Economists have convulsively maintained the assumption that humans are able to arrive at decisions by perfect deductive rationality, despite the fact empirical evidences are showing otherwise. The contradicting evidences have resulted in a personal view that instead of finding a unified theory about decision-making, a sound approach would be to study how humans in fact are reasoning in specific contexts. The context of interest for this paper is where it could be assumed humans’ persistence of acting rational is determined by the perceived burden of the problem. In this work, the inductive way of arriving at decisions plays an important role, and the paper will present a way of describing this process in a consistent way. The process will be denoted as the actual level of behavioral change, and represent the core property of this paper. Applying the presented theory is most appropriate for situations where it could be assumed the burden of a problem, expressed as a prevalence rate, will drive the behavioral change. The line of reasoning in this paper will therefore be applied to the important arena of fighting the spread of HIV.

abstract: This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

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Den här studien syftar till att undersöka vilka matematiska resonemang som framträder när elever löser uppgifter i grupp. Det resultat som framkommit baseras utifrån fallstudie där elevers användning av matematiska resonemang har observerats i två klassrum. I studiens resultat visade det sig att imitativa matematiska resonemang framträdde när de löste uppgifter i grupp. Det visade sig även att kreativt matematiskt resonemang användes, men endast vid ett fåtal tillfällen. Slutsatsen utifrån studiens resultat är att uppgifter som främjar kreativa matematiska resonemang borde få en given plats i undervisningen, utan att imitativa resonemang försvinner. När det kommer till studiens resultat och dess relevans för vårt yrke så är det viktigt att skapa en medvetenhet kring matematiska resonemang och att ge elever möjlighet till att använda såväl imitativa som kreativa resonemang. Vidare forskning utifrån studiens resultat kan vara att undersöka öppna uppgifters påverkan på imitativa och kreativa resonemang.

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The object of this research was to investigate young children's mathematical spatial reasoning in a Montessori classroom. Spatial reasoning is an important part of children's mathematical learning and development; however, opportunities for rich spatial reasoning are not readily available in the classroom. Rather, there is a focus on numeracy at the expense of geometry where activities for spatial development are usually found. Montessori designed a sensory curriculum around children's development, yet spatial reasoning in a Montessori classroom has not been fully investigated. This was a qualitative study using some tools of ethnography. The theoretical framework was Radford's sensuous cognition (2013, 2014) which allowed for an understanding of human development as cultural with the body essential to that development. The data, captured by video, were the children's semiotic traces (Bartolini Bussi and Baccaglini-Frank (2015, p. 393) which are the visible productions of the children's spatial reasoning such as their movements, text, drawings, and speech. The analysis found that the children had ample opportunities for engaging in challenging mathematical problems which required their spatial reasoning. These engaging activities resulted in the children using a wide range of spatial skills as they reasoned mathematically. The children's movement, the main semiotic trace generated by the children, was crucial to their spatial reasoning. This investigation concluded the pedagogical practices created a rich and dynamic environment for the children's spatial development. Practices included the use of well-designed mathematical manipulatives, engagement in the manner of guided play, co-operative learning with peers of mixed ages, extensive time for activities, and assessment based on observations of individual children.

Mathematics education reform has led to more conceptually focused instruction in the classroom. Yet, students with disabilities are receiving fewer chances than other students to engage in meaningful mathematics. Furthermore, a research divide between mathematics education and special education in mathematics has led to significant gaps in research on the individual and conceptual understanding of students with disabilities. Through task-based interviews and classroom observations, this study begins the process of closing this research gap through an examination of students' understanding of variational and covariational reasoning. Data suggest that the participants, two students with disabilities, increased their conceptual understanding in a reformed learning environment with support from teacher presence and questions. The students were able to increase their understanding of the difference between discrete and continuous functions, demonstrated an ability to self-correct, and improved their ability to choose appropriate levels of reasoning. The results suggest that conceptually oriented instruction with the presence and questioning of a teacher can support students with disabilities in developing a deep and rich understanding of complex mathematics.

There is much debate on the best way to educate students on how to generate hypotheses to enhance clinical reasoning in nursing education. To increase opportunities for nursing programs to promote the discovery of accurate and broad-level hypotheses, scholars recommend abductive reasoning which offers an alternative approach to hypothetico-deductive reasoning. This study explored the effects of abductive reasoning training on hypothesis generation abilities (accuracy, expertise, breadth) of first and second year baccalaureate nursing students in a problem-based learning curriculum. A quasi-experiment with 64 participants (29 control, 35 experimental) was conducted. Based on their allocation, study participants either took part in abductive reasoning training or informal group discussion. Three different test questionnaires, each with a unique care scenario, were used to assess participants’ hypothesis generation abilities at baseline, immediate post-test and one-week follow-up. Content validity for care scenarios and other study materials was obtained from content academic experts. Compared to control participants, experimental participants showed significant improvements at follow-up on hypothesis accuracy (p=0.05), expertise (p=0.006), and breadth (p=0.003). While control participants’ hypotheses displayed a superficial understanding of care situations, experimental participants’ hypotheses reflected increased accuracy, expertise and breadth. This study shows that abductive reasoning, as a scaffolding teaching and learning strategy, can allow nursing students to discover underlying salient patterns in order to better understand and explain the complex realities of care situations. Educating nursing students in abductive reasoning could enable them to adapt existing competencies when trying to accurately and holistically understand newer complex care situations. This could lead to a more holistic, person-based approach to care which will allow nursing students to see various health-related issues as integrated rather than separate. / Thesis / Doctor of Philosophy (PhD) / This study explored the effects of a training program on hypothesis generation abilities of nursing students. The training program aimed to teach students how to think more broadly about care situations. Student’s hypothesis generation abilities were measured through the use of three care scenarios, each of which was presented before, immediately after and one-week after the training program. Only first and second year nursing students were included in the study. About half of the students were provided with the training while the other half were provided with informal discussion about hypothesis generation. After one-week, it was discovered that students who received the training had improved significantly in their ability to generate broad hypotheses. These students also generated hypotheses that were more accurate than the other group of students who did not receive the training. Due to the training, students’ abilities in discovering the important aspects of the care situation also improved.

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