from theory to practice: digital technology use in the teaching and learning of university...

From Theory to Practice: Digital Technology Use in

the Teaching and Learning of University

MathematicsMike Thomas

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

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


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)

Critique of TPACK

(Graham, 2011)

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)

Pedagogical Technology Knowledge (PTK)(Thomas & Hong, 2005; Hong & Thomas, 2006)


Mathematical Knowledge for Teaching (MKT)


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 KnowledgeSubject Matter Knowledge

Figure 3.2 Comparison between MKT and PCK by Ball & Bass (2006)

(Ball & Bass, 2006)

Pedagogical Technology Knowledge (PTK)

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 role of confidence

• 42 female teachers from Auckland, New Zealand • All teaching mathematics in Years 9-13 (age 14-18

years)


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).


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)




• 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



• 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!



• 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)



• 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)

An example of a schemeMathematical 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)


Forming an Instrument

(Trouche & Drijvers, 2008, p. 368)

Overview

TasksTechniquesSchemes

Tool/Instrument

Mind Mathematics


Focus for technology

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 Lecturer’s Role

Mathematical task or activity

Epistemic mediation by technology

Pedagogy - lecturer

Orchestration of affordances


Focus here

The Instrumental Approach

Personal instrumental genesis – the

teacher can use the tool for personal

mathematical activity

Professional instrumental genesis – the teachers can use the tool as a

didactical teaching tool (and support

students’ instrumental genesis)


The tension of instrumental geneses

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 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)



• 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)


Types of Orchestration

(Drijvers, Tacoma, Besamusca, Doorman & Boon, 2013)


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)


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)


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


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


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 ofassignment and exam responses


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


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.


Technology screenshots taken from the lecture videos


Desmos screenshots from the lecture videos


These are examples of technical-demo orchestrations using the web-based Desmos graphing program

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


Wolfram Alpha screenshots from the lecture videos


All three screens were employed in explain-the-screen orchestrations.

One of the tutorial tasks


More on the task


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 29th and 12 o’clock on the 29th. 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…


Student Working from the Examination


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.


Lecture Recording Views

0

20

40

60

80

100

120Number Who Watched Which Module Recording


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”.


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.


Course Evaluation – Less positiveExtensive 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.


Technology Use Questionnaire


Some Questionnaire Questions


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.


Technology UsePositive 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.


Technology UsePositive 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


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.


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.

Subscale Mean* (Low-High) Cronbach Alpha

Attitude to maths ability 3.89 (3.33-4.56) 0.695

Confidence with technology 4.42 (4.33-4.44) 0.910

Attitude to instrumental genesis 4.40 (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 technology 4.11 (3.67-4.44) 0.872


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


A second study: An epistemological gapMathematics 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.


Pointwise

Find the rate of change of the function f, where f(x) = x3, at the point (2, 8).


Global

If the function f is such thatf(x) = x3, sketch the graph of y = f(x – 1) – 1.

Translate by (1, –1)


Interval

The function f is such thatf(x) = x3–3x2+2x+1. Find the interval (a, b) for which:(i) ’f (x) < 0 and(ii) ’’f (x) < 0.


Local

A local property is one that depends on the values of f in a neighbourhood of a specific point x0

The function f is such thatf(x) = x3–3x2+2x+1. Find an interval [x0–h, x0+h] for which ’f (x)→0 as h→0 for x in the interval [x0–h, x0+h]


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


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 differentiation module using CAS• Differentiation module based on learning

activities with 5 levels. Focus on average and instantaneous rate of change.

• Level 1CAS used for a numeric approximation

for

as h varied from 0.1 to 0.000001Aim: Symbolic process (and object) with local thinking leading to some idea of the limit as h→0

h

fhfhr

)2()2()(

2)( xxf


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.


Relationship between the slope function and the graphs of f and ’f


Level 5

Sketch the derivative using interval reasoning on gradient without being given an explicit function


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


Results and Analysis

• Final term test– Sketch the derivative for the given graphs


Case 1: Symbolic process algebraic thinking (30%)


• 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



• 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

Case 2: Embodied process interval thinking (56%)


• 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



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.

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


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.


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?


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].


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.

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 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.


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?


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 k2–6k+13 for k, you have to change it to x2–6x+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.


Instrumental orchestrationThe lecturer suggests drawing the graph of x2–6x+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.


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.


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


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


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?


Institutional considerations –what is the role of DT in…

The importan

ce of alignmen

t

Course curriculum

Examination and

assessment

The mathematics department

The mathematics lecture room

Students’ experiences in other subjects


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)


• Contact

[email protected]


from theory to practice: digital technology use in the teaching and learning of university...

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