The thesis consists of four articles and this summarizing part. All parts have focused on bringing some insights into how to design a didactical situation including dynamic software (GeoGebra) to support students’ mathematical problem solving and creative reasoning as means for learning. The four included articles are: I. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behavior, 37, 48-62. II. Olsson, J. (2017). The Contribution of Reasoning to the Utilization of Feedback from Software When Solving Mathematical Problems. International Journal of Science and Mathematics Education, 1-21. III. Olsson, J. Relations between task design and students’ utilization of GeoGebra. Mathematical Thinking and Learning. (Under review) IV. Olsson, J., & Granberg, C. Dynamic software, problem solving with or without guidelines, and learning outcome. Technology, Knowledge and Learning. (Under review) Background A common way of teaching mathematics is to provide students with solution methods, for example strategies and algorithms that, if followed correctly, will solve specific tasks. However, questions have been raised whether these teaching methods will support students to develop general mathematical competencies, such as problem solving skills, ability to reason and acquire mathematical knowledge. To merely follow provided methods students might develop strategies of memorizing procedures usable to solve specific tasks rather than drawing general conclusions. If students instead of being provided with algorithms, are given the responsibility to construct solution methods, they may produce arguments for why the method will solve the task. There is research suggesting that if those arguments are based on mathematics they are more likely to develop problem solving and reasoning-skill, and learn the included mathematics better. In such didactic situations, where students construct solutions, it is important that students have instructions and tasks that frame the activity and clarify goals without revealing solution methods. Furthermore, the environment must be responsive. That is, students need to receive responses on their actions. If students have an idea on how to solve (parts of) the given problem they need to test their method and receive feedback to verify or falsify ideas and/or hypotheses. Such activities could be supported by dynamic software. Dynamic software such as GeoGebra provides features that support students to quickly and easily create mathematical objects that GeoGebra will display as visual representations like algebraic expressions and corresponding graphs. These representations are dynamically linked, if anything is changed in one representation the other representations will be altered accordingly, circumstances that could be used to explore and investigate different aspects and relations of these objects. The first three studies included in the thesis investigate in what way GeoGebra supports creative reasoning and collaboration. These studies focus questions about how students apply feedback from GeoGebra to support their reasoning and how students utilize the potentials of GeoGebra to construct solutions during problem solving. The fourth study examine students’ learning outcome from solving tasks by constructing their methods. Methods A didactical situation was designed to engage students in problem solving and reasoning supported by GeoGebra. That is, the given problems were not accompanied with any guidelines how to solve the task and the students were supposed to construct their own methods supported by GeoGebra. The students were working in pairs and their activities and dialogues were recorded and used as data to analyse their engagement in reasoning and problem solving together with their use of GeoGebra. This design was used in all four studies. A second didactical situation, differing only with respect of providing students with guidelines how to solve the task was designed. These didactical situations were used to compare students’ use of GeoGebra, their engagement in problem solving and reasoning (study III) and students’ learning outcome (study IV) whether the students solved the task with or without guidelines. In the fourth study a quantitative method was applied. The data from study IV consisted of students’ results during training (whether they managed to solve the task or not), their results on the post-test, and their grades. Statistical analysis where applied. Results The results of the first three studies show qualitative aspects of students solving of task with assistance of GeoGebra. GeoGebra was shown to support collaboration, creative mathematical reasoning, and problem solving by providing students with a shared working space and feedback on their actions. Students used GeoGebra to test their ideas by formulating and submitting input according to their questions and hypotheses. GeoGebra’ s output was then used as feedback to answer questions and verify/falsify hypotheses. These interactions with GeoGebra were used to move the constructing of solutions forward. However, the way students engage in problem solving and reasoning, and using GeoGebra to do so, is dependent on whether they were provided with guidelines or not. Study III and IV showed that merely the students who solved unguided tasks utilized the potential of GeoGebra to explore and investigate the given task. Furthermore, the unguided students engaged to a larger extent in problem solving and creative reasoning and they expressed a greater understanding of their solutions. Finally study IV showed that the students who managed to solve the unguided task outperformed, on posttest the students who successfully solved the guided task. Conclusions The aim of this thesis was to bring some insights into how to design a didactical situation, including dynamic software (GeoGebra), to support students' mathematical problem solving and creative reasoning as means for learning. Taking the results of the four studies included in this thesis as a starting point, one conclusion is that a didactical design that engage students to construct solutions by creative reasoning supported by GeoGebra may enhance their learning of mathematics. Furthermore, the mere presence of GeoGebra will not ensure that students will utilize its potential for exploration and analysis of mathematical concepts and relations during problem solving. The design of the given tasks will affect if this will happen or not. The instructions of the task should include clear goals and frames for the activity, but no guidelines for how to construct the solution. It was also found that when students reasoning included predictive argumentation for the outcomes of operations carried out by the software, they could better utilize the potential of GeoGebra than if they just, for example, submitted an algebraic representation of a linear function and then focused on interpreting the graphical output. / Det övergripande syftet med avhandlingen har varit att nå insikter i hur man kan designa en didaktisk situation inklusive en dynamisk programvara (GeoGebra) för att stödja elevernas lärande genom matematisk problemlösning och kreativt resonemang. En bärande idé har varit att elever som själva konstruerar lösningsmetoder till problembaserade uppgifter lär sig matematik bättre än elever som får en metod att följa. Resultaten visar att GeoGebra är ett stöd vid konstruerandet av lösningsmetoder och att elever då också resonerar kreativt. Det vill säga, de skapar en för dem en ny resonemangssekvens som innehåller en lösningsmetod som stöds av argument förankrade i matematik. Idén med att elever på egen hand konstruerar lösningen på uppgifter har även belysts genom att jämföra med elever som löser uppgifter där de får vägledning till lösningsmetoden. Resultaten visar att elever som får en lösningsmetod inte resonerar kreativt, de utnyttjar inte GeoGebras potential att stödja ett undersökande arbetssätt, och de lär sig mindre av den matematik som ingår i uppgifterna. Denna avhandling består av 4 artiklar och en kappa. De fyra artiklarna är: I. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behavior, 37, 48-62. II. Olsson, J. (2017). The Contribution of Reasoning to the Utilization of Feedback from Software When Solving Mathematical Problems. International Journal of Science and Mathematics Education, 1-21. III. Olsson, J. Relations between task design and students’ utilization of GeoGebra. Mathematical Thinking and Learning. (Under review) IV. Olsson, J., & Granberg, C. Dynamic software, problem solving with or without guidelines, and learning outcome. Technology, Knowledge and Learning. (Under review) Artikel 2 och 3 är jag ensam författare till. Det innebär att jag designat studien, planerat och genomfört datainsamling, analyserat data och formulerat slutsatser, samt skrivit texten och korresponderat med tidskrifter. Artikel 1 och 4 har jag skrivit i samarbete med Carina Granberg. Vi bedömer att arbetet med artikel 1 fördelats lika. Allt skrivarbete har fortgått genom åtskilliga granskningar av varandras utkast och diskussioner om slutgiltiga formuleringar. I arbetet med artikel 4 har jag haft huvudansvaret för designen av studien och planering för datainsamlingen. Skrivarbetet har genomförts på samma sätt som i arbetet med artikel 1.

The thesis consists of four articles and this summarizing part. All parts have focused on bringing some insights into how to design a didactical situation including dynamic software (GeoGebra) to support students’ mathematical problem solving and creative reasoning as means for learning. The four included articles are: I. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behavior, 37, 48-62. II. Olsson, J. (2017). The Contribution of Reasoning to the Utilization of Feedback from Software When Solving Mathematical Problems. International Journal of Science and Mathematics Education, 1-21. III. Olsson, J. Relations between task design and students’ utilization of GeoGebra. Mathematical Thinking and Learning. (Under review) IV. Olsson, J., & Granberg, C. Dynamic software, problem solving with or without guidelines, and learning outcome. Technology, Knowledge and Learning. (Under review) Background A common way of teaching mathematics is to provide students with solution methods, for example strategies and algorithms that, if followed correctly, will solve specific tasks. However, questions have been raised whether these teaching methods will support students to develop general mathematical competencies, such as problem solving skills, ability to reason and acquire mathematical knowledge. To merely follow provided methods students might develop strategies of memorizing procedures usable to solve specific tasks rather than drawing general conclusions. If students instead of being provided with algorithms, are given the responsibility to construct solution methods, they may produce arguments for why the method will solve the task. There is research suggesting that if those arguments are based on mathematics they are more likely to develop problem solving and reasoning-skill, and learn the included mathematics better. In such didactic situations, where students construct solutions, it is important that students have instructions and tasks that frame the activity and clarify goals without revealing solution methods. Furthermore, the environment must be responsive. That is, students need to receive responses on their actions. If students have an idea on how to solve (parts of) the given problem they need to test their method and receive feedback to verify or falsify ideas and/or hypotheses. Such activities could be supported by dynamic software. Dynamic software such as GeoGebra provides features that support students to quickly and easily create mathematical objects that GeoGebra will display as visual representations like algebraic expressions and corresponding graphs. These representations are dynamically linked, if anything is changed in one representation the other representations will be altered accordingly, circumstances that could be used to explore and investigate different aspects and relations of these objects. The first three studies included in the thesis investigate in what way GeoGebra supports creative reasoning and collaboration. These studies focus questions about how students apply feedback from GeoGebra to support their reasoning and how students utilize the potentials of GeoGebra to construct solutions during problem solving. The fourth study examine students’ learning outcome from solving tasks by constructing their methods. Methods A didactical situation was designed to engage students in problem solving and reasoning supported by GeoGebra. That is, the given problems were not accompanied with any guidelines how to solve the task and the students were supposed to construct their own methods supported by GeoGebra. The students were working in pairs and their activities and dialogues were recorded and used as data to analyse their engagement in reasoning and problem solving together with their use of GeoGebra. This design was used in all four studies. A second didactical situation, differing only with respect of providing students with guidelines how to solve the task was designed. These didactical situations were used to compare students’ use of GeoGebra, their engagement in problem solving and reasoning (study III) and students’ learning outcome (study IV) whether the students solved the task with or without guidelines. In the fourth study a quantitative method was applied. The data from study IV consisted of students’ results during training (whether they managed to solve the task or not), their results on the post-test, and their grades. Statistical analysis where applied. Results The results of the first three studies show qualitative aspects of students solving of task with assistance of GeoGebra. GeoGebra was shown to support collaboration, creative mathematical reasoning, and problem solving by providing students with a shared working space and feedback on their actions. Students used GeoGebra to test their ideas by formulating and submitting input according to their questions and hypotheses. GeoGebra’ s output was then used as feedback to answer questions and verify/falsify hypotheses. These interactions with GeoGebra were used to move the constructing of solutions forward. However, the way students engage in problem solving and reasoning, and using GeoGebra to do so, is dependent on whether they were provided with guidelines or not. Study III and IV showed that merely the students who solved unguided tasks utilized the potential of GeoGebra to explore and investigate the given task. Furthermore, the unguided students engaged to a larger extent in problem solving and creative reasoning and they expressed a greater understanding of their solutions. Finally study IV showed that the students who managed to solve the unguided task outperformed, on posttest the students who successfully solved the guided task. Conclusions The aim of this thesis was to bring some insights into how to design a didactical situation, including dynamic software (GeoGebra), to support students' mathematical problem solving and creative reasoning as means for learning. Taking the results of the four studies included in this thesis as a starting point, one conclusion is that a didactical design that engage students to construct solutions by creative reasoning supported by GeoGebra may enhance their learning of mathematics. Furthermore, the mere presence of GeoGebra will not ensure that students will utilize its potential for exploration and analysis of mathematical concepts and relations during problem solving. The design of the given tasks will affect if this will happen or not. The instructions of the task should include clear goals and frames for the activity, but no guidelines for how to construct the solution. It was also found that when students reasoning included predictive argumentation for the outcomes of operations carried out by the software, they could better utilize the potential of GeoGebra than if they just, for example, submitted an algebraic representation of a linear function and then focused on interpreting the graphical output. / Det övergripande syftet med avhandlingen har varit att nå insikter i hur man kan designa en didaktisk situation inklusive en dynamisk programvara (GeoGebra) för att stödja elevernas lärande genom matematisk problemlösning och kreativt resonemang. En bärande idé har varit att elever som själva konstruerar lösningsmetoder till problembaserade uppgifter lär sig matematik bättre än elever som får en metod att följa. Resultaten visar att GeoGebra är ett stöd vid konstruerandet av lösningsmetoder och att elever då också resonerar kreativt. Det vill säga, de skapar en för dem en ny resonemangssekvens som innehåller en lösningsmetod som stöds av argument förankrade i matematik. Idén med att elever på egen hand konstruerar lösningen på uppgifter har även belysts genom att jämföra med elever som löser uppgifter där de får vägledning till lösningsmetoden. Resultaten visar att elever som får en lösningsmetod inte resonerar kreativt, de utnyttjar inte GeoGebras potential att stödja ett undersökande arbetssätt, och de lär sig mindre av den matematik som ingår i uppgifterna. Denna avhandling består av 4 artiklar och en kappa. De fyra artiklarna är: I. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behavior, 37, 48-62. II. Olsson, J. (2017). The Contribution of Reasoning to the Utilization of Feedback from Software When Solving Mathematical Problems. International Journal of Science and Mathematics Education, 1-21. III. Olsson, J. Relations between task design and students’ utilization of GeoGebra. Mathematical Thinking and Learning. (Under review) IV. Olsson, J., & Granberg, C. Dynamic software, problem solving with or without guidelines, and learning outcome. Technology, Knowledge and Learning. (Under review) Artikel 2 och 3 är jag ensam författare till. Det innebär att jag designat studien, planerat och genomfört datainsamling, analyserat data och formulerat slutsatser, samt skrivit texten och korresponderat med tidskrifter. Artikel 1 och 4 har jag skrivit i samarbete med Carina Granberg. Vi bedömer att arbetet med artikel 1 fördelats lika. Allt skrivarbete har fortgått genom åtskilliga granskningar av varandras utkast och diskussioner om slutgiltiga formuleringar. I arbetet med artikel 4 har jag haft huvudansvaret för designen av studien och planering för datainsamlingen. Skrivarbetet har genomförts på samma sätt som i arbetet med artikel 1.

Arbetet med olika representationsformer och transformationer mellan dessa har en viktig roll i elevers lärande i matematik och eftersom en stor del av matematikundervisningen i skolan utgår från olika läromedel är det viktigt att som lärare vara medveten om vilka lärtillfällen olika läromedel erbjuder. Syftet med denna läromedelsanalys är därför att ta reda på vilka lärtillfällen olika läromedel erbjuder och hur de skiljer sig åt i representationen av olika representationsformer inom det matematiska området linjära funktioner. För att ta reda på detta gjordes en läromedelsanalys av olika läromedel i kursen Matematik 1c, där läromedlets uppgifter analyserades utifrån vilken eller vilka representationsformer som fanns med i uppgiftsbeskrivningen och svaren till dessa uppgifter. Datan analyserades sedan utifrån etablerade teorier och metoder från semiotiken. Analysen påvisade att alla tre läromedel gav möjlighet att lära genom arbetet med olika representationsformer och transformationer. Det som framförallt skiljer de olika läromedlen åt är antalet lärtillfällen som erbjuds. Det går inte att entydigt rangordna läromedlen från sämst till bäst, men som aktiv pedagog bör man vara medveten om de olika läromedlens styrkor och svagheter. Eftersom det inte enbart är läromedlet som avgör vilka lärtillfällen eleverna stöter på under sin skolgång, utan även hur den enskilda pedagogen planerar och genomför sin undervisning, går det inte heller att säga något om hur elevernas förståelse för linjära funktioner kommer att utvecklas vid användningen av ett specifikt läromedel. Läromedelsanalysen ger således pedagoger möjligheten att se styrkor och svagheter för respektive läromedel och kompensera för dessa.

Inom den svenska matematikundervisningen på gymnasiet möter eleverna många olika matematiska uppgifter. Uppgifterna skiljer sig i att de testar olika förmågor men även hur uppgifterna är formulerade skiljer sig. Det kan bland annat röra sig om uppgifter som är textbaserade, grafiskt utformade eller som har en algebraisk representationsform. Denna studie avser att undersöka om representationsformen på matematiska uppgifter kan ha någon påverkan på i vilken utsträckning elever kan lösa uppgifterna samt redogöra för vilka de vanligaste misstagen kan vara inom de olika representationsformerna. Studien syftar även till att se om det är någon skillnad på svarsfrekvensen beroende på om uppgifterna testar elevernas förmåga att genomföra beräkningar (procedurell kunskap) eller förmågan att uppfatta begrepp och principer (konceptuell kunskap). Teorin som används vid framtagandet av uppgifter är Hallidayan-modellen om olika sätt att presentera matematik samt principen om procedurell- och konceptuell kunskap. Metoden som används inom studien är insamling av elevlösningar på tre olika prov som tar sin grund i var sin av de olika representationsformerna: textbaserat, grafiskt och algebraiskt samt att alla tre innehåller uppgifter som testar deras procedurella samt konceptuella kunskap. Resultatet visar att representationsformen på uppgifterna har betydelse för i vilken utsträckning eleverna kan lösa dem och att eleverna har speciellt svårt för grafiskt formulerade uppgifter. Resultatet visar även att eleverna är bättre på att genomföra beräkningar än att förstå matematiska principer. I diskussionen presenteras olika tankar och idéer till hur det kan komma sig att resultatet ser ut som det gör samt vad resultatet kan ha för påverkan på matematikundervisningen framöver. / In the Swedish mathematical education on upper secondary school level, the students face many different mathematical tasks. The tasks are being separated by testing different abilities and in how they are designed. They could differ in how they are presented, and they could for example be text based, graphical and algebraic. These are three different ways of form of representation. This study intends to examine if the form of representation could have an impact on to which extent the student can solve the tasks and elucidate the most common mistakes within the different form of representation. The study also aims to determine if there are any difference in the frequency of the response depending on if the task assess student’s ability to perform calculations (procedural knowledge) or the ability to recognize concepts and principles (conceptual knowledge). The theory used in developing the tasks is the Halliday’s model of different ways to present mathematics and the principles of procedural and conceptual knowledge. The method that is used in this study is collection of student’s answer in three different tests, each based on one of the three forms of representation: text based, graphical and algebraic. Additionally, all three tests contain two tasks which will test the students procedural and conceptual knowledge. The results show that the form of representation have an impact on the extent to which students can solve the tasks and that students particularly struggle with graphically formulated tasks. The results also reveal that students are better at performing calculations than understanding mathematical principles. The discussion presents various thoughts and ideas on why the results appear as they do and what impact the results may have on mathematical education in the future.

I det svenska skolsystemet sker olika stadieövergångar och övergången från högstadiet till gymnasiet är en sådan. I ämnet matematik visar tidigare forskning att en skillnad mellan dessa stadier är ett ökat studietempo och en förskjutning mot en mer formell matematik i det senare stadiet. Syftet med denna studie har varit att undersöka vilka innehållsmässiga likheter och skillnader det finns mellan dessa stadier inom området linjära funktioner. Detta har gjorts via en innehållsanalys av några läroböcker som används för respektive stadie då dessa kan ses som den potentiellt realiserade läroplanen. Vidare har studien baserats på Anna Sfards teori om operationell respektive strukturell begreppsuppfattning som ett sätt att få syn på och kontrastera det innehåll som behandlas i de olika läroböckerna. Resultatet visar att det finns likheter och skillnader mellan de olika stadierna, utifrån hur detta uttrycks via läroböckerna, och gymnasiet tenderar att fokusera mer på det strukturella i funktionsbegreppet medan högstadiet i högre grad betonar det operationella. Vidare finns det skillnader mellan de olika läroböckerna inom samma stadie där studien visar att beroende på kombination av läromedel för högstadiet respektive gymnasiet kan det bli olika grad av repetition på gymnasiet. Vissa kombinationer kan ge en större överlappning mellan innehållet i stadierna medan andra kombinationer riskerar att istället skapa ett glapp mellan stadierna. / In the Swedish school system, different stage transitions take place and the transition from lower secondary school to upper secondary school is one such. In the subject of mathematics, previous research shows that a difference in these stages is an increased pace of study and a shift towards more formal mathematics. The purpose of this study has been to investigate what content-related similarities and differences there are between junior high school and high school mathematics in the area of ​​linear functions. This has been done via a content analysis of some textbooks that are used for the different stages, as these can be seen as the potentially implemented curriculum. Furthermore, the study has been based on Anna Sfard's theory of operational and structural concepts as a way to gain insight into and contrast the content covered in the various textbooks. The result shows that there are similarities and differences in the different stages, from how this is expressed via the textbooks, and the upper secondary school tends to focus more on the structural concept of function, while the lower secondary emphasizes the operational aspects to a greater degree. Furthermore, there are differences between the different textbooks within the same stage, where this study shows that depending on the combination of textbooks for lower- and upper secondary school, there may be different degrees of repetition in the latter. Some combinations can provide a greater overlap between the content of the stages, while other combinations risk instead creating a gap between the stages.