Solving Probability Problems Research Articles

Arguments for the effectiveness of human problem solving

The question of how humans solve problem has been addressed extensively. However, the direct study of the effectiveness of this process seems to be overlooked. In this paper, we address the issue of the effectiveness of human problem solving: we analyze where this effectiveness comes from and what cognitive mechanisms or heuristics are involved. Our results are based on the optimal probabilistic problem solving strategy that appeared in Solomonoff paper on general problem solving system. We provide arguments that a certain set of cognitive mechanisms or heuristics drive human problem solving in the similar manner as the optimal Solomonoff strategy. The results presented in this paper can serve both cognitive psychology in better understanding of human problem solving processes as well as artificial intelligence in designing more human-like agents.

Students’ difficulties in probabilistic problem-solving

There are many errors can be identified when students solving mathematics problems, particularly in solving the probabilistic problem. This present study aims to investigate students’ difficulties in solving the probabilistic problem. It focuses on analyzing and describing students errors during solving the problem. This research used the qualitative method with case study strategy. The subjects in this research involve ten students of 9th grade that were selected by purposive sampling. Data in this research involve students’ probabilistic problem-solving result and recorded interview regarding students’ difficulties in solving the problem. Those data were analyzed descriptively using Miles and Huberman steps. The results show that students have difficulties in solving the probabilistic problem and can be divided into three categories. First difficulties relate to students’ difficulties in understanding the probabilistic problem. Second, students’ difficulties in choosing and using appropriate strategies for solving the problem. Third, students’ difficulties with the computational process in solving the problem. Based on the result seems that students still have difficulties in solving the probabilistic problem. It means that students have not able to use their knowledge and ability for responding probabilistic problem yet. Therefore, it is important for mathematics teachers to plan probabilistic learning which could optimize students probabilistic thinking ability.

Self-regulated learning of principle-based concepts: Do students prefer worked examples, faded examples, or problem solving?

Acquisition of principle-based concepts involves learning how and when to apply a specific principle to different instances of the same problem type. Within this domain, learning is best achieved when practice involves studying worked examples followed by problem solving. When given the choice to use worked examples versus problem solving, how do people regulate their learning? Furthermore, do they use faded examples effectively when given the opportunity during learning? In three experiments, participants learned how to solve probability problems under practice conditions involving either (a) a combined schedule of worked examples, partial examples (Experiments 2 and 3), and problem solving, (b) problem solving only, or (c) self-regulated learning in which participants could choose a worked example, a partial example (Experiments 2 and 3), or problem solving on each trial. Self-regulated learners chose to study worked examples on fewer than 40% of the trials and seldom did so prior to problem solving. However, participants did regulate their learning effectively when they could use partial examples during practice. Participants also demonstrated some sophisticated problem solving, such as by studying worked examples more often after failed versus successful problem-solving attempts.

The Investigation of Pre-service Elementary Mathematics Teachers’ Subject Matter Knowledge About Probability

Because of the change in the middle school mathematics curriculum in Turkey, it is necessary to examine subject matter knowledge of pre-service elementary mathematics teachers about probability. This study is significant in terms of the Turkish mathematics education literature since it contributes to the future curriculum efforts for elementary mathematics education programs. Data were collected through face-to-face interviews which were focused on explanations about basic probability concepts and an instrument which was developed for evaluating content knowledge for probability of elementary mathematics teachers. Since mathematics teachers should have both procedural and conceptual knowledge regarding the concept taught, researcher aimed to understand to what extent pre-service elementary mathematics teachers are capable of conceptual and procedural knowledge needed for probability teaching. Findings showed that the 23 participants needed to develop their conceptual knowledge regarding probability and they tended to behave computational oriented while solving probability problems which showed also their higher procedural understanding. Besides, it was concluded that they couldn’t make expected connections between probability and statistics concepts.

How do high school students solve probability problems? A mixed methods study on probabilistic reasoning

ABSTRACTWhen studying a complex research phenomenon, a mixed methods design allows to answer a broader set of research questions and to tap into different aspects of this phenomenon, compared to a monomethod design. This paper reports on how a sequential equal status design (QUAN → QUAL) was used to examine students’ reasoning processes when solving probability problems. A select clustered sampling resulted in the inclusion of 168 high school students in a first, quantitative phase, in which a questionnaire was used to assess how they solved probability problems. This questionnaire included probability items that were based on the outcome orientation, the representativeness misconception, and the equiprobability bias. In a second, qualitative phase, 18 students who were purposefully sampled from the first research phase were interviewed in order to conduct an in-depth study of their probabilistic reasoning processes. In this paper, we illustrate and discuss how several mixed methods research purposes were realized throughout our study: development, expansion, and initiation.

Problem Solving as Probabilistic Inference with Subgoaling: Explaining Human Successes and Pitfalls in the Tower of Hanoi.

How do humans and other animals face novel problems for which predefined solutions are not available? Human problem solving links to flexible reasoning and inference rather than to slow trial-and-error learning. It has received considerable attention since the early days of cognitive science, giving rise to well known cognitive architectures such as SOAR and ACT-R, but its computational and brain mechanisms remain incompletely known. Furthermore, it is still unclear whether problem solving is a “specialized” domain or module of cognition, in the sense that it requires computations that are fundamentally different from those supporting perception and action systems. Here we advance a novel view of human problem solving as probabilistic inference with subgoaling. In this perspective, key insights from cognitive architectures are retained such as the importance of using subgoals to split problems into subproblems. However, here the underlying computations use probabilistic inference methods analogous to those that are increasingly popular in the study of perception and action systems. To test our model we focus on the widely used Tower of Hanoi (ToH) task, and show that our proposed method can reproduce characteristic idiosyncrasies of human problem solvers: their sensitivity to the “community structure” of the ToH and their difficulties in executing so-called “counterintuitive” movements. Our analysis reveals that subgoals have two key roles in probabilistic inference and problem solving. First, prior beliefs on (likely) useful subgoals carve the problem space and define an implicit metric for the problem at hand—a metric to which humans are sensitive. Second, subgoals are used as waypoints in the probabilistic problem solving inference and permit to find effective solutions that, when unavailable, lead to problem solving deficits. Our study thus suggests that a probabilistic inference scheme enhanced with subgoals provides a comprehensive framework to study problem solving and its deficits.

Working Memory Capacity and Self‐Explanation Strategy Use Provide Additive Problem‐Solving Benefits

SummaryThe present study examined the impact of working memory capacity (WMC) on college students' ability to solve probability problems while using a self‐explanation strategy. Participants learned to solve probability problems in one of three conditions: a backward‐faded self‐explanation condition, an example problem pairs self‐explanation condition, or a control (no self‐explanation) condition. Even when accounting for the impact of WMC, learning to problem‐solve using self‐explanation led to superior problem‐solving performance. Conditions that prompted self‐explanation during problem‐solving resulted in significantly better problem‐solving performance than the control condition. These findings provide insight into the influence of individual differences on problem‐solving when strategies are provided, as well as information about the effectiveness of the self‐explanation strategy during mathematical problem‐solving. Copyright © 2016 John Wiley & Sons, Ltd.

Program-combinatorial approach to solving problems of error-free readout of random point images

A novel approach is proposed for solving probability problems arising in studying reliability of readout of random point structures. The essence of this approach is based on combining specialized program-analytical and discrete-combinatorial methods of research seldom used in academic practice. The efficiency of the proposed approach is demonstrated by an example of solving a particular problem that involves two-threshold readout of random point images.

Probabilistic problem solving in biomedicine

Probabilistic problem solving in biomedicine

The Cognitive Processes of Probability Estimation in Random Sampling from Discrete Uniform Spaces

The research examines how high school students solve simple probability problems, and the probability estimations of students, boys and girls, in a discrete and uniform probability space. In discrete uniform probability spaces problem solving using estimations of probabilities and strategies of solving problems in a variety of situations were described. Solving probability problems among elementary intermediate and high school students were studied, however, reviewing the literature in probability; I am not familiar with any research on estimating probabilities or characterizations of rules people use for estimating and solving probability problems relating specifically to numerically symmetric and asymmetric objects. The research describes the rules and heuristics that student use to solve probability problems and estimate probabilities. The research has shown that students at different levels of proficiency in solving probability problems take a different approach when solving and estimating probabilities. When students are asked to estimate the probability of selecting two beads with different colors from a bag containing beads of different colors some calculate probabilities and other estimate them using a variety of rules. The use of rules for estimating probabilities is directly related to numerical relations and symmetries of objects. I describe a variety of rules students use for estimating probabilities, e.g., the

A probability perspective

Abstract This paper draws together four perspectives that contribute to a new understanding of probability and solving problems involving probability. The first is the Subjective Bayesian perspective that probability is affected by one’s knowledge, and that it is updated as one’s knowledge changes. The main criticism of the Bayesian perspective is the problem of assigning prior probabilities; this problem disappears with our Information Theory perspective, in which we take the bold new step of equating probability with information. The main point of the paper is that the formal perspective (formalize, calculate, unformalize) is beneficial to solving probability problems. And finally, the programmer’s perspective provides us with a suitable formalism. To illustrate the benefits of these perspectives, we completely solve the hitherto open problem of the two envelopes.

The Process of Probability Problem Solving: Use of External Visual Representations

We investigate the role of external inscriptions, particularly those of a spatial or visual nature, in the solution of probability word problems. We define a taxonomy of external visual representations used in probability problem solving that includes pictures, spatial reorganization of the given information, outcome listings, contingency tables, Venn diagrams, trees, and novel graphical representations. We also propose a process model for probability problem solving (PPS) and use it as a framework to better understand how and why external visual representations are used. In a study of 34 novice probability problem solvers, participants worked to solve six probability word problems covering six probability subtopics. Both written and verbal structured interview protocols were analyzed to investigate when and how external visual representations are spontaneously used by problem solvers. Analyses of the coded transcripts showed that participants' probability problem-solving efforts move through the stages of PPS in a sequential but not always linear manner, sometimes exhibiting iterated attempts to represent the problem mathematically and to find a solution strategy. Results showed that use of specific external visual representations was associated with specific probability topics, and that certain choices of representation are associated with higher rates of solution success. These findings suggest that an external visual representation can facilitate probability problem solving, but only when an appropriate representation is chosen. Finally, we present evidence to show that external visual representations are usually created and first used during the stages of representing the problem mathematically and finding a solution strategy. However, pictures are often created during the initial stage of problem text understanding, and tables are sometimes created during computation of the solution.

Finding and fixing errors in worked examples: Can this foster learning outcomes?

Learning from worked examples is an effective learning method in well-structured domains. Can its effectiveness be further enhanced when errors are included? This was tested by determining whether a combination of correct and incorrect solutions in worked examples enhances learning outcomes in comparison to correct solutions only, and whether a mixture of correct and incorrect solutions is more effective when the errors are highlighted. In addition, the effectiveness of fostering self-explanations was assessed. In Experiment 1, the participants learned to solve probability problems under six conditions that constituted a 2 × 3-factorial design (Factor 1: correct and incorrect solutions with highlighting the errors vs. correct and incorrect solutions without highlighting the errors vs. correct solutions only; Factor 2: prompting written self-explanations vs. no prompts). An aptitude-treatment interaction was found: providing correct and incorrect solutions fostered far transfer performance if learners had favourable prior knowledge; if learners had poor prior knowledge correct solutions only were more favourable. Experiment 2 replicated this interaction effect. Thus, a mixture of correct and incorrect solutions in worked examples enhanced learning outcomes only for “good” learners. In addition, Experiment 2 showed that confronting learners with incorrect solutions changed the quality of their self-explanations: on the one hand, new types of effective self-explanations could be observed, but on the other hand the amount of the very important principle-based self-explanations was substantially reduced. A possible measure to prevent this negative side effect of incorrect solutions is discussed.

USE OF EXTERNAL VISUAL REPRESENTATIONS IN PROBABILITY PROBLEM SOLVING

We investigate the use of external visual representations in probability problem solving. Twenty-six students enrolled in an introductory statistics course for social sciences graduate students (post-baccalaureate) solved eight probability problems in a structured interview format. Results show that students spontaneously use self- generated external visual representations while solving probability problems. The types of visual representations used include: reorganization of the given information, pictures, novel schematic representations, trees, outcome listings, contingency tables, and Venn diagrams. The frequency of use of each of these different external visual representations depended on the type of probability problem being solved. We interpret these findings as showing that problem solvers attempt to select representations appropriate to the problem structure, and that the appropriateness of the representation is determined by the problem’s underlying schema. First published May 2007 at Statistics Education Research Journal Archives

Thinking Science 프로그램의 확률 활동이 중학생의 확률적 사고 형성에 미치는 효과

이 연구의 목적은 중학생의 인지수준과 확률적 사고수준을 측정하여 그 관계를 분석하고 Thinking Science (TS) 프로그램의 확률 활동을 적용하여 그 효과를 분석하는 것이다. 중학교 1학년 219명을 실험집단과 통제집단으로 나누어, 실험집단에는 TS 프로그램의 확률 활동을, 통제집단에는 전통적인 과학 수업을 적용하였다. 결과에 의하면, 중학생의 인지수준은 대부분 구체적 조작기에 해당하였고, 많은 학생들의 확률적 사고수준은 확률 문제 해결에 양적 전략을 사용하면서 주관적 전략도 함께 사용하는 과도기적인 수준이었다. 또한, 인지수준이 높을수록 확률적 사고 수준도 높았으며, 확률의 구성요소 중에서 표본 공간과 한 사건에 대한 확률이 확률 비교와 조건부 확률보다 먼저 발달하였다. TS 프로그램의 확률 활동은 학생들이 확률 문제 해결에 양적 전략을 사용하도록 하는 데에 효과적이었다. 특히 사전에 확률 문제 해결에 주관적 전략과 양적 전략을 혼용하던 중기 구체적 조작기인 학생들이 사후에 양적 전략을 사용하도록 하고, 한 사건에 대한 확률을 인식하도록 하는 데에 효과가 있었다. The purposes of this study were to investigate the correlation between the cognitive level and the probabilistic thinking level and to analyze the effects of the probability activities in Thinking Science (TS) program on the development of probabilistic thinking. The 219 7th grade students were sampled in the middle school and were divided into an experimental group and a control group. The probability activities in TS program were implemented to the experimental group, while only normal curriculum was conducted in the control group. The results of this study showed that most of 7th grade students were in the concrete operational stage and used both subjective and quantitative strategy simultaneously in probability problem solving. It was also found that the higher the cognitive level of the students, the higher the probabilistic thinking level of them. The sample space and the probability of an event in the constructs of probability were first developed as compared to the probability comparisons and the conditional probability. The probability activities encouraged the students to use quantitative strategy in probability problem solving and to recognize probability of an event. Especially, the effectiveness was relatively higher for the students in the mid concrete operational stage than those in any other stage.

Spreadsheets as a Simulation Tool for Solving Probability Problems

Summary This article illustrates the use of spreadsheets as a simulation tool for solving a collection of probability problems.

Keiyo Cattle Raiding, Kechui Mathematics and Science Education: What do They Have in Common?

In the context of globalization, indigenous educational systems of endangered cultures are faced with extinction. Biologists recognize that biodiversity is dependent upon sustaining and promoting knowledge and values of indigenous cultures. Education researchers have a similar responsibility — to promulgate educational diversity before several thousand years of valuable educational knowledge is extirpated. This paper presents the Keiyo of Kenya who have a rich educational system. Its focus is on an indigenous game, Kechui (using calculations to sharpen your mind), based on cattle raiding, and the Keiyos' use of the game for learning mixed strategy-based probabilistic problem solving. Kechui may also have international potential for teaching probabilistic problem solving in contemporary mathematics and science, especially genetics.

Understanding the Nature of Errors in Probability Problem-Solving

This study provides an investigation of relationships among different types of errors occurring during probability problem-solving. Fifty non-mathematically sophisticated graduate student subjects enrolled in an introductory probability and statistics course were asked to solve a set of probability problems, and their attempts at solution were analyzed for presence and type of errors. The errors contained within these solutions were categorized according to a coding scheme which identifies 110 specific kinds of errors in four categories: text comprehension errors, conceptual errors, procedural errors, and arithmetic/computation errors. Relationships among types of errors included in each category were investigated using hierarchical clustering via additive trees. Implications of these relationships for the teaching and learning of probability problem-solving are discussed.

Learning subgoals and methods for solving probability problems

We hypothesize that typical example problems used in quantitative domains such as algebra and probability can be represented in terms of subgoals and methods that these problems teach learners. The "quality" of these subgoals and methods can vary, depending on the features of the examples. In addition, the likelihood of these subgoal's being recognized in novel problems and the likelihood of learners' being able to modify an old method for a new problem may be functions of the training examples learners study. In Experiment 1, subjects who studied examples predicted to teach certain subgoals were often able to recognize those subgoals in nonisomorphic transfer problems. Subjects who studied examples demonstrating two methods rather than one exhibited no advantages in transfer. Experiment 2 demonstrated that if the conditions for applying a method are highlighted in examples, learners are more likely to appropriately adapt that method in a novel problem, perhaps because they recognize that the conditions do not fully match those required for any of the old methods. Overall, the results indicate that the subgoal/method representational scheme may be useful in predicting transfer performance.

Understanding and solving probability problems: A developmental study

Children's understanding of what variables and relations are important in problem structures, and their use of these variables and relations in problem solving, were examined. One hypothesis suggests that knowledge of relevant solution variables is a prerequisite for encoding those variables, which in turn is a prerequisite for learning new strategies that use those variables. An alternative hypothesis holds that knowledge of relevant variables is an outcome, rather than a precursor, of efforts to invent new strategies. In the current studies, children between the ages of 5 and 13 years were given Piaget and Inhelder's (1975, The origin of the idea of chance in children, New York: Norton) two-set, alternative choice probability problems. In Experiment 1, problem understanding was assessed by asking children to construct two-set problems that could test whether a learner understood how to solve a model problem type. In Experiment 2, understanding was assessed by asking children to modify model problems to make them harder for a learner to solve. In both experiments, children modified or reproduced only those properties of model problems used either correctly or incorrectly in solving the models. These results partially support both hypotheses, and suggest a mechanism by which problem solving knowledge develops.