From Theory to Practice: Digital
Technology Use in the Teaching and
Learning of University Mathematics
Mike Thomas
The University of Auckland
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
TPACK
(Koehler & Mishra, 2009)
“emphasises the
connections,
interactions, and
constraints between and
among content,
pedagogy and
technology” (Mishra &
Koehler, 2006, p.1025)
Koehler & Mishra, 2009
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)
The University of Auckland
Pedagogical Technology Knowledge (PTK)
(Thomas & Hong, 2005; Hong & Thomas, 2006)
The University of Auckland
Mathematical Knowledge for Teaching
(MKT)
Subject Matter Knowledge
Pedagogical Content Knowledge
Knowledge of Content
Common Content
and Students (KCS)
Knowledge (CCK)
Specialised Content
Knowledge of
Knowledge (SCK)
Curriculum
Knowledge at the
mathematical
horizon
Knowledge of Content
and Teaching (KCT)
Figure 3.2 Comparison between MKT and PCK by Ball & Bass (2006)
(Ball & Bass, 2006)
Pedagogical Technology Knowledge (PTK)
The University of Auckland
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
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
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
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
Instrumental Genesis
• 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
The University of Auckland
Instrumental Genesis
• 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!
The University of Auckland
Instrumental Genesis
• 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)
The University of Auckland
Instrumental Genesis
• 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)
The University of Auckland
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)
Forming an Instrument
(Trouche & Drijvers, 2008, p. 368)
The University of Auckland
Overview
Schemes
Mind
Techniques
Tasks
Tool/Instrume
Mathematics
nt
Focus for
technology
The University of Auckland
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
The Lecturer’s Role
Focus here
Pedagogy lecturer
Epistemic
mediation by
technology
Orchestration of
affordances
Mathematical
task or
activity
The Instrumental Approach
The University of Auckland
The tension of instrumental geneses
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 University of Auckland
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
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
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
Types of Orchestration
(Drijvers, Tacoma, Besamusca, Doorman & Boon, 2013)
The University of Auckland
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
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
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
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
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
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
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
Technology screenshots taken from the
lecture videos
The University of Auckland
Desmos screenshots from the lecture videos
These are examples of technical-demo orchestrations using the
web-based Desmos graphing program
The University of Auckland
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-toinvestigate, with students immediately encouraged to
use Desmos, or other technology in their possession,
to investigate further examples
The University of Auckland
Wolfram Alpha screenshots from the lecture
videos
All three screens
were employed in
explain-the-screen
orchestrations.
The University of Auckland
One of the tutorial tasks
The University of Auckland
More on the task
The University of Auckland
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
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
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
More on the task
The University of Auckland
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…
The University of Auckland
Student Working from the Examination
The University of Auckland
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.
5.5
5.4
5.3
5.2
5.1
4.10.
4.9
4.8
4.7
4.6
4.5
4.4
4.3
4.2
4.1
3.10.
3.9
3.8
3.7
3.6
3.5
3.4
3.3
3.2
3.1
2.8
2.7
2.6
2.5
120
2.4
2.3
2.2b
2.2a
2.1
The University of Auckland
Lecture Recording Views
Number Who Watched Which Module Recording
100
80
60
40
20
0
The University of Auckland
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
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
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
Technology Use Questionnaire
The University of Auckland
Some Questionnaire Questions
3.
5.
9.
13.
Do you think the lecturers made sufficient use of these technologies to help you
understand their use and value? If not, specify which you would have liked more
of.
Which technologies do you personally own or have easy access to? [list given]
Which mathematics learning technologies did you personally use in the course?
Please indicate your frequency of use, and whether this was the first time you had
used them.
Did you like the extensive use of technology in MATHS 102? Please explain.
The University of Auckland
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
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
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
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
Attitudes Survey
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
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.
The University of Auckland
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 CAScalculators
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
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
Pointwise
Find the rate of change of
the function f, where
f(x) = x3, at the point (2, 8).
The University of Auckland
Global
If the function f is such that
f(x) = x3, sketch the graph
of y = f(x – 1) – 1.
Translate by (1, –1)
The University of Auckland
Interval
The function f is such that
f(x) = x3–3x2+2x+1.
Find the interval (a, b) for
which:
(i) (x) < 0 and
(ii) (x) < 0.
The University of Auckland
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 that
f(x) = x3–3x2+2x+1.
Find an interval [x0–h, x0+h]
for which (x)→0 as h→0
for x in the interval
[x0–h, x0+h]
The University of Auckland
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
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
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
f (2  h)  f (2) for f ( x)  x 2
r ( h) 
h
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
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
Relationship between the slope function and
the graphs of  and 
The University of Auckland
Level 5
Sketch the derivative using interval reasoning on
gradient without being given an explicit function
The University of Auckland
Technique
• 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
Decreasing
• Produce a table of values on intervals (below)
• Constructing this table requires local or interval reasoning
to find properties of the function f’ for a function f
• Repeated embodied actions are required
The University of Auckland
Results and Analysis
• Final term test
– Sketch the derivative for the given graphs
The University of Auckland
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
The University of Auckland
Case 1: Symbolic process algebraic thinking (30%)
• 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
The University of Auckland
Case 1: Symbolic process algebraic thinking (30%)
The University of Auckland
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
The University of Auckland
Case 2: Embodied process interval thinking (56%)
The University of Auckland
Case 2: Embodied process interval thinking (56%)
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.
The University of Auckland
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
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
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
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 discussthe-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 University of Auckland
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
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
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
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 xaxis.
• Student: I see, I understand
why I don’t have real roots
looking at the graph.
The University of Auckland
Instrumental orchestration
The 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.
The University of Auckland
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
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
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
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
Institutional considerations –what is the role
of DT in…
The mathematics
department
Examination
and assessment
Course
curriculum
The mathematics
lecture room
The
importance
of
alignment
Students’
experiences in
other subjects
The University of Auckland
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
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
• Contact
moj.thomas@auckland.ac.nz