Igor' Kontorovich | The University of Auckland (original) (raw)

Papers by Igor' Kontorovich

Research paper thumbnail of Why Do Experts Pose Problems for Mathematics Competitions?

Freiburger Empirische Forschung in der Mathematikdidaktik, 2015

Inspired by the recurrent findings on the steady decrease in students' interest in mathematics, t... more Inspired by the recurrent findings on the steady decrease in students' interest in mathematics, this paper is concerned with sources of experts' motivation for posing problems for mathematics competitions. Twenty-six experts from nine countries participated in the study. The inductive analysis of the data suggests that experts utilise posing problems for mathematics competitions for fulfilment of their internal needs: an intellectual need for enriching their mathematical knowledge base and a socio-psychological need for belonging, recognition and appreciation. Educational implications of the findings for students are discussed and future research directions are presented.

Research paper thumbnail of How do experts pose problems for mathematics competitions?

DESCRIPTION Abstract of Dissection. Kontorovich, I. (2013). How do experts pose problems for math... more DESCRIPTION Abstract of Dissection. Kontorovich, I. (2013). How do experts pose problems for mathematics competitions? Abstract. Unpublished dissertation. Haifa: Technion - Israeli Institute of Technology.

Research paper thumbnail of Towards exploring expertise in mathematics educational research: What are the requirements and duties of the researcher?

While considerable resources are invested in mathematics educational research and in nurturing fu... more While considerable resources are invested in mathematics educational research and in nurturing future scholars, little is known about expertise in this field. In this paper, we explore the requirements and duties of mathematics educational researchers, as a preliminary step towards characterizing the components of their expertise. The data corpus of the study consisted of 57 position announcements for assistant and associate professor in 48 universities and 4 colleges located in thirty US states. An inductive content analysis revealed four types of requirements and duties: (1) required background in mathematics and mathematics education; (2) teaching and mentoring duties; (3) research and publishing obligations; (4) department and university duties. The implications of the findings are discussed from the perspectives of high-education programs and graduate students who are considering mathematics education research as their career track. The findings are used to formulate goals and ...

Research paper thumbnail of High-achieving students, prospective teachers and an expert acting on the same problem-posing task. In High-Achieving Students, Prospective Teachers and an Expert Acting on the Same Problem-Posing Task

The importance of problem posing in mathematics and mathematics education has been acknowledged b... more The importance of problem posing in mathematics and mathematics education has been acknowledged by many schools. However, one of the frequently reported findings in the studies on students' and teachers' problem posing is a poor quality of the created problems. This paper integrates findings of three studies concerned with the mechanisms and holistic processes that govern problem-posing performance potentially leading to creation of high-quality problems. The first study was conducted with high-achieving high-school students, the second study was on two prospective teachers and the third study was conducted with an expert in creating problems for mathematics competition. The posers in all the three studies were given the same problem-posing task. In this paper I overview and contrast the findings of these three studies. The observed similarities and differences converge to some ideas on the importance of posers' perceptions of the problem-posing situation they are involv...

Research paper thumbnail of What makes an interesting mathematical problems? A perception analysis of 22 adult participants of the competition movement

What makes an interesting mathematical problem? A perception analysis of 22 adult participants of... more What makes an interesting mathematical problem? A perception analysis of 22 adult participants of the competition movement. In B. Roesken The two aims of the paper are: (1) to identify the goals that the adult participants of the competition movement wish to achieve with an aim of the problems given at mathematics competitions, and (2) to identify characteristics of an "interesting competition problem" which contribute to achieving these goals. The data were collected from 22 adult participants of the competition movement from seven countries. The findings stress the proximity of the competition movement to the mainstreams in the field of mathematics education, and support adapting its successful practices for a broad range of students. Some practical and research applications are discussed.

Research paper thumbnail of Feeling of innovation in expert problem posing

Publications in this series have been peer reviewed. editorial board: liisa tainio (chair), kaisu... more Publications in this series have been peer reviewed. editorial board: liisa tainio (chair), kaisu Rättyä (secretary), kalle Juuti, Henry leppäaho, eila lindfors, Harry Silfverberg, Arja virta and eija Yli-Panula Studies in Subject Didactics 6 Current state of research on mathematical beliefs Xviii Cover and design: katja kontu layout: Mikko Halonen Printing: unigrafia oy, Helsinki iSSn 1799-9596 (printed) iSSn 1799-960X (pdf ) iSBn 978-952-5993-08-0 (printed) iSBn 978-952-5993-09-7 (pdf ) https://helda.helsinki.fi/ Helsinki 2013

Research paper thumbnail of Indicators of creativity in mathematical problem posing: How indicative are they

In nd di ic ca at to or rs s o of f c cr re ea at ti iv vi it ty y i in n m ma at th he em ma at ... more In nd di ic ca at to or rs s o of f c cr re ea at ti iv vi it ty y i in n m ma at th he em ma at ti ic ca al l p pr ro ob bl le em m p po os si in ng g: : H Ho ow w i in nd di ic ca at ti iv ve e a ar re e t th he ey y? ?

Research paper thumbnail of Towards a comprehensive framework of mathematical problem posing

This theoretical essay presents a consolidated theoretical framework for analysing mathematical p... more This theoretical essay presents a consolidated theoretical framework for analysing mathematical problem posing. The main feature of the suggested framework is that it builds upon broadly accepted problem solving models, and, simultaneously, includes theoretical constructs that are identified as specific for problem posing. The framework consists of four facets: resources, problem posing heuristics, aptness and social context in which problem posing occurs. The framework contributes to the existing research literature by consolidating findings on particular aspects of problem posing that have been explored so far and suggests a research agenda for further advancing the field.

Research paper thumbnail of Development of researcher knowledge in mathematics education: Towards a confluence framework

We present a framework of researcher knowledge development in conducting a study in mathematics e... more We present a framework of researcher knowledge development in conducting a study in mathematics education. The key components of the framework are: knowledge germane to conducting a particular study, processes of knowledge accumulation, and catalyzing filters that influence a researcher decision making. The components of the framework originated from a confluence between constructs and theories in Mathematics Education, Higher Education and Sociology. Drawing on a self-reflective interview with a leading researcher in mathematics education, professor Michèle Artigue, we illustrate how the framework can be utilized in data analysis. Criteria for framework evaluation are discussed.

Research paper thumbnail of Essential aspects for inclusion in future consolidated problem posing frameworks

After seventy years of research, remarkable body of theoretical and empirical knowledg... more After seventy years of research, remarkable body of theoretical and empirical knowledge on
Problem Posing is accumulated. Still, it is not enough to make problem posing a common activity in
a regular mathematics classroom. One of the ways to deal with this problem is to consolidate our
knowledge on PP in comprehensive theoretical framework(s). The paper presents four aspects that
may be included in future consolidate theoretical framework(s) of Problem Posing and argues the
aspects' essentiality.

Research paper thumbnail of Reviewing Mathematics & Mathematics Education: Searching for Common Ground

Canadian Journal of Science, Mathematics and Technology Education, 2014

A review of Michael N. Fried and Tommy Dreyfus (Eds.). (2014). Mathematics & mathematics educatio... more A review of Michael N. Fried and Tommy Dreyfus (Eds.). (2014). Mathematics & mathematics education: Searching for common ground. Dordrecht, the Netherlands: Springer, 402 pp. ISBN: 978-94-0077472-8 (Hardcover and e-book).

Research paper thumbnail of A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS COMPETITIONS

International Journal of Science and Mathematics Education, 2014

This paper is concerned with organizational principles of a pool of familiar problems of expert p... more 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.

Research paper thumbnail of An exploratory framework for handling the complexity of mathematical problem posing in small groups

The Journal of Mathematical Behavior, 2012

The paper introduces an exploratory framework for handling the complexity of students' mathematic... more The paper introduces an exploratory framework for handling the complexity of students' mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students' knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers' comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problemposing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students' decision making while posing problems.

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

Educational Studies in Mathematics, 2013

Success stories," i.e., cases in which mathematical problems posed in a controlled setting are pe... more 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.

Research paper thumbnail of Why Do Experts Pose Problems for Mathematics Competitions?

Freiburger Empirische Forschung in der Mathematikdidaktik, 2015

Inspired by the recurrent findings on the steady decrease in students' interest in mathematics, t... more Inspired by the recurrent findings on the steady decrease in students' interest in mathematics, this paper is concerned with sources of experts' motivation for posing problems for mathematics competitions. Twenty-six experts from nine countries participated in the study. The inductive analysis of the data suggests that experts utilise posing problems for mathematics competitions for fulfilment of their internal needs: an intellectual need for enriching their mathematical knowledge base and a socio-psychological need for belonging, recognition and appreciation. Educational implications of the findings for students are discussed and future research directions are presented.

Research paper thumbnail of How do experts pose problems for mathematics competitions?

DESCRIPTION Abstract of Dissection. Kontorovich, I. (2013). How do experts pose problems for math... more DESCRIPTION Abstract of Dissection. Kontorovich, I. (2013). How do experts pose problems for mathematics competitions? Abstract. Unpublished dissertation. Haifa: Technion - Israeli Institute of Technology.

Research paper thumbnail of Towards exploring expertise in mathematics educational research: What are the requirements and duties of the researcher?

While considerable resources are invested in mathematics educational research and in nurturing fu... more While considerable resources are invested in mathematics educational research and in nurturing future scholars, little is known about expertise in this field. In this paper, we explore the requirements and duties of mathematics educational researchers, as a preliminary step towards characterizing the components of their expertise. The data corpus of the study consisted of 57 position announcements for assistant and associate professor in 48 universities and 4 colleges located in thirty US states. An inductive content analysis revealed four types of requirements and duties: (1) required background in mathematics and mathematics education; (2) teaching and mentoring duties; (3) research and publishing obligations; (4) department and university duties. The implications of the findings are discussed from the perspectives of high-education programs and graduate students who are considering mathematics education research as their career track. The findings are used to formulate goals and ...

Research paper thumbnail of High-achieving students, prospective teachers and an expert acting on the same problem-posing task. In High-Achieving Students, Prospective Teachers and an Expert Acting on the Same Problem-Posing Task

The importance of problem posing in mathematics and mathematics education has been acknowledged b... more The importance of problem posing in mathematics and mathematics education has been acknowledged by many schools. However, one of the frequently reported findings in the studies on students' and teachers' problem posing is a poor quality of the created problems. This paper integrates findings of three studies concerned with the mechanisms and holistic processes that govern problem-posing performance potentially leading to creation of high-quality problems. The first study was conducted with high-achieving high-school students, the second study was on two prospective teachers and the third study was conducted with an expert in creating problems for mathematics competition. The posers in all the three studies were given the same problem-posing task. In this paper I overview and contrast the findings of these three studies. The observed similarities and differences converge to some ideas on the importance of posers' perceptions of the problem-posing situation they are involv...

Research paper thumbnail of What makes an interesting mathematical problems? A perception analysis of 22 adult participants of the competition movement

What makes an interesting mathematical problem? A perception analysis of 22 adult participants of... more What makes an interesting mathematical problem? A perception analysis of 22 adult participants of the competition movement. In B. Roesken The two aims of the paper are: (1) to identify the goals that the adult participants of the competition movement wish to achieve with an aim of the problems given at mathematics competitions, and (2) to identify characteristics of an "interesting competition problem" which contribute to achieving these goals. The data were collected from 22 adult participants of the competition movement from seven countries. The findings stress the proximity of the competition movement to the mainstreams in the field of mathematics education, and support adapting its successful practices for a broad range of students. Some practical and research applications are discussed.

Research paper thumbnail of Feeling of innovation in expert problem posing

Publications in this series have been peer reviewed. editorial board: liisa tainio (chair), kaisu... more Publications in this series have been peer reviewed. editorial board: liisa tainio (chair), kaisu Rättyä (secretary), kalle Juuti, Henry leppäaho, eila lindfors, Harry Silfverberg, Arja virta and eija Yli-Panula Studies in Subject Didactics 6 Current state of research on mathematical beliefs Xviii Cover and design: katja kontu layout: Mikko Halonen Printing: unigrafia oy, Helsinki iSSn 1799-9596 (printed) iSSn 1799-960X (pdf ) iSBn 978-952-5993-08-0 (printed) iSBn 978-952-5993-09-7 (pdf ) https://helda.helsinki.fi/ Helsinki 2013

Research paper thumbnail of Indicators of creativity in mathematical problem posing: How indicative are they

In nd di ic ca at to or rs s o of f c cr re ea at ti iv vi it ty y i in n m ma at th he em ma at ... more In nd di ic ca at to or rs s o of f c cr re ea at ti iv vi it ty y i in n m ma at th he em ma at ti ic ca al l p pr ro ob bl le em m p po os si in ng g: : H Ho ow w i in nd di ic ca at ti iv ve e a ar re e t th he ey y? ?

Research paper thumbnail of Towards a comprehensive framework of mathematical problem posing

This theoretical essay presents a consolidated theoretical framework for analysing mathematical p... more This theoretical essay presents a consolidated theoretical framework for analysing mathematical problem posing. The main feature of the suggested framework is that it builds upon broadly accepted problem solving models, and, simultaneously, includes theoretical constructs that are identified as specific for problem posing. The framework consists of four facets: resources, problem posing heuristics, aptness and social context in which problem posing occurs. The framework contributes to the existing research literature by consolidating findings on particular aspects of problem posing that have been explored so far and suggests a research agenda for further advancing the field.

Research paper thumbnail of Development of researcher knowledge in mathematics education: Towards a confluence framework

We present a framework of researcher knowledge development in conducting a study in mathematics e... more We present a framework of researcher knowledge development in conducting a study in mathematics education. The key components of the framework are: knowledge germane to conducting a particular study, processes of knowledge accumulation, and catalyzing filters that influence a researcher decision making. The components of the framework originated from a confluence between constructs and theories in Mathematics Education, Higher Education and Sociology. Drawing on a self-reflective interview with a leading researcher in mathematics education, professor Michèle Artigue, we illustrate how the framework can be utilized in data analysis. Criteria for framework evaluation are discussed.

Research paper thumbnail of Essential aspects for inclusion in future consolidated problem posing frameworks

After seventy years of research, remarkable body of theoretical and empirical knowledg... more After seventy years of research, remarkable body of theoretical and empirical knowledge on
Problem Posing is accumulated. Still, it is not enough to make problem posing a common activity in
a regular mathematics classroom. One of the ways to deal with this problem is to consolidate our
knowledge on PP in comprehensive theoretical framework(s). The paper presents four aspects that
may be included in future consolidate theoretical framework(s) of Problem Posing and argues the
aspects' essentiality.

Research paper thumbnail of Reviewing Mathematics & Mathematics Education: Searching for Common Ground

Canadian Journal of Science, Mathematics and Technology Education, 2014

A review of Michael N. Fried and Tommy Dreyfus (Eds.). (2014). Mathematics & mathematics educatio... more A review of Michael N. Fried and Tommy Dreyfus (Eds.). (2014). Mathematics & mathematics education: Searching for common ground. Dordrecht, the Netherlands: Springer, 402 pp. ISBN: 978-94-0077472-8 (Hardcover and e-book).

Research paper thumbnail of A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS COMPETITIONS

International Journal of Science and Mathematics Education, 2014

This paper is concerned with organizational principles of a pool of familiar problems of expert p... more 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.

Research paper thumbnail of An exploratory framework for handling the complexity of mathematical problem posing in small groups

The Journal of Mathematical Behavior, 2012

The paper introduces an exploratory framework for handling the complexity of students' mathematic... more The paper introduces an exploratory framework for handling the complexity of students' mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students' knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers' comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problemposing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students' decision making while posing problems.

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

Educational Studies in Mathematics, 2013

Success stories," i.e., cases in which mathematical problems posed in a controlled setting are pe... more 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.