Solving Probability Problems Research Articles

Transforming fifth graders’ misconceptions in solving probability comparison problems with purposefully designed instructional interventions

ABSTRACT Students often struggle to solve probability problems, and teaching probability is an international challenge. This study aimed to design and implement two rounds of test-based instructional interventions to support Chinese fifth graders in incrementally transforming their misconceptions when solving probability comparison problems. The intervention was purposefully designed based on the findings from two rounds of tests. The results of the single-group quasi-experiment showed that the instructional support was feasible and effective in transforming misconceptions and enhancing students’ understanding of probability concepts. The findings of this study raised community attention to Chinese students’ misconceptions that are obscured by their good performance on large-scale tests. In addition, it demonstrated the feasibility and effectiveness of purposefully designing incremental, individualized instructional interventions based on tests in promoting elementary student learning. Reflecting on the design of the instructional support, we provide pedagogical suggestions for enhancing students’ performance in probability problem-solving.

An Analysis of Students’ Mathematical Reasoning in Solving Probability Problems Judging from Learning Styles: The Converger

[Background] Several challenges are encountered by students when attempting to solve probability problems, one of which relates to reasoning about the given problem scenario. This phenomenon may arise due to variations in students’ learning styles. [Objective] Thus, the current study aimed to examine students’ reasoning abilities when dealing with probability problems, focusing on those with converging learning styles. [Method] This study employed a qualitative method, involving two students with a converging learning style who were invited to solve reasoning tasks. Three instruments were utilized: learning style questionnaire, mathematical reasoning tasks, and semi-structured interviews. The research data was analyzed through data reduction and display. [Results] The study’s findings revealed that, in solving probability problems, convergers (students with a converging learning style) primarily relied on imitative reasoning. These students demonstrated four patterns. First, they accurately and comprehensively articulated known information from a question and correctly identified the question being asked. Second, they possessed a conceptual understanding of probability but provided inaccurate explanations regarding arithmetic sequences. Although they could devise problem-solving strategies, their articulation of these strategies lacked precision. Third, they utilized the concept of probability and employed problem-solving strategies to solve the given problem and explain the steps involved in reaching the solution. Four, they derived answers from problem-solving strategies and drew accurate conclusions, often exhibiting a tendency to trust their answers and subsequently verify them. [Conclusions] Students with converging learning styles predominantly used imitative reasoning when solving probability problems.

Enhancing Students’ Numeracy Skills in Solving Probability Problems through Realistic Mathematics Education

Numeracy skills can help students to solve everyday problems. The low numeracy skills of students are caused by not understanding the problem, lack of strategy, and obstacles to evaluating and interpreting the solution to the context. The teacher needs to implement the learning process by using the RME approach, which uses local context and facilitates activity to think mathematically. This study aims to investigate the RME approach in enhancing numeracy skills and to describe the numeracy skills of students in solving problems according to numeracy indicators. The explanatory sequential mixed methods were used to analyze data from tests and interviews. The research subjects were 32 Grade 8 students, and four selected students chose to do the interviews based on each numeracy skills category. The results show a significant difference between the average numeracy skills of students before and after learning with the RME approach. Students with proficient, competent, and basic competencies fulfill the numeracy indicators. Meanwhile, students who need special intervention still have difficulty modeling problems into mathematical representations, incorrectly use the concept of probability, have inappropriate problem-solving strategies, and cannot interpret mathematical solutions in the form of arguments.

Exploring default-interventionist interaction of thinking activity types on probability problem-solving

This research aims to determine the thinking activity types dominated by a mental process in producing answers characterized by automatic, unconscious, and subjective-empirical processes (system 1) in solving problems so that the default-interventionist interaction occurs. This research novelty is the formulation of the contents and thinking activity arrangement adapted to students' thinking when solving problems. The problem used in this research is a mathematical problem that triggers students to produce answers quickly with confidence that the answers are correct at a high level. Another problem is about probability because the mode of occurrence of students' learning difficulties at the secondary school level occurs when learning the concept of probability. This is qualitative research with a case study approach. The research subjects were students of Mathematics Education in semester 1. The results showed that thinking activity one could condition the occurrence of type 1 default-interventionist interaction. Thinking activity two could condition the occurrence of type 2 default-interventionist interaction. Thinking activity three could condition the occurrence of type 3 default-interventionist interaction. This research concluded that the default-interventionist interaction occurred because the content and arrangement of the thinking activity conditioned the subjects to pay attention to information gradually and change the subjects’ beliefs. Lecturers were recommended to produce, develop, and research thinking activities on topics other than probability at various levels of education. The default-interventionist interaction was essential to be conditioned when system one dominated students' thinking, causing difficulties.

Research on visualization in probability problem solving

The importance of visual representations in education and mathematics is well known. Probabilities are a domain in mathematics that uses many visual representations since their theory consists of a variety of diagrams and graphs. In the past, many studies have shown that the use of various representations in teaching probabilities can greatly improve learning. Of course, the use of a visual representation or a visual tool when teaching or solving an exercise can have a variety of roles. The present work is based on the ancillary and informative role of the image. The following research examines the extent to which students, by solving a probability problem, have the need to use a visual representation or image. Additionally, the differences in student performance are investigated, given the role of the image in the activity. This knowledge can improve the teaching methods of probabilities and, with their appropriate use, school textbooks. The results show that there are more perspectives but also needs for research on the use of visual representations for the teaching of probabilities. The presence of the image works as a motivating factor for children to solve a problem with a lack of probabilistic knowledge, but the role of the image affects their final performance.

Exploring Pre-service Teachers’ Cognitive Process in Solving Probability Problems Related to the Maritime Context of the Riau Archipelago

This research aims to investigate how pre-service teachers think through the process of solving the probability problem related to the daily life context in the maritime region of the Riau Archipelago. The research methods used descriptive qualitative approach. The research subject is a pre-service teacher who have previously taken the Introduction to Mathematical Statistics course in the Mathematics Education program at UMRAH. Testing and interviews used as data collection methods. Qualitative data analysis carried out in three stages, reduction, data presentation, and drawing conclusions. The results show that there are four phases how the subject process the information to folmulate solution, formulate the travel routes through two different ports in Batam City, identifying an itinerary of possible events, calculate the number of samples space, and calculate the probability of event. The results show that the subjects were unable to accurately indentify the sample space based on context, which had an impact on the subjects’ failure to produce the right solution. The maritime context of the Riau Archipelago used was part of the subject’s prior knowledge. The knowledge helps them understand the information presented and construct an idea of how to formulate the itinerary.

Influence of material difficulty and strategy knowledge on worked example: learning strategy selection for solving probability problems

Influence of material difficulty and strategy knowledge on worked example: learning strategy selection for solving probability problems

데이터 기반 확률교육을 위한 수학과 교육과정 재구조화 방향 탐색

The aim of this study is to explore how to restructure the probability content standards in the national mathematics curriculum by pursuing the goals of developing probabilistic thinking competence and probabilistic problem solving. We named the perspective pursuing these goals as Data-Driven Probabilistic Education (DDPE) in that data sensitivity plays a significant role in probabilistic thinking and problem solving, and moreover relations between data and probability matter in statistics and machine learning fields. We further investigated into how to facilitate probabilistic thinking and probabilistic problem solving as well as what to adapt the meaning or grounding sense of the two major goals in the context of DDPE. Findings suggested some alterations to be made in the content standards and directions of teaching and learning in the curriculum document. First, to understand the probability concept based on data sensitivity, to understand randomness, and to solve probabilistic problems using modelling approach are suggested to be included in the contend standards. Second, we suggest to include contexts in which decisions about risk are made in the teaching and learning directions to enhance competence of dealing with volatility and uncertainty using probabilistic thinking competence. Third, rather than writing general statements on using technological tools, it is suggested to present specific ways or procedures of probability simulation using a computer in relation to understanding stages of probability concepts and solving probabilistic problems as guides to teaching and learning to realize DDPE.

Multi-focus Image Fusion Based on Random Walk

An effective multi-focus image fusion algorithm based on random walk is proposed in this paper. Random walk and guided filter have attracted extensive attention in image fusion. Random walk is usually used to solve probability problems and it has a good smoothing effect, and guided filter can preserve the gradient information of the image well. The combination of two algorithms can better retain the edge information of the input image. Six sets of source images and five existing methods are used in the experiment and the experimental results show that the proposed algorithm outperforms the existing methods in both subjective and objective evaluation.

INVESTIGATION OF KNOWLEDGE OF MATHEMATICS TEACHERS ABOUT PROBABILITY PROBLEMS

The probability subject and problems about this subject can cause some difficulties for teachers as well as for students. In this context, this study aims to investigate the knowledge and opinions of mathematics teachers about the subject of probability. This research was designed as a case study, which is one of the qualitative research designs. The sample of the study consisted of a total of 28 mathematics teachers, 18 of whom were females and 10 of whom were males, working in state schools in a southern city of Turkey. In addition, interviews were also conducted with a total of 8 mathematics teachers, 6 females and 2 males, among the teachers who participated in the study in order to investigate their general opinions about the subject of probability. In order to collect data, the probability problems test was used to determine the knowledge of mathematics teachers about probability problems, and a semi-structured interview form was used to investigate their general opinions on probability. The results of the research showed that the teachers’ rate of determining the type of probability problems correctly was 80.6 %. It was also seen that the 37.75% of the mathematics teachers used formulas, 29.6% of them used procedural methods and 31.60% of them used graphical methods in the calculations to solve probability problems. Article visualizations:

PEMBELAJARAN TEORI PROBABILITAS MENGGUNAKAN R

In learning probability theory, if a series of statistical experiments is carried out several times, identifying the possible samples produced is not an easy thing. If this happens, then a significant probability problem will arise. The objective of this study is how to learn probability theory using R software. Based on the simulation results it can be concluded that by combining the syntax in R can be used to solve probability problems such as identifying sample points from experiments, events, event operations, probabilities and conditional probabilities. This will help students learning in understanding of the probability theory material. The use of R will be very pronounced for experiments with a large enough scale that results in a large sample probability.

Mathematical Probabilistic Thinking Process Stages in Problems Solving Probability

Probabilistic thinking is one type of thinking skills which belongs to the Higher Order Thinking Skills (HOTS). Students need to have the probabilistic thinking ability to face the life which is full of uncertainty. The purpose of this research is to formulate the stages of mathematical probabilistic thinking processes in solving probability problems. It was a descriptive qualitative research involving eight students of the 9 th grade of SMP Muhammadiyah 3 Mlati Sleman Yogyakarta as the subjects. We administered a probabilistic thinking test and then observed and interviewed them to get the data. The data were then analyzed using triangulation method. This study resulted the five stages of mathematical probabilistic thinking process. They are: (1) understanding the problem of uncertainty that needs to be solved; (2) identifying all possibilities that will occur from a problem; (3) grouping the results of the identified event; (4) determining the probability of the occurred events; and (5) verifying the results.

Analysis of Student Thinking Ability in the Topic of Probability: A Descriptive Study of Gender and Math Ability

The probability is very closely related to everyday life that includes a probabilistic situation. This situation contains an element of uncertainty when someone is thinking about a problem that has various possibilities or consists of a part of luck, then that person is thinking probabilistically. The purpose of this study is to analyze the mathematical probability thinking skills of senior high school students in the response aspect with gender and mathematical abilities. This type of research is a qualitative descriptive study based on five stages, namely (1) natural design, (2) descriptive data, (3) attention, (4) data processing, and (5) the main focus of research. The research was conducted at Senior High School in Jayapura, Indonesia. The research subjects were students of class XI Science. The research instrument consisted of mathematical ability test sheets, probability problem-solving tasks, interviews. The results of the investigation show that the profile of male students' probabilistic thinking on mathematical abilities in solving probability problems includes providing statistical responses in solving probability. Issue is related to the sample space (representatives formation) because it uses a combination to calculate the number of members of the sample space by providing the right reasons. Meanwhile, for women, the profile of probabilistic thinking is using several strategies in solving probability problems related to the sample space (representatives formation). Based on the investigation of the results shows that there are differences in mathematical ability between male and female students probability of problem-solving.

Providing ability to probability: Reducing cognitive load through worked‐out examples

AbstractUndergraduate students tend to struggle with probability in their introductory statistics course. Probability problem solving requires several steps. First, students must make sense of the probability scenario, then determine the appropriate probability rules, and finally, execute the procedures to solve the problem. With no previous exposure to probability, this presents too great a cognitive load for many students. Using worked‐out problems then transitioning to partially worked‐out problems in an introductory statistics course at a large university helped students succeed at solving probability problems. The worked‐out problems included writing prompts to encourage self‐explanation of students' thinking through studying the worked‐out examples. This paper explains the use of these instructional principles and their implementation in an introductory statistics course for non‐STEM majors, resulting in higher student achievement and understanding.

Mathematical Communication Profile in Solving Probability Problems Reviewed by Self-efficacy of Prospective Mathematics Teachers

Mathematical communication plays a very important role in learning mathematics, so it needs to be developed in mathematics learning. Mathematical communication is the process of conveying mathematical ideas to obtain a shared understanding between those who give and receive information. This study aims to describe the mathematical communication profile of prospective mathematics teachers based on self-efficacy in solving probabilistic problems. By solving probabilistic problems, it can improve probabilistic thinking, so that probabilistic problem solving can improve the ability to think at a higher level or Higher Order Thinking Skill, which is urgently needed today. This qualitative research is a study case. Subjects in this study were prospective mathematics teachers selected based on self-efficacy. The instruments used in this study were written tests and guided interviews based on probabilistic problems. Then the subjects were interviewed based on their work on the assignment. Data collection techniques are done through observation, interviewing, and documentation, while the data analysis technique used by researchers uses the Miles and Huberman model, namely by the process of (1) reduction data, (2) presentation data, and (3) drawing conclusions and verification. The results showed that prospective mathematics teachers who had different self-efficacy in solving probabilistic problems had different mathematical communication skills.

Mathematical probability: student’s misconception in higher education

Probability is one of the mathematical concepts that are abstract for students. Probability concepts used in daily life. As a basis of statistics, students used the probability concept. This topic is one of the compulsory subjects learned in the Mathematics Education Study Program of Universitas Ahmad Dahlan. However, there are still many students who have difficulties in understanding the concept of probability. This study aims to describe students’ misconceptions in solving Probability problems. The research used descriptive method. The subjects in this study were the fourth-semester students of Mathematics Education Study Program Class C of Universitas Ahmad Dahlan. The results of this study found that there are four misconceptions made by students in solving Probability problems. First, students’ error in interpreting questions. Second, students’ errors in the procedure of proving the theorems of probabilities. Third, students’ misconceptions in the application of Bayes’ rules while they are solving problems. Fourth, students’ error in calculating the possibility of an event.

Grade 12 Students’ Proficiency in Solving Probability Problems Involving Contingency Tables and Tree Diagrams

Grade 12 Students’ Proficiency in Solving Probability Problems Involving Contingency Tables and Tree Diagrams

Students’ errors in solving probability problems viewed by learning style

The students’ error in this study is an error made by students in solving probability problems. This study aims to describe students’ errors in probability problems viewed by learning styles. The learning style is a person’s main potential in thinking and processing activities to find out information. To explain this problem, first, we developed and validated three probability test items. Then, 26 students were given one set of tests and continued with the giving one set of learning style questionnaire. Next, six students were selected as representatives of each learning style group with the purposive sampling technique. The interview was conducted to ascertain the types of students’ errors. Finally, the data were analyzed based on Newman’s theory. The results showed that the types of errors made by the visual group consisting of misunderstanding the problem, being wrong in writing the final answer, and being wrong in carrying out problem-solving procedures. The types of errors made by the auditory group involve an error in carrying out problem-solving procedures, not knowing the concept to solve the test, and not understanding problems. The types of errors made by the kinesthetic group include misunderstanding problems and not knowing concepts to solve the test. This finding suggests that the types of students’ errors for the visual group consist of comprehension, encoding, and process skills; for the auditory group consist of process skills and transformation; for the kinesthetic group consist of comprehension and transformation.

Interactive Learning Environment Supporting Visualization in the Teaching of Probability

AbstractThe probability is exceptional in the teaching of mathematics because students often have difficulties to understand the basic terms and the problem solving strategies. Understanding lacks of the probability concept and various types of misconceptions arise from the misleading intuition and misinterpretations of experience with the stochastic phenomena. The probability concept seems too abstract to some students therefore it is advisable to use mathematical problems based on real-life ideas, such as drug efficacy testing, tests for diagnosing of diseases in medicine, sports competitions, and games. By eliminating misconceptions and improving understanding of the problem solving strategies, it is possible to use various types of visualization to solve problems from this field of mathematics. Tables and different types of graphic diagrams can help students to understand the basic rules and problem solving techniques. This paper describes the main objectives and the structure of an interactive worksheet, prepared in spreadsheet environment, in which students are guided to use the visualization to solve probability problems. The implementation of an automatic feedback enables to evaluate the students’ answers. In the case of incorrect answer, solutions of the additional tasks using tree diagram or a tangram are recommended to the students. Students can decide which type of the visualization is more understandable for them to solve the probability of random events. Solving different task sequences using the selected types of visualization allows more learning paths for students. The final part of the paper contains an evaluation of the results and experiences of problem solving in the teaching of probability at grammar schools.

A Qualitative Research on the Difficulties and Failures about Probability Concepts of High School Students

This research aimed to examine the difficulties and failures of eleventh-grade students regarding probability concepts. With this aim, ten different open-ended probability problems were asked to the 142 eleventh-grade students. Each of these problems requires using different basic probability concepts. It is qualitative research, and the data gathered in this study were analyzed using by descriptive analysis method. The findings indicated that many of these students had difficulties in solving probability problems and failed to apply different probability concepts to these problems. Notably, it was stated that many students had difficulties in finding all the probabilities without any missing in these problems. Besides, it is essential to develop reasoning skills since the students usually used trial-and-error method and prediction to solve probability problems.