Modelling the derivative in Sketchpad: an ...

Modelling the derivative in Sketchpad: an instrumental and TPACK approach Mdutshekelwa Ndlovu Stellenbosch University Centre for Pedagogy (SOUTH AFRICA) Abstract: In this paper I illustrate the representational capabilities of Sketchpad that have the potential to enhance a deeper understanding of the derivative concept in introductory calculus if appropriate learning trajectories are designed. Sketchpad is dynamic mathematics software with Trouche’s instrumental theory affordances that can support multiple representations of mathematical concepts. The proliferation of digital technologies, under which dynamic mathematics software falls, challenges mathematics educators and teacher educators to accelerate the integration of these new tools into the classroom. To this end I present a hypothesized learning trajectory of the derivative for the instrumental geneses of the derivative as an instantaneous rate of change and as a rate of change function. Six forms of representation of the derivative emerge as a potential part of the mathematics teacher’s Technological Pedagogical Content Knowledge (TPACK). A recommendation is made to vigorously equip and capacitate pre-service and in-service mathematics teachers or risk them becoming an impediment. KEY WORDS: Sketchpad, dynamic mathematics software, modelling, instrumental theory, derivative, TPACK,

Introduction The integration of new Information and Communication Technology (ICT) tools in the teaching of mathematics and science is actively encouraged worldwide, more so on the back of rapidly expanding digital technology penetration rates even in developing country contexts. The purpose of this paper is to demonstrate the affordances (enablements, potentialities and constraints) of dynamic mathematics software in representing the derivative concept in introductory calculus. The ideas of a model and modelling in mathematics education are examined first and the potentialities of Sketchpad dynamic geometry software in modelling the derivative are explored from an instrumental/documentational genesis perspective before finally locating them in the technological pedagogical content knowledge expected of teachers of mathematics in a technology rich classroom. Modelling in mathematics education In physics, mathematics, chemistry or other physical sciences, a model is a system consisting of elements, relationships among elements, operations that describe how the elements interact and patterns or rules (e.g. symmetry, commutativity, transitivity, etc) are formed (Lesh & Doerr, 2000, p. 362). A model in this sense has a structure made up of components and relationships that connect the component elements to represent a physical reality. In the case of the derivative, in mathematics, the graphical representation yields two conceptualisations or models of a) the derivative as an instantaneous rate of change and b) the derivative as a rate of change function. Each of these models or encapsulations has its own elements that interact to make a representative system. As the gradient at a point, the components include the graph of the functional relationship and a tangent drawn at a point as Figure 1 shows. In contrast to the paper-and-pencil environment, the tangent itself can be viewed as a subsystem constructed through plotting two points (say A and B) on the graph, joining them with a line and animating them to coincide with each other to produce the tangent line.

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f(x) = x2 4

Move B -> A Animate Tangent Line 3

Slope AB = 1.34 A: (0.67, 0.45)

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B: (0.67, 0.45)

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A B -4

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-2

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Figure 1: The instantaneous rate of change as tangent at a point As a rate of change function, the components can include the graph of the (quadratic) functional relationship as before, Point C, as the slope value (1.34) plotted against the xvalue (abscissa) of a point of tangency (0.82, 0.67), and the locus of this point obtained by animating the tangent over an interval covering the viewing window as shown in Figures 2a and 2b. f(x) = x2

y 5

Move B -> A Animate Tangent Line

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A: (0.82, 0.67) B: (0.82, 0.67) xA = 0.82

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Slope AB = 1.64 2

C 1

A B -4

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2

x

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Figure 2: a) Plotting Point C as 1st step of setting up the derivative as a rate of change function nd b) Animating Point C as 2 step of the derivative as a rate of change function

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From a cognitive science perspective, English and Halford (1995 p. 13) contend that a model is a hypothesized knowledge structure and processes underlying the learning and application of mathematics. This view of a model emphasises modelling of thought processes, learning scenarios and applications of mathematical knowledge to solve problems and is consistent with Simon’s (1995) notion of a hypothetical learning trajectory (HLT). Theoretical frameworks supporting technological integration The addition of ICT tools shifts attention to theoretical frameworks that are able to explain their integration into the classroom in general and mathematics education in particular. The instrumental approach to technology rich classroom environments (Artigue, 2002; Trouche, 2004) and the Technological Pedagogical Content Knowledge (TPACK) by Koehler and Mishra (2009) are among such frameworks. The instrumental approach The instrumental approach acknowledges the unexpected complexity of technology integration into the mathematics classroom and proposes that the use of ICT tools involves instrumental genesis, a process during which the object or artefact is turned into an instrument (Drijvers et al, 2010). In this paper Sketchpad is the mathematical work tool by which students can be empowered to enhance their conceptual understanding of the derivative as demonstrated in Ndlovu (2008) and Ndlovu et al. (2011). Essentially the success of the concept building process depends on the software affordances 1, students’ knowledge in the form of utilisation schemes (usage schemes for hardware operation and instrumented action schemes for mathematical concept development), and the teacher’s instrumental orchestrations (didactical configurations for classroom arrangements, exploitation modes for teaching methods, and didactical performances for teacher interventions during learning). The modelling of the derivative is suited to the teacher’s exploitation modes or hypothetical learning trajectories (HLTs) to support learners’ instrumental genesis in a Sketchpad context. By analogy to Trouche’s (2004, p. 291) description of the humanmachine interaction in a symbolic calculator context, the following are some affordances of Sketchpad for the construction of the derivative as an instantaneous rate of change and as the rate of change function: 1) Internal constraints/affordances/enablements/affordances relating to the construction of a derivative as an instantaneous rate of change (what, by nature, the artefact can do): a) The plotting of functions: Sketchpad contains Computer Algebra Systems (CAS) properties by which it can plot a function given in three coordinate systems (polar, square or rectangular grid) from its Graph menu (an enablement – effectively making the user able to do something); b) It can additionally plot trigonometric functions in both radian and degree measures by prompting the user to make a choice when entering such functions (a constraint –obliging the user to choose one way or another.) c) Once plotted a function is automatically displayed in symbolic form (an affordance - a gesture favoured by the software), but leaving the option open for the user to hide the displayed symbolic expression; 1

A collective term which I occasionally use in this paper to refer to Trouche’s (2004) ‘constraints, enablements, affordances and potentialities’.

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d) It can plot a point on a function if the user chooses and can construct a segment, line or ray passing through two points (enablements) 2) Command constraints/affordances/enablements/potentialities relating to the construction of a derivative as an instantaneous rate of change (the available commands) – a) There is no command in Sketchpad menu for the direct computation of the derivative as an instantaneous rate of change, and yet there is no prohibition of the user to invent his/her own procedures – or instrumented action schemes (a potentiality and an enablement). b) There is also no command for the construction of a tangent to a function plot (a potentiality – as this is not in any menu) but it can be constructed through a specific sequence of instrumented action schemes ( an enablement) 3) Organisation constraints/affordances (how are the available commands organised?) a) The different applications (symbolic, graphical or numerical) allowing the study of functions are accessible from the menus b) To obtain the numerical representations of a function the user must first plot a point on the function, determine its (x,y) coordinates, animate it and tabulate sample points as the animation proceeds – a potentiality or unprescribed/voluntary sequence. (NB: This is in opposition to the paper and pencil procedures.) c) Animation does not exist in paper and pencil procedures (a potentiality of Sketchpad) d) More importantly, the teacher’s conjectured learning/construction sequence of Sketchpad tasks enabling learners to construct the derivative as an instantaneous rate of change can be organised in the following steps or instrumented action schemes: Step 1: Plot a function; Step 2: Construct two points on the function plot; Step 3: Construct a line joining the two points; Step 4: Label the points A and B; and, Step 5: Move point B to point A slowly (animation) to construct the tangent at point A. Some affordances of Sketchpad software for the construction of the derivative as a rate of change function after the construction of the derivative as an instantaneous rate of change include: 4) Internal constraints/affordances/enablements a) Sketchpad can plot a point provided that the corresponding “knowledge” (of the abscissa and the ordinate) have been entered (internal constraints); b) Sketchpad can also label a plotted point using the Display menu; c) Sketchpad can display the abscissa separately (as xA) as shown in Figure 2a. (enablement); 5) Command constraints/affordances/enablements/potentialities a) There is only one two-step command for the exact symbolic determination and graphing of the derivative of a function provided the appropriate function (symbolic or graph) has been selected (an affordance preferred by the software and an enablement respectively); b) However, using the available commands, Sketchpad can plot the slope of a tangent line against x-values as described for Point C in the modelling process above (a potentiality as software not originally meant to plot slope against xvalues)

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c) The animation of the tangent using available animation commands in the Display menu in turn animates the plotted Point C to produce a tracing/locus ( a potentiality ) as in Figure 2b (a combination of enablements resulting in a potentiality that the software designer did not anticipate) – the rate of change function notion of the derivative; d) A comparison of the locus of Point C (a potentiality) and the results of the two-step derivative command (an affordance and a constraint) can be made and the two mathematical objects can be verified to be the same as in Figure 3 where the graph of the derivative is the dotted line. e) The derivative can also be represented in tabular form through the Graph menu command (an enablement) as evident in Figure 3 ; f) The tracing of Point C by any colour or dot width as it animates is available on the Display menu as a choice a user can deliberately make (a potentiality and an enablement).

Figure 3: The derivative as a gradient function plot and trace of Point C 6) Organisation constraints/affordances/enablements/potentialities a) The conjectured procedure for modelling the derivative as a gradient function can be organised in the following sequence of steps or instrumented action schemes: Step1: Measure the tangent slope using the Measure menu; Step 2: Display the abscissa of Point A using the Measure menu; Step 3: Plot the tangent slope against the abscissa of Point A using the Plot as (x,y) command on the Graph menu to construct Point C; Step 4: Trace Point C; Step 5: Animate the tangent to produce the locus of Point C; Step 6: Determine and plot the derivative directly from the Graph menu; and, Step 7: Tabulate the coordinates of Point C as it animates. b) This organisation makes is possible to represent the derivative in both static and dynamic symbolic, graphic, and numeric forms (2 x 3 = 6). The technological pedagogical content knowledge framework The ideas of instrumental orchestration (didactic configuration, exploitation mode and didactic performance) as described in Trouche’s (2004) instrumental theory can complement Shulman’s (1986) model for teacher professional knowledge and skills

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which distinguishes content knowledge CK (mathematics), pedagogical knowledge PK and pedagogical content knowledge PCK (forming the intersection of CK and PK). Using the PCK model as a point of departure, Koehler and Mishra (2009) developed a framework consisting of technology, pedagogy and content knowledge, abbreviated as TPCK to start with and later changed to TPACK. TPACK was defined as the coherent body of knowledge and skills at the intersection of technological knowledge (TK), pedagogical knowledge (PK) and subject content knowledge (CK) required for the implementation of ICT in teaching. TK includes knowledge of operating systems, computer hardware and the ability to use the software being adapted for classroom use such as Sketchpad dynamic software in this case. TPACK includes an understanding of the representation of concepts using technologies, pedagogical techniques that use technologies in constructive ways to teach content, knowledge of what makes concepts difficult or easy to learn and how technology can help redress some of the problems that students face; knowledge of students’ prior knowledge and theories of epistemology, and knowledge of how technologies can be used to build on existing knowledge (Drijvers et al, 2013). As such, TPACK and the teacher’s instrumental orchestrations described in the instrumental approach can complement each other in designing instruction in technology rich classrooms in tandem with constructivist active learning approaches. For example, depending on availability of resources the teacher can: a) use a whole class approach where he/she demonstrates or uses learner as Sherpa while maintaining his/her guidance role; b) use a more learner-centred approach where learners work in pairs or in small groups guided by worksheets; or c) use a more individualised approach where each learner works on his/her own PC, laptop or ICT device guided by a worksheet. Conclusion The potential of improved student conceptual understanding and the consequent academic achievement brought by dynamic mathematics software suggest that these new tools must be vigorously integrated into both pre-service and in-service teacher education programmes. There is a huge challenge, especially in developing country contexts, to equip teachers and learners with the ICT resources and TPACK and instrumental genesis knowledge and skills necessary for effective classroom integration. Once in the field, many teachers have such limited time, resources and expertise that a real danger exists of them becoming an impediment rather than a catalyst in ICT integration if not adequately supported and capacitated timely enough to keep pace with the fast changing digital landscape. References Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. Drijvers, P., Doorman, M., Boon, P., Reed, H. & Gravemeijer, K. (2010). The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75, 213-234. Drijvers, P., Tacoma, S., Besamusca, A., Doorman, M. & Boon, P. (2013).Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations. ZDM Mathematics Education. DOI 10.1007/s11858-013-0535-1. English, L. D. & Halford, G. S. (1995). Mathematics Education: models and processes. Mahwah, NJ: Lawrence Erlbaum Associates. Koehler, M. J. & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9(1)

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Lesh, R. & Doerr, H. (2000). Symbolizing, communication and mathematizing: key components of models and modelling. In P. Cobb, E. Yackel, & K. McClain, (eds), Symbolizing and communicating in mathematics classrooms (pp. 361–383). Mahwah, NJ: Lawrence Erlbaum Associates. Ndlovu M., Wessels, D. & De Villiers, M. (2011). An instrumental approach to modelling the derivative in Sketchpad. Pythagoras, 32(2), 8–22. Ndlovu, M. (2008). Modeling with Sketchpad to enrich students’ concept image of the derivative in introductory calculus: developing domain specific understanding. Unpublished DEd thesis: Pretoria: University of South Africa. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerised learning environments: guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307.

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