Functions have an important place in the Swedish national curriculum. This study explores Swedish high school students’ conceptions and reasoning when solving mathematical problems involving functions. Two students were asked to solve three problems and reason aloud while describing how they approached their solutions. These problems dealt with function expressions, function representations, and the definition of the function concept. The study adopted a discourse analysis, where recorded discussions and explanations were examined to ascertain students’ strategies, abilities, challenges, conceptual understanding, and misconceptions. The main insights from the study are based on exploring how misconceptions and conceptual understanding underlie students’ challenges and abilities. Understanding representations seemed to increase general abilities, and the presence of correct conceptual definitions did not imply a good understanding of the concept of functions. The students’ algebraic knowledge did not hinder their function understanding, but rather was helpful when moving between representations and handling function expressions.

Technological enhancement of teaching and learning of mathematics has been an expanding topic of research. A considerable body of research explores the potential of technologies in teaching and learning mathematics, however, a review of the literature shows that there is little research on how students use these digital tools to solve mathematical tasks. This study follows students’ use of a digital tool (Desmos) when working with quadratic functions. The study analyses data collected from 28 Year 11 students from a school in Sweden focussing on the students’ usage of Desmos as they solved multiple tasks. Students worked in pairs, and their discussions were video-recorded and analysed using a content analysis framework. The analysis reveals four categories of approach; solve algebraically and verify with Desmos, solve with Desmos and verify algebraically, integrated method, where students use the tool differently for different problem-solving questions, and lastly using Desmos as an aide, a category including students that used Desmos to solve tasks completely after failed algebraic attempts. Our findings can thus be of great value in assisting teachers in recognising these differences and provide opportunities for students to explore the use of digital tools in enhancing both their learning experience and performance.

 This thesis explores Swedish high school students’ learning experiences with the

concept of functions, focusing on the challenges they encounter and the role of

digital tools in deepening their understanding. Despite its central role in mathematics,

the concept of functions remains one of the most challenging topics for many

students. As a foundation for mathematical modelling, logical reasoning, and

problem-solving, addressing these learning difficulties is essential for long-term

mathematical development. With the growing integration of technology in education,

digital tools have become increasingly significant in mathematics instruction. This

thesis examines how such tools enhance conceptual understanding by analysing how

students use them to solve function-related tasks. It draws on four interconnected

papers that together provide a comprehensive perspective on students’ learning of

functions and the pedagogical effects of digital tool usage. Paper I presents a

qualitative analysis based on observations of two high school students solving

function problems. It identifies misconceptions and highlights how strong graphical

knowledge and visual reasoning reduce learning difficulties. Paper II is a quasi-

experimental comparison between students using a digital tool and those employing

traditional methods. Results show that students using digital tools performed

significantly better. Paper III investigates how students independently use a digital

tool when solving mathematical tasks, identifying four key strategies: using it as the

primary solving method, for verification, as a support tool, or variably depending on

the task. Paper IV examines the digital tool usage of teachers and students, as well as

their respective perspectives. It further explores teaching strategies, teachers’

perspectives on student usage, and the instructions students receive about digital

tools. Findings across the studies indicate that visualisation through digital tools

enhances conceptual understanding by providing dynamic, interactive learning

experiences. Students can autonomously use these tools in diverse ways, influencing

their learning outcomes and engagement. The research underscores the importance of

teacher facilitation in guiding meaningful tool use and offers practical insights for

integrating technology into mathematics education. Overall, this thesis contributes to

the growing field of technology-enhanced mathematics education and provides

recommendations for improving students’ understanding of functions through digital 

means. Collectively, the studies demonstrate that digital visualisation makes abstract

concepts more tangible, enables multiple solution pathways, and supports student

understanding. However, they also emphasise that the realisation of the potential of

digital tools depends on their use and on intentional pedagogical integration.

 Denna avhandling undersöker svenska gymnasieelevers lärande av

funktionsbegreppet, med fokus på de utmaningar de möter och den roll digitala

verktyg spelar för att fördjupa deras förståelse. Trots sin centrala roll i matematiken

är funktionsbegreppet ett av de mest utmanande områdena för många elever.

Begreppet utgör grunden för matematisk modellering, logiskt resonemang och

problemlösning, och därför är det viktigt att bemöta dessa inlärningssvårigheter för

att främja långsiktig matematisk utveckling hos elever. I takt med att teknologin

integreras allt mer i utbildningen har digitala verktyg blivit en viktig del av

matematikundervisningen. Denna avhandling undersöker hur sådana verktyg kan

stärka den konceptuella förståelsen av funktionsbegreppet. Avhandlingen bygger på

fyra sammanhängande studier som tillsammans ger en omfattande bild av elevers

lärande om funktioner och de pedagogiska effekterna med användning av digitala

verktyg. Studie I presenterar en kvalitativ analys baserad på observationer av två

gymnasieelever som löser funktionsuppgifter. Den identifierar vanliga

missuppfattningar och visar hur starka grafiska kunskaper och visuellt resonemang

kan motverka inlärningssvårigheter av funktionsbegreppet. Studie II är en

kvasiexperimentell jämförelse mellan elever som använder ett digitalt verktyg och de

som använder traditionella metoder. Resultaten visar att elever som använde digitala

verktyg presterade signifikant bättre. Studie III undersöker hur elever på egen hand

använder ett digitalt verktyg vid lösning av matematiska uppgifter, och identifierar

fyra huvudsakliga strategier: att använda det som primär lösningsmetod, för

verifiering, som stödverktyg eller på olika sätt beroende på uppgiften. Studie IV

undersöker lärares och elevers perspektiv om och användning av digitala verktyg.

Vidare utforskar studien lärares lektionsplanering med digitala verktyg, deras

perspektiv på elevers verktygsanvändning samt instruktioner som elever får om

digitala verktyg. Resultaten från studierna visar att visualisering genom digitala

verktyg stärker den konceptuella förståelsen genom att erbjuda dynamiska och

interaktiva lärandeupplevelser. Elever kan självständigt använda dessa verktyg på

olika sätt, vilket påverkar deras läranderesultat och engagemang. Forskningen

betonar vikten av lärarens roll i att vägleda meningsfull användning av verktyg och

ger praktiska insikter för att integrera teknik i matematikundervisningen.Sammantaget bidrar avhandlingen till det växande forskningsområdet kring

teknikförstärkt matematikundervisning och erbjuder rekommendationer för att

förbättra elevers förståelse av funktioner genom digitala verktyg. Tillsammans visar

studierna att digital visualisering gör abstrakta begrepp mer konkreta, möjliggör flera

lösningsvägar och ökar elevers förståelse. Samtidigt betonas att förverkligandet av de

digitala verktygens potential beror på hur de används och på en medveten

pedagogisk integration.

This study examines the effect of using Desmos on students’ performance in learning functions. An embedded mixed methods design was applied and involved 98 students from an upper secondary school in Sweden. Students’ assessments (pre- and post-test) and opinion polls were the two main data collection instruments. The results show that both groups (experimental and control) experienced a significant improvement in their post-test scores. However, the experimental group had a statistically significant improvement in comparison with that of the control group across the five constructs used in this study. The qualitative data revealed that the majority of the students ascribed a positive effect of the use of Desmos on their general understanding of function concepts, their ability to analyze functions and check their answers through visualization, which are difficult when working using paper and pencil.

Students’ ability to solve problems that require higher order thinking skills is not just necessary but also a sufficient condition in our quest for training students who have the skills and competencies to compete effectively both at the national and international levels. Achieving SDG4 is underpinned by examining how UNESCO’s sustainable competencies with emphasis on systems thinking, anticipatory, normative, strategic, critical thinking, self-awareness, and integrated problem-solving competencies are integrated or understood within the classroom context. Despite the integration of problem-solving into the Swedish school curriculum, students’ performance in this area has not been as expected. This study, therefore, examines the problem-solving attitudes of Swedish students and its implications for teacher training and development. Using a survey design and the Problem-Solving Attitude Inventory (PSAI) data was collected from 432 (primary, lower and upper secondary) students. Descriptive and inferential (ANOVA) statistics analyses were conducted to examine the problem-solving attitudes among the different grade levels. The results show that two of the constructs (problem-solving tendency, personal control) were statistically significant., F(2, 429)=5.007, p=0.007 and F(2,429)=3.071, p=0.047 respectively. The constructs (problem-solving confidence and avoidance style) did not show any statistically significant difference, F(2, 429) = 1.609, p = 0.201 and F(2, 429) = 0.484, p = 0.616 respectively between school levels. The results show that primary school students ascribed more positively to the items in the four constructs as compared to their lower and upper secondary peers. Personal control was the main contributing construct and had a direct influence on the other constructs. The ANOVA also showed a significant difference between gender and problem-solving attitudes with males ascribing more positively to all the four constructs except personal control. The results provide critical implications for teacher training and development. The conceptual framework shows that there is a need for teacher training programmes to equip teachers on how to change or influence students' self-control characteristics. We suggest the use of contextual and challenging activities to help enhance students’ personal.-control construct.