| 1. |
- Bergwall, Andreas, 1972-
(författare)
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Proof-related reasoning in upper secondary mathematics textbooks : Characteristics, comparisons, and conceptualizations
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Proofs and proving are difficult to learn and difficult to teach. A common problem is that many students use specific examples as evidence for general statements. Difficulties with proofs are also part of the transition problems that exist between secondary and tertiary schooling in mathematics. As mathematics teaching often follows a textbook, the design of textbooks has been pointed out as one possible cause of the problems, and international textbook research suggests that proofs often have only a marginal place in textbooks.This thesis focuses on proofs and proving in upper secondary mathematics textbooks. It also addresses theoretical and methodological questions about what marks an opportunity to develop proving competence, and which properties of such opportunities are relevant to investigate and characterize. The thesis is based on data from four Swedish and Finnish textbook series for upper secondary school, and focuses on sections on logarithms, primitive functions, definite integrals, and combinatorics. It examines how addressed mathematical principles are justified, and whether the textbooks’ exercises offer opportunities to develop proof-related skills such as formulating and investigating hypotheses, developing and evaluating arguments, identifying and correcting errors, and finding counterexamples.The results show that just over half of the mathematical principles addressed in the analyzed textbook material are justified, and that only half of the justifications are general proofs. Few exercises are proof-related (10%), and those that include reasoning about general cases even fewer. General proofs are more common in the Finnish books, but proof-related tasks are more common and of a more varied nature in the Swedish ones. The most common form of proofs are direct derivations of calculation formulas, while reasoning about existence and uniqueness is unusual, as are contrapositive proofs and proofs by contradiction.Based on the results, explicit suggestions are offered as to what teaching can pay more attention to. For the analysis and design of proof-related activities, a framework consisting of four main categories is proposed: develop a statement, investigate a statement, develop an argument, and investigate an argument. Several properties that such activities may have, regardless of which category they belong to, are discussed. Finally, three areas for future research are suggested: how worked examples can support students’ learning of proof, how textbooks can be designed to stimulate formulation as well as the formal proving of hypotheses, and mapping of differences regarding proof between upper secondary and university textbooks.
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| 2. |
- Eriksson, Henrik, et al.
(författare)
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Conjugacy of Coxeter elements
- 2009
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Ingår i: The Electronic Journal of Combinatorics. - 1097-1440 .- 1077-8926. ; 16:2, s. R4-
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Tidskriftsartikel (refereegranskat)
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| 3. |
- Eriksson, Henrik, et al.
(författare)
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Exact expectations for random graphs and assignments
- 2003
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Ingår i: Combinatorics, probability & computing. - 0963-5483 .- 1469-2163. ; 12, s. 401-412
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Tidskriftsartikel (refereegranskat)abstract
- For a random graph on n vertices where the edges appear with individual rates, we give exact formulas for the expected time at which the number of components has gone down to k and the expected length of the corresponding minimal spanning forest.For a random bipartite graph we give a formula for the expected time at which a k-assignment appears. This result has a bearing on the random assignment problem.
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| 4. |
- Eriksson, Henrik, et al.
(författare)
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Expected inversion number after k adjacent transpositions
- 2000
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Ingår i: Formal Power Series and Algebraic Combinatorics. - 3540672478 ; , s. 677-685
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Konferensbidrag (refereegranskat)abstract
- We give expressions for the expected number of inversions after t random adjacent transpositions have been performed on the identity permutation in Sn+1 The problem is a simplification of a problem motivated by genome evolution. For a fixed t and for all n greater than or equal to t, the expected number of inversions after t random adjacent transpositions isE-nt = t - 2/n ((t)(2)) + Sigma(r=2)(t) (-1)(r)/n(r) [2(r)C(r)((t)(r+1)) + 4d(r) ((t)(r))]where d(2) = 0, d(3) = 1, d(4) = 9, d(5) = 69,... is a certain integer sequence. An important part of the our method is the use of a heat. conduction analogy of the random walks, which guarantees certain properties of the solution.
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| 5. |
- Eriksson, Henrik, et al.
(författare)
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Note on the lamp lighting problem
- 2001
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Ingår i: Advances in Applied Mathematics. - : Elsevier BV. - 0196-8858 .- 1090-2074. ; 27:03-feb, s. 357-366
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Tidskriftsartikel (refereegranskat)abstract
- We answer some questions concerning the so-called sigma -game of Sutner [Linear cellular automata and the Garden of Eden, Math. Intelligencer 11 (1989), 49-53]. It is played on a graph where each vertex has a lamp, the light of which is toggled by pressing any vertex with an edge directed to the lamp. For example, we show that every configuration of lamps can be lit if and only if the number of complete matchings in the graph is odd. In the special case of an orthogonal grid one gets a criterion for whether the number of monomer-dimer tilings of an m x n grid is odd or even.
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| 6. |
- Andrighetto, Giulia, et al.
(författare)
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Changes in social norms during the early stages of the COVID-19 pandemic across 43 countries
- 2024
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Ingår i: Nature Communications. - : NATURE PORTFOLIO. - 2041-1723. ; 15:1
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Tidskriftsartikel (refereegranskat)abstract
- The emergence of COVID-19 dramatically changed social behavior across societies and contexts. Here we study whether social norms also changed. Specifically, we study this question for cultural tightness (the degree to which societies generally have strong norms), specific social norms (e.g. stealing, hand washing), and norms about enforcement, using survey data from 30,431 respondents in 43 countries recorded before and in the early stages following the emergence of COVID-19. Using variation in disease intensity, we shed light on the mechanisms predicting changes in social norm measures. We find evidence that, after the emergence of the COVID-19 pandemic, hand washing norms increased while tightness and punishing frequency slightly decreased but observe no evidence for a robust change in most other norms. Thus, at least in the short term, our findings suggest that cultures are largely stable to pandemic threats except in those norms, hand washing in this case, that are perceived to be directly relevant to dealing with the collective threat. Tightness-looseness theory predicts that social norms strengthen following threat. Here the authors test this and find that, after the emergence of the COVID-19 pandemic, hand washing norms increased, but no evidence was observed for a robust change in most other norms.
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| 7. |
- Bernard, Mark, et al.
(författare)
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The subgroup problem : When can binding voting on extractions from a common pool resource overcome the tragedy of the commons?
- 2013
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Ingår i: Journal of Economic Behavior and Organization. - : Elsevier BV. - 0167-2681 .- 1879-1751. ; 91, s. 122-130
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Tidskriftsartikel (refereegranskat)abstract
- Using a common pool resource game protocol with voting we examine experimentally how cooperation varies with the level at which (binding) votes are aggregated. Our results are broadly in line with theoretical predictions. When players can vote on the behavior of the whole group or when leaders from each group can vote for the group as a whole, extraction levels from the common resource pool are close to the social optimum. When players extract resources individually, there is substantial overextraction. When players vote in subgroups, there is initially less overextraction but it increases over time. This suggests that in order for binding voting to overcome the tragedy of the commons in social dilemmas, it should ideally affect the group as a whole.
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| 8. |
- Bernard, Mark, et al.
(författare)
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The subgroup problem: When can binding voting on extractions from a common resource pool overcome the tragedy of the commons?
- 2013
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Ingår i: Journal of Economic Behavior and Organization. - : Elsevier BV. - 0167-2681 .- 1879-1751. ; 91, s. 122-130
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Tidskriftsartikel (refereegranskat)abstract
- Using a common pool resource game protocol with voting we examine experimentally how cooperation varies with the level at which (binding) votes are aggregated. Our results are broadly in line with theoretical predictions. When players can vote on the behavior of the whole group or when leaders from each group can vote for the group as a whole, extraction levels from the common resource pool are close to the social optimum. When players extract resources individually, there is substantial overextraction. When players vote in subgroups, there is initially less overextraction but it increases over time. This suggests that in order for binding voting to overcome the tragedy of the commons in social dilemmas, it should ideally affect the group as a whole.
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| 9. |
- de Barra, Mícheál, et al.
(författare)
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How feedback biases give ineffective medical treatments a good reputation.
- 2014
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Ingår i: Journal of medical Internet research. - : JMIR Publications Inc.. - 1438-8871. ; 16:8, s. e193-
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Tidskriftsartikel (refereegranskat)abstract
- BACKGROUND: Medical treatments with no direct effect (like homeopathy) or that cause harm (like bloodletting) are common across cultures and throughout history. How do such treatments spread and persist? Most medical treatments result in a range of outcomes: some people improve while others deteriorate. If the people who improve are more inclined to tell others about their experiences than the people who deteriorate, ineffective or even harmful treatments can maintain a good reputation.OBJECTIVE: The intent of this study was to test the hypothesis that positive outcomes are overrepresented in online medical product reviews, to examine if this reputational distortion is large enough to bias people's decisions, and to explore the implications of this bias for the cultural evolution of medical treatments.METHODS: We compared outcomes of weight loss treatments and fertility treatments in clinical trials to outcomes reported in 1901 reviews on Amazon. Then, in a series of experiments, we evaluated people's choice of weight loss diet after reading different reviews. Finally, a mathematical model was used to examine if this bias could result in less effective treatments having a better reputation than more effective treatments.RESULTS: Data are consistent with the hypothesis that people with better outcomes are more inclined to write reviews. After 6 months on the diet, 93% (64/69) of online reviewers reported a weight loss of 10 kg or more while just 27% (19/71) of clinical trial participants experienced this level of weight change. A similar positive distortion was found in fertility treatment reviews. In a series of experiments, we show that people are more inclined to begin a diet with many positive reviews, than a diet with reviews that are representative of the diet's true effect. A mathematical model of medical cultural evolution shows that the size of the positive distortion critically depends on the shape of the outcome distribution.CONCLUSIONS: Online reviews overestimate the benefits of medical treatments, probably because people with negative outcomes are less inclined to tell others about their experiences. This bias can enable ineffective medical treatments to maintain a good reputation.
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| 10. |
- Dorrough, Angela Rachael, et al.
(författare)
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Cooperation in the cross-national context
- 2022
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Ingår i: CURRENT OPINION IN PSYCHOLOGY. - : ELSEVIER. - 2352-250X. ; 44, s. 281-285
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Forskningsöversikt (refereegranskat)abstract
- In this article, we outline the current state of research concerning cooperation in the cross-national context. We present several theoretical approaches and empirical findings regarding national differences in cooperation, as well as how cooperation may depend on the national background of the interaction partner. In addition, we discuss the influence of (national) group norms, cultural similarity, and ingroup membership. This review concludes with a call for research on cooperation to include more non-WEIRD nations and more systematically cover national background as one important social category determining the willingness to cooperate.
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