Reasoning Method Based on Intervals with Symmetric Truncated Normal Density

Error parameters are inevitable in systems. In formal verification, previous reasoning methods seldom considered the probability information of errors. In this article, errors are described as symmetric truncated normal intervals consisting of the intervals and symmetric truncated normal probability density. Furthermore, we also rigorously prove lemmas and a theorem to partially simplify the calculation process of truncated normal intervals and independently verify the formulas of variance and expectation of symmetric truncated interval given by some scholars. The mathematical derivation process or verification codes are provided for most of the key formulas in this article. Hence, we propose a new reasoning method that combines the probability information of errors with the previous statistical reasoning methods. Finally, an engineering example of the reasoning verification of train acceleration is provided. After simulating the large-scale cases, it is shown that the simulation results are consistent with the theoretical reasoning results. This method needs more calculation, while it is more effective in detecting non-error’s fault factors than other error reasoning methods.

An introductory biology research-rich laboratory course shows improvements in students’ research skills, confidence, and attitudes

As part of a wider reform to scaffold quantitative and research skills throughout the biology major, we introduced course-based undergraduate research experiences (CURE) in sections of a large-enrollment introductory biology laboratory course in a mid-level, public, minority-serving institution. This initiative was undertaken as part of the in the National Science Foundation / Council for Undergraduate Research Transformations Project. Student teams performed two or three experiments, depending on semester. They designed, implemented, analyzed, revised and iterated, wrote scientific paper-style reports, and gave oral presentations. We tested the impact of CURE on student proficiency in experimental design and statistical reasoning, and student research confidence and attitudes over two semesters. We found that students in the CURE sections met the reformed learning objectives for experimental design and statistical reasoning. CURE students also showed higher levels of experimental design proficiency, research self-efficacy, and more expert-like scientific mindsets compared to students in a matched cohort with the traditional design. While students in both groups described labs as a positive experience in end-of-semester reflections, the CURE group showed a high level of engagement with the research process. Students in CURE sections identified components of the research process that were difficult, while also reporting enjoying and valuing research. This study demonstrates improved learning, confidence, and attitudes toward research in a challenging CURE laboratory course where students had significant autonomy combined with appropriate support at a diverse public university.

Is Automation of Statistical Reasoning a Suitable Mindware in a Base-Rate Neglect Task?

Until recently, studies within the dual-process approach were mainly focused on group differences in processing, and individual differences were neglected. However, individual differences have proven to be a significant factor in conflict detection efficiency and the overall success in base-rate neglect and similar tasks. This should be taken into consideration within the framework of the Hybrid Model of Dual Processing. New tendencies in the development of this model have focused attention on the degree of mindware instantiation as a predictor of base-rate neglect task efficiency. This study aimed to examine the relationship between mindware and base-rate neglect task efficiency and to test and explore the relationship between base-rate response frequency and conflict detection efficiency and the degree of mindware instantiation. All participants solved base-rate neglect tasks, made judgments of confidence in their responses, and solved the Statistical Reasoning Test, Cognitive Reflection Test and Numeracy Scale. We used the Statistical Reasoning Test as a measure of mindware instantiation. The degree of mindware instantiation was found to be the only significant predictor of base-rate neglect task efficiency and the results showed that participants with a higher degree of mindware instantiation generally made more base-rate responses. No correlation was found between the degree of mindware instantiation and conflict detection efficiency. These findings support the hypothesis that the power of logical intuition depends on the individual’s degree of mindware instantiation. Therefore, the results of this research indicate the importance of further research into the role of statistical reasoning in base-rate neglect task efficiency. However, we discuss that there are some methodological limitations in this research which might explain why the degree of mindware instantiation had no relationship with conflict efficiency.

Practices of the Random Variable Proposed in the Chilean Mathematics Curriculum of Secondary Education

Working with statistical data from real contexts has become fundamental in school-level statistics, because it enables the development of statistical reasoning. In this regard, the notion of the random variable is fundamental to statistical data analysis. Thus, the aim of this research was to characterise the meanings of the concept of the random variable that are promoted in the Chilean mathematics curriculum for secondary education. To achieve this, we examined the representativeness of the meanings of the random variable intended by the curriculum in relation to the meanings of reference for this concept, using theoretical and methodological notions from the onto-semiotic approach. The findings show that the meanings of reference were not fully represented in the textbooks and the national curriculum, and that the most promoted meaning was that of the random variable as a variable of interest. Regarding the types of representation, it was observed that the graphical and tabular representations were neglected. Lastly, it was noticed that definitions of the concept of variable, its classifications, and differences were omitted almost entirely from the textbooks.

Learning progressions and the Australian curriculum mathematics: The case of statistics and probability

Mathematics curricula have traditionally focused on content knowledge, often in the form of a scope and sequence of increasingly difficult mathematics. The importance of using and applying mathematics is recognised in the current Australian Curriculum Mathematics (AC: M) as ‘proficiencies’ that are intended to be integrated with the content. There is little support for teachers to develop these proficiencies – reasoning, understanding, problem solving and fluency. Learning progressions are sequences of learning that focus on cognitive processes, and thus provide a useful basis for curriculum development. Using an empirical Statistical Reasoning Learning Progression as an exemplar, a new approach to curriculum development is suggested that links content knowledge with the proficiencies. The outcome is a zone-based, rather than year level based, curriculum that allows teachers to target their teaching, so that students develop increasingly sophisticated understanding of statistics and probability.

From Comparative and Statistical Assessments of Liveability and Health Conditions of Districts in Hong Kong towards Future City Development

Liveability is an indispensable component in future city planning and is practically linked with the health status of individuals and communities. However, there was nor comprehensive and universal district-level framework for assessing liveability due to geospatial and social discrepancies among different countries. In this study, using Hong Kong, a highly dense and international city as an example, the Liveability and Health Index (LHI-HK) consisting of 30 indicators was established, with 21 of them related to education, economy, housing, walkability/transport, environment, and health facilities aspects, while the health conditions of citizens in individual districts were examined by other 9 indicators. Respective scoring allocation was determined by statistical reasoning, and was applied to quantify the connections between liveability and health among the 18 districts of Hong Kong in both 2016 and 2019. Temporal changes of spatial features could be traced by this quantitative framework, and obvious correlations between liveability and health were attained, with R values of 0.496 and 0.518 in 2016 and 2019, and corresponding slopes of 0.80 and 0.88, respectively. Based on the statistical results, it was found that Sai Kung and Kwun Tong are the most and the least liveable district of Hong Kong in 2019. The LHI-HK index was well-validated by renowned AARP liveability index and The California Healthy Places Index (HPI), with R values of 0.90 and 0.70, and the potential uncertainties due to data projection were less than 2.5% for all districts, which implicates its relevancy and appropriateness in conducting similar spatial assessments in international cities. Further, both favorable and unfavorable spatial arrangements of each of the 3 district types in Hong Kong were identified, namely residential, commercial, and industrial districts. This opens new windows in enhancing liveability and health status within communities, with the aim of promoting the sustainability of cities in the long run.