The role of visual thinking in the teaching and learning of mathematics has been acknowledged. Vinner (1991) describes the conflict between formal definitions and concept images while Dubinsky and Tall (1991) and Sfard (1991) researched visual imagery, all of which fundamentally led to visualisation as a construct. More recently, the debate has shifted to the role of visual representations (Nardi, 2014) in mathematical understanding. There is however consensus about more diverse roles of visualisation in the teaching and learning of mathematics yet to be endorsed, such as discovery and justification.
73 The exposition of Vinner (1991) that a deep understanding of a concept relies not only on the formal definition but also the formation of a concept image, was the stimulus for recent research on the integration of visualisation at university level. Kupferman (2014) used a visually rich approach to help first year students understand the definition of the limit concept. However, findings from this study indicate that students cannot reap the benefits of visualisation when they are not yet accustomed to it. Kupferman (2014:196) ascribes this to the “increasing gap between high school and university levels”. A disconnect between school and university mathematics is also acknowledged further afield. In the US, Quinn (2012:37, cited in Eisenberg, 2014:43) recognises that
school mathematics is still firmly located in the nineteenth century, so student success rates in modern [university level] courses has been very low. There is a great deal of pressure to improve this situation, but recent changes, such as use of calculators and emphasis on vague understanding over skills, have actually worsened the disconnect. Something has to change”.
However, “it is not enough to complain”, mathematics educators and researchers “have a moral obligation to do something about it” (Eisenberg, 2014:39). To address the apparent disconnect between school and university mathematics, it is necessary to take a step back and briefly review the status of visualisation with CAS in the teaching and learning in schools.
There are indeed copious accounts of the successful implementation of CAS in schools (i.e. Drijvers, 2003; 2015) that analyse the detailed techniques used by students and the development of their mental schemes. Soon after graphing calculators were introduced into classrooms, it became evident that students struggled to interpret graphical representations. On the one hand, Trouche (2005:141) realises that even the frequent use of software is no guarantee that students can “efficiently articulate”
between algebraic and graphic representations. On the other hand, Trouche (2005) has no illusions about the potentialities of graphic calculators regarding visualisation.
As a consequence, this “growing awareness of the constraints and the potentials of the CAS tool” (Drijvers, 2000:191) plays an important role in the dawn of a new theory.
In the early 2000’s, the theory of instrumental genesis was developed to accommodate the interplay between digital technology, paper-and-pen techniques and the
74 emergence of mental schemes (Artique, 2002). This framework has been well received for its guiding principles to design tasks and to analyse students’ conceptual understanding. Taking advantage of two philosophies, Drijvers (2000; 2015) mixes the instrumental genesis framework with the theory of Realistic Mathematics Education (RME), for the latter, see Chapter 3. From the design and task analysis point of view, Drijvers (2000; 2015:141) regards these two theories to be “complimentary” since RME can indicate certain caveats of CAS in the learning process.
The next decade witnessed considerable growth in educational practices in favour of CAS in school mathematics. In Australia, Goos and Bennison (2008) report that most schools in the state of Queensland are using graphic calculators, while computer algebra systems (CAS) software is available at 18% of schools. According to responses from teachers, some of the advantages of classroom technology are that it 1) facilitates real life applications, 2) assists with understanding of concepts, 3) reveals links between algebraic, numeric and graphic representations, 4) allows the study of
“sophisticated concepts”, 5) enables exploration of novel problems, 6) offers a
“dynamic feedback” system and 7) influences students’ perceptions and attitudes towards mathematics (Goos & Benison, 2008:115). Teacher concerns relate mostly to logistical problems; these include the unavailability of hard and software, the effort to learn new software and the lack of training and professional development opportunities. Between 90% - 95% of Grade 11 and Grade 12 students who will pursue mathematics at university level are learning with graphic calculators.
In contrast to school implementation, the commitment to CAS in higher education is sluggish. Caveats regarding visualisation in teaching and learning at university level are noted by several scholars. Since visualisation is not part and parcel of many students’ mathematical custom, they may not necessarily benefit from a visual approach (Nardi, 2014). In fact, “visualization is also a form of language that needs to be learned before one can take advantage of its strength” (Kupferman, 2014:198). As a language, visualisation not only involves the development of connections between different representations, but also needs to be expressed verbally (Hershkowitz, 2014).
The need for a visual language is aptly demonstrated with Hershkowitz’s (2014:200) example of the pattern concept. Figure 2.5 illustrates a pattern in which the building blocks are different geometrical forms, sizes, colours and spacing. These building
75 blocks collectively serve as a visual alphabet from which a visual word or visual sentence can be constructed. The repetition of the visual sequence creates a periodic pattern; both the thinking and the language are visual, without which the concept cannot be fully grasped. Visual thinking and visual language are intertwined: “to be expressed, visual thinking needs a language, visual or other; and visual language, to be meaningful, needs to be attached to some conceptual entity” (Nardi, 2014:193).
Figure 2.5: Visualisation as a language; the visual alphabet of a pattern (Hershkowitz, 2014:200).
There is an obligation to teach visualisation (Presmeg, 2014), especially when learning is situated in a computerised environment. In her summary on the current research trends vis-á-vis visualisation in mathematics education, Presmeg (2006) identifies curriculum development and appropriate teaching methods in dynamic computer environments to be prominent. Challenges include the need to 1) prioritise technological affordances that impact visualisation and the understanding of objects and actions; 2) acknowledge the impending “epistemological change” (Yerushalmy, 2014:205) inherent in the design of visually sensitive curricula and 3) implement curricular changes sanctioned by “in-depth” research findings and recommendations (ibid.).
In support, Nardi (2014:209) calls for a “new didactical contract” to unambiguously clarify the role of visualisation to students. Students get accustomed to rely on formally established facts through proofs, axioms and lemmas. Therefore, they are often unfamiliar with the legitimacy of visual argumentation and hesitant to depend thereon.
When facts were previously rejected if not formally established, it may be a difficult tradition to modify and could set a precedent for skepticism towards the validity of visual reasoning. Often, visual images lack proper explanations (i.e. in textbooks) or may seem detached from the accompanying descriptions which can be equally perplexing.
Nardi (2014) envisages a didactical contract that permits students the use of facts not Periodic
76 formally proven. These facts may be acquired from articulated accounts of students’
thinking, writing and other support bases such as CAS. To facilitate this contract, students need to be empowered with the efficacy of visualisation, realised by fostering
“a fluent interplay between analytical rigour and often visually based intuitive insight”
(ibid.:209).
Another concern is detected in research (i.e. Suwarsono, 1982, cited in Clements, 2014:186) indicating that certain students who excel in paper-and-pen visualisation tests are indeed reluctant to use “visual methods” to perform tasks. Eisenberg and Dreyfus (1991:25, cited in Clements, 2014:188) concur that students are “reluctant to use visual methods, preferring algorithmic over visual thinking”. More so, students who regard themselves as “visual thinkers” are often “weaker” performers (ibid: 186 & 187.).
Then again, Eisenberg (1994:110, cited in Clements, 2014:187) asserts that “a vast majority of students do not like thinking in terms of pictures”. This tug of war between analytical and visual thinking is not a new phenomenon. Eisenberg (1991:146) acknowledges that although it seems “natural” to view many mathematical concepts in a visual form, students are “tied to processing information and solving exercises analytically, not visually”. Indeed, this dichotomy has been “debated for hundreds of years, with the visual image eventually losing out” (ibid.)
The debate is taken up by Presmeg (2014) who questions the aptness of judging visualisation according to criteria applicable to conventional analytical approaches. Of concern is that visualisation is often regarded as an approach that students need to rather avoid on arrival at university. Such a disregard of a visual approach can brand visualisation as being “additional” or inferior to the favoured symbolic approach (ibid.:
216). Stemming from these challenges, Presmeg (2014:216) suggests that “one approach is to allow the curriculum to explore visualization for its own sake … [and]
acknowledge that it has a logic of itself”.
The importance of a learning environment as recorded by Bishop (1988) seems to sit well with Nardi’s (2014) envisaged didactical contract. Bishop (1988) narrates how many Jamaican children have few toys, yet two boys from a rural background whose fathers were respectively a mechanic and a stone mason, performed outstandingly on symmetry tasks. Bishop (1988:174) deduces that this is “evidence that a learning
77 environment in which structured and manipulative materials predominate can help to encourage the creation of visualisations and thus the visualisation process itself”. The role of the social environment in the form of a parent is as important as the availability of the mechanical and masonry materials used. More so, certain interpretation skills involving “figural information are trainable, given the appropriate experiences"
(ibid.:174).
Cognisant of challenges outlined in this section, the next section considers a feasible pedagogical map to empower visualisation in a CAS environment.
2.4.2 Pedagogical map for learning with CAS
Pierce and Stacey (2010) engineered a map of learning opportunities in a CAS environment as illustrated in Figure 2.6. These authors categorise ten pedagogical opportunities in three levels: the type of tasks (bottom row); the type of classroom intervention (centre) and the goals (top row) of these learning opportunities. A brief description is given of each of the ten opportunities and illustrated with examples.
Figure 2.6: Pedagogical map for learning with CAS adapted from Pierce and Stacey (2010:6).
78 2.4.2.1 Task level opportunities
The category in the bottom row of Figure 2.6 describes five diverse learning opportunities afforded by CAS tasks.
a) Paper-and-pen skills
Pierce and Stacey (2010) suggest using CAS tasks to develop paper-and-pen skills.
When solving equations with paper-and-pen, these authors propose that instant CAS answers can be used as cross-referenced feedback. For instance, to solve the quadratic equation x2b x Mathematica’s built-in ‘Solve’ command can reveal 1 0, the interim solution in surd form as x12
b b24
and x12
b b24
. Thiscan support the paper-and-pen obtained answer once the quadratic formula is applied.
With this symbolic, provisional Mathematica solution as backup, students can track side-by-side the correctness of their algebraic manipulations done with paper-and-pen.
Once a value for the parameter b is selected, Mathematica can be used to calculate the numeric solution
x 3.30
andx 0.30
which can in turn, be validated with a pocket calculator. This example illustrates how tasks that combine paper-and-pen work with CAS activities can offer learning opportunities to 1) develop algebraic manipulation as well as programming skills and 2) provide step-by-step feedback in new and/or complex situations (Pierce & Stacey, 2010).b) Use real life data
The use of real life data is the second initiative offered by Pierce and Stacey (2010) in designing tasks with CAS. The arrival of CAS in classrooms and lecture halls has sparked opportunities to explore real world data, notoriously regarded as “messy”
(ibid.:15). More often than not, classroom activities are carefully simulated by lecturers to suit the skills level of students and to teach specific concepts. Pierce and Stacey (2010) refer to such contexts as being cleaned-up. Paradoxically, real life data do not behave within neat pockets of isolated knowledge and skills. Literature is liberally sprinkled with examples of tasks exploring real life data (i.e. Ang, 2010; Klymchuk et al., 2008; Trouche, Drijvers, Gueudet & Sacristán, 2013). Real life contexts stimulate students’ interest in mathematics when they perceive the discipline to be relevant in solving real world problems (Pierce & Stacey, 2010; Zbiek & Conner, 2006).
79 c) Explore regularity and variation
Pierce and Stacey (2010) regard CAS tasks to be advantageous for exploring the shape and behaviour of functions when parameters are varied. To illustrate, consider the quadratic polynomial yx23x Any changes made in the parameter c will c. affect the graphical behaviour of the function. Since the parameter c is associated with the vertical intercept, a change in the value of c will affect a vertical shift of the graph.
Figure 2.7 illustrates Mathematica graphs that show how variations in the parameter c will affect the behaviour of the parabola yx23x Based on visual evidence, c. conjectures can be made by observing the regularity in functional behaviour as a result of variation in parameters (Pierce & Stacey, 2010).
Figure 2.7: A family of functions obtained for the parabola yx23x through c variations in the parameter c.
The dynamic affordances of CAS allow students to experiment with different parameters and instantly observe the effect thereof on the properties of a function or families of functions (Berger, 2012).
80 d) Simulate real situations
Dynamic software allows the user to create an interface that sets a model of a real world problem in motion, this is referred to as a simulation. Geometric drawings can be created that represent properties of the “real motion of a mechanism” (González-López, 2001:127). Such simulations can be analysed in terms of their functional behaviours which in turn, can provide an understanding of the motion of the model in space and time but fall beyond the scope of this study.
e) Link representations
A CAS environment offers the user a symbolic, numeric and graphic interface. In addition, a real world problem is often represented in the form of a narrative or as a
“word problem” (Verschaffel, Van Dooren, Greer & Mukhopadhyay, 2010:9). To illustrate the multiplicity of representations, a differential equation can be represented symbolically or narratively, whereas its solution can be represented symbolically, numerically and graphically. These multiple presentations are illustrated in Figure 2.8.
Figure 2.8: Multiple representations of a differential equation and its solution.
Symbolic differential equation
Symbolic solution Numeric solution
Narrative
A population has a constant rate of change dN/dt. If the number of individuals in the population is one at time t=0 and increases to six individuals after five days, find the number of individuals after three days.
1
t 0 2 3 4 6 N 1 3 4 5 7
Graphic solution
81 In a CAS environment, students need to develop a “representational fluency” in order to acquire deep understanding of a concept (Pierce and Stacey, 2010:8). It is essential to form links where different representations support and complement each other. In the same vein, Gagatsis and Nardi (2016) regard representational fluency as the flexibility to recognise, operate on and transform representations of the same mathematical concept. An alternative view is that of “representational versatility”
(Thomas, 2008:12) that requires not only a fluency in translating between representations, but also the “ability to interact procedurally and conceptually with individual representations”. By way of example, a typical interaction in Figure 2.8 would be for students to verify N(5)6 by means of the symbolic, numerical and graphical representations.
2.4.2.2 Classroom level opportunities
The category in the middle row of Figure 2.6 describes two opportunities afforded by CAS learning within the classroom setting.
a) Classroom social dynamics
Opportunities to overturn traditional classrooms dynamics, where lecturers talk and students listen, should involve group work (Pierce & Stacey, 2010). This is in contrast with the tradition whereby students mostly work on their own and in silence.
b) Classroom didactical contract
Brousseau (1984, cited in Balacheff, 1990:260) introduces a didactical contract as
the rules of social interaction in the mathematics classroom [which] include such issues as the legitimacy of the problem, its connection with the current classroom activity, and the responsibilities of both the teacher and pupils with respect to what constitutes a solution or to what is true.
The nature of the didactic setting is therefore fundamental to the construction of knowledge and meaning-making. CAS effectuates an additional “authority” into the classroom which can empower students to have greater control of their own learning (Pierce & Stacey, 2010:9). Such a shift in power could elevate the role of students to that of co-researchers and mitigate the role of the lecturer to that of a mentor; thus
82
“resulting in a change to the didactic contract” (ibid.). This contract can engender deeper levels of understandings as students negotiate their own unique solutions, as opposed to memorising preconceived or ready-made solutions. With different strategies and personal explorations, students not only develop conceptual understandings, but also learn to master the software and programming principles of CAS. A prototypical case of how the teaching and learning with CAS brought about several modifications in the didactical contract of Austrian classrooms is reported by Heugl (1997). Due to the use of CAS in classrooms, changes as indicated in Table 2.3 can be witnessed.
Table 2.3: Changes in didactical contract due to CAS (Heugl, 1997:147).
From To
Doing Planning, interpreting
Reproductive learning Active, experimental learning Teacher-oriented Learner-oriented
Knowledge about calculations Knowledge about strategies
Exercises Problems (modelling,
operating, interpreting)
Ross et al. (2010:22) credit a CAS environment for shared opportunities in serving as a tutor, a teaching assistant and a learning tool. Such a three-way obligation augments and enhances the traditional custom whereby lecturers are the main providers or authorities of knowledge. Drijvers (2015) regards the didactical role of technology in classroom dynamics as firstly a tool for doing laborious calculations, thus outsourcing work that ordinarily was done with paper-and-pen and secondly, a learning environment. The learning environment facilitates the practicing of skills and nurtures the development of conceptual understanding (see Figure 2.9). While CAS may provide a welcome relief from long calculations and the rehearsal of skills, concept development “is the most challenging [process] to exploit” (Drijvers, 2015:137).
83 Figure 2.9: Didactical functions of technology in mathematics education (Drijvers, 2015:137).
The learning process in a technologically-rich environment is of “increasing complexity”
and a “non-trivial” matter (Drijvers & Trouche, 2008:365). Therefore, an adapted contract should also reflect on didactical goals that no longer focus on “recipes or unequivocal ways given by the teacher to the pupils” (Heugl, 1997:144). Indeed, the didactical contract, structured tasks, social environment and the learning goals are influential aspects that will contribute to the value of CAS in learning mathematics (Drijvers & Gravemeijer, 2005).
2.4.2.3 Goals
a) Exploit contrast of ideal and machine mathematics
The pedagogical map of Pierce and Stacey (2010:10) suggests that learning with CAS affords lecturers to “deliberately capitalise on the constraints, anomalies or limitations of technology”. When unexpected outputs are encountered, students are exposed to the rift between ‘textbook-like’ mathematics and machine-produced mathematics.
Often, students have to interpret error messages when algorithms or syntactical programming codes are breached. In turn, lecturers can seize such opportunities “as catalyst[s] for rich mathematical discussion[s]” (Pierce & Stacey, 2010:7). CAS provides an enriched environment for students to discover nuanced differences between familiar (ideal) mathematics and unanticipated (machine) mathematics. As a result of new opportunities by way of experiential learning, learning goals can be adjusted to focus more on non-stereotyped mathematical thinking to stimulate and promote different skills.
Didactical functions of technology in
Mathematics education
Do Mathematics
Learn Mathematics
Practical skills
Develop concepts
84 b) Emphasise skills and real world applications
As a result of reformed goals, Pierce and Stacey (2010) conclude that less time is spent on practicing routine skills, leaving more time to focus on concepts and related applications. As such, a greater emphasis can be placed on mathematical thinking and understanding of concepts. Heid, Thomas and Zbiek (2013:623) show how the use of CAS can elicit graphical reasoning skills as “an alternative to symbolic reasoning”. CAS emboldens opportunities for students to connect their graphical and symbolic actions, processes and products, stimulating heightened reasoning skills through investigation
As a result of reformed goals, Pierce and Stacey (2010) conclude that less time is spent on practicing routine skills, leaving more time to focus on concepts and related applications. As such, a greater emphasis can be placed on mathematical thinking and understanding of concepts. Heid, Thomas and Zbiek (2013:623) show how the use of CAS can elicit graphical reasoning skills as “an alternative to symbolic reasoning”. CAS emboldens opportunities for students to connect their graphical and symbolic actions, processes and products, stimulating heightened reasoning skills through investigation