Fermi Method Research Articles

Sky smiley face: Solutions for Fermi Questions, November 2024

Sky smiley face: Solutions for Fermi Questions, November 2024

Better guesses

AbstractIt has recently become popular to analyze scenarios in which we guess, in terms of a trade‐off between the accuracy of our guess (namely, its credence) and its specificity (namely, how many answers it rules out). Dorst and Mandelkern describe an account of guessing, based on epistemic utility theory (EUT), in which permissible guesses vary depending on how one weighs accuracy against specificity. We provide a minimal formal account of guessing that: (i) does not employ EUT, but rests on how such trade‐offs are treated in the sciences; (ii) is relatively parsimonious; and (iii) is consistent with a variety of more specific models that describe what an agent is doing when they (rationally) guess. Our account also recovers patterns of guessing and predictions about typical outcomes of guessing, as identified by Dorst and Mandelkern. Furthermore, we focus on how permissible guesses can be improved upon, via changes in an agent's credence distribution. Such better permissible guesses can be generated in solving Fermi problems—guessing problems of a type that has received almost no attention in the philosophical literature—which we also analyze. Our account strengthens the case for understanding guessing (now, very broadly considered) in terms of accuracy‐specificity trade‐offs.

Pigeon strike!: Solutions for Fermi Questions, October 2024

Pigeon strike!: Solutions for Fermi Questions, October 2024

Shadow hands: Solutions for Fermi Questions, September 2024

Shadow hands: Solutions for Fermi Questions, September 2024

Solving a class of Thomas–Fermi equations: A new solution concept based on physics-informed machine learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas–Fermi differential equations. To tackle the inherent intricacies of solving Thomas–Fermi equations, we employ the quasi-linearization technique, which transforms the original non-linear problem into a series of linear differential equations. Our approach utilizes collocation least-squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas–Fermi problem types, including those featuring non-local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas–Fermi equation with an impressive 39-digit precision.

Refracting UFOs: Solutions for Fermi Questions, May 2024

Refracting UFOs: Solutions for Fermi Questions, May 2024

Cats, dogs, and roll waves: Solutions for Fermi Questions, April 2024

Cats, dogs, and roll waves: Solutions for Fermi Questions, April 2024

Snow spheres and ice cubes: Solutions for Fermi Questions, February 2024

Snow spheres and ice cubes: Solutions for Fermi Questions, February 2024

The weighty words of Lord Rayleigh: Solutions for Fermi Questions, January 2024

The weighty words of Lord Rayleigh: Solutions for Fermi Questions, January 2024

“Density” of light and pi: Solutions for Fermi Questions, December 2023

“Density” of light and pi: Solutions for Fermi Questions, December 2023

Ellipses, leaves, and solar crescents: Solutions for Fermi Questions, November 2023

Ellipses, leaves, and solar crescents: Solutions for Fermi Questions, November 2023

Dental Floss, Calculus, and Jail: Solutions for Fermi Questions, October 2023

Dental Floss, Calculus, and Jail: Solutions for Fermi Questions, October 2023

The Modelling Strategies Used by the Pre-service Teachers in Fermi Problems: The Task of Automatic Irrigation System

This study aims to explore pre-service teachers’ solution spaces and model strategies in the mathematical modelling tasks involving estimations. A case study method was employed in the research. 119 pre-service teachers participated in the study. The data were collected in two weeks. The pre-service teachers created groups consisting of four or five individuals according to their wishes. Thus, 26 different groups emerged. The modelling problem was designed by the researcher by considering containing the reality, authenticity, closeness, openness, and modelling competencies criteria. The data collection tools consisted of the worksheets and presentation records of the pre-service teachers. The deduction and induction analysis methods were used together in the data analysis. According to the obtained findings, it was determined that the pre-service teachers used five different modelling strategies. In the problem given within the scope of the number of elements that could fit in a surface, the “reference point” was used most; on the other hand, the “concentration measures” were used least. It is found that taking the area covered by a sprinkler model as a reference rather than the area occupied by sprinklers in the solution spaces produces more realistic results. The solutions spaces created within these strategies was used to detail the modelling process. It is significant for teachers who will apply a similar modelling problem to know the created different modelling strategies and solution spaces. It is thought that the solution spaces will provide a source for teachers to keep track of the information related to their students.

Determination of Mistakes Made by Pre-Service Mathematics Teachers in Fermi Problem-Solving Performance and Solutions

One of the reasons why students fail in mathematics courses is that the problems used in the lessons are abstract. In this context, it is important to embody the problems used in the lessons. Fermi problems are open-ended problems, a type of problem that reflects real-life situations that are solved by making a series of predictions. In the future, with these problems involving real-life situations, it is today’s pre-service mathematics teachers who are primarily responsible for carrying out an effective teaching process in school mathematics. This research was carried out within the scope of the case study design, which is one of the qualitative research designs, which aims to determine the errors made by primary school mathematics teacher candidates in their performance and solutions to solve the Fermi problem. The study group of the study consisted of 44 pre-service teachers studying in the 4th grade in the department of primary school mathematics teaching in a province located in the southern region of Turkey in the 2022-2023 academic year. As a result of the research, it was concluded that the scores obtained by the pre-service mathematics teachers from the solutions of Fermi problems were not at the desired level, and the performance of the teacher candidates was at a medium level. From the error detection, it was find that the source of this situation is mostly due to the weakness in the dimensional estimation process.

The Effectiveness of Fermi Problem-Based Learning on Improving Communication Skills among United Arab Emirates Cycle Two Students

Communication skills are necessary for students to empower them in facing new global challenges. Thus, this study employed Fermi Problem-based Learning to improve communication skills in teaching Science among the Cycle Two students in United Arab Emirates. It also examined the effectiveness of this strategy between grade 7 and grade 8 students. This study used quasi-experiment design. The instrument for research was pre and post-test using Communication Skills which was prepared by the researcher. The participants purposively selected according to a certain criterion, then divided for the experimental group, consisting of 15 grade seven and eight students, from Ramah Primary and Secondary School, they were taught Science units using the Fermi Problem-Solving method for 6-weeks, Whereas the control group consisted of 15 grade seven and eight students who were taught the same science units using traditional methods. The data were analysed using the Two Way ANCOVA to determine the effectiveness of the strategy on improving their communication skills and to determine the variation in the effectiveness of the strategy between grade 7 and grade 8 students. The findings revealed the effectiveness of the Fermi problem based learning strategy in developing the communication skills in teaching Science among the United Arab Emirates Cycle Two students. The findings also demonstrated that there were no significant differences in the effectiveness of the strategy amongst grade 7 and grade 8 students. Thus, Fermi Problem-Based learning is considered an effective strategy in teaching Science and in improving students’ communication skills and suitable for all Cycle Two students. In addition, this strategy can help teachers to integrate Fermi Problem-Solving into their teaching practice in order to achieve the objective of teaching science in the 21st century.

Another angle on perspective: Solutions for Fermi Questions, May 2023

Another angle on perspective: Solutions for Fermi Questions, May 2023

Of mice,* humans, and flies: Solutions for Fermi Questions, April 2023

First Page

Pre-service teachers’ flexibility and performance in solving Fermi problems

Fermi problems are real-context estimation tasks that are suitable for introducing open-ended problems in primary school education. To ensure their effective introduction in the classroom, teachers must have adequate proficiency to deal with them. One of the key aspects of problem-solving proficiency is flexibility, but there are few studies on flexibility in solving real-context problems. This study, based on an analysis of the errors made by 224 prospective teachers when solving a Fermi problem sequence, establishes performance levels. In addition, we define levels of flexibility in using multiple solutions across the sequence, which allows us to address the main objective: to study the relationship between performance and flexibility. We found that there are significant relationships between flexibility levels and the number and severity of errors made. Encouraging flexibility in prospective teachers may be an efficient way to improve their performance in solving real-context problems.

Analysis of the Relationship between Creativity in Fermi Problems Measured by Applying Information Theory, Creativity in Psychology, and Mathematical Creativity

Many educational institutions demand the development of creativity. However, it is still insufficient for encouraging creative work or thinking. One reason is the lack of simple tools to measure creativity in schools. This study focused on Fermi problems to solve the reason for this issue. Fermi problems have been suggested to be deeply related to creativity. However, few empirical studies have been conducted on their relationship. Therefore, this study conducted a survey and analyzed them by structural equation modeling. The results showed a moderate correlation between creativity in Fermi problems and creativity in psychology (r = 0.47, p < 0.01). Additionally, it was shown that there is a strong correlation between creativity in Fermi problems and mathematical creativity (r = 0.76, p < 0.01). Furthermore, regression analysis showed that creativity in Fermi problems is an important factor for measuring creativity in psychology and mathematical creativity.

Fermi problems as a hub for task design in mathematics and stem education

Abstract In this paper, we draw on recent research on so-called Fermi problems and situate the fundamental principles underlying this type of tasks and their use from a task design perspective. We use the models and modelling perspective on teaching and learning to elaborate on aspects related to the design of single-use, as well as sequences of, Fermi problems. In addition, we discuss a framework (called the Fermi problem activity template [FPAT] framework) for supporting the design and use of Fermi problems to facilitate students' learning within particular mathematics content areas and/or aimed at particular concepts or higher-order thinking skills. We also illustrate how the FPAT framework can be used to (i) facilitate interdisciplinary collaborations with other subjects such as the social sciences, but in particular with the other STEM subjects; and (ii) support teachers in adapting and implementing Fermi problems in their classrooms.