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 Teachers’ views on dynamically linked multiple representations and relational understanding of mathematics - an investigation into the use of TI-Nspire in Scottish secondary schools 

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Aberdeen University 


Duncan, Alan 




Aberdeen University 




Aberdeen, Scotland 


This qualitative study documents case studies on use of TI-Nspire by teachers who are using the new Scottish Curriculum for Excellence.  The main research question is: Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons or not, and what evidence do they provide to support their findings?




TI-Nspire, Scotland 

Document Content

he new Scottish Curriculum for Excellence encourages teachers to create learning
situations in which students can be open to new thinking and ideas, think creatively
and independently, learn independently and as part of a group, link and apply new
kinds of learning to new situations, communicate in different ways and in different
settings and use technology in their learning. Students are to be encouraged not only
to link aspects of their learning within any one subject but across other areas of the
curriculum. These ideas are being promoted simultaneously with the release of TINspire
technology which is specifically designed to allow dynamically linked multiple
representations which are designed to enable students to see and make connections
between images and concepts in mathematics and science. This research asks
whether the use of this technology enhances students’ understanding of the
mathematics now being experienced in multiple representations and goes on to
investigate whether teachers are changing their classroom practice in accordance with
the Curriculum for Excellence.
In May 2008 the project got underway when training was provided for 12 teachers from
6 Scottish secondary schools. The schools were chosen to represent a range of types
from a range of geographical locations and to be representative of most Scottish
secondary schools, each being fully comprehensive, ranging from rural to city and
covering a large range of socio-economic backgrounds. The teachers have a range of
background experience with one in her second year in teaching and others with many
years of classroom experience. There is also a considerable range of experience with
ICT in general and with mathematics software, especially in its use in classrooms. The
use of TI-Nspire with students was mainly focused from August 2008 to June 2009. All
of the teachers used TI-Nspire Teacher Edition software whereas all the students used
the handheld version. The teachers completed 66 detailed lesson evaluations which
were analysed for illustrations of the impact of the technology and changes in
classroom practice and teaching methods.
Introduction and Previous Related Research
A section detailing the reasons which provided the researcher with the motivation for
this particular study is followed by a review of literature relating to the use of multiple
representations generally and specifically with respect to ICT. A smaller section
considers the background literature relating to the wider aspects of this study and
associated teachers’ classroom practice.
Important Issues
The literature review is followed by a description of the Curriculum for Excellence
indicators and also a brief description of what is known in Scotland as Assessment is
for Learning, which is an important consideration for all teachers. This research looks
specifically at ‘relational understanding’ which is described and compared with the less
thorough ‘instrumental understanding’. These terms are defined before the actual
research questions are documented. A description of the TI-Nspire technology follows
these issues.
Research Methods and CPD for teachers
The descriptive research design involving a ‘mixed methods’ approach, whose central
premise is that the use of quantitative and qualitative approaches in combination
provides a better understanding of issues under investigation than either approach
alone, is justified in terms of valuing the thoughtful analytical lesson evaluation
comments received from a highly trained body of professional teachers as well as the
numerical data obtained in the study. A triangulation procedure is also used and
involves teacher lesson evaluations, student questionnaires and lesson observations
carried out by the researcher.
The teachers were provided with a total of 6 days of training throughout the period of
the research. These consisted of 2 days before beginning the teaching phase, 2 days
in mid-project (November 2008) and 2 days in March 2009. The nature and purpose of
these training days is described and justified in this section. This training was
extremely important and worthwhile and helped instil confidence in the teachers as
well as give them an opportunity to share experiences.
Research Questions
Do teachers find that the use of dynamically linked multiple representations enhances
their students’ relational understanding of the mathematics involved in their lessons or
not, and what evidence do they provide to support their findings?
The research also considers the following related issues:
• In what ways is the learning and teaching of mathematics changing as a result
of using the software plus handhelds?
• When using the technology, are teachers conscious of changing the way they
teach particular topics?
• When using the technology, are teachers conscious of changing the way they
teach in general? If so, what are these changes and how are they justified?
• When using the technology, what is the impact on students’ motivation and
• Is gaining mastery of the software/handhelds seen as a burden, barrier, or as
valuable and motivating?
• In what ways can the use of the handhelds assist teachers in the formative
assessment of their students?
The questions related to Curriculum for Excellence are:
• Do teachers think the use of the handhelds helps students to become
Successful Learners, Confident Individuals and Effective Contributors?
• Which Curriculum for Excellence indicators provide appropriate criteria against
which to measure students’ abilities in the above capacities?
The Findings
An initial survey was conducted to consider the teachers’ prior use of various forms of
technology, including graphing calculator, graph drawing software, spreadsheets and
dynamic geometry. It was found that there was a wide spread of experience among
the teachers. Most had made some use of a graphing calculator for teaching purposes,
most used spreadsheets for their own purposes but not for teaching, about half made
regular use of an interactive whiteboard. Ten of the twelve teachers had never used
dynamic geometry. By linking dynamic geometry with lists and spreadsheets with the
TI-Nspire, the teachers were using technology that they had not previously used for
teaching purposes.
The next section describes a short survey conducted six months into the project. It
confirmed that all lessons being evaluated did involve the use of linked multiple
representations using TI-Nspire, and that teachers evaluated these lessons for that
very reason. They expressed a willingness to evaluate such lessons whether they
were successful or not. The survey also elicited what teachers consider to be reliable
evidence of relational understanding, pupil motivation and successful learning.
The teachers in the study were representative of the highly qualified profession in
Scotland and they demonstrated typical commitment to the project. The response rate
for the return of completed lesson evaluation proformas was over 90% and the detail
they provided is extensive.
In 80% of the 66 lesson evaluations received, the teachers concluded that the use
of multiple representation with TI-Nspire enhances students’ relational
understanding of the mathematics involved and they were willing to provide
extensive evidence to support their argument. Only 3% contained a negative
response. Among the evidence which teachers considered when making their
decisions were; specific reference to the advantages of the use of multiple
representations for the particular lesson topic, verbal and written responses from
students, improved discussion, ‘seeing’ students’ understanding and improved
An equally large majority of the lessons involved a change of practice for the teachers.
It appears that by being asked to use multiple representations with TI-Nspire, the
teachers think of new and different ways to teach the particular topic, put more
emphasis on links within and across topics, expect more involvement from students
and in some instances teach topics earlier than what would be considered normal.
When considering changes to their practice in general, teachers highlighted a change
in classroom dynamics giving students more freedom to investigate, allowing more
discussion, making an effort to link topics and thinking of ways to use the technology to
help deepen students’ understanding.
In more than half of the evaluations the teachers emphasised the positive impact that
using TI-Nspire had on students’ motivation and engagement. A smaller number noted
the positive contribution to pace and amount of learning. A variety of other positive
comments were made and less than 10% of comments were negative.
A convincing majority of almost two thirds argued that gaining mastery of the
handhelds was not a problem for students and indeed that it was even perceived as
valuable and motivating. Less than 20% of comments related to the difficulties
experienced and some indicated the temporary nature of these. In contrast to the
students, it appears that gaining mastery of the software and handhelds is more of a
problem for the teachers but still a majority of 70% argued that it was not a problem
but valuable and motivating and worth the effort.
With regard to the impact of TI-Nspire handhelds on formative assessment, teachers
were almost unanimously positive in their comments. Other than direct observation of
handheld screens, teachers stressed issues such as increased discussion, more
questioning, more open questioning, more student self assessment and more instant
feedback both to students and teachers. Over 90% of the comments related to
positive, beneficial observations regarding the use of handhelds for ongoing formative
assessment purposes.
Teachers were asked to indicate which Curriculum for Excellence indicators featured
in each of their lessons. In more than 90% of the lessons, ‘Enthusiasm and motivation
for learning’ was chosen. More than three quarters highlighted ‘Openness to new
thinking and ideas’, ‘Learn independently and as part of a group’, ‘Make reasoned
evaluations’ and ‘Solve problems’. Close runners up were ‘Relate to others and
manage themselves’, ‘Think creatively and independently’ and ‘Apply critical thinking in
new contexts’.
Lesson Observations and Commentary
This section of the report provides a description of each of the lessons observed by the
researcher. Attention is focused on the use of dynamically linked multiple
representations and the associated issues of classroom practice, teaching
methodology and student engagement. The main points which arose were;
• emphasis on linking representations and the impact of change in one of these
• use of investigative approaches
• work being tackled significantly earlier than would normally be expected
• high level of questioning and teacher-student and student-student discussion
• high level of enthusiasm, enjoyment and interest displayed by students
• variety of approaches to help students gain mastery of the handhelds such as
‘lock-step’ teaching, use of worksheet, use of animated PowerPoint slides
• issues relating to management of the technology
• new approaches to the teaching of statistics
Conclusions and Recommendations from teachers’ lesson evaluations
The teachers involved in this study, no matter what their background, length of
experience as a teacher or extent of experience with ICT were convinced that the use
of multiple representations of mathematical concepts generally enhances their
students’ relational understanding of these concepts and were willing to provide
extensive evidence to support their arguments. Also by virtue of using TI-Nspire in
their classrooms and by using files involving dynamically linked multiple
representations which include geometry and spreadsheets, there was a significant
increase in the use of both dynamic geometry and spreadsheets by the teachers in the
sample schools during the project, in comparison to previous practice.
A majority of the TI-Nspire lessons involved a change from normal practice. It appears
that by being encouraged to think about possible multiple representations of the
mathematics involved and by using the technology to assist with this teachers were
more inclined to produce a different way of teaching each mathematics topic.
Not only did the teachers change the way they taught particular mathematics topics
but also the way they taught in general, moving largely from teacher exposition
followed by textbook exercises to more practical and investigative approaches
involving more active learning for the students together with more classroom
The teachers provided evidence of improved motivation and engagement among their
students. This may be a direct result of using the technology but may also be a
consequence of the changing classroom practice or even a by-product of improved
understanding. Further study would be required to attempt to apportion causal
A convincing majority of almost two thirds of the teachers’ comments regarding gaining
mastery of the handhelds argued that this was not a problem for students and was
even perceived as valuable and motivating. It appears that the teachers themselves
experienced some difficulty in gaining mastery of the technology but argued that it was
well worth the time and effort for the benefits which accrued. Some professional
development time is therefore necessary for teachers before they can make effective
use of the technology and feel comfortable and confident in its use in their classrooms.
Regarding the use of handhelds for ongoing formative assessment purposes, over
90% of the comments relate to positive, beneficial observations. Teachers can as per
normal observe students’ written work as well as what appears on their handhelds but
it seems to be the changing classroom practice and in particular the increased level of
discussion which is allowing teachers to ‘hear the children’s thinking’ and hence gain
access to their levels of understanding. Such practice should clearly be encouraged.
The teachers also indicated that using TI-Nspire technology met several of the aims of
the Scottish Curriculum for Excellence, especially in helping students to become
‘successful learners’ and ‘effective contributors’. In particular the following CfE
indicators featured greatly in the teachers’ lesson evaluations; enthusiasm and
motivation for learning, openness to new thinking and ideas, ability to solve problems,
learn independently and as part of a group and make reasoned evaluations.
Conclusions and recommendations from observed lessons
• Time needs to be found initially to introduce students to the technology. It is
recommended that the minimum necessary keystrokes are introduced for each
lesson thus building up an improved facility over time. Some lessons may
require little more than the ability to drag an object in a geometry page.
• Teachers should experiment with and evaluate a variety of ways of introducing
keystrokes to students. Some teachers used worksheets while others used
animated PowerPoint presentations. Having these available allows students
who fall behind to catch up again without interrupting others.
• Teachers should understand the meaning of the phrase ‘dynamically linked
multiple representations’ using the technology but should also be aware that
other representations may exist. These may include physical representations
such as ‘walking a graph’.
• When doing data logging using a motion detector it is recommended that
students learn from the experience of creating a graph before being asked to
interpret distance-time graphs.
• Teachers should try to emphasise the impact on one representation caused by
a change in another.
• Some lessons involved the teaching of topics earlier than would have been
expected in the Scottish mathematics syllabus. The technology made this
possible and did not appear to cause any problems for the students. Teachers
should bear this in mind when deciding how and when to teach each topic.
• Students appear to be able to accept new terminology or notation which is
used by the handhelds as long as they are reassured by the teacher that they
will get a fuller explanation at a later date.
• Students appear to benefit from “private investigation” with their own individual
handheld. This is not so accessible when working only from the teacher’s
edition software displayed on a whiteboard. Hence it is recommended that both
approaches be used as each has its own advantages.
• Teachers should consider the use of ‘sliders’ to assist students in situations
where they are expected to investigate “what happens if …..? or when…..?”
• Teachers should also consider carefully the interplay between handheld and
the recording of results or findings. Some teachers asked students to record
their results in a text page on the handheld while others preferred to have the
results written in the students’ jotters (notebooks).
• Teachers should be aware of file management (documentation) issues. If
handhelds are issued randomly to students each time the class meet then
students cannot be expected to store their work, their files. If the handhelds are
numbered they can be issued to particular individuals to overcome this
problem. However, if the handhelds are shared across a number of classes
then this management issue becomes more complex. This can be used as an
argument to encourage students to purchase their own handheld.
Recommendations for Further Research
Further research is needed to explore issues relating to teachers’ use and/or lack of
use of technology in mathematics classrooms. Despite the rich body of national (UK)
and international research literature providing evidence of the benefits of using
technology to enhance students’ learning in mathematics, it appears that many
teachers do not take advantage of this technology. We need to go beyond the obvious
financial restrictions to enquire what aspects of a teacher’s working environment and
conditions act as constraints to the introduction of specific technology and how might
teachers be supported in order to reduce the impact of these constraints.
We should also look at how teachers can be encouraged and supported to become
action researchers. The teachers in this study were involved in action research in that
they reflected on their practices and recorded their thoughts, experiences and findings
in their lesson evaluations. It would be useful to find out how individual teachers might
be assisted to produce further documentation of their experiences for the benefit of
fellow teachers and educational research in general.
Given the conclusion of the teachers in this study that the use of multiple
representations with TI-Nspire enhances students’ relational understanding, it would
be useful to investigate whether this is true for all such representations or whether
there are specific areas of the curriculum where this approach is most productive and
valued. This proved to be beyond the scope of this study.
The teachers also indicated that using TI-Nspire handhelds along with a more
investigative and enquiring teaching methodology allowed the use of formative
assessment strategies. It would seem sensible to conjecture that the introduction of
classrooms of networked handhelds could provide even better access to students’
thinking and hence to improved formative assessment. Research is needed to enquire
whether or not this hypothesis is correct.
Lastly, it would be useful to investigate whether the findings of this Scottish study can
be replicated internationally
AGDFinal Report 11Mar 2010.pdf    
Created at 10/10/2011 10:24 AM  by SP017\rfoshay 
Last modified at 10/10/2011 10:24 AM  by SP017\rfoshay