Helena Rocha

The teacher’s central role in technology integration and the challenges of that integration emphasise the need for a deeper understanding about the teacher’s knowledge required to teach with technology. Based on previous work and a systematic literature review, we identified three knowledge models often used: TPACK, KTMT and
PTK. The goal of this paper is to discuss the similarities and differences between these knowledge models and present a Global Model. This Global Model is not a new model. On the contrary, it is a model developed based on the existing models and intending to integrate in a single model the knowledge domains considered in the different existing models. The Global Model highlights the common domains considered and the common roots for the three models, but it also makes explicit the differences, mostly related to the understanding of the domains or even to the domains considered, and also to the way how the knowledge’s development is conceived.

The challenges of disciplinary integration in teaching practice, especially in the implementation of tasks that promote interdisciplinary approaches, highlight the importance of teachers' professional knowledge in this process. Using a naturalistic approach, this study aims to characterize the professional knowledge of a physics teacher when adopting an interdisciplinary approach using technology. The results of the research revealed that the teacher mobilizes different types of knowledge, showing a differentiated knowledge of the most appropriate mathematical application to teach a given piece of content, and knowledge of how pedagogical strategies can be aided through different applications using technology.

In this chapter, we examine the role of policies and other factors affecting digital technology (DT) integration in mathematics education. In particular, we develop a cross-national analysis of the impact on DT implementation in four countries: two countries in Europe (Italy and Portugal) and two countries in Latin America (Colombia and Mexico). We analyze the role that policies, political changes, reforms, curricula, educational organization and systems, sociocultural aspects, and teachers’ training, knowledge, and beliefs play toward possible DT implementations. We observe that there is a discourse in policies to promote digital technologies’ use, but in practice the availability and integration of such resources in mathematics classrooms is still scarce. We also note that the efforts done during the pandemic did not change this, promoting general ICT use, rather than DT resources that might enhance mathematics teaching and learning.

This study was conducted while 9th grade students learn to solve inequalities and seeks to understand their approach to solving problems with a real-life context. Specifically, the aim is to understand: (1) What are the main characteristics of the students’ approaches to the proposed problems? (2) What is the impact of the real context on the students’ resolutions? A qualitative and interpretative methodology is adopted, based on case studies, with data collected through documentary collection and audio recording of discussions between a pair of students while solving problems. The main conclusions suggest a trend to approach problems without establishing immediate connections with what was being done in the classroom, with students’ decisions being essentially guided by criteria of simplicity. The real context of the problems seems to have the potential to develop in students a more integrated mathematics, focused on understanding and not so much on the repetition of mechanical and meaning-independent procedures. The students’ familiarization with the context in question is one of the aspects highlighted by this study.

This study seeks reflection on the approaches of 11th grade students to Linear Programming problems, discussing the approaches taken at different moments of the teaching process. It aims to analyze:
How is the students’ mathematical competence characterized in relation to problemsolving;
What differences can be identified in the resolutions at different moments of the teaching and learning process.
We adopt a qualitative and interpretative methodology, analyzing the approaches of two pairs of students with different mathematical backgrounds. The analysis is guided by P´olya’s stages of solving a problem and aspects of the understanding of mathematical competence. The results show different approaches to the problems depending on the teaching moment and different competences. The mathematical background impacts the students’ success when they implement routine procedures, however it does not seem to determine the students’ competence to reason about a problem.

The potential of technology to enhance the learning of mathematics is widely acknowledged. However, realizing this potential requires teachers to possess the knowledge to effectively integrate it into their teaching practices. This study aims to characterize the mathematics teacher's knowledge when using different technologies in teaching and learning. It also aims to study the existence of relationships between the specific domains of the Knowledge for Teaching Mathematics with Technology – KTMT and the integration of graphing calculator and Excel. The methodology adopted is qualitative with an interpretative approach, using a case study of Mathematics teacher in the 11th grade (16-17 years old) with extensive experience in the use of technology. This study shows that different KTMT knowledge is mobilized, according to the technology chosen and the specific characteristics of each of the technologies used influenced the teacher's pedagogical choices.

This study aims to discuss similarities and differences between knowledge models focusing on technology, developing the Global Model. This is not a new model, but rather a contribution to the articulation between different conceptual frameworks.

Este estudo visa discutir semelhanças e diferenças entre modelos de conhecimento com foco na tecnologia, desenvolvendo a partir destes o Modelo Global. Este não é um novo modelo, mas antes um contributo para a articulação entre diferentes quadros conceptuais.

The potential of digital technologies for teaching and learning mathematics is widely recognized and teachers’ knowledge is one of the elements impacting their integration. Several authors have intended to characterize the teachers’ knowledge required and developed several models, being TPACK one of these models. In this study, we seek to conduct a systematic review of the research on the integration of digital technologies by mathematics teachers based on the TPACK model. Specifically, we intend to answer the following research questions: (1) What are the main methodological options adopted? (2) How is the framework operationalized/used in the studies? The review was based on a search in the Scopus database and resulted in the identification of 10 relevant documents. The analysis suggests a prevalence of qualitative approaches, but a strong use of questionnaires; and an integration of the model with other frameworks, namely the developmental model of TPACK.

Teachers’ knowledge plays a central role in technology integration. In this study we analyze situations, where there is some divergence between the mathematical results and the information offered by the graphing calculator (lack of mathematical fidelity), putting the focus in the teachers and in their approaches. The goal of this study is to analyze, in the light of knowledge for teaching mathematics with technology (KTMT) model, the teachers’ professional knowledge, assuming the situations of lack of mathematical fidelity as having the potential to reveal some characteristics of their knowledge. Specifically, considering the teaching of functions at 10th grade (age 16), we intend to analyze: (1) What knowledge do the teachers have of technology and of its mathematical fidelity? (2) What can the teachers’ options related to situations of lack of mathematical fidelity tell us about their knowledge in other KTMT domains? The study adopts a qualitative and interpretative approach based on the case studies of two teachers. Data were collected by interviews and class observation, being the analysis guided by the KTMT model. The main result points to the relevance of the mathematics and technology knowledge. However, there is evidence of some difficulties to integrate the information provided by the technology with the mathematics, and also of some interference of the teaching and learning and technology knowledge, and specifically of the knowledge related to the students. This suggests that the analysis of the teachers’ actions in relation to situations of lack of mathematical fidelity, can be useful to characterize their KTMT.

This study aims to understand the functional thinking of 10th-grade students while studying functions. Specifically, we intend to answer the following research questions: what are the functional thinking processes used by 10th-grade students when studying functions? What difficulties do students present while learning functions? In view of the nature of this research objective, we adopted a qualitative and interpretative approach. In order to answer these questions, data were collected from the written records produced by the students while solving the proposed tasks, from records of the oral interactions during discussions and from a questionnaire. The results show that functional thinking processes were implicit in the resolution of the tasks proposed to the students. The students expressed an understanding of how the variables were related, presenting evidence of their functional thinking while working on the new concepts represented by the functions addressed in the proposed tasks. Some students expressed difficulties in interpreting the different types of representations associated with the functions, in retaining the necessary information from a graphical representation that would help them to draw conclusions and establish correspondences, in explaining functional relationships, and in interpreting the information provided by algebraic expressions. These difficulties can reduce the recognition of the relationships between variables and their behavior in the different representations, becoming an obstacle to learning for some students.