Mathematics

Research

28 pages, 1672 KiB

Open AccessArticle

The Binomial Distribution: Historical Origin and Evolution of Its Problem Situations

by Jaime Israel García-García, Nicolás Alonso Fernández Coronado, Elizabeth H. Arredondo and Isaac Alejandro Imilpán Rivera

Mathematics 2022, 10(15), 2680; https://doi.org/10.3390/math10152680 - 29 Jul 2022

Cited by 4 | Viewed by 4153

Abstract

The increase in available probabilistic information and its usefulness for understanding the world has made it necessary to promote probabilistic literate citizens. For this, the binomial distribution is fundamental as one of the most important distributions for understanding random phenomena and effective decision [...] Read more.

The increase in available probabilistic information and its usefulness for understanding the world has made it necessary to promote probabilistic literate citizens. For this, the binomial distribution is fundamental as one of the most important distributions for understanding random phenomena and effective decision making, and as a facilitator for the understanding of mathematical and probabilistic notions such as the normal distribution. However, to understand it effectively, it is necessary to consider how it has developed throughout history, that is, the components that gave it the form and meaning that we know today. To address this perspective, we identify the problem situations that gave origin to the binomial distribution, the operational and discursive practices developed to find solutions, and the conflicts that caused a leap in mathematical and probability heuristics, culminating in what is now known as the binomial distribution formula. As a result, we present five historical links to the binomial phenomenon where problem situations of increasing complexity were addressed: a case study using informal means (such as direct counting), the formalization of numerical patterns and constructs related to counting cases, specific probability calculus, the study and modeling of probability in variable or complex phenomena, and the use of the distribution formula as a tool to approaching notions such as the normal distribution. The periods and situations identified correspond to a required step in the design of binomial distribution learning from a historical epistemological perspective and when solving conflicts. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

31 pages, 5491 KiB

Open AccessArticle

How to Train Novices in Bayesian Reasoning

by Theresa Büchter, Andreas Eichler, Nicole Steib, Karin Binder, Katharina Böcherer-Linder, Stefan Krauss and Markus Vogel

Mathematics 2022, 10(9), 1558; https://doi.org/10.3390/math10091558 - 5 May 2022

Cited by 10 | Viewed by 2794

Abstract

Bayesian Reasoning is both a fundamental idea of probability and a key model in applied sciences for evaluating situations of uncertainty. Bayesian Reasoning may be defined as the dealing with, and understanding of, Bayesian situations. This includes various aspects such as calculating a [...] Read more.

Bayesian Reasoning is both a fundamental idea of probability and a key model in applied sciences for evaluating situations of uncertainty. Bayesian Reasoning may be defined as the dealing with, and understanding of, Bayesian situations. This includes various aspects such as calculating a conditional probability (performance), assessing the effects of changes to the parameters of a formula on the result (covariation) and adequately interpreting and explaining the results of a formula (communication). Bayesian Reasoning is crucial in several non-mathematical disciplines such as medicine and law. However, even experts from these domains struggle to reason in a Bayesian manner. Therefore, it is desirable to develop a training course for this specific audience regarding the different aspects of Bayesian Reasoning. In this paper, we present an evidence-based development of such training courses by considering relevant prior research on successful strategies for Bayesian Reasoning (e.g., natural frequencies and adequate visualizations) and on the 4C/ID model as a promising instructional approach. The results of a formative evaluation are described, which show that students from the target audience (i.e., medicine or law) increased their Bayesian Reasoning skills and found taking part in the training courses to be relevant and fruitful for their professional expertise. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

16 pages, 4110 KiB

Open AccessArticle

Combinatorial and Proportional Task: Looking for Intuitive Strategies in Primary Education

by Maria Ricart and Assumpta Estrada

Mathematics 2022, 10(8), 1340; https://doi.org/10.3390/math10081340 - 18 Apr 2022

Cited by 4 | Viewed by 1301

Abstract

The development of probabilistic thinking at school requires enhancing combinatorial and proportional reasoning. For this reason, 190 sixth-grade elementary school students (11–12-year-old), without previous instruction in the topic, solve a task consisting of five questions that address both types of reasoning. This study [...] Read more.

The development of probabilistic thinking at school requires enhancing combinatorial and proportional reasoning. For this reason, 190 sixth-grade elementary school students (11–12-year-old), without previous instruction in the topic, solve a task consisting of five questions that address both types of reasoning. This study explores the problem-solving strategies used by schoolchildren. The results obtained indicate that, in general, the students do not show strategies in the answers to the combinatorial questions. In addition, it is observed that they have difficulties in understanding the proposed statements and arguing the issues that explicitly require a justification. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

16 pages, 384 KiB

Open AccessArticle

Algebraization Levels in the Study of Probability

by María Burgos, Carmen Batanero and Juan D. Godino

Mathematics 2022, 10(1), 91; https://doi.org/10.3390/math10010091 - 27 Dec 2021

Cited by 1 | Viewed by 2845

Abstract

The paper aims to analyze how the different degrees of mathematical formalization can be worked in the study of probability at non-university educational levels. The model of algebraization levels for mathematical practices based on the onto-semiotic approach is applied to identify the different [...] Read more.

The paper aims to analyze how the different degrees of mathematical formalization can be worked in the study of probability at non-university educational levels. The model of algebraization levels for mathematical practices based on the onto-semiotic approach is applied to identify the different objects and processes involved in the resolution of a selection of probabilistic problems. As a result, we describe the possible progression from arithmetic and proto-algebraic levels of mathematical activity to higher levels of algebraization and formalization in the study of probability. The method of analysis developed can help to establish connections between intuitive/informal and progressively more formal approaches in the study of mathematics. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

16 pages, 562 KiB

Open AccessArticle

What Mathematical Knowledge Do Prospective Teachers Reveal When Creating and Solving a Probability Problem?

by Marlén Alonso-Castaño, Pedro Alonso, Maria Mellone and Luis J. Rodríguez-Muñiz

Mathematics 2021, 9(24), 3300; https://doi.org/10.3390/math9243300 - 18 Dec 2021

Cited by 2 | Viewed by 2558

Abstract

The teaching of probability is conditioned by teachers’ mathematical knowledge. In this paper, an exploratory study is carried out with prospective teachers. A training task was designed requiring them to create and solve a probability problem using the values of euro coins, which [...] Read more.

The teaching of probability is conditioned by teachers’ mathematical knowledge. In this paper, an exploratory study is carried out with prospective teachers. A training task was designed requiring them to create and solve a probability problem using the values of euro coins, which was adapted to students aged 11 to 12. The study aimed at determining what mathematical knowledge prospective teachers show when dealing with the task. The data were collected through the Moodle online Campus. We framed the data analysis in the Mathematical Knowledge for Teaching model and we used content analysis as the methodological approach. The results indicate that, despite finding evidence of adequate common and specialised mathematical knowledge, in approximately half of the prospective teachers participating in the study, too many of them still show a lack of knowledge in both subdomains. There was also little evidence of knowledge of the curriculum. The main finding of the research is that, when prospective teachers get involved in complex creative tasks, they mobilised together specialised and common mathematical knowledge, working into different mathematical processes such as problem posing and solving, communication, and argumentation, which reinforces the need to continue working on these types of complex tasks. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

17 pages, 2742 KiB

Open AccessFeature PaperArticle

Secondary School Students’ Construction and Interpretation of Statistical Tables

by Jocelyn D. Pallauta, Pedro Arteaga and José A. Garzón-Guerrero

Mathematics 2021, 9(24), 3197; https://doi.org/10.3390/math9243197 - 11 Dec 2021

Cited by 7 | Viewed by 2615

Abstract

Understanding statistical tables is a main component of statistical literacy, although related research dealing with secondary school students is scarce. The purpose of this study was to investigate secondary school students’ performance when translating graphs to tables and then interpreting the resulting table. [...] Read more.

Understanding statistical tables is a main component of statistical literacy, although related research dealing with secondary school students is scarce. The purpose of this study was to investigate secondary school students’ performance when translating graphs to tables and then interpreting the resulting table. Using content analysis of the responses to a questionnaire given to a sample of Spanish students, we analysed the correctness of the constructed table, the semiotic conflicts that appeared in their construction, the interpretation of the table elements, the ability to argue on the basis of the information in the table, and the reading level exhibited by the student in this task. Most students correctly translated a pictogram to a frequency table, and a half of them a double bar graph to a two-way table. The main semiotic conflicts were misinterpretation of the icon in the pictogram and incorrect computation of marginal frequencies. About 40% of the sample correctly justified a statement using the data in the graph and then reached the upper reading level of the graph and table, but only a minority achieved that level when the question required knowledge of the context. The findings of the study suggested points to reinforce the teaching of statistical graphs and tables. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

15 pages, 740 KiB

Open AccessArticle

Prospective Mathematics Teachers Understanding of Classical and Frequentist Probability

by Carmen Batanero, Nuria Begué, Rocío Álvarez-Arroyo and Silvia M. Valenzuela-Ruiz

Mathematics 2021, 9(19), 2526; https://doi.org/10.3390/math9192526 - 8 Oct 2021

Cited by 1 | Viewed by 2576

Abstract

Strengthening the teaching of probability requires an adequate training of prospective teachers, which should be based on the prior assessment of their knowledge. Consequently, the aim of this study was to analyse how 139 prospective Spanish mathematics teachers relate the classical and frequentist [...] Read more.

Strengthening the teaching of probability requires an adequate training of prospective teachers, which should be based on the prior assessment of their knowledge. Consequently, the aim of this study was to analyse how 139 prospective Spanish mathematics teachers relate the classical and frequentist approaches to probability. To achieve this goal, content analysis was used to categorize the prospective teachers’ answers to a questionnaire with open-ended tasks in which they had to estimate and justify the composition of an urn, basing their answers on the results of 1000 extractions from the urn. Most of the sample proposed an urn model consistent with the data provided; however, the percentage that adequately justified the construction was lower. Although the majority of the sample correctly calculated the probability of an event in a new extraction and chose the urn giving the highest probability, a large proportion of the sample forgot the previously constructed urn model, using only the frequency data. Difficulties, such as equiprobability bias or not perceiving independence of trials in replacement sampling, were also observed for a small part of the sample. These results should be considered in the organisation of probabilistic training for prospective teachers. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

21 pages, 12354 KiB

Open AccessArticle

Analysing Probability Teaching Practices in Primary Education: What Tasks Do Teachers Implement?

by Claudia Vásquez and Ángel Alsina

Mathematics 2021, 9(19), 2493; https://doi.org/10.3390/math9192493 - 5 Oct 2021

Cited by 5 | Viewed by 4578

Abstract

This study analyses probability tasks proposed by primary education teachers to promote probabilistic literacy. To this end, eight class sessions at various levels of the Chilean educational system were recorded on video and analysed through the ”probability tasks” dimension from the “Observation Instrument [...] Read more.

This study analyses probability tasks proposed by primary education teachers to promote probabilistic literacy. To this end, eight class sessions at various levels of the Chilean educational system were recorded on video and analysed through the ”probability tasks” dimension from the “Observation Instrument for Probability Classes” (IOC-PROB), which includes five components: use of resources, probabilistic contexts, cognitive challenge, procedures and strategies, and probability meanings. The results show that probability tasks focus mainly on technical knowledge, causing the probability class to become an arithmetic class in which only formulas are applied, mechanically and with no meaning. As a result, we see no use of technological resources, a low use of physical materials, and an absolute predominance of solving decontextualised exercises. We conclude that it is necessary to enhance the probability teaching practices based on lesson plans that consider a wide variety of resources and contexts to gradually advance towards a representation of probabilistic knowledge that relies on conventional procedures and notations. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

26 pages, 13631 KiB

Open AccessArticle

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

by Patricia Belén Carrera, Luis R. Pino-Fan, Hugo Alvarado and Jesús Guadalupe Lugo-Armenta

Mathematics 2021, 9(19), 2435; https://doi.org/10.3390/math9192435 - 1 Oct 2021

Cited by 2 | Viewed by 2160

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

20 pages, 6335 KiB

Open AccessArticle

Inferential Reasoning of Secondary School Mathematics Teachers on the Chi-Square Statistic

by Jesús Guadalupe Lugo-Armenta and Luis Roberto Pino-Fan

Mathematics 2021, 9(19), 2416; https://doi.org/10.3390/math9192416 - 28 Sep 2021

Cited by 3 | Viewed by 2203

Abstract

Statistics education has investigated how to promote formal inferential reasoning from informal inferential reasoning. Nevertheless, there is still a need for proposals that explore and progressively develop inferential reasoning of students and teachers. Concerning this, the objective of this article is to characterize [...] Read more.

Statistics education has investigated how to promote formal inferential reasoning from informal inferential reasoning. Nevertheless, there is still a need for proposals that explore and progressively develop inferential reasoning of students and teachers. Concerning this, the objective of this article is to characterize the inferential reasoning that secondary school mathematics teachers show in the practices that they develop to solve problems regarding the Chi-square statistic. To achieve this, we use theoretical and methodological notions introduced by the onto-semiotic approach of mathematics knowledge and instruction. In particular, we have taken a theoretical proposal of levels of inferential reasoning for the Chi-square statistic. Based on the results, the main conclusion was that the proposal above effectively predicted the teachers’ practices, allowing us to distinguish characteristic elements of the levels of inferential reasoning. Full article

(This article belongs to the Special Issue Statistics Education: An Immediate Need in a Changing World)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Statistics Education: An Immediate Need in a Changing World

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (10 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI