1. From Theory to Practice: Digital Technology Use in the Teaching and Learning of University Mathematics Mike Thomas

2. Overview Some theoretical perspectives on digital technology (DT) use: PTK, TPACK and instrumental orchestration Some recent university DT research projects. Focus on orchestration Outcomes and issues The University of Auckland

3. The role of the lecturer We see that for use of DT the teacher or lecturer has a key role In attempts to outline what would assist a teacher or lecturer with DT use some frameworks have been developed Consider TPACK and PTK Developed with schools in mind – but appear to transfer to the tertiary sector The University of Auckland

4. Technology Pedagogy and Content Knowledge Koehler & Mishra, 2009 TPACK (Koehler & Mishra, 2009) “emphasises the connections, interactions, and constraints between and among content, pedagogy and technology” (Mishra & Koehler, 2006, p.1025)

5. Critique of TPACK (Graham, 2011)

6. Pedagogical Technology Knowledge - PTK Mathematical Knowledge for Teaching (MKT) –(Ball & Bass, 2006) Instrumental Genesis - (Rabardel & Samurcay, 2001) Orientations - dispositions, beliefs, values, tastes and preferences (Schoenfeld, 2011), attitudes and confidence in using DT (Thomas & Hong, 2005)

7. Pedagogical Technology Knowledge (PTK) (Thomas & Hong, 2005; Hong & Thomas, 2006) The University of Auckland

8. Mathematical Knowledge for Teaching (MKT) The University of Auckland Common Content Knowledge (CCK) Knowledge at the mathematical horizon Specialised Content Knowledge (SCK) Knowledge of Content and Students (KCS) Knowledge of Content and Teaching (KCT) Knowledge of Curriculum Pedagogical Content Knowledge Subject Matter Knowledge Figure 3.2 Comparison between MKT and PCK by Ball & Bass (2006) (Ball & Bass, 2006)

9. Pedagogical Technology Knowledge (PTK)

10. Comparison of Pedagogical Technology Knowledge (PTK) and TPACK TPACK, framework (Mishra & Koehler, 2006; Koehler & Mishra, 2009) has similarities to PTK, but –More generic, not focussed on mathematics –Little emphasis on epistemic value. TPACK relates to “knowledge of the existence, components and capabilities of various technologies as they are used in teaching and learning settings, and conversely, knowing how teaching might change as a result of using particular technologies.” (Mishra & Koehler, 2006, p. 1028) –No inclusion of the personal orientations of the teacher. These dispositions, beliefs, values, tastes and preferences shape the way we see the world, direct the goals we establish and prioritise the marshalling of resources, such as knowledge used to achieve the goals (Schoenfeld, 2010) The University of Auckland

11. The role of confidence 42 female teachers from Auckland, New Zealand All teaching mathematics in Years 9-13 (age 14-18 years) The University of Auckland

12. The role of confidence Results indicate a correlation between confidence in using technology in the mathematics classroom and teacher use of digital technology in a pedagogical manner facilitating learning of mathematical concepts (as well as procedures). Those with higher levels of confidence benefited from being part of a school-based group that shared and reflected on their instrumental genesis, practical classroom activities and ideas about the technology, especially in the early stages of learning about technology use. cf the argument that an individual’s development of mathematics teaching practice “is most effective when it takes place in a supportive community through which knowledge can develop and be evaluated critically” (Jaworski, 2003, p. 252). The University of Auckland

13. Instrumental Genesis Rabardel distinguishes between the use of technology as a tool, or artefact, and as an instrument. Transforming a technological tool into an instrument involves actions and decisions based on adapting it to a particular task via a consideration of what it can do and how it might do it. Implication: one tool can give rise to multiple instruments depending on the task This process of learning to use a tool as an instrument is called instrumental genesis, and it has two dimensions, namely instrumentalisation and instrumentation. (Rabardel & Samurcay, 2001) The University of Auckland

14. Instrumental Genesis The University of Auckland Instrumentalisation This charts the emergence and evolution of the artefact’s components for a particular task, such as the selection of pertinent parts, choice, grouping, elaboration of function, transformation of function, etc. This may be summarised as the subject adapting the tool to himself. Example: driving a car Task: get to work or school, go grocery shopping, transport furniture or rallying Each driver has to: adjust the mirrors, seat position to suit them, tune the engine For each task the settings differ: choose the radio channel, empty the boot, fold down the rear seats, add roll bars, etc

15. Instrumental Genesis The University of Auckland Instrumentation Involves the emergence and development of private schemes and the appropriation of social utilisation schemes for a particular task. The subject adapts himself to the tool. Example: driving a car Techniques: change gear, parallel park, three point turn, overtaking Each driver has to develop personal mental schemes to be able to carry out these techniques. Knowing and doing are not the same!

16. Instrumental Genesis The University of Auckland Technique: a set of rules, methods or procedures that is used for solving a specific type of problem An instrumented technique has a technical side that consists of an integrated series of machine acts that has become a routinized way of dealing with a specific type of regularly occurring task. Techniques and schemes co-evolve, consisting of means for using the artifact in an efficient way to complete the intended types of tasks. An instrument consists of both the artefact and the accompanying mental schemes that the user develops (Drijvers, 2003; Trouche & Drijvers, 2008)

17. Instrumental Genesis The University of Auckland Instrumental genesis: developing utilization schemes and instrumented techniques A utilization scheme integrates the technical skills for using the machine, and the conceptual meaning that is attached to these manipulations, including both mathematical understanding and insight into the way the technological tool deals with the mathematics. These schemes give meaning to the use of the tool. (Drijvers, 2003)

18. An example of a scheme Mathematical focus: conceptions of parameter in systems of equations Technique: isolate a variable in one equation, substitute it into a second and then solve that equation Scheme: Isolate-Substitute-Solve (ISS) instrumentation scheme for a CAS calculator. It was found that students had many unforseen problems with it Drijvers and van Herwaarden (2000) The University of Auckland

19. Forming an Instrument (Trouche & Drijvers, 2008, p. 368)

20. Overview Tasks Techniques Schemes Tool/Instrume nt Mind Mathematics The University of Auckland Focus for technology

21. DT use Epistemic mediation—oriented towards an awareness of the object [of the activity], its properties, and its changes in line with the subject’s actions Pragmatic mediation—oriented towards action on the object [of the activity], transformation, regulation management, etc The University of Auckland

22. The Lecturer’s Role Mathematical task or activity Epistemic mediation by technology Pedagogy - lecturer Orchestration of affordances The University of Auckland Focus here

23. The Instrumental Approach

24. The University of Auckland The tension of instrumental geneses

25. Instrumental Orchestration A didactical configuration - arrangement of artefacts in the environment An exploitation mode - the way the teacher decides to exploit the arrangement Orchestration can be: intentional and systematic management of artefacts aiming at the implementation of a given mathematical situation in a given classroom or a didactical performance - ad hoc decisions taken by the teacher (See Trouche, 2004; Drijvers, Boon, Reed & Gravemeijer, 2010) The University of Auckland

26. Instrumental Orchestration The notion of orchestration itself evolves through several steps: individual and static conception (orchestrations seen through didactical configurations and exploitations modes of the mathematical situation) a social perspective (orchestrations seen as the result of teachers’ collaborative work) a dynamic view (including the didactical performance, teachers’ adaptation on the fly and teacher adaptation over time) The University of Auckland

27. Instrumental Orchestration A primary goal of lecturer orchestrations is to engage students in activity producing techniques with both epistemic value, providing knowledge of the mathematical object under study, and ‘productive potential’ or pragmatic value (Trouche, 2004; Drijvers, Boon, Reed & Gravemeijer, 2010) The University of Auckland

28. Types of Orchestration (Drijvers, Tacoma, Besamusca, Doorman & Boon, 2013) The University of Auckland

29. Conjecture Strengthening teachers’ PTK (TPACK) will enhance their ability to use DT in teaching. How do we strengthen PTK? –Provide a focus on the mathematics before the technology –Build mathematical content knowledge –Assist with instrumental genesis to investigate conceptual understanding of mathematics (as well as procedural skills) –Encourage positive teacher orientations about the use of technology, especially confidence in its use –Work on task design (See ICMI study) The University of Auckland

30. Task design considerations with technology Take students beyond the routine Address a mathematical concept or idea (ie epistemic focus rather than pragmatic) Examine the role of language and ask students to write about how they interpret their work Consider dynamic multiple linked representations, involving treatments and, especially, conversions between representations (Duval, 2006) Build in the need for versatile interactions with representations (Thomas, 2008) Integrate technological and by-hand techniques Aim for generalisation Encourage students to think about explanations, proof and development of mathematical theory (See Kieran & Drijvers, 2006) The University of Auckland

31. Research project 1: UoA MATHS 102 Course - Intensive Technology (Essentially BYOD) Initial design Principles Lecturers model DT extensively. Students encouraged to use e.g. Desmos, Wolfram Alpha, Autograph, CAS calculators, Kahn Academy, Applets; Youtube; Smartphones and tablets All lectures recorded and available to students via online resource program (Cecil) DT integral to assessment: each student registered and enrolled into MathXL – a web-based homework, tutorial and assessment system, which was used for five skills quizzes (1% each) and the mid semester test (10%). Written assignments and tutorials also required DT, e.g. graphs, programming The University of Auckland

32. Research project 1: UoA MATHS 102 Course - Intensive Technology (Essentially BYOD) Initial design Principles Students encouraged to use any technology platform they had access to, including all calculators, mobile phones, computers, tablets, etc. and any e-resources they could access with these Technology should be actively used in the one-hour weekly tutorials that all students were expected to attend, and received credit for The University of Auckland

33. What we are Doing in the Study Data collection Pilot Sem 2, 2013. Full study Sems 1, 2 2014 Exit questionnaires: One looking at Attitudes; other at experiences with technology in the course Standard Course Evaluation Observations of volunteer groups working on specially designed active technology tasks in tutorials Interviews with volunteer participants Data from student use of MathXL, Cecil, inspection of assignment and exam responses The University of Auckland

34. Phase 2 Mathematical Focus Chose average and instantaneous rate of change as the mathematical focus Instrumental genesis aimed at epistemic mediation of this Lecturer has good instrumental genesis Students varied in their instrumental genesis Instrumental orchestration had to consider: –Lecturer’s computer, overhead display, internet access for Desmos, Wolfram Alpha, etc, computer program use for GeoGebra, etc, lecture video –Variety of student platforms in use: smartphones, tablets, computers The University of Auckland

35. Phase 2 Orchestrations The concept of average rate of change (AROC) of a function was introduced using a board-instruction orchestration Following the introduction of AROC a GeoGebra program, written by the lecturer, was displayed. Using dynamic dragging in this program, and an explain-the-screen orchestration, the lecturer was able to present examples of the AROC between two points both a variable and a fixed distance apart, and link the screen view to mathematical constructs. The University of Auckland

36. Technology screenshots taken from the lecture videos The University of Auckland

37. Desmos screenshots from the lecture videos The University of Auckland These are examples of technical-demo orchestrations using the web-based Desmos graphing program

38. Desmos in lectures 50% of the questionnaire respondents said that they used Desmos during the lectures The kind of orchestration that usually followed a technical-demo we have called a guide-to- investigate, with students immediately encouraged to use Desmos, or other technology in their possession, to investigate further examples The University of Auckland

39. Wolfram Alpha screenshots from the lecture videos The University of Auckland All three screens were employed in explain-the-screen orchestrations.

40. One of the tutorial tasks The University of Auckland

41. More on the task The University of Auckland

42. More on the task It didn’t take Sonja long to suggest a method. She said “You take the point at which the rate of change is greatest and take an x interval of 1 either side of it.” What do you think of her method? Is she right? Investigate the greatest average rate of increase over an x interval of 2 for this graph. Where does it occur? What about an x interval of 3? The University of Auckland

43. More on the task If the t interval is 1 instead, where does greatest average rate of increase occur then? If the t interval is k instead, where k ≥ 0.5, for what value of k does the greatest possible average rate of increase occur? If the t interval is k again, what happens to the average rate of increase as k gets smaller and smaller, i.e. as k → 0? Describe in detail a method that would help Raj and Sonja find greatest average rates of change for graphs like this one. The University of Auckland

44. Task engagement This task, written especially with active technology use in view, generated a lot of discussion and group work among the students and they investigated this task in more depth than they did previous tutorial tasks The progress of some students was limited by their lack of instrumental genesis They tended to use Desmos due to its relative ease of use rather than other programs such as GeoGebra that would have allowed a greater array of techniques to be used on the task Students tended to favour the computer over calculators No one solved all the problem but they did engage with mathematical concepts The University of Auckland

45. More on the task The University of Auckland

46. Concept engagement They knew how to calculate AROC: So you work out the average rate of change between that point and that point which is going to be 3.2 take away 0.1, which is pretty much that bottom point there. Between those two. And there’s only a difference of one. So you’ve got an average rate of change of 3.1. Are we good on that? They demonstrated some idea of local properties So that will give you the steepest line there. The other one is that one, which is pretty close, between the 29 th and 12 o’clock on the 29 th. But it’s not quite as good. But as your k gets smaller, so as your k interval gets smaller and smaller and smaller, that one will become your steepest line. But then it will swap to that one. …so m gets smaller and smaller…As m gets smaller, the greatest rate of change is going to effectively be steeper. Until you get to the stationary points. So the stationary points will remain the same, but as you get closer and closer… The University of Auckland

47. Student Working from the Examination The University of Auckland

48. Other Results Access to Online Resources: 107 (135) accessed recorded lectures to some extent, the majority up to 20 times, but 11 students accessed more than 40 (one student 115 times) Can also look at the module/lecture they viewed the most (e.g. differentiation lectures viewed more than integration, which is interesting) Number of times looked at online course book; past tests; past exams; etc. The University of Auckland

49. Lecture Recording Views The University of Auckland

50. Course Evaluation 77.1% overall satisfaction (lower than usual, 50 out of 135 students completed) Helpful: Access to web, some very helpful sites; MathXL-examples, quizzes, homework : 19 specific comments from 46 in total Specific comments about other technology: recorded lectures (7); Khan academy (5); Desmos (4) Example: “Utilization of MathXL, as well as being prompted during lectures of other sources of information available such as Desmos and Khan Academy to be able to be used concurrently with MathXL's resources”. The University of Auckland

51. Course Evaluation - Positive MathXL was extremely helpful for my learning. Being able to check my answers instantly was a great encouragement and stimulant. The weekly quizzes are a great way of keeping my skills up...MathXL is more productive and enables me to get feedback quickly on what it is I need to work more on. Khan Academy (website) was also extremely helpful. I found myself getting lost during the early lectures at University, and felt it necessary to go through the material again at a slower pace with lots of practice examples. Khan Academy allowed me to do this. I would say that throughout the semester, the lectures informed me of what it was I needed to learn and that I actually learned it through Khan Academy. Desmos was very useful for experimenting with functions to see how they appeared in graph form…I had to research on the web (mostly Khan Academy and Desmos) so I could answer most of the questions. The University of Auckland

52. Course Evaluation – Less positive Extensive use of technology made it very difficult to study content and do well in assessments, particularly if the student is not used to learning through computer-based content. Having the course book online made it highly inaccessible. Most students like to study with hard copies, i.e. paper and pen and having to print a whole course book is both time consuming and cost inefficient. As a student whom normally does well I have struggled with the extensive use of technology and computer based assessments in this course and struggled to fully learn the material and as a result have found my results to be rather poor. It is unfair to assume that our generation learns better through technology as everyone learns differently and many of us have always used textbooks etc. Thus the course did not provide adequate material suited to all learning styles and as a result has greatly disadvantaged some students. The University of Auckland

53. Technology Use Questionnaire The University of Auckland

54. Some Questionnaire Questions The University of Auckland

55. Technology Use – Phase 1 (Based on 13 responses from 131 students) All used MathXL, seven almost daily and six once or twice a week; 11 used Desmos, six of them daily, two once or twice a week; Six used Wolfram Alpha, five of them daily. Khan Academy was used daily by five students, Autograph by two and GeoGebra by one. In addition ten students made daily use of a graphic or CAS calculator. All used MathXL for the assessment quizzes with a mean of 4.72 out of five quizzes. Similarly, all used it for homework, ten at least once or twice a week and twelve for revision, ten at least once or twice a week. Furthermore, nine used it in their study plan and ten for help with solving problems, mostly at least once or twice a week. The University of Auckland

56. Technology Use Positive Comments I learnt a lot from this course through the many technologies made available to me. I spent several hours each week practicing using various websites, apps and online tutorials, as well as recorded lectures. Highly recommended. Being able to continue to interactively learn outside the classroom has helped significantly. MathXL helped me to focus on areas of maths I needed help with. There was a broad use of mathematical technology throughout this course, enabling students to feel supported in the learning process. Maths can be an intimidating subject to study, so by introducing technology to be enable visual learners like myself maths seems less daunting. The University of Auckland

57. Technology Use Positive Comments Particularly in year one mathematics, the use of technology has helped me gain a quicker and deeper understanding as to how various equations behave and being able to quickly look up a mathematics problem on the internet also assisted greatly. [It should be used in future] Because it is really useful for understanding concepts, for practising them and learning them The University of Auckland

58. Technology Use Negative Comments: MathXL was a disastrously unfair method of assessment as it was difficult to formulate your thoughts when a test is in such a different format to what you have always done. I have personally always been rather good at maths but I have done very poorly in this course as I have struggled with everything being computer/technology based. …too reliant on technology without understanding the core foundations of mathematics. It is like designing a bridge without first knowing fundamental engineering principles. The University of Auckland

59. Attitudes Survey Notable responses: Indication that even those who see themselves as good at maths may be less confident of achieving good results; I use the technology to find more than just the basic answer to the question (mean 4.11). Goals such as “to improve learning and understanding”, “to apply mathematics in the real world” explicitly mentioned, without any leading. SubscaleMean* (Low-High)Cronbach Alpha Attitude to maths ability3.89 (3.33-4.56)0.695 Confidence with technology4.42 (4.33-4.44)0.910 Attitude to instrumental genesis4.40 (4.11-4.56)0.820 Attitude to learning mathematics with technology 3.93 (3.11-4.22)0.838 Attitude to versatile use of technology4.11 (3.67-4.44)0.872 The University of Auckland

60. Issues/Results No clear differences in achievement rates between the research semester and previous Low participant response rate in spite of repeated encouragement Still need to resolve curricular consistency- would prefer students to have access to all technologies during the exam, especially since more now use tablets, laptops, smart-phones than have access to graphics or CAS- calculators Marking/Evaluating of assessments: How to interpret or evaluate the value of a solution; For computer-aided marking (other than just multiple choice), accuracy of interpretation of the solution and marking Multiple available technologies: Which ones should be used? Instrumental Genesis: Limited time available in a 12-week course Lecturer orchestration dependent on personal instrumental genesis and that of students The University of Auckland

61. A second study: An epistemological gap Mathematics students need the ability to move between point-wise, local and global perspectives of function (Artigue, 2009) “…working at university level on functions implies that students can adopt a local perspective on functions whereas only point-wise and global perspectives are constructed at the secondary school.” (Vandebrouck, 2011, p. 2095). Mathematical Principle: Need to develop interval and local views of function. The University of Auckland

62. Pointwise Find the rate of change of the function f, where f(x) = x 3, at the point (2, 8). The University of Auckland

63. Global If the function f is such that f(x) = x 3, sketch the graph of y = f(x – 1) – 1. Translate by (1, –1) The University of Auckland

64. Interval The function f is such that f(x) = x 3 –3x 2 +2x+1. Find the interval (a, b) for which: (i)  ’ (x) < 0 and (ii)  ’’ (x) < 0. The University of Auckland

65. Local A local property is one that depends on the values of f in a neighbourhood of a specific point x 0 The function f is such that f(x) = x 3 –3x 2 +2x+1. Find an interval [x 0 –h, x 0 +h] for which  ’ (x) → 0 as h → 0 for x in the interval [x 0 –h, x 0 +h] The University of Auckland

66. Method Pre-calculus course at a university in Korea –required study for those wanting to major in a mathematically related subject –entry grades are mixed –Avoided for as long as possible; many students have little interest in mathematics for its own sake 143 students in three classes, 136 students took the final term test 15 weeks; one two-hour session per week None of the students had used any digital technology before in mathematics – instrumental genesis problems The University of Auckland

67. Course content and delivery Linear, quadratic, cubic, exponential and logarithmic functions, differentiation, integration, probability and matrices Lecturer with good instrumental genesis demonstrated with GSP, Autograph and a TI-Nspire CAS calculator. Due to a lack of available technology students were not able to use a CAS calculator themselves During exercises involving sketching different functions students were able to use the graphical software Autograph Targets interval and local thinking The University of Auckland

68. The differentiation module using CAS Differentiation module based on learning activities with 5 levels. Focus on average and instantaneous rate of change. Level 1 CAS used for a numeric approximation for as h varied from 0.1 to 0.000001 Aim: Symbolic process (and object) with local thinking leading to some idea of the limit as h → 0 The University of Auckland

69. Level 3 Generalise to the rate of change symbolic process and encapsulate as a symbolic object. The CAS calculator was used to introduce students to a method of obtaining the derivative at a general point x = a by defining a function slope(h)=avgRC(f(a), a, h), a={–1, 0, 1, 2, 3}, the average rate of change over an interval of width h. The University of Auckland

70. Relationship between the slope function and the graphs of  and  ’ The University of Auckland

71. Level 5 Sketch the derivative using interval reasoning on gradient without being given an explicit function The University of Auckland

72. Technique Constructing this table requires local or interval reasoning to find properties of the function f’ for a function f Repeated embodied actions are required Locate the points where the gradient of the tangent line is zero: at x=0 and approximately x=1.5 Divide the real line into intervals whose endpoints are the critical numbers 0, 1.5, as above Produce a table of values on intervals (below) Decreasing The University of Auckland

73. Results and Analysis Final term test –Sketch the derivative for the given graphs The University of Auckland

74. Case 1: Symbolic process algebraic thinking (30%) The University of Auckland Students whose thinking is dominated by symbolic algebra may find such a question difficult since there is no algebra to work with. The modelling technique employed by these symbolic process-oriented students was: –assume the graph is a polynomial and determine its order –try to fit it to the general formula for such a polynomial function, using y=a(x-b) 2 +c or y=a(x-b)(x-c)(x-d) and information from the given graph to find the parameters and model the function –differentiate the symbolic function obtained and then draw its derived function from this

75. Case 1: Symbolic process algebraic thinking (30%) The University of Auckland For example, here they often used a polynomial function y=a(x+1)(x-2)(x-3) They then used the point (0, 2) to find a = 1/3 The brackets were then expanded The function was differentiated symbolically They completed the square to find the vertex The graph of the derivative was drawn

76. Case 1: Symbolic process algebraic thinking (30%) The University of Auckland

77. Case 2: Embodied process interval thinking (56%) The University of Auckland 80 (56%) students correctly drew the derived function graphs by a consideration of interval thinking They understood the technique and built the mental scheme Some of their comments were: –“if f(x) is increasing, f'(x)>0, if f(x) is decreasing, f'(x)<0” –“If the slope values change from positive to negative, then the values of the derivative change from positive to negative. If the slope values change from negative to positive, then the values of the derivative change from negative to positive” This employs embodied process thinking and links between symbolic and graphical representations

78. Case 2: Embodied process interval thinking (56%) The University of Auckland

79. Case 2: Embodied process interval thinking (56%) The University of Auckland This student was one of only two who realised that the point of inflection corresponded to the greatest negative gradient, and hence the local minimum on the derived function graph.

80. An example of instrumental orchestration A case study of 134 students in two pre-calculus classes of the same course at a university in Korea. Content: polynomial functions, trigonometry, logarithmic and exponential functions, limits, differentiation and integration. Taught using mainly lecturer demonstration with GeoGebra, Geometer’s Sketchpad and graphic calculator apps on a smartphone, which the students downloaded during the class. Students also used KakaoTalk on the SNS (Social Network Service), which allows one to send and receive messages on the screen of a smartphone. Lecturer has good PTK, including instrumental genesis and orientations The University of Auckland

81. An example of instrumental orchestration Smartphone and Kakaotalk allowed students to transfer the graphing calculator working to pen and paper, take a snapshot with smartphone and send it to the lecturer who could then give feedback This is an innovative approach requiring considerable instrumental genesis and orchestration on the part of the lecturer. The University of Auckland

82. Instrumental orchestration Student: Miss, I am going to sketch the conditional graph using GeoGebra, it is cut out when I put 2x–1(–1≤x≤1). How do I define the interval, please? The University of Auckland

83. Instrumental orchestration Entering f(x)=2x–1(–1≤x≤1) into GeoGebra the student was surprised by the discontinuous graph obtained, what she called ‘cut out’. Realising this was incorrect since she wanted the graph of 2x – 1 to display on the interval [–1, 1]. The University of Auckland This instrumentation problem was dealt with by the lecturer. The individual orchestration could be classified as an ad hoc didactical performance involving both discuss- the-screen, due to the need to explain why the graph was not as expected, and technical-support, where the correct input was provided. The lecturer did not take the opportunity to engage the student further.

84. The response Lecturer: Did you solve your problem of the interval? You have to enter the following in the input window: if(–1≤x≤1, 2x–1) Student: That’s what I wanted to know. The University of Auckland

85. The response Discussing what GeoGebra might do with an input such as f(x)=(2x–1)(–1≤x≤1) might have helped her to focus on the mathematical logic behind the placement of the interval and hence construct a suitable scheme for using them. This kind of orchestration, which does not appear to be covered by the taxonomy of Drijvers et al. (2013), could be classified as guide-to-investigate. The University of Auckland

86. A second example Student: The answer to question 5 in chapter 2, is k^2–6k+13=0, isn’t it? Lecturer: Yes, so a value satisfying this does not exist. Student: How do I represent the graph of k^2–6k+13=0? The answer doesn’t look clear. The value of k doesn’t have an exact value, right? The University of Auckland

87. A second example Student: Then, I don’t have to use the quadratic formula for the roots? Lecturer: To see the status of k, sketch the graph of k 2 –6k+13 for k, you have to change it to x 2 –6x+13 instead of k. Try it. Then you can see that the value of k does not exist on the x- axis. Student: I see, I understand why I don’t have real roots looking at the graph. The University of Auckland

88. Instrumental orchestration The lecturer suggests drawing the graph of x 2 –6x+13. The student responds “I see, I understand why I don’t have real roots looking at the graph.” The change of representation has provided epistemic insight. Lecturer’s orchestration: firstly, pragmatic, technical-support, assisting the student to see that the GC will only plot graphs in terms of x not k. The orchestration is helping the student develop an appropriate mental scheme with genuine epistemic value. It may produce the knowledge that the particular variable used in a function is irrelevant, leading to a technique whereby it may be substituted by any other variable. The University of Auckland

89. Instrumental orchestration Secondly, the orchestration encourages experimentation in order to learn (“Try it. Then you can see that the value of k does not exist on the x-axis”). In this case it involves having the versatility to link the function across two representations, with the mathematical outcome much easier to see from the graph than the algebra, and, this could be classified as a guide-to-investigate orchestration. The University of Auckland

90. What do we learn? A focus on the mathematical ideas/concepts is to be encouraged Instrumental orchestration requires a high level of lecturer PTK Students may be engaged but learning may not be enhanced There will be some student resistance to DT The University of Auckland

91. Lecturer Implications and questions How does the extent to which a lecturer has mastered a mathematical digital tool support them to transform it into a didactical professional instrument? (i.e what is the relationship between personal and professional instrumental geneses?) Professional development should take account of these two very different geneses It takes time to become instrumented – and lecturers need repeated cycles of lecture room practice for instrumental genesis Some digital technologies (and their inherent tasks) are more complex than others and require enhanced instrumental orchestration The University of Auckland

92. Lecturer Questions How can lecturer PD be organised to encourage a level of PTK that will promote instrumental genesis and instrumental orchestration? Which other theories might inform the design of PD activities that aim to introduce lecturers to digital technologies for teaching mathematics? How do we assist lecturers to construct suitable tasks with digital technology that focus on concepts? The University of Auckland

93. Institutional considerations –what is the role of DT in… The University of Auckland

94. Final words on technology use Pragmatic versus epistemic use “I think that calculators and CAS are great pedagogical tools, but are ineffectively used. Unfortunately students use them as computational devices. Most college discussions on using them or not is centered on students computational use and not as a pedagogical tool.” (Thomas et al., 2012) It can help calibrate the balance and interplay of procedural and conceptual knowledge if different concepts are emphasised, concepts studied more deeply, investigations of procedures extended, and increased attention placed on structure. (Heid, Thomas & Zbiek, 2013) The University of Auckland

95. Final words on technology use Pragmatic versus epistemic use “I think that calculators and CAS are great pedagogical tools, but are ineffectively used. Unfortunately students use them as computational devices. Most college discussions on using them or not is centered on students computational use and not as a pedagogical tool.” (Thomas et al., 2012) It can help calibrate the balance and interplay of procedural and conceptual knowledge if different concepts are emphasised, concepts studied more deeply, investigations of procedures extended, and increased attention placed on structure. (Heid, Thomas & Zbiek, 2013) The University of Auckland

96. Contact moj.thomas@auckland.ac.nz The University of Auckland