In the curriculum (Læreplanverket for kunnskapsløftet 2020) (Utdanningsdirektoratet, 2019) exploration and problem solving is one of the core elements in mathematics. This shows the importance of being able to facilitate problem solving in the mathematics teaching. We therefore wanted to investigate what experiences teachers have with facilitating problem solving in the mathematics teaching. For that reason, we designed the following research problem:

What experiences do six teachers have with adapting for problem solving in the mathematics teaching?

The master’s thesis is divided into three research questions that is going to help us address the research problem. The research questions focus on how the teachers understand problem solving, how the teachers structure the teaching when they facilitate problem solving, and what possibilities and challenges the teachers experience while facilitating problem solving in the mathematics teaching.

To address the research problem, a qualitative study has been conducted that included semi-structured interviews with six teachers. The teachers were experienced mathematics teachers with experience in facilitation problem solving. We conducted a theme-centred analysis to highlight the different experiences among the six teachers. The theme-centred analysis is divided into our three research questions.

The analysis showed both similarities and differences between the teachers within the three research questions. Several of the teachers expressed similar understandings of problem solving, but there were also some distinct differences in how the teachers explained their understanding of problem solving. We identified five characteristics with the teachers’ understanding of problem solving. How the teachers structure the teaching are mainly about how they introduce a problem, how they guide the pupils when they solve problems and how they end the lesson with problem solving. The teachers shared some similar experiences, but something that vary between the teachers is whether they facilitate discussion before and after the pupils solve problems or not. Time pressure is something several of the teachers describes as a challenge. Some teachers described differentiated teaching (norsk: tilpasset opplæring) and the pupils’ cooperation as a possibility, and some of the teachers saw it as a challenge when facilitating the teaching for problem solving.

This master's thesis examine how Tinkercad Codebloks can be used in mathematics. Forsström and Kaufmann (2018) and Dolonen et al. (2019) point to the need of more research related to programming in schools. This study wants to contribute experiences on how programming can be taught in mathematics, what reflections have been made around them, what potentials can be linked to programming and how to utilize them.

I have chosen to research the following thesis: How does Tinkercad and teachers facilitate students' argumentation, and what potential is there for developing argumentation through Tinkercad? To answer these research questions a case study was conducted. The study is based on two pairs of students at 7th grade and their work with programming pencil sharpener containers in Tinkercad Codebloks. This lesson was the students first meeting with Tinkercad. I have implemented participatory observation in my case-study. This was recorded with a video camera, as well as screenrecording. Based on the arguments of the students and teachers, a model inspired by Conner et al. (2014) for collective argumentation has been used. The models are further analyzed using a framework of three categories inspired by Lavys (2006) and Mason et al. (2011). The framework sees the student's argumentation in the light of algebraic thinking. The three categories are visual argument, arithmetic argument, and function argument.

This study shows signs of developing the student's arguments. The visual feedback in Tinkercad supported the students´ argumentation in some cases. The visual feedback in Tinkercad Codebloks was in other cases insufficient. This may have led the students to feel the need to argument arithmetically. The lack of visual support also led to some misconceptions. My study sheds light upon the fact that the teacher can contribute as support as a more knowledgeable other together with Tinkercad. The teachers supported the students to make mathematical arguments. The teacher helped the students when they were stuck in different ways, so that they were able to move on. The study also includes suggestions on how to utilize the potential in the situations that arise. In the potential it is suggested to let students evaluate their own statements. In the potential, emphasis is placed on letting the students evaluate their own statements, as well as how the teacher facilitates reflection when visual arguments are not sufficient and which methods are most suitable for solving the problem in different cases.

Norway is becoming more and more multilingual. Which reflects student diversity on the school bench. The multilingual aspect is influenced by people immigrating to Norway, with other language backgrounds. The aspect is also influenced by society's technological changes, where students have access to the whole world at their fingertips. The technological changes have brought the phenomenon known as gaming. Majority of children and teens today game in their spare time. This could help us adults at school to create relevant math problems for students. In 2020 the curriculum was renewed. The curriculum is supposed to be in line with the society we live in now, and with the future society the students will enter the work force in. In the new curriculum there is something called core elements in mathematics, some of the core elements include the usage of language in mathematics. For this reason, this assignment will answer the topic question How do multilingual students in the seventh grade use mathematical language, while using Minecraft: edu as a learning tool? To answer this topic question, two research questions have been formulated. The first research question is: How do multilingual students use communication, while using Minecraft: edu? The second research question is: What kind of mathematical argument do multilingual students use, while using Minecraft: edu?

To research these questions, Toulmin's argumentation model was used to analyse the arguments students came up with. Bishop's six activity forms were also used as a framework to look at the mathematical aspect of the students' conversations. Various literatures were also found, that were linked to multilingualism and multilingual students in mathematics, mathematical language, and argumentation, but also literature about using Minecraft and gaming in classroom teaching. This research is in collaboration with the LATACME project at Western Norway University of Applied Sciences. The data in this thesis is a video recording of two student groups. Access to the video recording was provided by the LATACME project. This qualitative research method makes me a complete observer. The students in the video are working with Minecraft as a learning tool. The students use mathematical language to communicate ideas and methods, and in building figures. Students also use mathematical language to argue with themselves and fellow students. Students used argumentation while calculating, in constructions and while filling in the TNT blasted areas.The different assignments led to variations in how the argumentation and communication were constructed and composed by the students during the observation.

The purpose of this thesis is to search for similarities between the teaching materials published by programming tools that can be used to apply computational thinking in schools. I consider this research valuable since the teachers’ competency on the field has been frequently disputed, especially since the new curriculum was released in 2020. I hope my analysis can help other teachers or educational institutions in the process of evaluating and choosing the most suitable tool to implement in their classrooms.

My thesis is: What are the characteristics of the teaching material of programming tools in mathematics and how do they apply computational thinking?

To answer my thesis, I have chosen to analyze nine teaching materials belonging to three separate tools. I will do so by using a framework based on Benton et al. (2016) and Brennan and Resnick (2012). These two frameworks combined with Charalambous et al. (2010) makes it so that I can analyze the tool both widely and deeply. This helps me with getting the point of the framework to come across in an appropriate way.

The tools I am looking in to is Kodeskolen, KUBO and Sphero. My findings show a lot of variation between them, but there are some common grounds considering computational thinking. All of them apply social cultural learning, even though they show different approaches. They are similar in the way that they can be used as an introduction to computational thinking, but they differ when it comes to how versatile they are, in the sense that some include several mathematical topics. The teaching materials also differ when it comes to how and to what extent they convey information and the teacher’s role.

The fact that the mathematics lessons take place in a language different from the students’ mother tongues is not a new phenomenon. However, the Medium of Instruction (MoI) may vary from country to country, depending on what is politically adopted. English is often used, which is also the case of this project. By focusing on Tanzanian secondary school students, the purpose of the project is to elucidate how the students work with mathematics when the MoI is different from their mother tongues. Additionally, it has been of interest to investigate how the students’ other languages, hence mother tongue or another language used at home, are included, or excluded in the mathematics teaching and learning.

In an African context, the research field has mainly focused on how the MoI may influence the students’ benefit from the mathematics lessons when the language used is different to the students’ home languages. However, less research is aimed specifically at the Tanzanian mathematics classroom. The project thus contributes to the development of the research field by examining how languages are used in mathematics teaching in Tanzania. The data material consists of ten qualitative interviews, of which eight were conducted with Grade 10 students and two with mathematics teachers. Additionally, 48 students solved two mathematical word problems, where only one of the tasks took the students’ experiences into account.

The data was analyzed using Modes of Belonging, a theoretical framework developed by Wenger (1998), which considered learning to take place in a social community of practice. Tanzania is linguistically diverse. English is, however, the only allowed MoI in secondary school. Nevertheless, Kiswahili, the MoI at primary level, was considered necessary in the mathematics lessons. This to make sense of English words and phrases, as well as what the mathematics teacher had explained in English. Both students and teachers considered English to be useful. Kiswahili was nevertheless considered an important tool, as it made it easier to explain and understand the mathematics. Furthermore, to take part of the community of practice of those who learned mathematics in English as an additional language, the students considered it necessary to use English. Both because it was expected of them, and because the language was considered important for further studies and careers. However, it emerged that the students used their mother tongue when practicing mathematics outside the classroom if they were to explain something to a fellow student, or if English or Kiswahili it was considered challenging.

This case study as sought an insight into how using Minecraft in lessons in elementary school can contribute to solving mathematical tasks, with a focus on what type of problem-solving strategies Minecraft can facilitate. I have chosen to write about Minecraft because lessons placed in digital games is something that really interests me, and about problem-solving because of the new focus this subject has gotten in the math curriculum.

The thesis is written from a student perspective, and also with a wish of contributing to how teachers see the use of digital games like Minecraft in math lessons. The data used in my case study is screen recordings with sound of seventh grade students doing tasks in Minecraft. The data has been coded and analyzed to identify what types of problem-solving strategies show up when using Minecraft in lessons.

Because of limited research on problem-solving strategies in Minecraft the theory in this study is built on previous research in lessons in Minecraft that I have tied together with different defined problem-solving strategies, and on research done on problem-solving strategies in digital games. The final patchwork of previous research is used to analyze my findings.

The analyzing and discussion part of my thesis shows problem-solving strategies appearing when students work with mathematical tasks. This happens both when the tasks ask for that strategy, and unexpectedly. Typical strategies that appeared are, among other things, logical reasoning, concretizing, and solving part of the problem. The analysis and the discussion points to the quality and selection of problem-solving strategies being dependent on the tasks given by the teacher, and the students experience in problem-solving.

In this thesis, we have performed a document analysis of the teaching material Dragonbox School with focus on the new core element, problem solving, in LK20. The aim of the study has been to figure out how Dragonbox School facilitates students’ problem solving in mathematics.

In order to find an answer to this question we have used an analysis tool developed by Charalambous et al. (2010) which is made for doing textbook analysis. This is the base for the analytic framework we have used for the entire thesis, and we have made some adjustments for it to fit our research question better. We have used the analysis tool to perform a horizontal analysis of the entire teaching material for 4th grade, and in addition we have performed a vertical analysis of parts of the teaching material. As we performed the analysis we found out that we had to nuance the concept of problem solving in order to capture as many tasks as possible with problem solving potential, and thus be able to answer the research question in the best possible way. Because of this we have used theory and previous research to figure out which elements problem solving contains, and then developed the following categories based on our findings: no problem solving, a few elements of problem solving, several elements of problem solving and pure problem solving.

The horizontal analysis shows how the teaching material is structured, but it doesn’t say anything about how problem solving is emphasized in the different parts of the teaching material. What we can draw from this is that the chapters are evenly distributed in the analog elements, and that the digital platform has a predominance of tasks that are used for practicing what the students already have learned. As we have only gone in depth on a few chapters, we do not have enough data to be able to say anything about what this has to say about how Dragonbox School facilitates problem solving.

The vertical analysis shows that there is a predominance of tasks that are not problemsolving in the chapters we have examined, and that all the pure problem-solving tasks are located in a separate book. With our requirements for problem solving and our analyzes, we have concluded that 90% of the tasks were not problem solving, 6.5% had few elements of problem solving, 3% had several elements of problem solving and 1% were pure problem solving tasks. It is nevertheless important to emphasize that whether a task is perceived as a problem-solving task or not depends on the problem-solver. It is therefore the case that what some students will consider as problem-solving tasks can be seen as routine tasks for others, and it is thus not possible to get a result that will accommodate all 4th graders' views on the tasks. To show how we have analyzed, we have included sample tasks from the categories not problem solving, a few elements of problem solving, many elements of problem solving and pure problem solving. These sample tasks are also an important part of the vertical analysis.

During our work on this assignment we have reflected on what the ultimate proportion of problem-solving tasks in a textbook is. There is no key to this question, and after all problem solving is only one half of six core elements in mathematics. It is therefore not correct to expect that all tasks will contain elements of problem solving. During the writing process, we have also become aware that problem solving is a nuanced concept that can be perceived in different ways, and this is also something that makes it challenging to say something specific about how a textbook should facilitate problem solving.

In 2020, the new curricula took effect in Norwegian schools. Due to a society that is constantly changing, there is a need to change the school in line with the societal changes. The goal of this is to equip students for the jobs of the future. However, which skills that will be relevant is not certain, but a majority believe that ICT and programming will be important. Therefore, programming has become part of the mathematics subject in the new curriculum. For this reason, this study will answer the question: What possibilities are there when programming Micro:Bit for mathematics teaching in the 4th and 5th grade? To delimit the task, two research questions have been designed 1. What mathematical competences are expressed in students’ work with Micro: Bit in the 4th grade? 2. How can one facilitate for progression within these mathematical skills in working with Micro: Bit in the 5th grade? To investigate the problem, I have found it relevant to first look at the mathematical aspect of programming a Micro:Bit. The students’ mathematical competence was identified according to Kilpatrick, Swafford and Findell’s (2001) theory model. The same theoretical background was used to provide an answer to the second research question, where an interview with the teacher gave us an opportunity to see how to facilitate progression within these mathematical competences of the students when in 5th grade. The identification of mathematical competences in the students’ work, and the teacher’s statements provide a basis for saying something about the possibilities that lie in the use of Micro: Bit in the 4th and 5th grade.

The purpose of this thesis has been to investigate how a selection of newly educated mathematics teachers facilitate the use of number stories when working with mathematical problems, in order to gain insight into pupils' understanding. The aim of this thesis has been to enlighten the research question:

What experiences do three newly educated mathematics teachers have with facilitating the use of number stories when working with mathematical problems at the lower primary school?

To enlighten this problem, a qualitative study has been implemented. Interviews were implemented with three newly educated mathematics teachers, with some knowledge and experience of facilitating the use of number stories when working with mathematical problems. Based on the teachers' statement in the interviews, a thematic analysis was carried out to identify and find patterns in the data material (Braun & Clarke, 2006). Through the analysis, I have found that the three newly educated mathematics teachers get to use the experiences they have from teacher's education as a teacher. Among other things, by facilitating the use of number stories when working with mathematical problems. A strength that emerges in the analysis is that the teachers, by facilitating the use of number stories when working with mathematical problems, can gain insight into the pupils' understanding. The study also shows that facilitating can give pupils opportunities to develop their mathematical understanding by learning from each other, connecting their everyday lives to the subject of mathematics and choosing their own strategies. More knowledge about the experiences of newly educated mathematics teachers at the lower primary school, especially in facilitating the use of number stories when working with mathematical problems, can contribute to mathematics teachers who work more purposeful to gain insight into pupils' understanding. Such a facilitation can also give pupils at the lower primary school the opportunity for deeper insight into their mathematical understanding, by connecting their everyday language to the more mathematical language.

This master’s thesis compares two textbooks, a paper-based textbook and a digital textbook, regarding assignments within the mathematical subject of functional learning. Considering the two research questions for the thesis, I have looked at how the two textbooks facilitate learning of transitions between representations in functional learning. In addition, the thesis has investigated the way in which students have the potential to acquire knowledge in Niss and Jensen’s (2002) four mathematical competencies; representation competence, symbol and formalism competence, assistive technology competence and communication competence, through work with numbered tasks in two textbooks. The data material has been examined and analyzed through a combination of qualitative and quantitative document and textbook analysis. The data material has been analyzed based on theory considering the didactical tetrahedron (Rezat & Sträßer, 2012), mathematical competencies (Niss & Jensen, 2002), high and low cognitive requirements in task formulations and Janvier’s table (1978) for transitions between forms of representation.

Even though this study only examined two textbooks, the results have nevertheless shown that there are clear trends common to both textbooks in functional theory, and that there are only smaller differences between the textbooks. The students get the same opportunities to practice their competencies in both representation competence, symbol and formalism competence and communication competence by working with the numbered tasks in both textbooks. Both
textbooks offer a potential for the students to practice representation competence in several ways, for example through multiple-choice tasks and tasks that place students on equal or different requirements for representation competence in level-divided tasks. There is a possibility that students acquire knowledge of symbol and formalism competence in functional
learning in all the numbered tasks in the teaching materials, but there is a general lack of task formulations with the question words how and why. The biggest difference between the teaching aids is the disagreement regarding the recommendation of aids regarding the digital graph designer GeoGebra. In addition, analyzes show that in order to be able to fulfil the core
element in mathematics, which is about “representation and argumentation”, one is dependent on more components than just the teaching materials. This is because none of the 124 assignments in the textbooks encourage students to collaborate.

According to my observations, there are no unified answer explaining what In-depth Learning really is. This fact creates challenges connected to how In-depth Learning shall be implemented and used. Professional renewal that includes changed subjects, must include improved facilitation for development of In-depth Learning. In comparison to traditional learning methods, both the development and changed progress of a subject are likely to give rise to the establishment of new ideas and creative learning programs linked to In-dept Learning philosophy.

This thesis discusses and evaluates In-depth Learning related to Mathematics as a subject. My approach has been to identify the perception amongst teachers teaching Mathematics considering In-depth Learning as a tool, and what methods can be used to achieve a high level of understanding of In-depth Learning in Mathematics. The aim of this thesis is to identify the most important factors of individual and collective In-depth Learning, from the point of view of teachers. My thesis does, however, not identify a strict differentiation in this regard. Through semi-structured interviews with 4 teachers (informers), I find somehow different experiences. These interviews can be perceived as the main findings of this research, and are thus discussed in line with the theoretical perspective of this thesis.

The findings of this study shows that the informers acknowledge the differences between the In-depth learning term and the more traditional teaching methods. It seems that they interpretate the two different learning methods as opposites. Their common perception was that In-depth Learning relates to the development of a deeper understanding of the Mathematics’ terms and theory, structures, and contexts. Also, I find that the informers argue that In-depth Learning includes the use of already existing knowledge to build and create new knowledge for use in different, developed contexts. Relatedly, In-depth Learning involves the exploration of the value of Mathematics and to use Mathematics as a tool also outside the learning situation. Learning processes requires motivation and efforts in the student’s learning situations/processes. To motivate and inspire, teachers must adapt learning in each specific situation and contribute to positive dynamics in the learning environment in line with accepted frames. The use of dialog, motivating for creativity, movement (physically), challenges, plays etc. are suggested. Further, the informers emphasized the use of learning tools as positive contributes to the learning processes. All the informers underlined the need for focus on activities that will promote critical thinking, evaluation, discussions/arguments etc. amongst the students. My thesis shows that informers generally have a common and unified understanding of how to create a good environment for In-depth Learning, despite small and normal variations observed. My goal through the interviews was to establish awareness of how In-depth Learning can be positive, and how the informers themselves can contribute to an improved learning environment amongst both students and colleagues.

This master’s thesis is research-based within mathematics didactics. It is part of the LATACME project which is affiliated with the University College of Western Norway. The overarching theme is the role of computational thinking within school mathematics. The aim of the thesis is to investigate which competencies computational thinking has as intention to promote within school mathematics. There are three main questions asked in this master’s thesis. The first research question examines the background for the implementation of computational thinking in school mathematics. The second research question assesses which aspects of computational thinking that emerge in three teachers’ understandings and perceptions of computational thinking. Finally, the third research question evaluates the aspects that might emerge through activities addressed by the three teachers.

In order to illuminate the research questions, a qualitative approach is utilized. Relevant reports have been examined to investigate the intention of computational thinking in mathematics. Subsequently, with an approved application from NSD, a qualitative research interview was conducted with three mathematics teachers about computational thinking. In order to investigate which aspects of computational thinking might emerge in the three teachers’ individual understandings and perceptions of computational thinking, a framework for computational thinking was used. This framework addresses core skills, approaches, and dimensions, in which computational thinking was used.

Findings in this master’s thesis show that the reports link the intention with computational thinking in school mathematics to expectations of a more general understanding of the role of technology in society. Computational thinking in school mathematics can facilitate development of skills and competencies as well as qualifying for the future needs of the society. Computational thinking plays a significant role in this process.

Results from the analysis show that there is a relationship between aspects of computational thinking that emerge in activities, and what the teachers state about computational thinking. However, the study reveals that there is no apparent correspondence between the intention of computational thinking in school mathematics, and what emerges through activities in mathematics, and in the three teachers’ understandings and perceptions of computational thinking.

The implementation of the new subject curriculum has resulted in several changes that are be more compatible with the social development. Among them, programming is one of the new topics. It is included as part of the mathematical education and considered an important skill of the 21-century. Still, several researchers mention a gap in the research on how this can affect students’learning in mathematics. Along with programming there has also been given more attention to students’ communication, and their abilities to contribute in mathematical conversations. In light of this, the following research question has been addressed: What potential does a programmable robot have to facilitate mathematical conversations
in primary school?
To investigate this research question, a case study was conducted. It was based on two teaching sessions with students from grade 6 and 7. Further,six students, distributed in two groups, were videotaped while working with the programming tool Sphero ball. Anna Sfard’s (2007) commognitive framework,with the basis of the four discursive elements, was used to analyse the students’ interactions. This made it possible to examine the contribution of the Sphero ball’s role in students’ production of mathematical narratives.

Based on the analysis, three aspects of the robot’s potential have been found: The Sphero ball’s offering and facilitating of several visual mediators, its ability to give students practical and real-life experiences, and its potential to take the role as the more knowledgeable other. As researchers have stressed, working with a robot in mathematical education is not a key to success by itself. The teaching design and the context of learning is of big importance. The intention of this study was therefore to investigate the potential the Sphero ball can have to facilitate students’ mathematical conversations. To know what possibilities a programming tool like this can have, can help teachers to developmathematicaltasksthat takes advantage of these possibilities. The Sphero ball also has similarities to both screen- and robot programming, which probably also make these findings in some way relevant for the use of other programming tools

Mathematical modeling is prominent in the new curriculum. Despite some research in the field that focuses on mathematical modeling, there is a lack of research aimed at validation within the field of mathematical modelling. Previous research points to a lack of agreement on what validation is and how it occurs. Based on this, I researched the following: What characterises the students validation and what role does the validation play in work including a modelling task for 10th grad mathematics? To shed light on the problem, a qualitative method was used where six groups of students in the 10th grade were observed with sound recordings when they worked on a modeling task during a lesson. The theoretical frame were Czocher´s validation activities, based on the modelling process of Blum and Leiß and the conception of Borromeo Ferri. The students validation was then seen in the context of their modelling process. The study shows that the students validated throughout the modelling process by comparing different parts of the process. In particular, they validated the real result, and their variables by changing the value, replacing or adding new variables. Furthermore, they validated the relations of the variables, their calculations and interpretations of the mathematical result. The findings show that some students validated against the task and not the real situation, as they experienced it as a pseudo-reality. Often the students invited validation from each other, where it was either received or not. When the students validated they either used prior knowledge and experiences from the real world, or by trusting a feeling without justifying why. Some also validated by asking an authority, either the researcher or the teacher. The validation plays an important role as it either gives the students a confirmation that their work is correct, that the model is insufficient so they reject it or helps them revise their model so it can better describe the situation. The findings also show that it is possible for the students to engage in validating activity without arriving at a correct model or real result. Through this study, mathematics teachers can gain insight to what validation is, where and how it occurs, and why it is important to focus on this in work with mathematical modeling. This knowledge will make teachers able to support the students in their work, help them develop good strategies to review and monitor their own modelling process and develop a mathematical model that is sufficient in relation to the reality they are modelling.

We live in a technology-rich society that is constantly developing. This has resulted in certain demands for the educational sector which is supposed to provide pupils with necessary skills for the future. From the autumn of 2020, a new curriculum has been gradually introduced in Norwegian schools, in which programming is included as part of the mathematics subject. There are several arguments for including programming in mathematics. For example, the use of programming in inquiry and problem solving can be a useful tool for developing mathematical understanding (Utdanningsdirektoratet, 2020). This is further investigated in this assignment, through the following research question: How can inquiry through the programming language Python help pupils develope their structural and operational conceptions of geometry? To answer the research question, this study focuses on six pupils in 10th grade who have worked in pairs to solve open geometry tasks in Python. One work process has been selected from each pair of pupils and presented as dialog extracts and screenshots from Python. To examine the pupils’ structural and operational conceptions, a new framework has been designed. The framework consists of four concepts: thinking competency, problem handling competency, algorithmic thinking, and inquiry. The four concepts have been selected because they contain elements that have something in common with Sfard’s (1991) description of structural and/or operational conceptions. In addition, the four concepts can be used to describe the pupils’ approach to the programming tasks, since all the concepts in one way or another are regarding the practice of mathematics. The analysis, which is based on the four concepts, indicates that operational and/or structural conceptions of geometry can be developed through inquiry in Python. The discussion examines Python’s contribution in this development. It appears that the trial-and-error method, along with the fact that one must solve the tasks step-by-step, can contribute to the development of the two mathematical conceptions. The analysis and discussion also indicate that pupils can benefit from programming in pairs.

Autumn 2020 a new set of guidelines is going to be implemented in the Norwegian curriculum. The aim is to improve deep learning knowledge across all subjects in primary school. In mathematics, six objectives are developed to help pupils improve their deep learning skills with mathematical modelling being one of these objectives.

The current study investigates how deep learning occur in mathematics when working with a mathematical modelling activity and how teachers can facilitate this. Thus, there are two research questions formulated:

1. What mathematical learning processes can be recognised when pupils are working
   with a mathematical modelling task?
2. How does Model-Eliciting activities (MEA) contribute to the quality of deep learning
   among pupils?

The pupils in the current study firstly worked with MEA as a preparatory work for the modelling activity task. All the tasks for the pupils were statistical tasks, and the modelling task was a part of the data collection for the current study. Tape recordings and questionnaires were collected from six pupils in 7^th grade. The data collection and modelling activity were
prepared by the researcher, who is also the teacher for the 7th grade. The data was analysed using qualitative methodology (Kvale & Brinkmann, 2015; Thagaard, 1998).

To examine the first research question the current study used a five-component model sat up by Nosrati and Wæge (2018). This model was used to analyse and discuss which learning processes the pupils were exposed to when working with mathematical were modelling. The current study found that the pupils were able to recognise conceptual understanding,
procedural knowledge, application, reasoning, metacognition and self-regulation, during a modelling task.

After examining the second research question, the results indicate that MEA contributes to improve the modelling activity amongst the pupils. Moreover, this indicates that using MEAs and modelling facilitates deep learning in mathematics. In addition, the current study found a relationship between MEA and how the pupils worked with the modelling activity. Another find in this study is that teachers can facilitate deep learning by working with modelling, when MEA are used as a preparatory work for the modelling activity task.

In this master’s thesis I have researched 7th grade students in Norway demonstrating reasoning in engagement with rich mathematical tasks, which conditions the students set to the tasks and how these conditions affected their mathematical reasoning. The study is based on Hagland’s, Hedrén’s and Taflin’s (2005) definition on rich mathematical tasks. Among other things, these tasks are detected by introducing mathematical ideas, being easy to work with, stimulate to mathematical conversation based on different solutions and can lead students and teachers to generate new interesting mathematical problems. These traits may lead students to demonstrate reasoning and ask questions to the task that may become conditions.

The theoretical framework about reasoning is based on a mix from Baroody’s (1993), Hana’s (2013), Lithner’s (2006; 2008) and Reid’s (2002) definitions about different forms of reasoning. These definitions have been used to elucidate some of the students’ reasoning, mainly deductive reasoning. In addition, Balacheff’s (1988) study about aspects of proof in pupils’ practice has been used to suggest some possible explanations for the results from this thesis.

The theoretical framework about looking at the conditions set by the students is inspired by Silver’s (1993) and Stoyanova’s and Ellerton’s (1996) definitions about problem posing. The main point is that a student, based on their mathematical experience, constructs personal interpretations of situations, and formulates meaningful mathematical problems. This is linked together with what Stylianides (2016) calls “proving tasks with ambiguous conditions”. These types of tasks are intentionally ambiguous – and thus subject to different legitimate assumptions by students – with a purpose to lead students to ask questions. These questions, which may become conditions, can affect the mathematical content that students work with – and this can affect the mathematical reasoning in some way.

The results show that 7th grade students often use deductive reasoning linked to arithmetic rules. There were also cases where students reasoned with assumed premises to reach a conclusion, which means they used hypothetically deductive reasoning. I have also seen one example of a creative mathematically founded reasoning, which was generated by a couple of students who spent time, effort and thought activity while working on a task.

Furthermore, I saw that the students – when facing rich mathematical tasks – not only set conditions that affected the mathematical content, but also spent time suggesting how to include conditions that seemed derogatory from the mathematical content. I see that the teacher’s input in such cases is a key variable to put students back on a mathematical track.

This master thesis examines how two teachers from the same school in Brazil work with mathematical reasoning. Existing research on mathematical reasoning shows that it should be implemented when teaching mathematics, and in the fall of 2020 the curriculum in Norway will be reformed, with a greater focus on mathematical reasoning. Therefore, I found it
interesting to examine how teachers from another country reflect on reasoning in their mathematics lessons. Based on this, the following research question has been prepared:

A case study: Mathematical reasoning in a Brazilian school. How do two teachers define and reflect on mathematical reasoning and how do they work to establish and develop norms in mathematical reasoning?

To answer the question, I traveled to Brazil and conducted interviews and observations with my two informants. I analyzed the collected data to examine what reflections both teachers had about reasoning, and how they worked to establish norms.

The findings of the study show that both teachers reflect differently on mathematical reasoning. One teacher defines mathematical reasoning as part of the learning process, where students learn how to argue to gain understanding. In addition, she had clear reflections on reasoning in general and did not separate between reasoning in mathematics and other subjects. The same teacher has a conscious relationship with the establishment of norms, and applies specific measures in the classroom, both individually and collectively. The second teacher had a clear theoretical mathematical perspective on mathematical reasoning, which she related to proofs and mathematical truths. She reflects to a greater extent on mathematical reasoning and how it differs from argumentation in other subjects. This teacher also reflects on socio-mathematical norms, rather than social norms in the classroom. The study shows that she has an unconscious relationship with how to establish norms, and which with approaches she practices in the classroom. My findings show that both teachers even though they work at the same school have different definitions and reflections, and that they have different approaches when it comes to how they go about establishing norms in mathematical reasoning.

The purpose of this study has been to investigate how pupils formulate and use their language to understand mathematics. There has been a particular focus aimed at the multilingual pupils, and the following two research questions has been highlighted and discussed:

1. How do pupils change and relate different registers when discussing and working with
   a mathematical modelling task?
2. Which socio-mathematical norms can be identified that can influence students’ choice
   of registers?

To answer these questions, I chose a qualitative approach, with case study as my research design. The primary data material was collected by observing and recording four pairs of pupils in the seventh grade, while they were discussing and working on a modelling task. In order to strengthen the validity of the data material, especially in the identification of sociomathematical norms, an interview with the pupils’ mathematics teacher was also implemented. In the analysis I’ve used Prediger’s (2016) concepts of the everyday language register, the school language register and the technical language register. Based on the pupils’ pair-wise discussion and individual written answers in the modeling activity, I have identified how the change and relation between different modes of expression can help create understanding. Among other things, the everyday language register contributes with adaptation and meaning making. The school language register helps the pupils stay focused on the mathematical problem that is to be solved, and the technical language register clarifies relationships between given information an what the task actually asks for. As little research has been done in Norway on the subject of interaction between the different language registers in mathematics education, this thesis is a contribution to gain insight into how students naturally use the different registers in their language, and what significance the change between registers can have for pupils1 understanding of mathematical concepts. Increased awareness and knowledge in this subject area could open up for more language positive mathematics classrooms, where both everyday language and academic language are seen as an important resource for learning.

(This study was undertaken at OsloMet University but was connected to LATACME and Tamsin Meaney was the supervisor alongside Vidgis Flottorp, who is on our national advisory board.)

The purpose of this research is to examine why minority-language parents send their children to Saturday school to learn mathematics. To take a closer look at this, a qualitative study has been conducted, in which three minority parents were interviewed. The three parents send their children to Russian, Tamil and Japanese Saturday schools. In order to gain insight into their thoughts and justifications for participation in Saturday school, a semi-structured interview was conducted. Given the quite limited research in this, and especially in Norway, this initial research has implications for both researchers and teachers.

This research has used Etienne Wenger’s (1998) theory of community of practice and modes of belonging. The community of practice in this case is minority parents who want to send their children to Saturday school to learn mathematics. The data from the interviews with minority parents were therefore analyzed using the modes of belonging, where there were findings in the categories with associated subcategories: engagement (relationships), imagination (pictures of ourselves and pictures of opportunities) and alignment (discourse and complexity). Theories from Anna Sfard (2001, 2007, 2009) and Richard Skemp (1976) were used highlight the different types of mathematical understanding which were seen as part of the discourse aspect of the alignment mode in the communities of practice.

The main findings of this research showed that it was a complex situation to say something about the community of practice, where they had different opinions about what was important to them. However, a future education turned out to be important for this community of practice, where there were various nuances of how they argued about using mathematics education in Saturday school to achieve this. The community of practice also seemed to have specific aspects that they preferred in mathematics teaching in Saturday school. This could be about what mathematics is and how it should be taught. Examples such as instrumental learning, critical thinking and generally better teaching methods from the teacher. Mathematics learning was often a secondary reason, behind the need for their children to learn their home language, for why parents in the community of practice chose to send their children to Saturday schools, but the many comments about it showed it was important. The results also showed that the children themselves did not want Saturday school attendance to learn math to the same degree as their parents, where within the community of practice there were different approaches to why the children participated and how their participation was supported.

In the autumn of 2020 a new curriculum, L20, will be introduced in Norwegian schools. A renewal of the curriculum may place new demands on teachers’ knowledge and competence. The mathematics course in L20 contains five core elements, which describe the most important academic content the students will work with in education throughout the primary school. One of the core elements is modeling and applications. This means that mathematical modeling appears clearer and earlier in the training course than it does in today’s curriculum. The increased focus on mathematical modeling in L20 is due to the fact that it is considered a relevant competence for participation in the continuous development of society. Mathematical modeling is about translating between the real world and mathematics, in both directions. Mathematical modeling is about incorporating real and realistic situations in the mathematics field. Increased use of realistic contexts can make the subject more accessible to students. In this paper I explore how teachers in grades 1 through 7 express themselves about realistic contexts and mathematical modeling in their teaching, and how they express themselves about the increased focus on mathematical modeling in L20. The data was collected by interviewing six teachers in grades 1-7, who teach mathematics. The data material is processed through a combination of conventional content analysis and categorization based on central aspects from theory of realistic contexts, mathematical modeling and associated didactic perspectives. The findings indicate that the teachers’ competence associated with the concept of realistic contexts was more extensive than that of mathematical modeling. The study also revealed relevant statements that can be linked to didactic perspectives of both realistic contexts and mathematical modeling, without the teachers always being aware of which of the concepts they were talking about. The teachers in the sample showed positive attitudes towards the introduction of L20 and the core element of modeling and applications. Statements emerged about a need to increase both their own and other teachers’ competence related to mathematical modeling in connection with the introduction of a new curriculum.

From the autumn of 2020, a renewed version of the Norwegian mathematics curricula will be implemented in practice. Along with the digitalization of our society’s, the education should contribute to strengthening students’ digital skills – areas of competence described as valuable for the future that awaits them. In the subject of mathematics, students should be provided the opportunity to develop digital skills in activities that promote exploration, problem solving and collaboration. Teachers are considered to be an important prerequisite for achieving these goals. Studies that examine how teachers integrate ICT into their teaching practice, often emphasize a focus on whether teachers’ competence is sufficient or not. The aim of this thesis relies on teachers’ beliefs in order to build an understanding of how ICT is integrated in their mathematics teaching practice. This is based on the approved literature that explains a clear relationship between teachers’ beliefs and their teaching practice. The research question is formed as an open focus. I seek insight into teachers’ beliefs about technology affordances based on their beliefs about effective teaching of mathematics. The direction of the research question takes the following direction:

1. What are teachers’ beliefs about effective teaching of mathematics?
2. What are the teacher’s beliefs about technology affordances to facilitate effective teaching of mathematics?

To examinate the research questions, four mathematics teachers participate in qualitative interviews. The analysis seeks insight of the teachers’ beliefs through principles of hermeneutics, along with the use of theoretical analysis tool. The framework of the study consists of a renewed and inspired version of Hadjerrouits (2017) affordance model. The purpose of the model is to investigate whether teachers see affordances with digital tools in the light of their beliefs about effective mathematics teaching. Key findings from the study indicate that an instrumentalist view is absent among the participants, which is different from the research literature. The Platonic view is most prominent among the teachers, while the problemsolver view emerges as clear in especially one teacher. This teacher is also the only participant that hold beliefs about technology affordances that is recognized by a problemsolver view. Findings also indicate that teachers’ beliefs about effective mathematics teaching, do not always correspond with the beliefs they hold about technology affordances.  

This study has investigated how procedural and conceptual knowledge was expressed in the written and oral mathematical argumentation of six 3rd grade students. To study this, a qualitative study has been conducted. The six students each wrote their own number story and were interviewed about what they have written. The focus has been on how the students used different forms of expressions, to form their arguments. The students were filmed during the data collection so that both the verbal and nonverbal forms of expression were able to be analysed. ‘Resonnering og argumentasjon’ is one of the core elements of the new curriculum (Kunnskapsdepartementet, 2018) that comes into effect in 2020, thus this study is very relevant to teachers. The students’ written number stories and the oral responses in the interviews were analyzed based on Toulmin’s (2003) argumentation model and four linguistic indicators inspired by Bills’ (2001; 2002) og Bills og Gray’s (2001) research. An analysis of the student’s use of personal pronouns, use of past and present tense, linguistic pointers and their multimodal argumentation indicated which elements of procedural and conceptual knowledge the students expressed. The students’ use of linguistic pointers and their multimodal argumentation proved most useful in determining procedural or conceptual knowledge. In the study, multimodal argumentation was found both in the written number stories and the oral responses in the interview. Toulmin was used as a tool to identify how the grammatical features/pictures are used to make their mathematical arguments. It is important to emphasize that this study looked at elements of procedural and conceptual knowledge, instead of categorize their knowledge as either procedural or conceptual. This is because it is not appropriate to say that students have one form for knowledge, but not the other (Rittle-Johnson, Siegler and Alibali, 2001, s. 347). In the study, elements of both procedural and conceptual knowledge were found in the written and oral argumentation of all six students. Nevertheless, the student’s oral argumentation gave more information for identifying the elements, as the argumentation was expressed more explicitly. Awareness of the recipient can influence the written argumentation that is expressed as thoughts can remain in the head when students write. The researcher’s question can make the students provide more information, and thus their argumentation is challenged (Yackel, 1995; Weber, Maher, Powell og Lee, 2008, s. 249).

In the core themes of the new curriculum, the Curriculum for the Knowledge Promotion 2020, which comes next autumn, shows that argumentation becomes a larger part of mathematics teaching. The content of the curriculum for the subject of mathematics is characterized by these core elements, which means that the argumentation becomes more distinct in the subject. This is the reason why I chose to correct my thesis to argumentation. In addition, I have chosen to participate in a project that deals with number stories, which is a story that includes mathematical problems (Botten, 2011, p. 183). The focus has thus been on linking narrative and mathematics together. The purpose of this study has been to show how students in the third grade argue when they work with number stories. In order to develop insight into this, number stories have been collected which the students have written, and we had interviews with some of the students to talk about their number story. I chose to have individual interviews with six students in order to go deeper into how they argued in writing and orally in the work of number stories. In the analysis of the arguments of the students, I have used Toulmin’s model of argumentation, where I have looked at how the argumentation was and what function the student’s argument have. The results have shown that the students argue differently in the written and oral argumentations, the analysis has shown that the oral argument brings out several of the elements in Toulmin’s model. This shows that the students argue more in conversation with an adult than in the written number story.

Society is constantly changing, and education must keep up with its development. By 2020, new curricula will be introduced in Norwegian schools, with development and revisions of the subjects that will prepare students for the future. One of the developments is that programming becomes a part of the mathematics curriculum. In this study, it has been investigated how programming can promote mathematical competence. The study is limited by the following two research questions: 1) How does mathematical competence express in pupils’ conversations when they program? 2) How do students construct mathematical knowledge when programming?

As a backdrop for the study, theories I consider relevant to investigating the research questions have been used. Based on Kilpatrick, Swafford & Findell’s (2001) framework on mathematical components and Niss & Højgaard Jensen’s (2002) division of mathematical competence, I have designed a framework that has been used to elucidate the first research question. Seymour Papert’s (1991, 1993) learning perspective of constructionism is used as a theoretical basis for the second research question, about how students construct mathematical knowledge when programming. In this context, mathematical knowledge is regarded as part of a comprehensive mathematical competence.

The study has been based on four pairs of students at grade 7, and their work on programming a pentagon in Scratch. The method for analyzing the student conversation has a qualitative hermeneutic approach. The analysis has proven that programming can contribute positively to promoting mathematical competence. Together, the pupils have come up with a solution to the task through an educational process, in which mathematical competence has been promoted in different areas. Collaboration on such mathematical tasks has provided insight on how students construct mathematical knowledge. Collaboration, creativity, reasoning, commitment, trial and error are some of the qualities that have characterized the learning process.

The purpose of this study is to contribute to increased knowledge about how programming can promote pupils’ mathematical competence. The aim is that the study will be of useful knowledge for teachers and other people in the education sector. I also hope that it will contribute to greater interest in how programming can strengthen mathematics teaching. The work on the study has given me a greater insight into how I, as a future math teacher, can combine mathematics and programming in an educational way. I have also gained a deeper knowledge of research in this field.

In this study, the purpose has been to seek insight into the interaction between argumentation and numerical understanding. In order to strive for insight, a qualified study has been conducted which four students from the 3rd grade has participated in. The young students have worked together in pairs, and tried to make a counting story together based on their interests. The two pairs were separately filmed and recorded under the process. According to the curriculum, students should be able to discuss and argue in mathematics. Furthermore, numerical understanding is drawn as a central part of mathematics teaching. This study contributes to insight into acceptances regarding the students’ argumentation and numerical understanding.

The study does not intend to create an integrated theory between argumentation and numerical understanding, but to get a closer look at the interaction between the two elements. For this study, theory is therefore attached to argumentation and numerical understanding as well number story. The students’ argumentation is analysed based on Toulmin (2003) model of work and the qualities of numerical understanding are analysed based on McIntosh, Reys and Reys (1992) descriptions of components that are central to numerical understanding.

From the analysis, it appeared that the students ‘interaction affected the students’ argumentation. Among other things, one could see how the numerical understanding influenced the young student’s argumentation. On several occasions, the numerical understanding appears as a basic element. Students argue in this study with their numerical understanding. One of the tendencies shows that students through argumentation have the opportunity to think about numbers and their applications in several ways. The argumentation influences the understanding of numbers by allowing students to explore, visualize and relate to different ways of thinking.

Number story gives students several different approaches and solutions. In addition, the analysis shows how number story’s challenge students to argue for the choices made. It has been concluded that the students use their numerical understanding in the argumentation and vice versa. The results show how the students use the numerical understanding to argue for use and choice of calculation operation. Through the analysis, it emerges how the students understand, calculate, apply, discuss and engage. In this way, students get the opportunity to assess strategy and answers in light of the problem in question. This helps form the basis for the understanding of numbers and the mathematical text. The results of the study indicate that the use of argumentation affected on their mathematical understanding of numbers.

In 2020, a new curriculum comes into effect which includes programming in the mathematics subject (Kunnskapsdepartementet, 2018a). Because of this, there has been a great debate on how programming should be written into the curriculum and how teachers can work with programming in mathematics teaching (Bærland og Gilje, 2017; Flote, 2016; Klovning, 2015; Waage, 2018). Based on the new curriculum that comes into effect in 2020, I have chosen to research the following problem: What qualities of mathematical argumentation can be identified when pupils program? This considers mathematical argumentation to be a social process that can be characterized by both everyday communication and formal evidence. To elucidate the issue, a qualitative research study has been made in which data material of children that worked with programming has been analyzed and discussed. The data material has been collected in collaboration with another master’s student where we studied programming and mathematical argumentation and developed a teaching program for the data collection. The students were given the task of programming a pen to draw some geometric figures in a programming program called Scratch. The data material has been presented in the form of dialog extracts and screenshots of the computer. With the help from theory on programming, mathematical argumentation and reasoning, qualities of the students’ mathematical argumentation have been identified and discussed to answer the problem. In this study, especially the researchers Lavy (2006) and Lithner (2000; 2006; 2008) have played a major role in which they have contributed with analysis tools. Both researchers wrote in their articles on different qualities of student mathematical argumentation, where Lavy researched various types of arguments when pupils programmed and Lithner researched various forms of reasoning. Analysis and discussion of the data showed that students who had the least challenges in the collaboration often argued mathematically for different proposals and solutions. The programming program made the work process detailed, which opened up to opportunities for mathematical argumentation to be created. Through the process of illuminating the problem, the thesis shows how teachers can work with programming without losing the focus on mathematics. At the same time, the thesis is also a proof that one can create teaching programs for programming without being an expert in the subject, and therefore, with a little effort, it is possible for all teachers to familiarize themselves with programming.

 

 

I have participated in a research group called LATACME, were I have been a part of the smaller research group called Produksjon av regnefortellinger for å fremme matematisk forståelse (Production of number stories to promote mathematical understanding). The research group examines argumentation in number stories. Based on this, I have collected data. The purpose of this study is to examine pupils’ argumentation in self-made number stories. This can be linked to two of the new core elements in the new curriculum, starting in 2020. These core elements are reasoning and argumentation, and representation and communication (Utdanningsdirektoratet, 2018a). Previous research on pupils’ argumentation in their self-made number stories, is a small field of research. This study can therefore contribute to more research on the field. This thesis is written from a pupils perspective, but can be used by teachers to gain insight into pupils’ argumentation. The research questions in this thesis is: How do pupils argue in their own number stories in mathematics? To answer this, the issue is divided into two more questions: (1) How do pupils argue in their written number story? (2) How do pupils verbally argue for the answer in the written number story? The argumentation of 20 pupils in third grade, is analyzed using the Toulmin model to show how much the pupils argue in the number stories and how much they argue in the interview. This is presented in a bar graph. The bar graphs show two findings that are highlighted in this thesis. One of the findings is that some pupils argue little in the interview, while others argue a lot. In order to show contrasts in the data, the analysis is based on two pupils who argue little and two pupils who argue a lot in the interview. The other finding that the bar graphs show, is that most pupil argue more in the interview than they do in the number stories. These two findings are discussed to find possible reasons for this

With implementation of the Knowledge Promotion Reform in 2006, writing was introduced as one of five basic skills. In Norwegian school writing as a basic skill is incorporated in the competence aims in all subjects, which means that all teachers have a joint responsibility for the pupils writing education. On their website, the Norwegian Directorate for Education and Training emphasize that the demands for good linguistic skills are increasing (Utdanningsdirektoratet, 2014). Writing is a priority area in Norwegian school, but according to Opsal (2013) not much research is done in the field. Previous research says that pupils written explanations can provide robust accounts of their mathematical reasoning (Moskal & Magone, 2000). Work on argumentation and written proof in mathematics often focus on higher grades (Hovik & Solem, 2013). Hovik and Solem (2013) argues for introducing argumentation and proof for the pupils earlier, to make them feel comfortable in explaining their reasoning. This study research how pupils at 4th and 7th grad write and argue in mathematic. To research this, the pupils wrote a text and argued for what happens if you (1) add two odd numbers and (2) add an odd number and an even number. The analysis is based on 59 texts written by pupils in 4th grade and 60 texts written by pupils in 7th grade. The analysis of the texts written by the pupils has a quantitative and qualitative approach. Different qualities in the texts are identified by using Balacheff (1988) and Toulmin (2003) as framework. Based on Balacheffs levels of argumentation and proof, the texts are categorized in five levels, and tells something about how the pupils argue. Toulmins model of argumentation is used to study argument in some selected texts. The result shows that the three intermediate levels have an equal distribution in both 4th an 7th grade. I found bigger difference between the grades on the highest and lowest level. In my analysis of single arguments, I identified different qualities in the arguments on the same level. The difference of quality within the same level was most prominent in 7th grade. In addition, my analysis shows that not all arguments can be categorized by using Balacheffs framework, because the arguments are too complex.