, delays introducing students to logarithms in their mathematics education until around three years after they have been used in general science, such as in the PH scale or the Richter Scale.
Current Practice in Teaching Mathematical Functions and Transformations
Judging current classroom practice is difficult without direct observation, however, current popular textbooks can be used as a quasi-measure of how mathematical functions and transformations are taught in Victorian classrooms. The textbooks from three publishers have therefore been selected to act as this quasi-measure of classroom practices, based upon their popularity in the Victorian education context, namely Cambridge University Press, Jacaranda (John Wiley & Sons Australia, Ltd.) and Nelson (Cengage Learning Australia Pty Ltd.). Only the VCE series for Nelson will be included as their texts tend not to be used at lower levels as much as the Cambridge Essential series and Jacaranda MathQuest series.
The presentation of functions in all three texts follows a similar form in that each function is presented with little to no linkage to previous functions. The presentation of the transformation equation, , differs across the three texts. The Nelson text presents the “Combination of transformations” as being of the form (Garner, Neal, Dimitriadis, Kouris, & Swift, 2016, p. 61). The Cambridge text follows the study design and presents the transformation as (Evans, Greenwood, Lipson, & Jones, 2015, p. 122). The Jacaranda text, on the other hand, presents transformations more akin to how this author would argue they should be presented, i.e. (Swale, Michell, Morris, & Rozen, 2016, p. 142). Unfortunately, all three texts suffer from the same problem of not consistently returning to and reinforcing the concepts of transformations when working with functions later in the text.
Criticism of the Current Popular Textbooks
The lack of consistency in the way functions are presented across textbooks and especially within textbooks is another major problem. For example, in the Cambridge text for Mathematical Methods Units 3&4 (Evans et al., 2015) presents the general transformation function, in the manner described in the VCAA study design (p. 122) albeit with the description in violation of the operational precedence of mathematics, but then presents, for example, circular functions with differing symbology, i.e. . It is no wonder then that students seem to perceive that each function requires different mathematical rules to be learnt.
Quadratics in turning point form, , initially demonstrate to students how transformations work, excluding the horizontal dilations and reflections. It makes perfect sense, therefore, to scaffold upon this and therefore present to students all of the other functions (mostly Power functions) that follow the same pattern prior to introducing the concepts of the horizontal dilation and reflection with the functions, such as circular functions, that require this parameter. Alternate forms of various functions, for example cubics in factorised form such as those given by , should then be introduced to students as exceptions to the rule, rather than trying to introduce multiple forms of each function simultaneously.
Suggested Presentation of Functions and Transformations
This author would suggest that the ideal way to present functions and their transformations to students would be to start with the presentation of Quadratics in turning point form, preferably in middle to late primary or at worst in very early secondary, followed immediately by the presentation of the inverse function, i.e. the square root function. Next would be to introduce cubics in the form of and their inverse, i.e. the cube root function. Following this would be the introduction of odd and even power functions and their respective inverses. Once the general power function has been dealt with, the natural progression, in this author’s view, is to then present the general solution for all functions of the form (see Appendix A), followed by the introduction of other functions such as the circular functions.
Once students have mastered the concept of what this author refers to as the “grand-daddy equation” (i.e. ), students should be encouraged to find, algebraically, the rule for the inverse of such functions, i.e. (see Appendix A). Once students have achieved mastery of the transformation equation and the remainder of the functions required to be learnt under the current Victorian curricula, as demonstrated throughout Appendix A. It should be noted that most functions studied, even through to the tertiary level, follow the “grand-daddy equation” and so working with students on this formulation does not hinder them in achieving a level of proficiency required to reach a high level of understanding; in fact, this author would argue that it greatly assists students in the development of their understanding of how all functions work.
This paper sought to argue that a fundamental change to Victorian curricula is required, however, some readers may view this as more of a cathartic exercise for the author. Whilst there may be some catharsis in the writing of this paper, the main purpose, from this author’s perspective, is to highlight that there is a missed opportunity to increase student understanding and engagement under the current curricula. It is incumbent upon those of us charged with educating the next generation that we imbue our students with a true passion for our disciplines. Indeed, it is the next generation that will be responsible for managing the society within we will hope to enjoy our retirement; we should therefore ensure that students within our care are given all possibility to attain a high level of mastery of our discipline.
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Last updated: 15 March 2019