Worked Examples Research Articles

Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint

<abstract><p>In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.</p></abstract>

Online Biostatistics Teaching to Medical Undergraduates by Comparing Google Classroom Module with Online Lecture during COVID-19 Pandemic: An Educational Interventional Study

Introduction: Coronavirus Disease-2019 (COVID-19) lockdown was an opportunity for starting online medical education in Kerala, India. It was a challenge to teach the core competency biostatistics online considering the unfamiliarity and possible network issues. Aim: To assess the learning and student satisfaction of biostatistics teaching, by comparing Google classroom module and online lecture, while teaching phase-2 MBBS students. Materials and Methods: This was a randomised educational interventional study done in the Department of Community Medicine, Government Medical College, Manjeri, Kerala, India, from December 2020 to September 2021. A total of 89 students of phase-2 MBBS were randomly divided into two groups as online lecture group and Google classroom group and five sessions were conducted. For the lecture group, classes were conducted via Google Meet using PowerPoint presentations and for the other group, PowerPoint with audio and workedout examples were provided in Google classroom. Immediate and retention scores of academic performance and satisfaction level were assessed using Multiple Choice Questions (MCQs). Mann-Whitney U test using Statistical Package for the Social Sciences (SPSS) software version 28.0 were done for analysis. Results: Of total 89 students participated, the mean age of the study population was 21.2±0.73, and there were 62 (69.7%) females and 37 (30.3%) males. Academic performance was similar for both groups in most of the immediate assessments, except activity on making graphs (p-value=0.001). In delayed assessment using median scores, the Google classroom group (110, IQR: 100-140) performed significantly better compared to the online lecture (100, IQR: 60-130) with (p-value=0.042). Regarding satisfaction, 68.3% of Google classroom students were satisfied, compared to 52.4% of the online lecture group. The flexibility to learn at own place, pace, and time was the main advantage of the Google classroom, while the need for self-motivation was the disadvantage. Direct interaction was the advantage of online lectures and inability for simultaneous doubt clarification and to attend at fixed timing was the disadvantage. Conclusion: Online teaching of biostatistics was effective and student satisfaction level was good. Google classroom module can be considered as a supplement for traditional teaching.

Tools for overcoming reliance on energy-based measures in chemistry: a tutorial review.

The vast majority of literature in the chemical sciences describes fundamental chemical and physical phenomena using scalar measures, such as the energy, even though many phenomena are beyond the scope of scalar-based considerations. This problem exists no matter how accurately the associated energies are calculated. The solution that is explained in this work is to remove the reliance on scalar quantum chemical measures and instead utilize the vector-based and full symmetry-breaking nature of next generation quantum theory of atoms in molecules (NG-QTAIM). The connection with experiment on neutral chiral molecules is explained. A selection of non-energy-based explanations are provided: the functioning of molecular devices, why the cis-effect is the exception rather than the rule, stereochemical phenomena including chiral discrimination, quantifying chiral character of formally achiral molecules, mixed S and R stereoisomer character and the effect of an applied electric field. Current and future developments along with suggestions for future avenues of investigation are discussed. This tutorial review provides the practical details required to implement NG-QTAIM for a range of phenomena that are not accessible with energy-based measures. Step-by-step worked examples are included with data sets and instructions for use of commercial and open-source software along with examples of how to interpret the results.

Creativity, metacognition, and self-efficacy in error detection in worked-out problem examples

La utilización de problemas verbales totalmente resueltos es práctica habitual en las aulas de ciencias de la educación secundaria. Dada la escasez de estudios sobre la detección de errores en problemas resueltos de ciencias, los objetivos de esta investigación se centraron en la habilidad de detectar errores en problemas resueltos y en los efectos sobre esta habilidad de la creatividad científica, las destrezas metacognitivas, la autoeficacia en el aprendizaje las ciencias, el nivel académico y el género. Para ello, se llevó a cabo una investigación cuantitativa ex post facto de carácter transversal. Participaron 139 estudiantes (74 mujeres y 65 hombres) de 3º y 4º de Educación Secundaria Obligatoria (ESO) , y 1º de Bachillerato (14-17 años). A todos ellos se les administró un cuestionario sobre destrezas metacognitivas y autoeficacia, otro cuestionario sobre creatividad científica, y una prueba de detección de errores en dos problemas verbales resueltos. Los análisis de correlaciones, de regresión múltiple y de mediación sugieren que: a) la capacidad de detección de errores fue baja en general; b) las variables que más influyeron en la variabilidad en la detección de errores en los problemas fueron el nivel académico, la autoeficacia y la creatividad científica; y c) la autoeficacia ejerció un papel de mediadora entre las destrezas metacognitivas y la detección de errores, lo que evidenció el efecto indirecto de las destrezas metacognitivas sobre la detección de errores en los problemas.

The Effects of a Mathematical Word Problem Program Based on WOE (Worked-out Example) for Underachieving Student

본 연구는 수학 학습부진 학생을 대상으로 WOE 기반 수학 문장제 원격 학습 프로그램을 개발 및 실시하고, 그 효과를 분석하였다. 이를 위하여 중학교 1학년 3명을 대상으로 12번의 중재로 구성된 WOE 기반 수학 문장제 원격 학습 프로그램을 실시하였다. 각 연구 참여자의 수학 문장제 해결 능력은 중재 전, 중재 중, 3주 후에 총 18번 측정되었다. 교육 프로그램은 학습자가 답안을 참고하여 스스로 학습하는 자율 학습 단계, 답안이 일부 소거된 WOE를 참고하여 교사와의 질의응답을 통해 문제를 해결하는 보충 학습 단계로 구성되었다. 세 학생 모두 문장제 문제 유형에 숙달하였으며, 중재 및 유지 단계에서 수학 문장제 해결 능력이 향상 및 유지된 것으로 나타났다. 본 연구는 WOE 기반 원격 학습 프로그램이 수학 문장제의 숙달도와 수학 문장제 해결 능력 향상에 효과적이라는 것을 보여준다.

Metrological advancements in cognitive measurement: A worked example with the NeuroMET memory metric providing more reliability and efficiency

Better metrics of cognition can be formed by carefully combining selected items from legacy short-term memory tests so as to enhance coherence in item design while not jeopardizing validity. In this paper, we report on how Rasch Measurement Theory and Construct specification equations (CSE) have been brought together when composing the NeuroMET Memory Metric (NMM). The NMM is guided by: i) entropy-based equivalence criteria; ii) a comprehensive understanding of the construct purported to be measured; and iii) how a collection of items works together. CSEs play a major role in ensuring the metrological legitimacy of the NMM in a way analogous to certified reference materials in more established areas of metrology. The resulting NMM for short-term memory recall has up to a five-fold reduction in measurement uncertainties for memory ability compared with an individual legacy test, and the entropy-based CSEs should enable more efficient and valid assessment.

Cronbach's alpha in mathematics education research: Its appropriateness, overuse, and alternatives in estimating scale reliability.

Critiques of coefficient alpha as an estimate of scale reliability are widespread in the literature. However, the continuous overuse of this statistic in mathematics education research suggests a disconnection between theory and practice. As such, this article argues, in a non-technical way, for the limited usefulness of coefficient alpha, its overuse, and its alternatives in estimating scale reliability. Coefficient alpha gives information only about the degree of the interrelatedness of a set of items that measures a construct. Contrary to the widely circulated misconceptions in mathematics education research, a high coefficient alpha value does not mean the instrument is reliable, and it does not imply the instrument measures a single construct. Coefficient alpha can only be dependable as an estimate of reliability under verifiable and restrictive conditions. I expose these conditions and present steps for their verification in empirical studies. I discuss some alternatives to coefficient alpha with references to non-technical articles where worked examples and programming codes are available. I hope this exposition will influence the practices of mathematics education researchers regarding estimation of scale reliability.

Expanding Formal School Curricula to Foster Action Competence in Sustainable Development: A Proposed Free-Choice Project-Based Learning Curriculum

A key determinant and outcome of successful environmental education is ‘pro-environmental behavior’, i.e., behavior that involves conscious action to mitigate adverse environmental impacts at personal or community level, e.g., reducing resource consumption and waste generation, avoiding toxic substances, and organizing community awareness initiatives. However, some theorists have sought to move away from rationalist models of behavioral modification, towards holistic pedagogical initiatives that seek to develop action competence. In light of the global push towards achievement of the Sustainable Development Goals (SDGs), emerging evidence suggests that education initiatives should foster action competence so students may be equipped to contribute to sustainable development as part of their education. The UNESCO Education for Sustainable Development (ESD) Roadmap 2030 has also identified key priority areas to strengthen ESD in formal curricula. This article reports two informal environmental education initiatives for promoting action competence and pro-environmental behaviors in school-aged children. The authors recommend that formal education settings (e.g., schools) should incorporate self-directed, free-choice project-based learning to augment environmental education programs and promote students’ action competence for contribution to attainment of SDGs. To this end, we propose a Free-Choice Project-based Learning for Action Competence in Sustainable Development (ACiSD) Curriculum, comprising six implementation dimensions, namely: (1) project duration and teaming arrangements, (2) topic selection, (3) student support, (4) teacher support, (5) learning environments, and (6) digital access and equity. For each implementation dimension, we recommend action steps to help educators implement this curriculum in their own educational settings, with the aid of an illustrative worked example.

The impact of working memory capacity on collaborative learning in elementary school students.

Working memory capacity may be a critical factor that influences the effectiveness of collaborative learning; however, no studies have directly explored this effect. Using worked examples as learning tasks, Experiment 1 used a 2 (working memory capacity) × 2 (learning format) factorial design to examine the effects of collaborative learning versus individual learning of 4th-grade Chinese elementary school students with different working memory capacities. High-capacity learners displayed less working memory resource depletion and better transfer performance during collaborative learning than individual learning. In contrast, no differences were found among the low-capacity learners. Collaborative learning benefited high-capacity learners but not low-capacity learners, per our observations. To further optimize collaborative learning for low-capacity learners and expand the findings to heterogeneous collaborative learning, Experiment 2 adopted a 2 (member capacity) × 2 (group capacity) factorial design to explore the effects of member and group working memory capacity on collaborative learning in heterogeneous groups. High-capacity members displayed less working memory resource depletion and better far transfer performance in high-capacity groups compared to low-capacity groups. Simultaneously, all members had better near transfer performance in high-capacity groups compared to low-capacity groups. Both member and group working memory capacities influenced the effect of heterogeneous collaborative learning. However, low-capacity members only partially benefited from collaborative learning in high-capacity heterogeneous groups.

Considerations of a thick lens formula for intraocular lens power calculation

BackgroundIn recent years, some lens manufacturers have committed to providing lens shape data for some of their lens models. The purpose of this study is to present a strategy for prediction of intraocular lens power and residual refraction based on a pseudophakic model eye containing 5 refractive surfaces and to show its applicability using worked examples. MethodsA pseudophakic model eye with a thin spectacle correction, a thick cornea (radius of curvatures for both surfaces and central thickness) and a thick IOL (either radius of curvatures RLa and RLp for front and back surface or equivalent power PL and Coddington factor CL; and either central thickness LT or edge thickness and optic diameter) was set up. Calculations were performed based on linear Gaussian optics (vergence formulae). Formulae were provided to derive the lens power/shape and the residual equivalent spectacle refraction SEQ. From the lens shape the location of the haptic plane HP, the image sided principal plane of the lens HL, and the ocular magnification OM were extracted. ResultsThe calculation of a thick intraocular lens and the prediction of residual refraction is presented with reference to 3 working examples: A) lens varied in PL and shifted with its haptic plane keeping the CL constant, B) lens varied in CL and shifted with its haptic plane keeping PL constant, and C) CL and PL of the lens varied keeping its haptic plane position in the eye constant. For each combination of parameters (PL, CL, or haptic plane shift) the parameters influencing SEQ, OM and HL-HP were analysed. ConclusionSome modern optical biometers currently on the market provide the radii of curvature of both corneal surface and all relevant distances in the eye. With additional data on the lens shape, it would be possible to improve lens power calculations by switching from thin to thick lens models for the cornea and for the lens. This would overcome one of the major drawbacks of current lens power calculation methods.

Exemplification process in online education: a longitudinal study of mathematics teachers.

The examples used in the teaching-learning process of mathematics have a crucial role in fostering conceptual understanding, and some variables can affect instructors' qualified example usage. This longterm study focused on mathematics teachers' exemplification process in face-to-faceand online learning environments. In this regard, the change in examples used by mathematics teachers were evaluated in terms of content preparation and presentation during the shift from face-to-face lectures to online classes. A longitudinal design was used in the study and the teaching processes of 14 middle-school mathematics teachers were observed over two semesters. Observation notes, course documents, and semi-structured interview data were analyzed, and content analysis findings were presented through descriptive statistics in order to compare content preferences in two different learning environments. The use of worked examples decreased, while the use of conceptual examples increased with the shift from face-to-face lectures to online classes. Also the length of time devoted to examples in online classes decreased, and examples were more teacher-centered. The interview revealed that mathematics teachers need support in terms of example preparation and presentation aspects in online learning environments. The other technological-pedagogical competencies that teachers might need to choose qualified examples in different teaching-learning environments are discussed in the light of relevant literature.

Novice teachers’ knowledge of effective study strategies

This survey research, assessed whether novice secondary school teachers knew and understood the effectiveness of empirically-supported learning strategies, namely spaced practice, retrieval practice, interleaved practice, using multimodal representations, elaborative interrogation and worked-out examples. These ‘proven’ strategies can be contrasted with frequently used learning strategies that have been found to be less effective, such as re-reading, taking verbatim notes, highlighting/underlining, summarizing, and cramming. This study broadens previous research on teachers’ knowledge of learning strategies by both refining and extending the methodology used in the scenario studies, and by administering it to a different, previously unexplored population. Novice teachers enrolled in a teacher training program (N = 180) in Flanders, Belgium were presented with a three-part survey, consisting of open-ended questions, learning scenarios and a list of study strategies. The results show that misconceptions about effective study strategies are widespread by novice teachers and suggests that they are unaware of several specific strategies that could benefit student learning and retention. While popular but less effective strategies such as highlighting and summarising were commonly named by them in open-ended questions, this was not the case for proven effective strategies (e.g., studying worked-out examples, interleaving, and using multi-modal representations) which were not or hardly mentioned. We conclude that this study adds to the growing literature that it is not only students, but also novice teachers who make suboptimal metacognitive judgments when it comes to study and learning. Explicit instruction in evidence-informed learning strategies should be stressed and included in both teacher professional development programs and initial teacher training.

CcaR: A package for assessing primary study overlap across systematic reviews in overviews.

An overview of reviews aims to collect, assess, and synthesize evidence from multiple systematic reviews (SRs) on a specific topic using rigorous and reproducible methods. An important methodological challenge in conducting an overview of reviews is the management of overlapping data due to the inclusion of the same primary studies in SRs. We present a free, open-source R package called ccaR (https://github.com/thdiakon/ccaR) that provides easy-to-use functions for assessing the degree of overlap of primary studies in an overview of reviews with the use of the corrected cover area (CCA) index. A worked example with and without consideration of chronological structural missingness is outlined, illustrating the steps involved in, calculating the CCA index and creating a publication-ready heatmap. We expect ccaR to be useful for overview authors, methodologists, and reviewers who are familiar with the basics of R and contribute to the discussion on different methodological approaches for implementing the CCA index. Future research and applications could further investigate the functionality or potential limitations of our package and other potential uses.

Using Incorrect Worked Examples to Investigate the Consistency of First and Third Graders’ Measurement Conceptions

Using Incorrect Worked Examples to Investigate the Consistency of First and Third Graders’ Measurement Conceptions

Let’s be rational: worked examples supplemented textbooks improve conceptual and fraction knowledge

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students’ fraction number line acuity, rational number concepts and procedures. Finally, the study assessed whether individual differences in prior knowledge of rational numbers moderate the effect of the textbooks. The experimental textbooks resulted in significantly greater improvements overall in fraction number line estimation skills than control textbooks. Further, example-supplemented textbooks were more effective than control textbooks at improving understanding of rational number concepts for those with low prior conceptual knowledge. Both textbooks were equally effective at improving procedural skills.

Using Social Media Pedagogies to teach School Mathematics in a selected Zimbabwean school during the COVID-19 Pandemic

The COVID-19 pandemic tested teachers’ and learners’ resilience to continue teaching and learning with the limited use of traditional classrooms. What became apparent was the need to find alternative ways to maintain the interactions between the teachers and the learners. This study explored the teaching and learning of secondary mathematics during the COVID-19 pandemic based on the community of inquiry framework. An explorative single case study design with forty learners and seven teachers from a rural school in Zimbabwe as participants was used. The purposive selection of the participants was based on their experiences of emergency remote learning and classes where COVID-19 protocols were observed. Data were collected through open-ended and closed questionnaires given to learners and structured and semi-structured interviews conducted with teachers. The qualitative data were analysed through thematic content analysis techniques. The findings show that in the wake of limited access to sufficient educational technologies and internet connectivity, teachers had to find alternative ways to teach mathematical problems by using already worked-out examples and voice notes, thereby reducing the teaching presence and teacher-learner interactions. However, the reduction of the teaching presence was compensated by increased self-regulated learning as learners tried to make sense of the examples shared through social media tools. The study recommends building teachers’ technological and pedagogical knowledge to teach mathematics using cheap social media applications to improve the teaching, social and cognitive presence. Keywords: COVID-19 pandemic, Educational technologies, School mathematics, Social media pedagogies

A RESPONSE TO COMMUNITY QUESTIONS ON THE MARINE20 RADIOCARBON AGE CALIBRATION CURVE: MARINE RESERVOIR AGES AND THE CALIBRATION OF 14C SAMPLES FROM THE OCEANS

ABSTRACTRadiocarbon (14C) concentrations in the oceans are different from those in the atmosphere. Understanding these ocean-atmospheric 14C differences is important both to estimate the calendar ages of samples which obtained their 14C in the marine environment, and to investigate the carbon cycle. The Marine20 radiocarbon age calibration curve is created to address these dual aims by providing a global-scale surface ocean record of radiocarbon from 55,000–0 cal yr BP that accounts for the smoothed response of the ocean to variations in atmospheric 14C production rates and factors out the effect of known changes in global-scale palaeoclimatic variables. The curve also serves as a baseline to study regional oceanic 14C variation. Marine20 offers substantial improvements over the previous Marine13 curve. In response to community questions, we provide a short intuitive guide, intended for the lay-reader, on the construction and use of the Marine20 calibration curve. We describe the choices behind the making of Marine20, as well as the similarities and differences compared with the earlier Marine calibration curves. We also describe how to use the Marine20 curve for calibration and how to estimate ΔR—the localized variation in the oceanic 14C levels due to regional factors which are not incorporated in the global-scale Marine20 curve. To aid understanding, illustrative worked examples are provided.

Mimicking tracing actions enhances young children's mathematical learning

AbstractLearning mathematical concepts and procedures typically requires extended cognitive effort, presenting a challenge for many children. People can make tracing actions with the index finger, as well as mimic another's movements, with little or no conscious effort. From the perspective of cognitive load theory, such biologically primary actions may facilitate learning biologically secondary concepts and skills requiring extensive cognitive effort, such as mathematics. The present study investigated effects on learning processes and outcomes of students mimicking a teacher's tracing actions from the perspective of an evolutionarily informed cognitive load theory. One hundred and thirteen Grade Two children learned about number lines, either observing a teacher tracing out elements of worked examples with her index finger, or mimicking the teacher's tracing actions with their own index finger. In accord with hypotheses, results indicated enhanced performance due to mimicking on cognitive load, motivation, and post‐test performance. Directions for future research are discussed.

Privacy Goals for the Data Lifecycle

The introduction of Data Protection by Default and Design (DPbDD) brought in as part of the General Data Protection Regulation (GDPR) in 2018, has necessitated that businesses review how best to incorporate privacy into their processes in a transparent manner, so as to build trust and improve decisions around privacy best practice. To address this issue, this paper presents a 7-stage data lifecycle, supported by nine privacy goals that together, will help practitioners manage data holdings throughout data lifecycle. The resulting data lifecycle (7-DL) was created as part of the Ideal-Cities project, a Horizon-2020 Smart-city initiative, that seeks to facilitate data re-use and/or repurposed. We evaluate 7-DL through peer review and an exemplar worked example that applies the data lifecycle to a real-time life logging fire incident scenario, one of the Ideal-Cities use cases to demonstrate the applicability of the framework.

The influence of the order and congruency of correct and erroneous worked examples on learning and (meta-)cognitive load.

Several studies highlight the importance of the order of different instructional methods when designing learning environments. Correct but also erroneous worked examples are frequently used methods to foster students' learning performance, especially in problem-solving. However, so far no study examined how the order of these example types affects learning. While the expertise reversal effect would suggest presenting correct examples first, the productive failure approach hypothesizes the reversed order to be learning-facilitating. In addition, congruency of subsequent exemplified problems was tested as a moderator of the effect of order on learning. For example, with arithmetic tasks, congruent problems target exactly the same calculation while incongruent problems refer to different calculations. Following cascade theory, a model of cognitive skill acquisition, presenting correct examples first should be more effective when the subsequent exemplified problems are different. To test the (conflicting) hypotheses, 83 university students were assigned to one of the four conditions in a 2 (correct vs. erroneous example first) × 2 (same vs. different exemplified problems) between-subject design. Learners navigated through a slideshow on the topic of Vedic mathematics consisting of explicit instruction, worked examples differing in terms of the experimental condition, and transfer problems. Although no main or interaction effects were found regarding students' learning performance, mediational analysis offered support for the expertise reversal effect, as it indicated that there is a significant indirect effect of order via mental load on learning. Presenting correct examples first and erroneous examples second resulted in a lower mental load, which in turn was associated with better learning performance. In contrast, presenting erroneous examples first and correct examples second resulted in a more accurate self-assessment of learning performance. These findings offer first insights into the question of how the presentation order of different example types impacts learning and provide practical recommendations for the design of educational media. Results are discussed in light of the ongoing debate regarding the question if less guided instructional methods should precede or succeed more guided methods.