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Rapporter och studier visar att elever som får traditionell matematikundervisning har ytliga kunskaper som snabbt glöms bort och att de ofta är oengagerade inför skolmatematiken. Kreativa resonemang i matematikundervisningen handlar om hur lärare kan planera och genomföra en mer elevaktiv undervisning, som leder till mer bestående och djupgående matematikkunskaper hos eleverna. Boken visar hur undervisning som bygger på problemlösning och kreativa resonemang ger läraren goda möjligheter att genom feedback och frågor stödja elever att utveckla alla de matematiska förmågor som nuvarande läroplan tar upp. Boken inleds med en teoretisk bakgrund med illustrerande exempel. Därefter presenteras ett ramverk för planering och genomförande av undervisning genom kreativa resonemang. Avslutningsvis diskuteras olika aspekter av hur undervisning som planerats enligt ramverket har fungerat i praktiken.Kreativa resonemang i matematikundervisningen vänder sig till lärare, skolledare och lärarstuderande som vill utveckla matematikundervisningen enligt aktuell forskning om lärande i matematik.

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.

This study contributes insights into how task design with different elements of guidancemay influence students’ utilization of dynamic software for problem solving andreasoning. It compared students’ solving of two tasks with different designs supportedby the dynamic software GeoGebra. Data analysed examined students’ approaches toutilizing GeoGebra, the characteristics of their reasoning and their ability to prove thevalidity of their solutions after solving the problems. The results showed that studentswho solved the task with less guidance (without instructions about a specific solvingmethod) were better able to utilize GeoGebra’s potential to support their reasoning andproblem solving. These students reasoned more creatively and presented more advanced proofs for their solutions than the more guided ones.

This study investigates how students’ reasoning contributes to their utilization of computer-generated feedback. Sixteen 16-year-old students solved a linear function task designed to present a challenge to them using dynamic software, GeoGebra, for assistance. The data were analysed with respect both to character of reasoning and to the use of feedback generated through activities in GeoGebra. The results showed that students who successfully solved the task were engaged in creative reasoning and used feedback extensively.

The present study investigates how to support students’ creative reasoning when they need assistance in solving non-routine tasks. Two groups of 11–12-year-old students solved the same tasks, one group receiving feedback directed at the task solution and the other feedback directed at their thinking processes. The results showed that stu- dents who received feedback directed at their thinking processes expressed reason- ing based on their attempts to solve tasks while the other group often repeated the researcher’s suggestions for solutions. However, there were some instances in which feedback on task level entailed students engaging in creative reasoning.

The present study contributes to knowledge about how to design tasks that benefit from dynamic software in math education, comparing practice performance and learning outcomes among 129 students practicing on two different task designs using GeoGebra. The task designs differed with respect to the presence or absence of guidelines on how to solve the task. One student group practiced on the guided task while the other student group practiced on the unguided task, and 1 week later a posttest was conducted. Data were statistically analyzed and showed significant differences with regard to success during practice for students solving the guided task. Among the students who succeeded in solving the task (guided or unguided) during practice, however, the analysis showed significant differences in the posttest performance in favor of the unguided students.

The present study investigates how a dynamic software program, GeoGebra, may support students' collaboration and creative reasoning during mathematical problem solving. Thirty-six students between the ages of 16 and 17 worked in pairs to solve a linear function using GeoGebra. Data in the form of recorded conversations, and computer activities were analyzed using Lithner's (2008) framework of imitative and creative reasoning in conjunction with the collaborative model of joint problem space (Roschelle & Teasley, 1994). The results indicated that GeoGebra supported collaboration and creative reasoning by providing students with a shared working space and feedback that became the subject for students' creative reasoning. Furthermore, the students' collaborative activities aimed toward sharing their reasoning with one another enhanced their creative reasoning. There were also examples of students using GeoGebra for trial-and-error strategies and students engaging in superficial argumentation.

Studies have shown that learning mathematics through programming can be complex and that the programming itself might even hamper students’ learning. However, few studies have focused on the role of the teacher and the teacher-student interaction that aims to support students’ learning when using programming. The present study examines a didactic design in which a teacher uses well-prepared questions and suggestions that focus on students’ reasoning to solve mathematical problems using Scratch, a block-based programming environment. Forty students, 10-11 years old, solved a geometry problem using Scratch supported by their teacher. The students’ screen activities and teacher-student interactions were recorded. The results indicate that well-prepared general and task-specific questions targeting students’ creative reasoning can help to overcome some of the complexities of learning mathematics while programming. The study furthermore shows that when students have sufficient programming skills, Scratch could even have the potential to support their reasoning and at the same time guide the teacher to give timely feedback.

The aim of the study was to explore teacher support to students’ mathematicalreasoning when they retrieve and use prior knowledge to learn new mathematicalcontent. Two lessons were designed, one with the purpose to engage students inretrieving prior knowledge of angles and shares, and the following to learn how toconstruct pie-charts. The teacher’s interactions with students were recorded and theresult showed that the teacher could support students’ reasoning by showing inte-rest in their thinking and asking challenging questions regarding clarification andjustification of their mathematical reasoning.

This is a report on the discussions (and post-re ections) of the MADIF10 symposium ”Conceptualizing a local instructional theory in design research”. Linking the discus- sion to Koeno Gravemeijer’s keynote at MADIF9 and drawing on di erent ongoing research projects, the aim of the symposium was to discuss [examples of] the opera- tionalization of design principles in order to deepen the understanding of some theo- retical concepts in design research. The contribution of the symposium is the inter- pretation of how local instruction theory interrelates with other concepts in design research, for instance, the hypothetical learning trajectory. The role of the concepts as both design tools and as outcomes was presented and discussed.

This paper presents the way pilot studies underpin a design for a future project investigating how formative feedback can be designed in order to support students’ creative reasoning when constructing solutions to mathematical tasks. It builds on the idea that creative reasoning is beneficial to students’ mathematical learning. Four pilot studies have been performed with the purpose of creating an empirical base for the preparation of formative feedback to students in mathematics classrooms. The results represent a development of general theoretical guidelines for formative feedback. Our specific and empirically based guidelines will act as a starting point for further intervention studies investigating the design of formative feedback aimed at supporting students’ creative reasoning.