Problem Posers Research Articles

Identifying competent problem posers and exploring their characteristics

In a study involving 66 masters and 60 sixth-grade students, we conducted Principal Component Analysis to identify more-and-less competent problem posers based on performance criteria rather than, as in prior research, relying on participants’ mathematical experience or background. Also, to cast light on characteristics of competent posers, we explored possible patterns in the problem-posing process based on the two identified groups’ self-reports and eye-movements. The results showed that: masters students had a significantly lower proportion of the more-competent group and a higher proportion of the less-competent group than sixth graders; more-competent posers perceived a better understanding of the problem-posing tasks than less-competent posers; more-competent posers exhibited significantly more fixation time on completing the entire problem-posing activities than less-competent posers, though this pattern disappeared across particular stages of the problem-posing process; and more-competent posers appeared to engage in a more purposeful search and processing to construct their problems than less-competent posers.

Line segment visibility with sidedness constraints

We study a family of line segment visibility problems, related to classical art gallery problems, which are motivated by monitoring requirements in commercial data centers. Given a collection of non-overlapping line segments in the interior of a rectangle, and a requirement to monitor the segments from one side or the other, we examine the problem of finding a minimal set of point guards. Guards may be placed anywhere in the interior of the rectangle but not on a line segment. We consider combinatorial bounds of problem variants where the problem solver gets to decide which side of the segments to guard or the problem poser gets to decide which side to guard, and many others. We show that virtually all variants are NP-Hard to solve exactly, and then provide heuristics and experimental results to give insight into the associated practical problems. Finally we describe a program for using experiments to guide the search for optimal combinatorial bounds.

Engagement and affect patterns in high school mathematics classrooms that exhibit spontaneous problem posing: an exploratory framework and study

Spontaneous problem posing (SPP) is presented as a phenomenon central to mathematical thinking, wherein learners generate problems without formal prompting, and where posed problems originate from the learner’s motivation to improve their knowledge. Because of this, they may serve as important markers of productive mathematical engagement—in particular, affective engagement—both for problem posers and their classroom communities (D’Mello & Graesser in Learning and Instruction, 22(2), 145–157, 2012). We report on a study which utilized mixed methods to examine SPP and associated affective engagement of students in four early high school mathematics classrooms in two geographic regions of the USA. For each classroom, we used observational and experience sampling methods to examine the patterns of affect problem posers and their peers experienced at individual and group levels, respectively, on a day with at least one instance of SPP observed compared with a day with no observed instances of SPP. Results show evidence of positive affect among problem posers, while their peers reported fewer negative emotions about mathematics tasks on days with SPP than on days without SPP. Moreover, a detailed analysis of two spontaneous problem posers revealed that they showed a desire to extend their own mathematical thinking, expressed dissatisfaction with their current knowledge, and directed their problems primarily towards the teacher. Results are discussed regarding the need to support students’ confidence in their original mathematical ideas in class, and for reducing negative emotional responses to mathematics tasks.

Problem-posing triggers or where do mathematics competition problems come from?

Spurred by Kilpatrick’s (1987) “Where do good problems come from?”, this study explores problem-posing triggers of experienced problem posers for mathematics competitions. Triggers are conceived as instances of noticing, where an impulse draws a poser’s attention and “triggers off” a mathematical re-action, one of the outcomes of which is a creation of a problem that gets accepted to some mathematics competition. The data were collected from 26 problem posers residing in nine countries, and who had experience in creating problems for national, regional, and international competitions. Three types of triggers emerged from the analysis: (i) Instances where the participants extracted mathematical phenomena from activities that are replete with modern elementary mathematics. These triggers were described in emotionally loaded terms that resonated with appreciation, surprise, challenge, and feeling of innovation. (ii) Cases where the participants abstracted mathematical phenomena from common everyday-life tasks in which mathematical optimization was beneficial. These situations emphasized participants’ desire to compensate for self-disappointment with an initially made decision and the value of finding a better alternative. (iii) Situations where the participants were asked to pose a problem ‘here and now’. The posers were unanimous in their dislike of such situations. They elaborated on how difficult it is to pose ‘here and now’ and highlighted the low quality of the resulting problems. The findings are situated in the body of knowledge on expert problem posing and educational literature on school students and teachers.

Studying the role of human agency in school mathematics

Mathematical discourse is often described as abstract and devoid of human presence, yet many school curricula espouse an aim to develop active, creative mathematical problem posers and solvers. The project The Evolution of the Discourse of School Mathematics(EDSM) developed an analytic scheme to investigate the nature of school mathematics discourse through the lens of high-stakes examinations in England. Following an overview of the scheme, this article ‘zooms in’ on the development of the sub-component addressing the question of how the origin of mathematical knowledge is construed, allowing investigation of the potential for students to see a role for themselves as active, creative agents in mathematical practices. Analytical tools operationalising this component are presented and their application illustrated. Results of analysis of examinations over a period of three decades suggest some increase in human agency, though some other aspects characteristic of higher-level mathematics may have reduced.

Expert Recurrence of Linear Problem Posing

According to Pelczer & Gamboa (2009) the Mathematical problem posing process of the skilled problem posers (teachers, mathematically gifted students) is usually cyclic whilst the novices usually pose problems in a linear way. (Cyclic model means that the skilled problem poser is able to switch between the phases of problem posing, to learn from his/her mistakes, to transform the idea which is not suitable. Linear model is just a simple straight way.)In my study I set expert category. (Expert = professional in problem posing, preparing problems for Mathematical competitions such as Mathematical Olympiad.) With respect to previous statements the hypothesis would be that the experts will show nearly no signs of linear problem posing process. But this is not true – there is some recurrence of linear problem posing in the expert group. The signs of the problem posing process are seemingly the same as in the novices’ case. But the background is different. The expertness lies in the fact that the expert is so experienced that he/she is able to come with an idea that needs no transformations. During the problem posing trajectory the experts usually perform the Final assessment stage. The experts also do not perform multiple Setup stages. (They may return to the Setup stage in order to do Transformations but they usually do not restart their work by the use of totally new Setup.)

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS COMPETITIONS

This paper is concerned with organizational principles of a pool of familiar problems of expert problem posers and the ways by which they are utilized for creating new problems. The presented case of Leo is part of a multiple-case study with expert problem posers for mathematics competitions. We present and inductively analyze the data collected in a reflective interview and in a clinical task-based interview with Leo. In the first interview, Leo was asked to share with us the stories behind some problems posed by him in the past. In the second interview, he was asked to pose a new competition problem in a thinking-aloud mode. We found that Leo’s pool of familiar problems is organized in classes according to certain nesting ideas. Furthermore, these nesting ideas serve him in posing problems that, ideally, are perceived by Leo as novel and surprising not only to potential solvers, but also to himself. Because of the lack of empirical research on experts in mathematical problem posing, the findings are discussed in light of research on experts in problem solving and on novices in mathematical problem posing.

Teachers’ Problem Posing in Mathematics

Abstract The article deals with problem posing as a teachers’ activity. Three categories of problem posers are considered according to their experience level – novices, specialists and experts. Three idea types – trial-error, semi-intentional and intentional ideas are analyzed in order to demonstrate differences between problem posing trajectories of the three groups. The idea types are illustrated by means of a case study.

THE QUALITY OF MATHEMATICAL PROBLEMS - EVALUATION AND SELF-EVALUATION

The research presented in the article consists of two parts. Firstly, opinions on mathematical problem quality are explored within four groups of participants (novices, specialists and experts in problem posing; high school students who never posed their own problems). Secondly, self-reflections written by the participants who have some experience in problem posing (novices, specialists and experts) are explored and compared with the general view of problem quality received in the first part of the research. The more experienced problem posers have more requirements on problem quality (both as general requirements and within their own work on posing problems). There is a slight decrease in ability to notice important features of mathematical problem quality after the first experience in problem posing. Experts lay stress on mathematical features of the problem whilst novices and specialists more on problem – student interaction.

Dissecting success stories on mathematical problem posing: a case of the Billiard Task

“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.

NOTES AND PROBLEMS: A new format for the PROBLEMS AND SOLUTIONS SERIES

In establishing the Problems and Solutions (P&S) series of Econometric Theory, the primary objective was pedagogical—to provide an intellectual resource for students, instructors, and researchers through the publication of student exercises and research-level problems in econometrics. Over the past 20 years, the P&S series has successfully served this goal. Large numbers of worked exercises and problems have now been published, and the series has promoted interaction between problem solvers, problem posers, and the ET readership, thereby helping to bring the wider econometric community together in a common purpose.

Roles and knowledge management in online technology communities: an ethnography study

The internet is a heterogeneous network of millions of computers that is continuously evolving. The interaction among people around the world on the internet has led to the formation of communities. Technical communities are groups who share a common interest in a technology. The literature on technology communities lacks a conceptual understanding of the roles of various players in the online community. An understanding of the different roles the members of the community assume at different phases, and the impact of the roles on knowledge management is crucial to manage and sustain such online technical communities. This study based on an ethnographic analysis of two technical communities, identifies seven distinct roles: core organiser, experts, problem poser, implementer, integrator, institutionaliser, and philosopher. The impact of each of the roles on knowledge management activities is discussed.

Roles and Knowledge Management in Online Technology Communities: An Ethnography Study

The Internet is a heterogeneous network of millions of computers that is continuously evolving. The interaction among people around the world on the Internet has led to the formation of communities. Technical communities are groups who share a common interest in a technology. The literature on technology communities lacks a conceptual understanding of the roles of various players in the online community. An understanding of the different roles the members of the community dawn at different phases, and the impact of the roles on knowledge management is crucial to manage and sustain such online technical communities. This study based on an ethnographic analysis of two technical communities identifies seven distinct roles: core organizer, experts, problem poser, implemented integrator, institutionalizer, and philosopher. The impact of each of the role on knowledge management activities is discussed.

Paul Erdõs. 26 March 1913 — 20 September 1996

In the first part of the twentieth century, Hungary produced an unusually large number of world-class mathematicians. They included, most notably, L. Fejér, A. Haar, F. and M. Riesz, J. von Neumann, G. Pólya, G. SzegÕ , P. Turán and perhaps, above all, the subject of this memoir, Pál (Paul) ErdÕs. As Ernst Straus put it, ErdÕs was ‘the crown prince of problem solvers and the undisputed monarch of problem posers’. ErdÕs was born in Hungary but left his native land when he was 21; from then on he lived in England, the USA, Canada, Israel and many other countries but frequently visited Hungary and had many Hungarian friends. Although he never had a ‘proper’ academic job, through his prodigious output, his host of co-authors, his constant travels and his amazing body of unsolved problems, he has greatly influenced mathematics today. He proved fundamental results in number theory, combinatorics, probability and approximation theory, as well as in set theory, elementary geometry and topology, and real and complex analysis. He was instrumental in the birth of probabilistic number theory and was the main advocate of the use of probabilistic methods in mathematics in general. He was also one of the originators of modern graph theory. He had an exceptional ability for joint work and many of his best results were obtained in collaboration; he wrote altogether about 1500 papers, perhaps five times as many as other prolific mathematicians, and he had about 500 collaborators.

Brian's Question

It has been my observation that children are not only good problem solvers, but also good problem posers. The following episode is a good example.

IFIAS: the top people's problem posers

The International Federation of Institutes for Advanced Studies aims to mobilise a global intellectual community to examine problems which have not been officially recognised and subjected to political convenience. Wendy Barnaby reports.