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Preparing Math Teachers for the Future:
Analysing the Use of Technology in a Master’s
Program
José Manuel Dos Santos Dos Santos1, Jaime Carvalho e Silva2, and Zsolt
Lavicza 3
1Department of Mathematics, University of Coimbra, Coimbra, Portugal, and inED - Centro de
Investigação e Inovação em Educação, Porto, Portugal
2Department of Mathematics, University of Coimbra, Coimbra, Portugal, and Center of Mathematics
of University of Coimbra, Coimbra, Portugal
3School of Education - Johannes Kepler University, Linz, Áustria
Abstract
Current guidelines for the mathematics curriculum from primary to secondary educa-
tion in Portugal include Computational Thinking as a cross-curricular theme and advo-
cate for the systematic use of technology. Official mathematics program documents pro-
vide methodological suggestions for teachers to develop computational thinking through the
use of dynamic geometry environments, such as GeoGebra, internet applets, Scratch, and
Python. In this context, it is important to understand how future teachers interpret the use
of these technological tools in the first year of a Master’s degree in Teaching Mathematics
for Basic and Secondary School Teachers in Portugal. This study analyses the work of 12
students during the “Computational Tools for the Teaching of Mathematics” course unit.
Supported by the research-based design methodology, the study examines the work carried
out over two school years, as well as the adjustments made between the first and second
cycles of this intervention, using qualitative data analysis techniques. The results indicate
that the integration of content knowledge and technological knowledge among the partici-
pants is complex throughout the two intervention cycles. Combining pedagogical content
knowledge with technological and content knowledge presents new challenges for the initial
training of mathematics teachers. This study offers insights to improve the course unit for
future editions and identifies issues to consider for continuous teacher training in Portugal.
1 Introduction
Since the onset of the 21st century, integrating Information and Communication Technolo-
gies (ICT) into mathematics education has yielded significant benefits for both educators and
students. ICT have been shown to enhance student interest, motivation, and academic perfor-
mance, while also fostering lifelong learning and positive interpersonal interactions [1]. These
technologies have transformed traditional teaching methodologies, encouraging more active stu-
dent participation [2]. Furthermore, ICT integration extends beyond the educational sector into
technological and industrial domains [3]. Research consistently demonstrates that ICT-based
teaching is more effective than traditional approaches, particularly at the secondary education
level [4]. However, obstacles such as educators’ limited ICT proficiency, inadequate training,
and insufficient technical support persist [1]. Addressing these challenges necessitates enhanced
training programs and improved technical support systems [1, 3].
ICTs are driving a global transformation in education, reshaping how students learn and
aligning education with the demands of the digital workplace [5]. These technologies facili-
tate active, real-world connected learning and enable personalised instruction through adaptive
learning systems [6]. The integration of ICTs has significantly influenced teaching content,
methodologies, and administrative processes, redefining the roles of both teachers and students
[7]. Successful ICT integration is contingent upon factors such as funding, access to equipment,
software availability, and personnel expertise [7].
Recent studies highlight the importance of integrating Computational Thinking (CT) into
mathematics education to enhance problem-solving skills and prepare students for the digital
era [8, 9, 10]. CT fosters mathematical reasoning, but its implementation faces challenges,
primarily due to teachers’ limited understanding of CT and its application in mathematics
[11, 12]. Effective teacher training and the development of appropriate assessment tools and
learning platforms are crucial for maximising the benefits of CT integration [8]. This study seeks
to explore how future teachers perceive the use of these technological tools during the first year
of a Master’s degree in Teaching Mathematics for Basic and Secondary School Teachers in the
Department of Mathematics at the University of Coimbra, within the context of a curriculum
supportive of technology and CT integration.
2 Framework
This section outlines key sources to understand the integration of CT and technological tools
in training in-service and pre-service mathematics teachers, framed by educational theories and
models. Driskell et al. identified a significant gap in research on professional development
in mathematics education focused on technology, noting that from 1975 to 2015, only 21 out
of 1,210 studies addressed this area, despite longstanding advocacy from bodies such as the
National Council of Teachers of Mathematics [13]. During this period, lesson study emerged
as a promising approach for collaborative planning, observation, and reflection among teacher
networks, with increasing interest in its application to professional development in technology-
enhanced mathematics teaching [14, 15].
Follmer et al. recently developed a theory-based evaluation framework for lesson study
programs, which demonstrates how pragmatic constraints can be turned into opportunities in
theory-driven evaluations [16]. Liu et al. found that hands-on experiences with technology in
teacher training programs in the United States accelerated classroom adoption and integration
[17]. Large-scale initiatives, such as the cluster model in Australia, have successfully built
curriculum knowledge and enthusiasm for teaching mathematics, with sustainability supported
by professional development content, collective action, and leadership at multiple levels [18].
Integrating various technologies into pre-service mathematics teacher education has been
shown to enhance professional knowledge, improve subject knowledge, and foster positive atti-
tudes toward technology use [19, 20]. Incorporating technology-rich methods courses and field
experiences can develop Technological Pedagogical Content Knowledge (TPACK), shifting pre-
service teachers’ perspectives to view technology as a tool for deepening student understanding
rather than mere reinforcement [21, 22, 23]. However, preparing pre-service teachers to ef-
fectively facilitate student thinking, reasoning, and problem-solving with technology remains
challenging and requires further research [20]. Ranellucci et al. [24] found that pre-service
teachers’ readiness and intentions to use technology are strongly influenced by perceptions of
its usefulness, ease of use, supportive conditions, and social norms.
Theodorio’s study in South Africa emphasizes the importance of essential support for ef-
fectively integrating technology into teacher professional development programmes, including
digital and non-digital tools, technical support, and collaborative engagement among educators
and IT staff [25]. Furthermore, integrating Education for Sustainable Development (ESD) into
mathematics teacher training is highlighted as crucial, with a focus on contextualisation, ecolog-
ical and mediational competencies, and understanding the relationship between mathematics
and the environment [26].
2.1 Models to study acceptance of technology and Innovation
Rogers’ Diffusion of Innovations theory [27] provides a framework for understanding the adop-
tion of new technologies in educational settings, encompassing stages such as knowledge, persua-
sion, decision, implementation, and confirmation. Teacher beliefs and attitudes play a crucial
role in this process.
The Technology Acceptance Model (TAM), developed by Davis in 1989, is another widely
used framework that explains how users accept and use new technologies [28]. TAM posits that
perceived usefulness and ease of use are key factors influencing technology adoption decisions.
However, while TAM has high validity and is extensively applied in information systems re-
search, it should be used cautiously in multinational contexts due to cultural variations [28].
Despite its widespread use, TAM has faced critiques and has been modified to address its
limitations [29].
In the context of pre-service mathematics teachers, TAM has been employed to study their
intentions to use technology, revealing that perceived usefulness, ease of use, and attitude
significantly influence this intention [30]. Extended TAM models, incorporating variables such
as subjective norm, facilitating conditions, and Technological Pedagogical Content Knowledge
(TPACK), have shown a good fit, explaining significant variance in technology use intentions
in mathematics education [31]. Nonetheless, actual practice may be hindered by factors such
as accessibility and professional development needs [32].
2.2 Instrumental orchestration in pre-service mathematics teachers
Instrumental orchestration in mathematics education refers to the strategic use of technological
tools by teachers in their lessons, involving didactical configuration, exploitation mode, and
didactical performance [33]. Research on instrumental orchestration has identified five main
clusters: managing teaching complexity, designing living resources, teaching with technology,
adult learners, and interacting with computers [34]. Recent studies have adapted this framework
for online contexts [35] and extended it to instrumental meta-orchestration for teacher education
[36], highlighting the importance of carefully planning and arranging digital artefacts in teaching
environments.
In mathematics teacher training, instrumental orchestration has been shown to enhance
both content knowledge and pedagogical skills. Studies suggest that pre-service teachers’ use
of technology is linked to their endorsed sociomathematical norms, with a variety of norms
indicating successful technology integration [37]. Additionally, combining instrumental meta-
orchestration with the Lesson Study methodology has proven effective in developing pedagogical
competencies in pre-service teachers [38].
Nine types of class orchestrations have been identified [39], with further subdivisions in
the work-and-walk-by orchestration, including technical-demo, guide-and-explain, link-screen-
paper, discuss-the-screen, and technical-support categories [40]. Instrumental orchestrations are
particularly effective in ICT-supported environments, especially when using dynamic mathe-
matics software like GeoGebra [41, 42]. These findings underscore the potential of instrumen-
tal orchestration frameworks to improve technology integration and pedagogical knowledge in
mathematics teacher training programs.
2.3 Technology in teaching maths with a favourable curricula
The integration of technology in mathematics education has garnered significant research inter-
est globally, examining its impact on teacher professional development, student achievement,
and classroom practices. Longitudinal studies, such as those on the MathForward program,
indicate that technology integration can significantly improve students’ mathematics test scores
[43]. However, successful implementation in classrooms hinges on factors including teach-
ers’ professional identities and their capacity to adopt innovative practices [44]. Pre-service
teacher education programs are crucial in preparing educators for effective technology integra-
tion, though challenges remain regarding perceived difficulty and inefficiency [45].
In technologically rich mathematics classrooms, student interactions with digital tools can
be categorised as Master, Servant, Partner, or Extension-of-self, underscoring technology’s
potential to empower students both individually and collaboratively [46]. Digital resources can
enhance mathematics teaching and learning, yet they may also challenge teachers’ practices
[39]. In Portugal, technology integration is part of the Essential Learning of Mathematics
curriculum for basic education and will be included in secondary education from 2024/2025
[47]. Initial teacher training in technology use presents added challenges, as preconceptions
shaped by personal educational experiences and professional aspirations influence the adoption
of new pedagogical approaches [48].
3 Methods
The curricular unit “Computational Tools for the Teaching of Mathematics” (CTTM) is a
second-semester unit, following two units on specific didactics of mathematics and the History
of Mathematics. This study, conducted over the 2022-2023 and 2023-2024 academic years,
aims to explore two research questions: how students, as future teachers, perceive the use of
technology in mathematics education, and how strategies can be tested to enhance teaching
practices.
Across two cycles, students’ work in the CTTM was analysed—eight projects in the first cy-
cle (ST1 to ST8) and four in the second cycle (ST9 to ST12). In both cycles, various technologies
were integrated into first-semester subjects related to specific didactics in analysis and geom-
etry. These technologies aimed to improve teaching practices by facilitating future teachers’
instructional activities, thereby enhancing student learning. Students engaged with a range
of tools, including GeoGebra, Scratch, Python Blocks, Graphic Calculators, ASYMPTOTE,
and MathCityMaps apps. These tools were applied in tasks designed for both primary and
secondary education students, demonstrating their potential to support mathematics teaching
and learning.
3.1 First Cycle
In the CTTM, during the second semester of 2022/2023, students were required, for their fi-
nal assignment, to create a web page that included: i) a brief presentation of their academic
background; ii) evidence of having attended a MOC; iii) examples of relationships between
mathematics and art from the collection at https://www.europeana.eu/pt; iv) a cartoon
on a mathematical topic to celebrate International Mathematics Day; v) software they would
consider using in their future roles as mathematics teachers; vi) trails created with the Math-
CityMap application. These items were analysed and discussed in the following section. It is
important to note that changes were made to the assignments proposed to students in 2023/2024
based on this analysis. These web pages were individual projects, except for the last item, the
creation of a trail in the MathCityMap application, which was a group effort.
3.2 Second Cycle
In CTTM in the second semester of the 2023/2024, students were required to produce a reflec-
tive portfolio as their final assignment. This portfolio documented collaborative work conducted
in pairs, aiming to enhance learning and contribute to their final grades. Each student reflected
on the development of their collaborative work on an assigned theme, with the portfolio com-
prising both collaborative and individual components, including planning and reflection tools
for simulated lessons.
Each student simulated two lessons in the presence of at least one course instructor. The
topics were drawn from four available options, such as using Python for visualisation in Real
and Complex Analysis or Scratch for teaching geometry and algebra in essential mathematics
learning. Pairs collaborated during thematic sessions, receiving support in preparing lesson
plans. The simulated lessons, lasting 45 minutes, were presented with the instructor and peers
acting as learners, followed by a feedback session, were be filled an evaluation rubric for the
student acting as teacher and the supervisor.
The planning models and evaluation rubrics for the lessons were previously discussed among
all participants. The analysis of the reflective portfolio was conducted post-course, ensuring
the anonymity of the collected data and adhering to ethical research standards.
4 Results
4.1 First cycle
The analysis of results from eight students (ST1 to ST8) reveals diverse approaches to using
computational tools in mathematics education. Each student demonstrated engagement with
various digital resources, contributing to projects aimed at enhancing teaching practices.
ST1 exhibited proficiency in tools such as GeoGebra, Desmos, Kahoot, and Poly, and ac-
tively participated in the Europeana project, creating comics and celebrating Mathematics
International Day. ST2 focused on using Poly, mathematics-related comics, images, and com-
pleted an online course on active learning. ST3 used Desmos, participated in Europeana, and
completed an eTWINNING course, emphasising comic creation.
ST4 integrated GeoGebra, Desmos, Kahoot, and Poly into teaching, contributing to Euro-
peana and developing a MathCityMap resource, while also engaging in social network train-
ing. Similarly, ST5 utilised multiple tools, engaged with Europeana, and created a comic.
ST6 focused on applying these tools and collaborating in the Europeana project through an
eTWINNING course.
ST7 demonstrated advanced engagement by using GeoGebra, Desmos, Kahoot, and Poly,
developing a MathCityMap resource, creating a comic, and presenting a lesson plan incorporat-
ing Python. ST8 concentrated on developing detailed lesson plans with Python, significantly
contributing to the Europeana project, MathCityMap, and an eTWINNING project.
The students’ engagement with computational tools and interdisciplinary projects like Eu-
ropeana and MathCityMap demonstrates their capacity to integrate ICT into mathematics
teaching. Their involvement in creating comics, collaborative projects, and continuous profes-
sional development through platforms like eTWINNING and NAU highlights a commitment to
innovative and effective educational practices.
4.2 Second cycle
The analysis of the results from the four students, ST9 to ST12, are in progress. However con-
sidering the aim this study were analised the results getted by the dyads, ST9&ST10 portfolio
and ST11&ST12. The portfolios elaborated by students demonstrate a focused integration of
computational tools, specifically Python programming (ST9&ST10) to teach secondary math-
ematics, and Scratch (ST11&ST12) to teach basic mathematics . The students worked both
collectively in pairs and individually, with their efforts documented in detailed lesson plans and
reflective practices.
In the Portfolio elaborated by dyads ST9&ST10, collaborative part includes justifications
for lesson plans on ST9 “Functions Defined by Branche” and “Cubic and Quartic Functions”
and ST10 “Quadratic Functions” and “Complex Numbers,” emphasising their importance and
integration into the curriculum. Individual reflections address simulated lessons, highlighting
strengths and weaknesses based on peer and supervisor feedback. The lesson plans progress from
basic to complex topics, ensuring alignment with students’ prior knowledge. Initial lessons use
block programming tools like EduBlocks to introduce Python, while subsequent lessons tackle
complex topics, requiring higher programming proficiency (see Figure 1, at left).
The lesson plans effectively integrate technological tools (Python programming), pedagogi-
cal strategies (progressive complexity, visual aids), and mathematical content knowledge (func-
tions, polynomial equations, complex numbers). They are designed to enhance CT, algorithmic
processes, and logical reasoning. Hypothetical learning trajectories anticipate student difficul-
ties, such as challenges in defining piecewise functions, and suggest strategies like focusing on
the function’s vertex for better understanding.
Pre-service teachers critically reflect on their lessons, recognising areas for improvement
based on feedback. They emphasise the alignment of computational goals with mathemati-
cal learning objectives and the importance of clear, step-by-step instructions in programming
tasks. The portfolios of dyads ST9 & ST10 demonstrate a thoughtful integration of technology
in teaching, reflecting a deep understanding of how programming can elucidate mathematical
concepts, fostering CT and problem-solving skills. The reflections underscore a commitment
to continuous improvement, vital for effective teaching practice, with Python’s use as a teach-
ing tool being particularly significant given its increasing relevance in both educational and
professional domains.
The portfolio created by dyads ST11&ST12 includes lesson plans with justifications centred
on the organisation of lessons, integrating technological, pedagogical, and mathematical content
knowledge, designing hypothetical learning trajectories, and reflections from pre-service teach-
ers. ST11’s lesson plans target 8th-grade linear functions, employing Scratch and GeoGebra
to enhance understanding through problem-solving and collaboration. ST12’s plans focus on
the“Pythagorean Theorem” and “Isometries,” using Scratch to create scenarios that apply the
theorem and develop isometric transformation tasks. Both sets of plans aim to cultivate CT and
problem-solving skills. ST11 integrates technology with mathematical concepts, using Scratch
for visualising linear functions and GeoGebra for dynamic representations. ST12’s approach
utilises Scratch to craft interactive problems, emphasising CT and ensuring comprehension of
the theorem’s application (see Figure 1, at wright).
Figure 1: At left, Task 1 of 1st simulated lesson of ST10 - Resolution included in the lesson
plan. At wright, scratch scenarios proposed in lessons plans of ST12.
The Hypothetical Learning Trajectories (HLTs) developed by ST11 and ST12 reflect a
strategic approach to addressing potential student difficulties in understanding linear functions
and the Pythagorean Theorem. ST11 anticipates misconceptions in linear functions, incor-
porating scaffolded support to mitigate these challenges, while ST12 engages students with
the Pythagorean Theorem through storytelling, problem-solving, and Socratic dialogue using
Scratch. Both trajectories utilise interactive tools to enhance learning.
The reflections of ST11 and ST12 reveal diverse insights. ST11 highlights the effectiveness
of Scratch and GeoGebra in improving student engagement and understanding, while acknowl-
edging areas for improvement based on peer and supervisor feedback. ST12 reflects on the
impact of computational tools on students’ problem-solving skills and stresses the importance
of continuous instructional improvement.
The analysis of the portfolios of dyads ST11&ST12 demonstrates a robust integration of
computational tools in mathematics education, emphasising student engagement and under-
standing. The portfolios showcase distinct approaches: ST9&ST10 focus on programming
proficiency and algorithmic thinking, whereas ST11&ST12 prioritise interactive learning and
practical applications. Both dyads exhibit reflective practices and a commitment to refining
their teaching strategies, underscoring the transformative potential of computational methods
in mathematics education.
4.2.1 Computational Thinking in portfolio
The portfolio of dyads ST9&ST10 emphasises CT as fundamental for mathematical problem-
solving. Their approach utilises algorithmic processes, focusing on step-by-step instructions
and systematic problem-solving with Python and EduBlocks. This methodology fosters logical
thinking, particularly in quadratic functions and complex numbers. They emphasise decompo-
sition, breaking down complex concepts to build on existing knowledge. Abstraction is achieved
by modelling mathematical problems in Python, enabling generalisation. Automation is pro-
moted through Python for repetitive calculations, enhancing understanding and efficiency. Data
representation is facilitated by visual aids and dynamic programming tools, making abstract
concepts more tangible.
Conversely, the portfolio of dyads ST11&ST12 integrates CT through interactive problem-
solving using Scratch and GeoGebra. These tools enhance logical reasoning and iterative re-
finement. GeoGebra provides dynamic visualisations, aiding comprehension of mathematical
concepts like the Pythagorean theorem and isometries. Decomposition and abstraction guide
students to break down complex problems and model real-world scenarios. Pattern recogni-
tion is developed by identifying relationships within mathematical problems, while iterative
development fosters continuous improvement through solution refinement.
Both portfolios underscore the importance of CT in mathematics education, with ST9&ST10
focusing on algorithmic processes and systematic problem-solving, and ST11&ST12 prioritising
interactive problem-solving and visualisation.
5 Discusion
Considering the results of 1st cycle, the integration of ICT in mathematics education, as evi-
denced by the students’ use of GeoGebra, Desmos, and other tools, supports the enhancement
of student interest, motivation, and performance [1, 2]. The students’ engagement with projects
like Europeana and MathCityMap reflects the transformative potential of ICT in making learn-
ing more interactive and connected to real-life scenarios [6].
The emphasis on continuous professional development, highlighted by the students’ comple-
tion of online courses, resonates with the findings of Driskell et al. [13] regarding the necessity
of professional development in the successful integration of educational technologies. The cre-
ation of comics and collaborative projects showcases the students’ creative and collaborative
efforts, supporting the notion that ICT can foster positive interactions and lifelong learning
[3, 7] .
The advanced lesson planning by ST7 and ST8, involving Python, aligns with the research
suggesting that integrating CT can enhance problem-solving skills and mathematical think-
ing [8, 9, 10]. This demonstrates the potential of well-designed ICT integration in preparing
students for the digital age.
The students’ work, in the first cycle, demonstrates a strong engagement with various
technological tools and projects, supporting the integration of ICT in mathematics education.
Two students reveal more commitment and present lesson planning highlight the potential
for enhancing both teaching practices and student outcomes. Further research and continued
emphasis on professional development are essential for overcoming the challenges associated
with ICT integration [1].
The use of the Technology Acceptance Model (TAM) was not feasible in this study, due
to the small number of participants. It should be noted that research conducted with this
model indicated that the original framework proposed by Davis in 1987 [49] needed to be
expanded with the introduction of Technological Pedagogical Content Knowledge (TPACK)
[31]. As the results of the first cycle demonstrate, only two students presented final lesson
plans that addressed didactic concerns regarding the use of technology and were sensitive to
the cross-cutting theme of CT.
The analysis of these results led to a reformulation of the tasks requested in the CTTM
curricular unit, including a mandatory collaborative work, the elaboration of detailed lesson
plans, simulations of classes with the entire group, joint analysis of all actions that took place
during these classes, and the development of a reflective portfolio in dyads containing both
individual and group work. Moreover, this strategy was implemented in all subjects of the
second semester of the Master’s degree in Teaching Mathematics for Basic and Secondary
School Teachers, under the responsibility of the Department of Mathematics.
In fact, the 2nd cycle present substantial differences, the analysis of the portfolios devel-
oped by students ST9 to ST12 underscores their integration of technology within the context of
established educational theories and practices. This discussion draws upon key frameworks, in-
cluding Rogers’ Diffusion of Innovations theory, the Technological Pedagogical Content Knowl-
edge (TPACK) model, and the concept of instrumental orchestration in technologically enriched
mathematics classrooms.
The collaborative efforts of ST9 and ST10 reflect a methodical approach to teaching ad-
vanced topics in secondary mathematics, specifically functions and complex numbers. Their
lesson plans demonstrate a clear alignment with curriculum standards and a thoughtful justifi-
cation for the selection of topics. This aligns with Rogers’ Diffusion of Innovations theory [27],
which highlights the importance of understanding how new educational practices and technolo-
gies are adopted within teaching contexts. By incorporating Python programming into their
plans, these students exemplify a commitment to integrating innovative tools into their teaching
practices.
The structured progression from simpler functions to more complex concepts illustrates the
pedagogical strategies outlined in the TPACK framework [21]. The inclusion of hypothetical
learning trajectories indicates a proactive approach to addressing potential student misconcep-
tions, which is essential for effective teaching practice. For instance, ST9 and ST10’s focus on
the vertex of piecewise functions as a critical learning point aligns with educational models that
advocate for anticipatory teaching strategies.
The portfolio of dyads ST9 & ST10 also reflect the principles of instrumental orchestra-
tion, particularly the Technical-demo and Guide-and-explain orchestrations. By demonstrating
Python techniques and guiding students through complex mathematical concepts, they en-
hance students’ understanding and engagement. This approach is consistent with findings that
emphasize the importance of careful planning and arrangement of digital artifacts in teaching
environments [37]. Also, The reflective practices displayed in portfolio of dyads ST9 & ST10 fur-
ther enhance their pedagogical development. Both students engage in critical self-assessment,
identifying strengths and areas for improvement. This reflective process is vital, as research
indicates that teacher beliefs and attitudes significantly impact the adoption of educational
technologies [13]. By emphasizing the alignment of computational goals with mathematical
learning objectives, ST9 and ST10 highlight the necessity for clarity and coherence in lesson
planning.
The ST11 & ST12 portfolio dyad focus on foundational mathematics topics for 8th-grade
students, specifically linear functions and the Pythagorean theorem. The integration of Scratch
and GeoGebra into their lesson plans showcases a dynamic approach to teaching, facilitating
student engagement through interactive problem-solving. This approach align with the find-
ings of recent studies indicating that engaging with various technologies can enhance pre-service
teachers’ subject knowledge and attitudes towards technology [19]. The emphasis on collabo-
rative learning strategies in ST11’s lesson plans is indicative of a pedagogical philosophy that
values active participation. The use of Scratch to visualize linear functions allows students to
interact with mathematical concepts in a tangible way, thereby fostering deeper comprehen-
sion. This aligns with the educational theories advocating for the integration of technological
tools to enhance student learning experiences. Similarly, ST12’s focus on the Pythagorean
theorem and isometries reflects an innovative pedagogical approach. The incorporation of sto-
rytelling in conjunction with Scratch fosters an exploratory learning environment, encouraging
critical thinking and problem-solving skills. The application of Socratic dialogue not only pro-
motes student engagement but also aligns with current educational theories that advocate for
dialogue-rich classrooms.
The focus on hypothetical learning trajectories in ST11&ST12 dyad portfolio is crucial for
identifying and addressing potential student misconceptions. This anticipatory strategy is sup-
ported by the literature on teacher professional development, which underscores the importance
of preparing educators to facilitate student thinking and reasoning [20]. Their lesson plans also
demonstrate the use of various orchestrations such as Discuss-the-screen and Link-screen-board,
which facilitate a deeper understanding of mathematical concepts through interactive and visual
methods.
Considering the integration of CT, portfolio of ST9&ST10 dyad underscore the significance
of CT in mathematics education, albeit through different lenses. The emphasis on algorithmic
processes and systematic problem-solving in ST9 and ST10’s work illustrates their understand-
ing of the cognitive processes involved in programming and mathematics. Their approach aligns
with the TPACK framework, as they effectively integrate technology to foster students’ logical
reasoning and problem-solving skills [50]. Conversely, portfolio ST11&ST12 dyad highlight in-
teractive problem-solving and visualization, utilizing tools that encourage students to engage
deeply with mathematical principles. Their work exemplifies the key components of CT, includ-
ing decomposition and pattern recognition, which are essential for mastering complex mathe-
matical concepts. This is consistent with recent findings that suggest integrating technology
in teacher education can significantly enhance pre-service teachers’ professional knowledge and
teaching effectiveness [18].
6 Final Remarks
In comparing the two cycles, the integration of technology and CT in mathematics education
revealed both advancements and challenges. The initial cycle demonstrated the effective use
of ICT tools such as GeoGebra and Desmos, which enhanced student engagement and per-
formance. Nonetheless, only two students developed lesson plans that adequately addressed
the integration of technological and CT, indicating a need for further refinement and profes-
sional development. The second cycle showed significant progress, with portfolios reflecting a
robust integration of technology through established educational theories, including TPACK
and instrumental orchestration. Students adeptly employed Python programming, Scratch,
and GeoGebra to teach various mathematics topics, demonstrating comprehensive lesson plan-
ning and pedagogical strategies that foster CT. These advancements align with the Portuguese
mathematics curricula, underscoring the systematic use of technology and the development of
CT as integral components. The second cycle highlighted a more refined approach to tech-
nology integration and emphasised the importance of ongoing professional development and
reflective practice.
Future research should track students and future teachers through their induction year
after completing the CTTM course to analyse their classroom practices. Simulated classes
offered valuable insights, yet actual student reactions in mathematics classrooms may differ
significantly.
Acknowledgments
This research was supported by the Centre for Research and Innovation in Education (inED)(https:
//doi.org/10.54499/UIDP/05198/2020), and the Center for Mathematics, University of Coim-
bra (https://doi.org/10.54499/UIDB/00324/2020), through the FCT - Fundação para a
Ciência e a Tecnologia, I.P..
References
[1] Nur Afiqah Zakaria and Fariza Khalid. The benefits and constraints of the use of infor-
mation and communication technology (ict) in teaching mathematics. Creative Education,
07(11):1537–1544, 2016.
[2] Nchimunya Chaamwe. Notice of retraction: Integrating icts in the teaching and learn-
ing of mathematics: An overview. In 2010 Second International Workshop on Education
Technology and Computer Science. IEEE, March 2010.
[3] Kaushik Das. Role of ict for better mathematics teaching. Shanlax International Journal
of Education, 7(4):19–28, September 2019.
[4] Amina Safdar. Effectiveness Of Information And Communication Technology (Ict) In
Mathematics At Secondary Level. PhD thesis, International Islamic University Islamabad,
Islamabad, Pakistan, 2013.
[5] Nibedita Boruah. Impact of ict in education. International journal of health sciences, page
1818–1822, April 2022.
[6] Saïd Assar. Information and Communications Technology in Education, page 66–71. El-
sevier, 2015.
[7] Xinran Wang. An overview of ict and educational change: Development, impacts, and
factors. Journal of Contemporary Educational Research, 6(8):1–8, August 2022.
[8] Ilham Muhammad, Husnul Khatimah Rusyid, Swasti Maharani, and Lilis Marina Angraini.
Computational thinking research in mathematics learning in the last decade: A bibliomet-
ric review. International Journal of Education in Mathematics, Science and Technology,
12(1):178–202, October 2023.
[9] Erick John Fidelis Costa, Livia Maria Rodrigues Sampaio Campos, and Dalton Dario
Serey Guerrero. Computational thinking in mathematics education: A joint approach to
encourage problem-solving ability. In 2017 IEEE Frontiers in Education Conference (FIE).
IEEE, October 2017.
[10] Maria Kallia, Sylvia Patricia van Borkulo, Paul Drijvers, Erik Barendsen, and Jos Tol-
boom. Characterising computational thinking in mathematics education: a literature-
informed delphi study. Research in Mathematics Education, 23(2):159–187, January 2021.
[11] Siri Krogh Nordby, Annette Hessen Bjerke, and Louise Mifsud. Primary mathemat-
ics teachers’ understanding of computational thinking. KI - Künstliche Intelligenz,
36(1):35–46, February 2022.
[12] Janice Teresinha Reichert, Dante Augusto Couto Barone, and Milton Kist. Computational
thinking in k-12: An analysis with mathematics teachers. Eurasia Journal of Mathematics,
Science and Technology Education, 16(6), March 2020.
[13] Shannon O Driskel, Sarah B Bush, Margaret L Niess, David Pugalee, Christopher R
Rakes, and Robert N Ronau. Research in mathematics educational technology: Trends
in professional development over 40 years of research. In T. G. Bartell, K. N. Bieda,
R. T. Putnam, K. Bradfield, and H. Dominguez, editors, North American Chapter of the
International Group for the Psychology of Mathematics Education, pages 656–662, East
Lansing, MI: Michigan State University, June 2015. PME-NA, ACM Press.
[14] Jodie Hunter and Jenni Back. Facilitating sustainable professional development through
lesson study. Mathematics Teacher Education and Development, 13(1):94—-114, March
2013.
[15] Jan D. Vermunt, Maria Vrikki, Nicolette van Halem, Paul Warwick, and Neil Mercer.
The impact of lesson study professional development on the quality of teacher learning.
Teaching and Teacher Education, 81:61–73, May 2019.
[16] D. Jake Follmer, Randall Groth, Jennifer Bergner, and Starlin Weaver. Theory-based
evaluation of lesson study professional development: Challenges, opportunities, and lessons
learned. American Journal of Evaluation, 45(2):292–312, August 2023.
[17] Feng Liu, Albert D. Ritzhaupt, Kara Dawson, and Ann E. Barron. Explaining technology
integration in k-12 classrooms: a multilevel path analysis model. Educational Technology
Research and Development, 65(4):795–813, October 2016.
[18] Merrilyn Goos, Anne Bennison, and Robin Proffitt-White. Sustaining and scaling
up research-informed professional development for mathematics teachers. Mathematics
Teacher Education and Development, 20(2):133–150, May 2018.
[19] Helena Rocha. Pre-service Teachers’ Knowledge and the Use of Different Technologies to
Teach Mathematics, page 505–515. Springer Singapore, November 2021.
[20] Rongjin Huang and Rose Mary Zbiek. Prospective Secondary Mathematics Teacher Prepa-
ration and Technology, page 17–23. Springer International Publishing, October 2016.
[21] Matthew J. Koehler, Punya Mishra, and Kurnia Yahya. Tracing the development of teacher
knowledge in a design seminar: Integrating content, pedagogy and technology. Computers
& Education, 49(3):740–762, 2007.
[22] Melike Yigit. A review of the literature: How pre-service mathematics teachers develop
their technological, pedagogical, and content knowledge. International Journal of Educa-
tion in Mathematics, Science and Technology, 2(1), March 2014.
[23] Shannon O. Driskell, Sarah B. Bush, Robert N. Ronau, Margaret L. Niess, Christo-
pher R. Rakes, and David K. Pugalee. Handbook of Research on Transforming Mathe-
matics Teacher Education in the Digital Age, chapter Mathematics education technology
professional development: Changes over several decades., pages 107–136. IGI Global, 2016.
[24] John Ranellucci, Joshua M. Rosenberg, and Eric G. Poitras. Exploring pre-service teachers’
use of technology: The technology acceptance model and <scp>expectancy–value</scp>
theory. Journal of Computer Assisted Learning, 36(6):810–824, June 2020.
[25] Adedayo Olayinka Theodorio. Examining the support required by educators for successful
technology integration in teacher professional development program. Cogent Education,
11(1), January 2024.
[26] Chia Shih Su, Danilo Díaz-Levicoy, Claudia Vásquez, and Chuan Chih Hsu. Sustainable
development education for training and service teachers teaching mathematics: A system-
atic review. Sustainability, 15(10):8435, May 2023.
[27] Everett M. Rogers, Arvind Singhal, and Quinlan. Margaret M. Diffusion of innovations.
In Don W. Stacks, Michael B. Salwen, and Kristen C. Eichhorn, editors, An Integrated
Approach to Communication Theory and Research, pages 415–433. Routledge, 3 2019.
[28] Patrícia Silva. Davis’ Technology Acceptance Model (TAM) (1989), page 205–219. IGI
Global, 2015.
[29] Joseph Bradley. If We Build It They Will Come? The Technology Acceptance Model, page
19–36. Springer New York, August 2011.
[30] Timothy Teo and Paul Van Schalk. Understanding technology acceptance in pre-service
teachers: A structural-equation modeling approach. The Asia-Pacific Education Re-
searcher, 18(1), June 2009.
[31] Timothy Teo, Verica Milutinović, Mingming Zhou, and Dragić Banković. Traditional vs.
innovative uses of computers among mathematics pre-service teachers in serbia. Interactive
Learning Environments, 25(7):811–827, June 2016.
[32] Seyum Getenet. Mathematics teacher educators’ and pre-service teachers’ beliefs about the
use of technology in teaching in an african university. International journal of innovative
interdisciplinary research, 2:9–20, 01 2013.
[33] Paul Drijvers, Michiel Doorman, Peter Boon, and Sjef van Gisbergen. Instrumental orches-
tration: theory and practice. In Proceedings of the sixth congress of the European Society
for Research in Mathematics Education, pages 1349–1358, 2010.
[34] Paul Drijvers, Sebastian Grauwin, and Luc Trouche. When bibliometrics met mathematics
education research: the case of instrumental orchestration. ZDM, 52(7):1455–1469, May
2020.
[35] Matheus Souza de Almeida and Elisângela Bastos de Mélo Espíndola. Online instrumental
orchestration: perspectives for the initial education of pre-service mathematics teachers in
the context of the supervised teaching practice. Em Teia | Revista de Educação Matemática
e Tecnológica Iberoamericana, 14(1):280, April 2023.
[36] Rosilângela Lucena, Verônica Gitirana, and Luc Trouche. Teacher education for integrating
resources in mathematics teaching: contributions from instrumental meta-orchestration.
The Mathematics Enthusiast, 19(1):187–221, January 2022.
[37] Ruya Savuran and Hatice Akkoç. Examining pre-service mathematics teachers’ use of tech-
nology from a sociomathematical norm perspective. International Journal of Mathematical
Education in Science and Technology, 54(1):74–98, October 2021.
[38] Francisco Eteval da Silva Feitosa, Sonia Barbosa Camargo Igliori, and Luc Trouche. The
teaching professional knowledge formation by an instrumental meta-orchestration. PNA.
Revista de Investigación en Didáctica de la Matemática, 18(1):35–56, October 2023.
[39] Paul Drijvers, Sietske Tacoma, Amy Besamusca, Michiel Doorman, and Peter Boon. Digital
resources inviting changes in mid-adopting teachers’ practices and orchestrations. ZDM,
45(7):987–1001, August 2013.
[40] Qi Tan and Zhiqiang Yuan. A professional development course inviting changes in preser-
vice mathematics teachers’ integration of technology into teaching: the lens of instrumental
orchestration. Humanities and Social Sciences Communications, 11(1), July 2024.
[41] Gülay Bozkurt and Candas Uygan. Lesson hiccups during the development of teaching
schemes: a novice technology-using mathematics teacher’s professional instrumental gen-
esis of dynamic geometry. ZDM, 52(7):1349–1363, July 2020.
[42] Dustin Schiering, Stefan Sorge, Steffen Tröbst, and Knut Neumann. Course quality in
higher education teacher training: What matters for pre-service physics teachers’ content
knowledge development? Studies in Educational Evaluation, 78:101275, September 2023.
[43] Ali Bicer and Robert M. Capraro. Longitudinal effects of technology integration and
teacher professional development on students’ mathematics achievement. EURASIA Jour-
nal of Mathematics, Science and Technology Education, 13(3), December 2016.
[44] Merrilyn Goos. Technology Integration in Secondary School Mathematics: The Develop-
ment of Teachers’ Professional Identities, page 139–161. Springer Netherlands, September
2013.
[45] Donna F. Berlin and Arthur L. White. A longitudinal look at attitudes and perceptions re-
lated to the integration of mathematics, science, and technology education. School Science
and Mathematics, 112(1):20–30, January 2012.
[46] Vincent Geiger. LEARNING MATHEMATICS WITH TECHNOLOGY FROM A SO-
CIAL PERSPECTIVE: A STUDY OF SECONDARY STUDENTS’ INDIVIDUAL AND
COLLABORATIVE PRACTICES IN A TECHNOLOGICALLY RICH MATHEMATICS
CLASSROOM. PhD thesis, University of Queensland Library, 2008.
[47] Ministério da Educação /Direção-Geral da Educação, Portugal. Base nacional
comum curricular; educação é a base. https://www.dge.mec.pt/noticias/
aprendizagens-essenciais-de-matematica, 2021. Accessed: 2022-02-03.
[48] José Manuel Dos Santos, Jaime Carvalho e Silva, and Zsolt Lavicza. Geogebra classroom na
formação inicial de professores de matemática. In Ana Amélia A. Carvalho, Eliane Schlem-
mer, Manuel Area, Célio Gonçalo Marques, Idalina Lourido Santos, Daniela Guimarães,
Sónia Cruz, Idalina Moura, Carlos Sousa Reis, and Piedade Vaz Rebelo, editors, Atas do
6ºEncontro Internacional sobre Jogos e Mobile Learning, pages 162–172, Coimbra, May
2024. Universidade de Coimbra, Centro de Estudos Interdisciplinares.
[49] Fred Davis. A technology acceptance model for empirically testing new end-user informa-
tion systems : theory and results. PhD thesis, Sloan School of Management, Massachusetts
Institute of Technology, Massachusetts, USA, 1986.
[50] Shuchi Grover and Roy Pea. Computational thinking: A competency whose time has come.
Computer science education: Perspectives on teaching and learning in school, 19:1257–
1258, 2018.
