| 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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| 11. |
- Enquist, Magnus, et al.
(författare)
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One cultural parent makes no culture
- 2010
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Ingår i: Animal Behaviour. - : Elsevier BV. - 0003-3472 .- 1095-8282. ; 79:6, s. 1353-1362
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Tidskriftsartikel (refereegranskat)abstract
- The ability to acquire knowledge and skills from others is widespread in animals and is commonly thought to be responsible for the behavioural traditions observed in many species. However, in spite of the extensive literature on theoretical analyses and empirical studies of social learning, little attention has been given to whether individuals acquire knowledge from a single individual or multiple models. Researchers commonly refer to instances of sons learning from fathers, or daughters from mothers, while theoreticians have constructed models of uniparental transmission, with little consideration of whether such restricted modes of transmission are actually feasible. We used mathematical models to demonstrate that the conditions under which learning from a single cultural parent can lead to stable culture are surprisingly restricted ( the same reasoning applies to a single social-learning event). Conversely, we demonstrate how learning from more than one cultural parent can establish culture, and find that cultural traits will reach a nonzero equilibrium in the population provided the product of the fidelity of social learning and the number of cultural parents exceeds 1. We discuss the implications of the analysis for interpreting various findings in the animal social-learning literature, as well as the unique features of human culture.
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| 12. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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A formal model accounting for measurement reliability shows attenuated effect of higher education on intelligence in longitudinal data
- 2024
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Ingår i: Royal Society Open Science. - : Royal Society Publishing. - 2054-5703. ; 11:5
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Tidskriftsartikel (refereegranskat)abstract
- The effect of higher education on intelligence has been examined using longitudinal data. Typically, these studies reveal a positive effect, approximately 1 IQ point per year of higher education, particularly when pre-education intelligence is considered as a covariate in the analyses. However, such covariate adjustment is known to yield positively biased results if the covariate has measurement errors and is correlated with the predictor. Simultaneously, a negative bias may emerge if the intelligence measure after higher education has non-classical measurement errors as in data from the 1970 British Cohort Study that were used in a previous study of the effect of higher education. In response, we have devised an estimation method that used iterated simulations to account for both classical measurement errors in the covariate and non-classical errors in the dependent variable. Upon applying this method in a reanalysis of the data from the 1970 British Cohort Study, we find that the estimated effect of higher education diminishes to 0.4 IQ points per year. Additionally, our findings suggest that the impact of higher education is somewhat more pronounced in the initial 2 years of higher education, aligning with the notion of diminishing marginal cognitive benefits.
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| 13. |
- Eriksson, Kimmo, 1967-
(författare)
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A Note on the Exact Expected Length of the kth Part of a Random Partition
- 2010
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Ingår i: Integers. - 1867-0652. ; 10, s. 309-311
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Tidskriftsartikel (refereegranskat)abstract
- Kessler and Livingstone proved an asymptotic formula for the expected length of the largest part of a partition drawn uniformly at random. As a first step they gave an exact formula expressed as a weighted sum of Euler's partition function. Here we give a short bijective proof of a generalization of this exact formula to the expected length of the kth part.
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| 14. |
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| 15. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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An Exponential Limit Shape of Random q-proportion Bulgarian Solitaire
- 2018
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Ingår i: Integers. - 1553-1732. ; 18
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Tidskriftsartikel (refereegranskat)abstract
- We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn, qn ≤ 1), playedon n cards distributed in piles. In each pile, a number of cards equal to the propor-tion qn of the pile size rounded upward to the nearest integer are candidates to bepicked. Each candidate card is picked with probability pn, independently of othercandidate cards. This generalizes Popov’s random Bulgarian solitaire, in whichthere is a single candidate card in each pile. Popov showed that a triangular limitshape is obtained for a fixed p as n tends to infinity. Here we let both pn and qnvary with n. We show that under the conditions q2npnn/log n → ∞ and pnqn → 0 asn → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limitshape.
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| 16. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Appropriateness ratings of everyday behaviors in the United States now and 50 years ago
- 2023
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Ingår i: Frontiers in Psychology. - : Frontiers Media S.A.. - 1664-1078. ; 14
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Tidskriftsartikel (refereegranskat)abstract
- Introduction: A crucial aspect of social norms pertains to determining which behaviors are considered appropriate. Here we consider everyday behaviors. Some everyday behaviors are rated as more appropriate than others, and ratings of the appropriateness of a given behavior may vary over time. The objective of this study is to elucidate the reasons behind variation in appropriateness ratings of everyday behaviors in the United States. Our theory focuses on how the evaluation of the appropriateness of a behavior is influenced by its potential for externalities and internalities, and how this influence may cause a change in norms over time.Method: Employing a preregistered design, we asked American participants to rate 37 different everyday behaviors based on their appropriateness in a range of common situations, as well as their potential negative externalities (e.g., being loud, being aggressive, taking up space) and positive internalities (e.g., pleasurability). Changes over time were calculated as the difference between mean ratings obtained in this study and ratings of the same behavior in a similar study conducted 50 years ago.Results: As expected, overall appropriateness ratings of everyday behaviors are associated both with their externalities and their internalities, so that the least appropriate behaviors tend to have considerable potential for negative externalities and little potential for positive internalities. Moreover, behaviors that have considerable potential for negative externalities are perceived as less appropriate now than 50 years ago.Discussion: By describing how social norms for everyday behaviors depend on the externalities and internalities of behaviors, this study contributes to theories about the emergence and change of social norms.
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| 17. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Are people really conformist-biased? : An empirical test and a new mathematical model
- 2009
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Ingår i: Journal of Evolutionary Psychology. - 0737-4828. ; 7:1, s. 5-21
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Tidskriftsartikel (refereegranskat)abstract
- According to an influential theory in cultural evolution, within-group similarity of culture is explained by a human conformist-bias, which is a hypothesized evolved predisposition to preferentially follow a member of the majority when acquiring ideas and behaviours. However, this notion has little support from social psychological research. In fact, a major theory in social psychology (Latan and Wolf (1981) argues for what is in effect a nonconformist-bias: by analogy to standard psychophysics they predict minority sources of influence to have relatively greater impact than majority sources. Here we present a new mathematical model and an experiment on social influence, both specifically designed to test these competing predictions. The results are in line with nonconformism. Finally, we discuss within-group similarity and suggest that it is not a general phenomenon but must be studied trait by trait.
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| 18. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Author Correction : Perceptions of the appropriate response to norm violation in 57 societies (Nature Communications, (2021), 12, 1, (1481), 10.1038/s41467-021-21602-9)
- 2021
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Ingår i: Nature Communications. - : Nature Research. - 2041-1723. ; 12:1
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Tidskriftsartikel (refereegranskat)abstract
- The original version of this Article contained an error in the author affiliations. Cecilia Reyna was incorrectly associated with ‘Universidad Nacional de Córdoba (UNC). Facultad de Psicología (UNC), Ciudad Universitaria, Bv. de la Reforma esquina, Enfermera Gordillo s/n, Córdoba, Argentina.’ instead of the correct ‘Instituto de Investigaciones Psicológicas (IIPsi), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), CABA, República Argentina.’ This has now been corrected in both the PDF and HTML versions of the Article. © The Author(s) 2021
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| 19. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Biases for acquiring information individually rather than socially
- 2009
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Ingår i: Journal of Evolutionary Psychology. - 0737-4828. ; 7:4, s. 309-329
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Tidskriftsartikel (refereegranskat)abstract
- We discuss theoretical and empirical arguments for a human bias to acquire information individually rather than socially. In particular, we argue that when other people can be observed, information collection is a public good and hence some of the individual variation in the choice between individual and social learning can be explained by variation in social value orientation. We conducted two experimental studies, based on the game Explore & Collect, to test the predictions that (1) socially and individually acquired information of equal objective value are treated differently, and (2) prosocial subjects tend to spend more effort than selfish subjects on individual acquiring of information. Both predictions were supported.
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| 20. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Cognitive ability and ideology join forces in the culture war : A model of opinion formation
- 2023
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Ingår i: PNAS Nexus. - : National Academy of Sciences. - 2752-6542. ; 2:6
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Tidskriftsartikel (refereegranskat)abstract
- We propose a model of moral policy opinion formation that integrates both ideology and cognitive ability. The link from people's ideology to their opinions is assumed to go via a semantic processing of moral arguments that relies on the individual's cognitive ability. An implication of this model is that the relative quality of arguments that justify supporting vs. opposing a moral policy-the policy's “argument advantage”-is key to how opinions will be distributed in the population and develop over time. To test this implication, we combine polling data with measures of the argument advantage for 35 moral policies. Consistent with the opinion formation model, the argument advantage of a moral policy accounts for how public opinion moves over time, and how support for the policy ideologies varies across different ideological groups and levels of cognitive ability, including a strong interaction between ideology and cognitive ability.
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| 21. |
- Eriksson, Kimmo, 1967-
(författare)
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Comment on “The Evolution of Cultural Complexity : Not by the Treadmill Alone” by Andersson & Read
- 2016
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Ingår i: Current Anthropology. - 0011-3204 .- 1537-5382. ; 57, s. 275-276
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- At the end of their thoughtful target article, Andersson and Read conclude that formal models of cultural evolution are “useful but must be kept in perspective.” As a mathematician with a great interest in social science, I have some experience of working with such models. Based on this experience, I very much agree with the “but” part of the above conclusion. I see a clear tendency in the cultural evolution literature to put too much trust in the value of formal models.
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| 22. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Cultural variation in the SES-gender interaction in student achievement
- 2023
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Ingår i: Frontiers in Psychology. - : FRONTIERS MEDIA SA. - 1664-1078. ; 14
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Tidskriftsartikel (refereegranskat)abstract
- IntroductionIs the socioeconomic gap in academic achievement larger among boys than girls? Several scholars have proposed such an interaction between socioeconomic status (SES) and gender. Prior empirical studies have yielded mixed evidence, but they have been conducted almost exclusively in Western countries. Here we propose the hypothesis that the SES-gender interaction is stronger in less gender-equal societies.MethodsWe estimated the SES-gender interaction in 36 countries using data from two international large-scale assessments (PIRLS and TIMSS). The degree of gender equality was measured by the Global Gender Gap Index.ResultsConsistent with the hypothesis, the SES-gender interaction was stronger in societies with less gender equality.DiscussionOur findings suggest that cultural factors determine how the socioeconomic achievement gap differs between boys and girls.
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| 23. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Deception and price in a market with asymmetric information
- 2007
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Ingår i: Judgment and Decision Making. - 1930-2975. ; 2:1, s. 23-28
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Tidskriftsartikel (populärvet., debatt m.m.)abstract
- In markets with asymmetric information, only sellers have knowledge about the quality of goods. Sellers may of course make a declaration of the quality, but unless there are sanctions imposed on false declarations or reputations are at stake, such declarations are tantamount to cheap talk. Nonetheless, in an experimental study we find that most people make honest declarations, which is in line with recent findings that lies damaging another party are costly in terms of the liar’s utility. Moreover, we find in this experimental market that deceptive sellers offer lower prices than honest sellers, which could possibly be explained by the same wish to limit the damage to the other party. However, when the recipient of the offer is a social tie we find no evidence for lower prices of deceptive offers, which seems to indicate that the rationale for the lower price in deceptive offers to strangers is in fact profit-seeking (by making the deal more attractive) rather than moral.
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| 24. |
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| 25. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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Domain-specific tightness : Why is Sweden perceived as tighter than the United States?
- 2022
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Ingår i: Current Research in Ecological and Social Psychology. - : Elsevier BV. - 2666-6227. ; 3, s. 100049-100049
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Tidskriftsartikel (refereegranskat)abstract
- The tightness of a society is defined as the strength of social norms and the degree of sanctioning within thesociety. However, a society’s tightness may vary across behavioral domains. A recent global survey found thatSweden is generally perceived as relatively tight, even though it is known to be very permissive with respectto sexual relations and gender roles. Here we examine perceptions of the tightness of Sweden and the UnitedStates in six other domains. We find that Sweden is perceived as tighter than the US specifically with respect tonorms about how people may talk about other groups and norms about considerate behavior in public. These domain-specific differences partially mediate the country difference in perceived overall tightness. In sum, this study demonstrates how domain-specific tightness may be measured and highlights the value of such measures to obtain a more nuanced picture of how tightness varies across countries
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