Teachers’ Experiences of the Integration of 21st Century Learning in
the Mathematics Classroom – The Bridge21 Model in Action
Aibhin Bray
Centre for Research in IT in Education (CRITE), School of Education and School of Computer Science & Statistics,
Trinity College Dublin, Dublin, Ireland
Keywords: Continuous Professional Development, 21st
Century Learning, Post-Primary Education, Mathematics
Education.
Abstract: Current research indicates a need for ongoing support and continuous professional development (CPD) for
teachers in order to facilitate the development of 21st Century pedagogies and the integration of technology,
as well as to scaffold their changing role in the classroom. This article describes a particular model of 21st
Century teaching and learning and an associated approach to CPD, with a particular focus on mathematics
education. A qualitative, case study approach has been taken in order to explore the teachers’ experiences of
using the model of teaching and learning, as well as their perceptions of the students’ experiences. A total of
15 teachers who attended a Contextual Mathematics module on the CPD course provided consent for their
data to be used in this study and a constant comparative analytic technique has been used to analyse their
written reflections. Results indicate that the approach has the potential to address many of the difficulties
associated with 21st Century teaching and learning identified in the literature review.
1 INTRODUCTION
There is no single, universally recognised definition
of 21
st
Century skills or of the types of teaching and
learning required to achieve them. However, in their
comparative analysis of international frameworks for
21
st
Century competences, Voogt and Roblin (2012)
identify a common recognition of the development of
skills relating to communication and collaboration,
problem-solving and creativity, as well as
technological fluency, as being fundamentally
important. Many of these skills can be defined as
higher-order thinking and learning skills, or “life-
skills”, and they are seen as being transversal (not
subject-specific) and multi-dimensional, impacting
on attitudes and knowledge (Dede, 2010a; Voogt and
Roblin, 2012).
Although a general recognition of the benefit of
these skills is not new, an approach to education that
emphasises the importance of acquiring them in an
integrated manner throughout curricular activities,
combined with the potential of technology to assist in
their realisation, can be viewed as innovative
(Conole, 2008; Dede, 2010a; 2010b; Voogt and
Roblin, 2012). Meaningful incorporation of such 21
st
Century Learning (21CL) skills in mainstream
curricula however, may require a change in
pedagogic focus. Such a change would require a shift
in teaching and learning approaches, de-emphasising
the more traditional, procedural activities still
common in educational practice, and increasing
emphasis on the more complex skills that require an
understanding of ‘why’ as well as ‘how’ the routine
procedures should be used (Conneely et al., 2013;
Dede, 2010b; Fullan and Langworthy, 2014; Voogt
and Roblin, 2012). The role of technology in 21CL is
perceived as important, in that it requires the
development of specific competences regarding the
effective use, management and evaluation of
information across many different platforms (Martin
and Grudziecki, 2006; Voogt and Roblin, 2012).
It is generally recognised that 21CL can be best
supported through pedagogic approaches such as
Inquiry-Based Learning (IBL), Problem-Based
Learning (PBL), and collaboration, as well as a more
formative approach to assessment (Conneely et al.,
2013; Conole, 2008; Fullan and Langworthy, 2014;
Voogt and Roblin, 2012). Some of these approaches
as well as barriers to, and possible facilitation of, their
implementation are discussed in the following
sections.
Bray, A.
Teachers’ Experiences of the Integration of 21st Century Learning in the Mathematics Classroom - The Bridge21 Model in Action.
In Proceedings of the 8th International Conference on Computer Supported Education (CSEDU 2016) - Volume 2, pages 219-230
ISBN: 978-989-758-179-3
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
219
2 LITERATURE REVIEW
European research (Commission of the European
Communitities, 2008; Euler and Maaß, 2011) has
highlighted a need for appropriate structure, support
and scaffolding in order to create opportunities for
young people to develop key 21CL competences
within the school environment. It is recognised
however, that this is not a straightforward task.
Indeed, the PRIMAS report (Euler and Maaß, 2011)
has acknowledged that although many teachers wish
to develop teaching and learning strategies that
incorporate skills associated with 21CL, they are
hampered by numerous factors outside of their
control, including the restraints of current practices in
curriculum and assessment, and a lack of relevant
continuous professional development (CPD) for
teachers.
Euler and Maaß (2011) identify three levels of
problems associated with the implementation of a
21CL approach to teaching and learning: the
overarching school system; a lack of resources
including CPD; and teachers’ beliefs. The macro-
level issues identified by Euler and Maaß, relating to
policies, curriculum, and assessment, while
fundamental, are not addressed in detail in this
research.
At the meso- and micro-levels, problems relating
to classroom management and the difficulties that
teachers may have in redefining their role, have been
identified as contributing to the gap between the
intended curricula, which tend to recognise the
importance of 21
st
Century skills, and that which is
actually implemented (Conneely et al., 2013; Euler
and Maaß, 2011; Voogt nd Roblin, 2012). Not only
are teachers expected to facilitate the acquisition of
21
st
Century skills among their students, but they are
also expected to possess the skills themselves (Voogt
and Roblin, 2012). Discussion alone is not sufficient
to address these issues, rather a shift in the beliefs and
practices of policy-makers and practitioners is
required (Dede, 2010b). Educators need to be
provided with adequate support and CPD in order to
master the necessary skills and teaching strategies,
but also to ‘unlearn’ the beliefs and assumptions that
underpin the traditional industrial-model of
classroom practice (Conneely et al., 2013; Dede,
2010b; Voogt and Roblin, 2012).
It has been suggested that an approach to CPD that
presents teaching as a problem-solving activity, or
‘research-in-action’, may be particularly appropriate
for linking teaching practice with children’s learning
(Commission of the European Communitities, 2008).
This has clear links with the approach described in
this research.
3 BRIDGE21 AND CPD
One of the findings that has emerged throughout the
analysis of the literature, is a need for ongoing
support and continuous professional development
(CPD) for teachers in order to facilitate the
development of 21
st
Century pedagogies and the
integration of technology, as well as to scaffold their
changing role in the classroom (Conneely et al., 2013;
Dede, 2010a; Maaß nd Artigue, 2013; Voogt and
Roblin, 2012). This paper describes the
implementation of a structured CPD module that has
been incorporated into a larger Postgraduate
Certificate (PG Cert) course in 21
st
Century Teaching
and Learning for post-primary school teachers,
coordinated by the School of Education in Trinity
College Dublin (Bridge21, 2014). This course began
in September 2014 and the first cohort of teachers
have recently completed the programme.
Bridge21 offers a structured pedagogic model for
the integration of 21CL in classrooms, and supports
an innovative approach to CPD, strongly influenced
by the Japanese model of Lesson Study that uses an
iterative cycle of goal setting, planning, teaching and
observation, review, and revision (Lewis et al., 2009).
Groups of teachers form communities of practice to
engage in a process of systematic examination of their
practice, with the goal of becoming more effective
teachers and optimising their lessons (Maaß and
Artigue, 2013; Takahashi nd Yoshida, 2004).
The initial experience for teachers engaging with
the Bridge21 CPD model involves active
participation in immersive and authentic activities,
which enables them to understand the power of the
approach at a personal level. Throughout the process,
participants are provided with the resources, practical
designs and collegial support that Donnelly et al.,
(2011) highlight as necessary conditions to motivate
change amongst teachers.
The Contextual Mathematics module on the PG
Cert requires each of the attending teachers to
complete an assignment that involves the creation and
implementation of a 21CL activity in their school,
with a standard class. The activity should use the
Bridge21 model of 21CL (Section 4) and a set of
design heuristics (Section 5) developed for the
creation of contextual, 21CL mathematics learning
activities (Bray et al., 2013; Bray and Tangney,
2013b; 2014; Tangney et al., 2015). The written part
of the assignment involves a description of the
activity, highlighting the basic and transversal skills
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220
that are covered, as well as a reflection on their own,
and the students’ experiences of the process.
A total of 22 teachers attended the contextual
mathematics module which forms the basis of this
study. These teachers came from a wide range of
schools and had levels of teaching experience ranging
from 3 to 19 years. In a number of cases, two or three
of the teachers came from the same school, which
promoted greater levels of collaboration in the design
and implementation of their activities, and also
strengthened the communities of practice within the
schools. Fifteen of the teachers provided consent for
their work to be included in this research. This paper
provides an analysis of their work, placing particular
emphasis on their reflections on the process, and their
experiences with it.
4 THE BRIDGE21 MODEL
Bridge21 (www.bridge21.ie) is a model of
collaborative, project-based learning that has been
developed at the authors’ institution (Lawlor et al.,
2010; Lawlor et al., 2015). Initially established in
2007 as part of an outreach programme, the Bridge21
model has been developed throughout the intervening
years and is currently being trialled in a number post-
primary schools as part of a systemic process of
reform of the Irish education system (Conneely et al.,
2015; Department of Education and Skills, 2012;
Lawlor et al., 2015). The Bridge21 activity model is
inspired by the concept of Design Thinking (Brown,
2008), and brings together many of the elements of
Figure 1: The Bridge21 Activity Model.
21CL in a structured manner, scaffolding teachers’
introduction of the pedagogical approach. A Bridge21
learning experience involves a number of steps
(Figure 1):
1. Set-Up: Ice breaker and team formation.
2. Warm-Up: Divergent thinking activity.
3. Investigate: Explanation of the problem context.
4. Plan: Group planning.
5. Create: An iterative process
a. Exploration with resources.
i. In the field.
ii. In the classroom.
b. Modelling and Calculation:
i. Analysis and Synthesis.
6. Present: Competition and/or Presentations.
7. Reflect: Reflection and Discussion.
In addition to the activity process, Bridge21 observes
a particular model of collaboration and group work
influenced by the World Organisation of the Scout
Movement (Bénard, 2002), which emphasises mixed-
ability groups and individual, as well as group
reflection. The role of the teacher is to act as a guide
and mentor, scaffolding and orchestrating the
learning. The physical learning space is configured to
support an inquiry-based, technology-mediated, and
collaborative approach to learning. The Bridge21
approach to CPD involves teacher participation in a
full cycle of the activity model prior to the
collaborative development of their own activities and
their delivery in the classroom.
5 THE DESIGN HEURISTICS
In the Contextual Mathematics module, the use of a
particular set of Design Heuristics for the creation of
contextual 21CL mathematics activities is advocated.
These heuristics have been developed and refined by
the author over the course of the last three years. The
theoretical foundations of the set of Design Heuristics
were developed from an extensive literature review
(Bray and Tangney, 2013a; 2014; Tangney et al.,
2015). Particular attention was paid to the Realistic
Mathematics Education, or RME (Freudenthal, 1991)
approach to mathematics education, which, since its
inception in the 1960s has become internationally
influential in curriculum design (Clements et al.,
2013).
In conjunction with the theoretical foundations
stemming from the literature, an iterative process of
activity design, pilot activities and in-school
interventions were used to refine the Design
Heuristics (Bray et al., 2013; Bray et al., in press;
Bray and Tangney, 2013a; 2014). The primary
Teachers’ Experiences of the Integration of 21st Century Learning in the Mathematics Classroom - The Bridge21 Model in Action
221
concepts that underpin the heuristics can be
summarised as follows:
1. “Activities should follow a 21CL model such as
Bridge21: they should be collaborative and team-
based in accordance with a socially constructivist
approach to learning.
2. They should make use of a variety of technologies
(digital and traditional) suited to the task, in
particular, non-specialist mobile technology such
as smartphones and digital cameras that students
have to hand. Emphasis should be placed on the
transformative, as well as the computational,
capabilities of the technology.
3. Task design should prioritise guided-discovery,
involving problem-solving, investigation and
sense-making, and a move from concrete to
abstract concepts. Tasks should be open-ended,
allowing for different trajectories and solutions;
they should have a “low-floor” and “high-
ceiling”, such that all students will be able to
engage meaningfully with the problem, with the
potential for more interested/able students to push
its boundaries.
4. The context of the problem, and the learning
experience, should be interesting and
immersive/real, adapting the environment and
class routine as appropriate;
5. Presentation, competition and reflection can be
used for assessment purposes.” (Bray & Tangney,
In Press)
Activities designed in accordance with these
heuristics, and implemented using the Bridge21
approach have been associated with increases in
students’ levels of engagement with mathematics, and
their attitudes to using technology for its learning
(Bray et al., in press; Bray and Tangney, In Press).
6 METHODOLOGY AND
RESEARCH QUESTIONS
The work presented here is framed as an exploratory
case study, which aims to investigate teachers
experiences of the creation and implementation of
mathematics activities designed in accordance with
the Design Heuristics and the Bridge21 methodology.
The research design is a single case study, with
multiple embedded units, each consisting of one of
fifteen teachers’ implementations of an activity in
their school, and subsequent reflection on the process.
The context is Post-Primary Education – Authentic
Setting (the researcher is not an observer), and the
case relates to teacher experiences and their
perceptions of their students’ engagement (Figure 2).
The written assignments, which include a description
and rationale for the design of the learning
experience, content and skills to be covered, evidence
to demonstrate student learning, and a personal
reflection on the experience, form the basis of the data
for analysis. As described in section 7, a constant
comparative analytic technique was used to highlight
the emerging themes (Strauss and Corbin, 2008).
Figure 2: Case Study Model.
The specific aims of this exploratory case study are:
1. To explore the experiences of teachers in the
creation and implementation of such activities,
with particular emphasis on their perceived
barriers to, and benefits of, the approach.
2. To explore the teachers’ perceptions of their
students’ experiences with the activities.
The data that has been collected for this exploratory
study is purely qualitative and comes from the written
reports of the teachers. Not all of the teachers
provided authorisation for their work to be included
in this research - the total number of assignments that
have been analysed is fifteen.
7 THE MATHS LEARNING
ACTIVITIES
A number of the teachers worked collaboratively on
the design of the activities, and joint implementation
was permitted for teachers at the same school. In total,
11 different activities were created by the 15 teachers.
The teacher-designed activities were conducted with
students across four different year groups, ranging in
age from 12 – 16. All of the activities were creative,
contextual and transformative in their use of
technology (Puentedura, 2006).
Context: Post-Primary Education -
Authentic Setting
Case: Teacher Experience
Teacher 1
Teacher
2
Teacher
3
Teachers
...
Teacher
15
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222
Table 1: Teachers' Activities.
Activity Class Description
Heights with
Helium
Transition Year
1
(age 15/16)
A helium balloon and technology was used to find the measure of certain heights around our school.
This meant dealing with only two variables, Height and Time and being able to use the free video
analysis software Kinovea (www.kinovea.org) to obtain these variables and a Spreadsheet to graph
the data.
Functions in
context: analysing
trajectory.
2
nd
Year
(age 13/14)
Each team of students take video clips of attempts to throw a ball into a basket. They then use
appropriate software to analyse the trajectory of a successful shot. Using a suitable graph, they
compare successful and unsuccessful shots.
Distance, Speed and
Time
3
rd
Year
(age 14/15)
Students ask themselves “how fast am I running?” Based on their introduction to Kinovea and their
knowledge of Microsoft Excel, they are asked to answer this question and illustrate their answers
in the form of graphs and tables.
Statistics/
Height/Distance,
Speed and Time
1
st
Year
(age 12/13)
Working in groups, students are tasked with comparing the speed of the shortest and tallest
members of their group over a specified distance. The data collected, and analysis of their findings,
will be done using Kinovea.
Egg Drop
Challenge
Transition Year
(age 15/16)
Teams of four students work to design a method of safely dropping an egg from a first floor window.
They use smart phones, digital camera and iPads to visually record the activity. They generate data
from the activity and use a video App and maths analysis software to provide mathematical evidence
for their approach.
Quadratic
equations, functions
and algebra.
Transition Year
(age 15/16)
The students are asked to plot the quadratic function for the flight of their shot in a football crossbar
challenge. The students are dived into groups of 3-4 students. Each group works with the tracker
software to analyse the shot that is closest to hitting the crossbar. The students use the software
Kinovea and excel to find and plot the flight of their shot. The students are asked to analyse the graph
produced.
Children’s Birthday
Party
2
nd
Year
(age 13/14)
Given an advertisement for a party hire company, the students use GeoGebra to explore different
combinations of tables etc. to get the best value for money.
Speed, Distance,
Time
Transition year
(age 15/16)
The students are presented with the problem ‘Who is the fastest in the class?’ In their groups they must
produce a method of experiment and a Microsoft excel presentation of their results.
Shoot a basket!
2
nd
Year
(age 13/14)
In groups, the students develop different ways to analyse and make ‘real’ quadratic functions through
group work and peer teaching and learning. Students learn to select, create, and use many new forms
of technology, such as GeoGebra and Tracker (physlets.org/tracker). The groups will be briefly
introduced to the programmes but need to decide if it will help answer the question, “What makes a
successful shot successful?” As the students gain experience working with the programmes, they
become more aware of the technology available around them.
Speed\Distance\
Time, Statistics
1
s
t
Year
(age 12/13)
The students undertake a study to determine if the speed of the ball affects the chances of scoring goal.
Speed Camera
2
nd
Year
(age 13/14)
In groups students are required to use technology to analyse the speed of cars passing by the school.
They represent the data using an appropriate chart and come up with a hypothesis as to whether
different coloured cars are more prone to breaking the speed limit based on their data.
1
Transition year is a one-year school programme in which the focus is on personal, social, vocational and educational development,
providing opportunities for students to experience diverse educational inputs in a year that is free from formal examinations.
The majority of the activities focussed on the
mathematics involved in data collection and
representation, patterns, and linear and quadratic
functions. In all cases this was reported as being the
first experience that the students had of generating
their own functions using “real” data that they had
collected empirically. Brief descriptions of the
activities are provided in Table 1.
Of particular relevance to this research are the
written reflections from the teachers’ assignments,
which provide insight into their experiences with the
implementation of contextual mathematics learning
activities and into the barriers to, and benefits of, the
approach.
8 ANALYSIS OF RESULTS
The results are drawn from a qualitative analysis of
the teachers’ reflections and a constant comparative
approach to the analysis of the data has been taken.
The steps in this process follow the procedure
outlined by Glaser (1965) and Strauss and Corbin
(2008). Constant comparison is a method of reducing
qualitative data to codes emerging from within the
original source, while retaining much of the richness
of the original data. Thus, the results of the analysis
can be used to create a rich picture of the teachers
experiences, potentially identifying any common
themes or categories.
8.1 Generation of Initial Codes and
Categories
NVivo10 software was used to facilitate the process
of coding and theming. After the first five
assignments were analysed, a total of 23 codes were
identified. These fell into the two main categories of
Teachers’ Experiences of the Integration of 21st Century Learning in the Mathematics Classroom - The Bridge21 Model in Action
223
Barriers, with five associated codes, and Benefits,
with 18 associated codes. All segments of text
associated with each of these codes were re-examined
and compared before moving on to the next set of
assignments, of which the next four led to the addition
of five new codes, four under the category of
Benefits, and one under Barriers. At this point, the
process of memoing – keeping detailed notes on the
thought process involved behind the coding – was very
useful for highlighting areas that could potentially
benefit from re-organisation (Figure 3). In particular,
the codes associated with the category of Benefits
seemed to be developing into a number of
subcategories, some relating to teachers and some to
students, some to the development of key skills, and so
on.
The remaining six assignments only led to the
generation of two more codes, leading to the tentative
conclusion that a reasonable level of saturation of
codes may have been reached (Strauss and Corbin,
2008).
Figure 3: Sample Memo.
All of the text was re-examined after each session
of analysis, and particularly after the addition of new
codes, in order to compare the coded text within their
assigned nodes and also to identify whether they
could be associated with any other codes. This
process of constant evaluation and comparison has
led to a rigorous association of codes and text.
8.2 The Reduction of Codes
Once the initial development of codes and categories
was completed, the process of reducing and merging
the codes, and developing sub-categories began. This
involved an examination of the codes and the coded
segments in order to determine whether there was any
crossover of themes.
Figure 4: Barriers Category.
The Barriers category had significantly fewer
references than Benefits, and included student
abilities, teams, technical difficulties (at individual
and school level), and time constraints (Figure 4).
Table 2: Matrix Coding of Task Design and Perceived Benefits.
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224
(In Figures 4, 5 and 6, Sources refers to the
number of individual teachers whose data were
categorised at the related code, while References
relates to the overall number of coded segments.)
The category of Benefits had a total of 295
references, in comparison to only 37 in the Barriers
category. At this point in the analysis, a number of
subcategories were confirmed in the Benefits
category. These related to benefits to the students
(key skills, other outcomes, associated task attributes)
and benefits to the teachers (change in beliefs, teacher
as facilitator, and teacher as learner).
8.3 Analysis of Relationships
The process of analysis of relationships used the
coding matrix facility of NVivo10. Analysis focused
on the relationships between the teachers’ perceptions
of the task design elements that had an impact on
themselves and on their students, and their perceived
benefits. No associations between task attributes and
barriers were identified. Table 2, above, provides a
numerical analysis of the number of times that
segments of text were co-coded with a particular
aspect of task design and a perceived benefit. The
most significant elements (most frequently coded) of
the task design columns and perceived benefits
columns have been highlighted.
Using the sum functionality at the end of each row
and column, it is clear that the fact that the tasks were
student-led has had the most significant impact on
perceived benefits, particularly on the sense of
student ownership or autonomy, on their conceptual
understanding, and on engagement. The student-led
approach also seems to be significant in effecting a
change in the role of the teacher in the classroom.
Peer learning and the contextual nature of the task
design appear to have had beneficial effects on the
students and teachers, particularly in the areas of
collaboration, communication and engagement.
In terms of perceived benefits, it appears that the
task design has had most impact on student
engagement, with the tasks set in contexts that were
meaningful to the students and the student-centred
nature of the activities appearing to have the greatest
effect.
Conceptual understanding is highlighted as the
second highest co-coded perceived benefit, and this
seems to be related to tasks that are set in contexts that
are meaningful to the students, as well as the student-
led nature of the learning
9 FINDINGS
The findings that have emerged through analysis of
the relationships between task design and the
perceived benefits of the approach, go some way to
confirm the link between the approach to
mathematics teaching and learning and increases in
student engagement emerging from earlier research
(Bray et al., in press; Bray and Tangney, In Press). In
particular, there is an apparent link between the
student-led, contextual and meaningful approach to
activity design, and a perception of increased
engagement, conceptual understanding, and
confidence. However, in addition to these
relationships, a number of findings have emerged
relating to the teachers’ perceptions of the barriers to
the implementation of activities of this kind, and also
of the benefits that engagement with these tasks can
engender.
9.1 Barriers
The CPD model presented here, has addressed some
of the barriers to the integration of technology and the
implementation of new teaching and learning
strategies highlighted in the literature review, such as
a need for a structured and supportive approach
(Conneely et al., 2013; Dede, 2010a; Euler and Maaß,
2011; Means, 2010; Voogt and Roblin, 2012).
However, many of the more systemic barriers remain,
and have been identified by the teachers. The most
significant of these relates to time constraints and the
difficulty that implementing a project-based, inquiry
activity in a series of 40 minute classes, which was
identified as a problem in 10 of the 15 assignments
(Note: teacher initials have been used to code
quotations): “Having a longer block of time would
have been more productive, having to stop after 40
minutes and then pick up again a day or two later was
inconsistent, especially when we were running into
problems” (AH)
Technical barriers were an issue for nine of the
teachers, with five identifying personal difficulties
with the technology, which would be easily
rectifiable on a re-run of the project: “The camera we
were using ran out of battery power during the
penalty shoot outs... More cameras would need to be
made available, especially if more teachers were to
start working with this approach.” (WMI)
Eight of the teachers identified technical barriers
at the school level, which primarily related to
inadequate access to the technology: “Resourcing
fully functioning laptops could be a challenge - I need
to ensure that the limited number of laptops are
Teachers’ Experiences of the Integration of 21st Century Learning in the Mathematics Classroom - The Bridge21 Model in Action
225
available for at least three class periods.” (IS)
Other barriers that were identified by the teachers
referred to lower than expected levels of students’
technical expertise, and difficulties relating to the
development of well-functioning teams.
9.2 Benefits
The perceived benefits associated with the approach
far outweigh the barriers, and can be broken down
into benefits for teachers and benefits for students.
9.2.1 Benefits to Teachers
The teachers perceived a number of changes to their
beliefs and to their role in the classroom.
Figure 5: Key Skills.
Two of the teachers in particular discussed the
impact that teaching in this way has had on their
beliefs about mathematics teaching:
“After trying this, my eyes have been opened to the
possibilities of covering the curriculum, but by
changing the setting of the learning, you can teach a
lot more effectively to an audience who are stimulated
and engaged.” (JPF)
“I will be honest that I found it more difficult to
change my teaching style when it came to Maths. I
was teaching the way I was taught, which was with
very little understanding.” (MC)
It appears that the role of the teacher in the
classroom is significantly affected through the
implementation of these activities. The change in role
from transmitter of information to facilitator of
learning was not a comfortable one for some of the
teachers; however, in all cases, it was hailed as a
positive development, empowering the students to
take ownership of their own progress. “I decided to
tell the students of how this was as much of a learning
curve to me as it was to them. This was because I
really did feel that they would lose confidence in me
if they felt that I was trying to teach them rather than
facilitate them. This seemed to empower them as they
felt that even though I wasn’t part of their team, I was
learning and teaching with them.” (MC)
In addition to the change in role from teacher to
facilitator, six of the teachers also identified
themselves as co-learners in the classroom, both in
terms of learning about the technology with and from
the students, and learning about how to make
activities of this kind more successful in the future.
9.2.2 Benefits to Students
The benefits to the students have been deconstructed
into the subcategories of ‘key skills’, ‘other
outcomes’ and ‘associated task attributes’. The
relationships between the task attributes and the
perceived benefits of the approach have already been
discussed in section 7.3. This section will therefore
focus on the perceived benefits of the approach to
students, without dwelling on their associations with
the task design.
The key skills subcategory is made up of the
codes listed in Figure 5. It is clear from this figure that
the most common skills that were developed relate to
collaboration and communication, technological
confidence and creativity and problem-solving. The
students generally seemed to enjoy working in teams
and learning with and from their peers. Many of the
teachers recognised the potential that technology has
to facilitate a deeper understanding of the
mathematics involved in the activities, as well as
increasing the students’ technological skills.
“The resounding theme of the [student] reflection
was that they could really engage with one another
and more importantly that they could engage more
with the abstract topics of maths because of their
ability to use technology in everyday maths.” (DR)
In addition to the development of key skills, a
number of other beneficial outcomes emerged
through students’ participation in the activities
designed by the teachers. These outcomes are listed
in figure 6.
Figure 6: Other Beneficial Outcomes.
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An increase in student engagement relating both
to how they felt about the subject (affective
engagement) and how they behaved in the classroom
(behavioural engagement) (Pierce, Stacey, &
Barkatsas, 2007), was evident through the teachers’
reflections. Comments such as those provided below,
clearly illustrate the sense of engagement and
motivation experienced by students and teachers
alike.
“All the team members were fully engaged in the
activity; their pride in and ownership of their
learning was clearly expressed… It’s really
heartening to encounter such a level of motivation
and commitment.” (DD)
“Please let’s do more of this stuff! It’s brought Maths
to life! I really get it now! J” (Student)
“This project was a thoroughly enriching experience
for both the students and teachers assisting them.”
(DOC)
“After this contextual Maths workshop, they asked for
a Maths club. To me that is success!” (MC)
There is a high level of cross-coding of segments
of text coded as engagement and as enjoyment.
However, a deeper analysis of the text coded at
enjoyment indicates that this code is particularly
closely related to affective engagement. Any
segments that are coded at enjoyment and not at
engagement relate specifically to the idea of having
fun in the class, both from the point of view of the
students, and the teachers:
“This project has highlighted one of the most
enjoyable pieces of technology that I have used in my
teaching career” (IB)
“I feel that the students enjoyed this realistic,
contextualized activity and by taking part they have
taken a step forward in developing their
technological skills, becoming better problem solvers
and gaining attributes in working as part of a team.”
(AH)
“The students also had fun, which they said that they
thought they would never be able to say about
Maths.” (MC)
An increase in students’ conceptual understanding
and confidence was identified in nine of the analysed
reports. This appears to be particularly closely
associated with the contextual and meaningful nature
of the tasks, a relationship that is clearly captured in
the following: “I am sure that none of these students
will ever forget how they deepened their
understanding of quadratic functions: the next time
they video a friend kicking a football or teeing off in
golf they will visualise that ball moving across the
Cartesian plane, describing a smooth parabola.”
(DD)
In addition, the open-ended task design and the
student-led approach within the classrooms appears
to have led to a deepening of the students’
understanding: “The open-ended nature of the activity
produced a new energy in the teams: they were not
working to find one answer (already known to me) but
were engaged in a meaningful exploration of the
topic.(DD)
Seven of the reports refer to the increased sense
of student ownership of their work, leading in turn to
pride, engagement and motivation.
“Students came into their own when given the
opportunity to work as a group and they seemed to
grow as individuals even in the short space of time
while working in groups with their peers” (DR)
“Moreover, I feel that if I had taken over this aspect
of the project… I would be impacting on their self-
efficacy.” (DF)
By handing the responsibility for the learning to
the students, they were seen to develop as individuals
and as members of a group, with the apparent increase
in levels of motivation and pride in their learning
leading to higher levels of conceptual understanding.
“All the participants felt that they had created their
own quadratic function and understood that it could
be mathematically analysed.” (DD)
In summary, these findings provide a
compellingly positive picture of the approach to the
development and implementation of mathematics
learning activities that correspond to the design
heuristics described in this research.
10 DISCUSSION
The analysis of teachers’ reflections described in this
paper has provided an opportunity to explore various
aspects of the participants’ experiences of the
Contextual Mathematics module on the Postgraduate
Certificate, thereby addressing the research aims
identified in section 5. In particular, analysis of the
data has permitted:
Examination of the experiences of teachers in the
creation and implementation of such activities,
paying particular attention to the barriers to, and
benefits of, the approach.
Exploration of the teachers’ perceptions of their
students’ experiences with the activities.
These topics have been explored throughout this
research. This discussion will explore aspects of the
reflections that mirror concerns that emerged in the
literature review, and will also set out the primary
limitations of the exploratory study.
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227
10.1 Addressing the Issues
Throughout the analysis of the teachers’ reports, it
was interesting to see that many of the problems
associated with mathematics education that had been
identified through the literature review, were also
highlighted by the teachers taking part in this module.
The predominantly formulaic approach to text-book
questions (Boaler, 1993) was identified by one
teacher as an area that the approach advocated in the
CPD module, had the potential to address.
“These problems involved being given the function,
algebraically or graphically, and all the information
required to answer some fairly predictable questions.
There was never any redundant information either:
just enough and not too much to apply the usual
procedures… I considered that setting the students
the task of creating their own quadratic curve would
give them a real sense of ownership and a greater
insight into the nature of quadratic functions.” (DD)
The teachers’ reflections indicate a belief that this
approach may go some way to address the
fragmented, and de-contextualised nature that
frequently pervades school mathematics (Albert and
Kim, 2013; Dede, 2010a).
“It was useful for students to see different aspects of
Maths used in one place rather than the disjointed
treatment that they usually receive in a text book.”
(WMI)
In addition, as observed by Oldknow (2009), the
use of personal devices, such a mobile phones, to
generate mathematical models, contextualised the
mathematics for the students, providing a relevance
and meaning to the topic:
“For students, to discover that they can take their
ubiquitous phone out of their pocket and create a
mathematical model of an everyday event grounds
Maths in the real world.” (DD)
The issues surrounding teachers’ beliefs and their
changing role in the classroom can also be seen to be
addressed through the structured, immersive and
supportive nature of the CPD program. The provision
of a specific pedagogical structure (Bridge21) and set
of lesson design heuristics provide the teachers with
an approach that has been tested and shown to work.
The teachers all seemed to have been empowered by
this, and were confident to approach their classes in a
different way. The results appear to have been
beneficial for both teachers and students.
“I have worked with this particular class group on
two other 21
st
Century Teaching and Learning
Assignments. Their development throughout the
course of the year has been astounding. The flair with
which they now competently and confidently use
technology to gather and analyse information, and
present their findings is very impressive. This project
was a thoroughly enriching experience for both the
students and teachers assisting them.” (DOC)
10.2 Limitations
It is clear from this analysis that the approach to the
creation and implementation of mathematics learning
activities that has been developed in this research has
the potential to address many of the issues that were
highlighted in the literature review. However, it is
important to identify the limitations of this study.
Firstly, the sample that is used in this exploratory
study consists of teachers who have opted to be a part
of the research, and who are participants on a CPD
course that they have chosen to attend. It is therefore
a self-selecting sample and cannot be seen as
representative.
Another point that needs to be highlighted is that
the reflective pieces provided by the teachers were all
submitted for assessment purposes. There is a
possibility that the participants therefore emphasised
the positive aspects of their experiences more than the
negative. This is a limitation of the study to date,
which future work will aim to overcome through
interviews with participants and their students and
non-participant observation of the classes.
Another drawback of this exploratory study is its
small size. The analysis of fifteen teachers’ reports is
unlikely to permit the generation of any substantive
theory. However, the consistency of the results do
allow the generation of hypotheses and research
questions to follow up on the initial, very promising,
findings.
11 CONCLUSIONS
This study will require further expansion in order to
fully examine the emerging themes. It is a very
encouraging however, to see such positive results
emerging from the work with teachers. In particular,
the following quote from one of the attendees on the
Contextual Mathematics module highlights the
teachers’ understanding of the intention behind this
research.
“The importance of 21
st
Century teaching and
learning and indeed the B21 model can be seen by
Green and Hannon who state, “In an economy driven
by knowledge rather than manufacturing, employers
are already valuing very different skills, such as
creativity, communication, presentation skills and
team building. Schools are at the frontline of change
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and need to think about how they can prepare young
people for the future workplace” (2007, p. 15). As
such a huge emphasis is being placed on STEM
subjects/activities in schools, RME in conjunction
with the B21 model helps to contextualise maths for
our students, increasing their engagement and
allowing them to use technology in a meaningful
way.” (MC)
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