<p>Students’ use of superficial reasoning seems to be a main reason for learning difficulties in mathematics. It is therefore important to investigate the reasons for this use and the components that may affect students’ mathematical reasoning development. Assessments have been claimed to be a component that significantly may influence students’ learning.</p><p>The purpose of the study in Paper 1 was to investigate the kind of mathematical reasoning that is required to successfully solve tasks in the written tests students encounter in their learning environment. This study showed that a majority of the tasks in teacher-made assessment could be solved successfully by using only imitative reasoning. The national tests however required creative mathematically founded reasoning to a much higher extent.</p><p>The question about what kind of reasoning the students really use, regardless of what theoretically has been claimed to be required on these tests, still remains. This question is investigated in Paper 2.</p><p>Here is also the relation between the theoretically established reasoning requirements, i.e. the kind of reasoning the students have to use in order to successfully solve included tasks, and the reasoning actually used by students studied. The results showed that the students to large extent did apply the same reasoning as were required, which means that the framework and analysis procedure can be valuable tools when developing tests. It also strengthens many of the results throughout this thesis. A consequence of this concordance is that as in the case with national tests with high demands regarding reasoning also resulted in a higher use of such reasoning, i.e. creative mathematically founded reasoning. Paper 2 can thus be seen to have validated the used framework and the analysis procedure for establishing these requirements.</p><p>Paper 3 investigates the reasons for why the teacher-made tests emphasises low-quality reasoning found in paper I. In short the study showed that the high degree of tasks solvable by imitative reasoning in teacher-made tests seems explainable by amalgamating the following</p><p>factors: (i) Limited awareness of differences in reasoning requirements, (ii) low expectations of students abilities and (iii) the desire to get students passing the tests, which was believed easier when excluding creative reasoning from the tests.</p><p>Information about these reasons is decisive for the possibilities of changing this emphasis. Results from this study can also be used heuristically to explain some of the results found in paper 4, concerning those teachers that did not seem to be influenced by the national tests.</p><p>There are many suggestions in the literature that high-stake tests affect practice in the classroom. Therefore, the national tests may influence teachers in their development of classroom tests. Findings from paper I suggests that this proposed impact seem to have had a limited effect, at least regarding the kind of reasoning required to solve included tasks. What about other competencies described in the policy documents?</p><p>Paper 4 investigates if the Swedish national tests have had such an impact on teacher-made classroom assessment. Results showed that impact in terms of similar distribution of tested competences is very limited. The study however showed the existence of impact from the national tests on teachers test development and how this impact may operate.</p>

Students’ use of superficial reasoning seems to be a main reason for learning difficulties in mathematics. It is therefore important to investigate the reasons for this use and the components that may affect students’ mathematical reasoning development. Assessments have been claimed to be a component that significantly may influence students’ learning. The purpose of the study in Paper 1 was to investigate the kind of mathematical reasoning that is required to successfully solve tasks in the written tests students encounter in their learning environment. This study showed that a majority of the tasks in teacher-made assessment could be solved successfully by using only imitative reasoning. The national tests however required creative mathematically founded reasoning to a much higher extent. The question about what kind of reasoning the students really use, regardless of what theoretically has been claimed to be required on these tests, still remains. This question is investigated in Paper 2. Here is also the relation between the theoretically established reasoning requirements, i.e. the kind of reasoning the students have to use in order to successfully solve included tasks, and the reasoning actually used by students studied. The results showed that the students to large extent did apply the same reasoning as were required, which means that the framework and analysis procedure can be valuable tools when developing tests. It also strengthens many of the results throughout this thesis. A consequence of this concordance is that as in the case with national tests with high demands regarding reasoning also resulted in a higher use of such reasoning, i.e. creative mathematically founded reasoning. Paper 2 can thus be seen to have validated the used framework and the analysis procedure for establishing these requirements. Paper 3 investigates the reasons for why the teacher-made tests emphasises low-quality reasoning found in paper I. In short the study showed that the high degree of tasks solvable by imitative reasoning in teacher-made tests seems explainable by amalgamating the following factors: (i) Limited awareness of differences in reasoning requirements, (ii) low expectations of students abilities and (iii) the desire to get students passing the tests, which was believed easier when excluding creative reasoning from the tests. Information about these reasons is decisive for the possibilities of changing this emphasis. Results from this study can also be used heuristically to explain some of the results found in paper 4, concerning those teachers that did not seem to be influenced by the national tests. There are many suggestions in the literature that high-stake tests affect practice in the classroom. Therefore, the national tests may influence teachers in their development of classroom tests. Findings from paper I suggests that this proposed impact seem to have had a limited effect, at least regarding the kind of reasoning required to solve included tasks. What about other competencies described in the policy documents? Paper 4 investigates if the Swedish national tests have had such an impact on teacher-made classroom assessment. Results showed that impact in terms of similar distribution of tested competences is very limited. The study however showed the existence of impact from the national tests on teachers test development and how this impact may operate.

In recent years, advanced machine learning (ML) models have revolutionized industries ranging from the healthcare sector to retail and E-commerce. However, these models have become increasingly complex, making it difficult for even domain experts to understand and retrace the model's decision-making process. To address this challenge, several frameworks for explainable AI have been proposed and developed. This thesis focuses on counterfactual explanations (CFEs), which provide actionable insights by informing users how to modify inputs to achieve desired outputs. However, computing CFEs for a general black-box ML model is computationally expensive since it hinges on solving a challenging optimization problem. To efficiently solve this optimization problem, we propose using Bayesian optimization (BO), and introduce the novel algorithm Separated Bayesian Optimization (SBO). SBO exploits the formulation of the counterfactual function as a composite function. Additionally, we propose warm-starting SBO, which addresses the computational challenges associated with computing multiple CFEs. By decoupling the generation of a surrogate model for the black-box model and the computation of specific CFEs, warm-starting SBO allows us to reuse previous data and computations, resulting in computational discounts and improved efficiency for large-scale applications. Through numerical experiments, we demonstrate that BO is a viable optimization scheme for computing CFEs for black-box ML models. BO achieves computational efficiency while maintaining good accuracy. SBO improves upon this by requiring fewer evaluations while achieving accuracies comparable to the best conventional optimizer tested. Both BO and SBO exhibit improved capabilities in handling various classes of ML decision models compared to the tested baseline optimizers. Finally, Warm-starting SBO significantly enhances the performance of SBO, reducing function evaluations and errors when computing multiple sequential CFEs. The results indicate a strong potential for large-scale industry applications. / Avancerade maskininlärningsmodeller (ML-modeller) har på senaste åren haft stora framgångar inom flera delar av näringslivet, med allt ifrån hälso- och sjukvårdssektorn till detaljhandel och e-handel. I jämn takt med denna utveckling har det dock även kommit en ökad komplexitet av dessa ML-modeller vilket nu lett till att även domänexperter har svårigheter med att förstå och tolka modellernas beslutsprocesser. För att bemöta detta problem har flertalet förklarbar AI ramverk utvecklats. Denna avhandling fokuserar på kontrafaktuella förklaringar (CFEs). Detta är en förklaringstyp som anger för användaren hur denne bör modifiera sin indata för att uppnå ett visst modellbeslut. För en generell svarta-låda ML-modell är dock beräkningsmässigt kostsamt att beräkna CFEs då det krävs att man löser ett utmanande optimeringsproblem. För att lösa optimeringsproblemet föreslår vi användningen av Bayesiansk Optimering (BO), samt presenterar den nya algoritmen Separated Bayesian Optimization (SBO). SBO utnyttjar kompositionsformuleringen av den kontrafaktuella funktionen. Vidare, utforskar vi beräkningen av flera sekventiella CFEs för vilket vi presenterar varm-startad SBO. Varm-startad SBO lyckas återanvända data samt beräkningar från tidigare CFEs tack vare en separation av surrogat-modellen för svarta-låda ML-modellen och beräkningen av enskilda CFEs. Denna egenskap leder till en minskad beräkningskostnad samt ökad effektivitet för storskaliga tillämpningar.  I de genomförda experimenten visar vi att BO är en lämplig optimeringsmetod för att beräkna CFEs för svarta-låda ML-modeller tack vare en god beräknings effektivitet kombinerat med hög noggrannhet. SBO presterade ännu bättre med i snitt färre funktionsutvärderingar och med fel nivåer jämförbara med den bästa testade konventionella optimeringsmetoden. Både BO och SBO visade på bättre kapacitet att hantera olika klasser av ML-modeller än de andra testade metoderna. Slutligen observerade vi att varm-startad SBO gav ytterligare prestandaökningar med både minskade funktionsutvärderingar och fel när flera CFEs beräknades. Dessa resultat pekar på stor potential för storskaliga tillämpningar inom näringslivet.