Roza Leikin

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• D.Sc., Technion
• Dean at University of Haifa

Mathematical modelling (MM) and problem posing (PP) are two creativity-directed mathematical activities highly effective in the realization of students' mathematical potential and the development of teachers' proficiency. MM and PP are intrinsi-cally interconnected: MM processes require the formulation of mathematical problems, while situation-base...

本文以臺灣資賦優異高中生與一般生為對象，探究其在函數圖形與方程式表徵轉換解題行為和腦波表現。以瑞文氏空間智力測驗篩選出33位資賦優異生及38位一般生，檢測兩組學生的行為表現和腦波反應。就解題行為分析顯示資優生答題正確性顯著高於一般生；兩組學生在答題的反應時間則無顯著差異。就腦波分析顯示，所有學生在函數圖形與方程式轉換階段（S2）的腦區反應顯著高於函數圖形閱讀階段（S1），顯示函數圖形和方程式表徵轉換需要較高的認知資源。另從P300分析顯示，函數圖形和方程式表徵轉換會造成側腦化現象。就兩組學生比較，其在閱讀函數圖形階段，事件相關電位（event-related potential）成分均無顯著差異，顯示兩組學生在函數圖形閱讀階段腦區反應並無顯著差別。但在函數圖形與方程式表徵轉換階段，資賦優異學...

The goal of this study was to identify conceptions towards early academic studies in computer science. We focus on a program which offers high school students the unique opportunity to earn a B.Sc. degree in parallel to their studies, resulting in them holding a prestigious degree at an early age. Activity theory framed the design of this study. Fi...

Leikin 所發展的測驗 工具瞭解臺灣學生創造力的表現，並以數學成就測驗 SAT-M 將學生區分為高、中、低三群，探 究數學成就是否影響數學創造力表現，最後，探究數學創造力與一般創造力是否存在關聯性。 研究樣本為新竹地區三所國中共 313 位八年級學生。研究結果包括： (1)八年級學生在四題數 學創造力問題流暢性、靈活性、獨創性、和創造力面向的表現均呈現顯著差異。其中，距離問題 在流暢性、靈活性、獨創性和創造力表現最佳。而果醬問題則在流暢性、靈活性、獨創性和創造 力表現最差。進一步將四題數學創造力問題區分為幾何和代數，分析顯示學生在流暢性、靈活 性、獨創性、創造力，幾何均顯著高於代數。 (2)不同數學成就的學生在流暢性、靈活性、獨創 性、創造力等部分皆達顯著性差異，此現象顯示學生的數學成就...

This study explored the unique connections between music and mathematics as perceived by four groups of experts: professional mathematicians and musicians, as well as teacher educators in these two fields. Using 2 × 2 study design, we studied four groups of participants, comprising theorists and educators from various Israeli universities. During s...

Studies on mathematical enrichment courses are rare. Even more unique are studies that examine the influence of enrichment courses on students’ achievements in curricular studies. This study attempted to fill this lacuna. To this end, 142 9th grade students took part in a one-year enrichment course in which they were asked to solve challenging extr...

The notion of open mathematical problems that appears in the mathematics education literature includes a variety of mathematical questions and tasks. Observation of the distinction between open-end and open-start problems leads us to draw a distinction between Multiple Solution Strategies Tasks (MSTs) and Multiple Solution Outcomes Tasks (MOTs). Wh...

Thius document includes a collection of the position documents presented at the International Workshop of Israel Science foundation. https://curiosity-isf.edu.haifa.ac.il/

Research on technology and mathematics education has been a longstanding interest of the PME community. In this paper we revisit the interplay between technology and conjecturing within the process of problem-solving with an intention to capture different aspects of the processes in which students make and explore mathematical conjectures, and role...

Mathematics educators argue that open-ended tasks as a powerful tool for the development of students’ creativity in mathematics, while it is well known that solving open-ended tasks is challenging for students. Recently we argued that not every open-ended task is fully open, as even when a task has a multiplicity of solution outcomes completeness o...

PME confrence proceedings Plenary Lectures Working Groups Seminar Colloquium National Presentation Oral Communications Posters

Volume 4 Research Reports P – Z

Volume 2 https://pme46.edu.haifa.ac.il/ Research Reports A-G

Volume 3 https://pme46.edu.haifa.ac.il/ Research Reports H-O

The study adopted an ERP methodology to examine two psychological constructs-field-dependence-independence (FDI) and symmetry (SYM)-on geometry problem solving. Based on a newly developed instrument, analyses of Taiwanese high school students showed that both FDI and SYM significantly influenced students' response accuracy (Acc). The FDI also deter...

This Edited Book recognizes the centrality of mathematical challenge in teaching and learning mathematics, addresses cognitive and socio-emotional components of mathematical challenge, and provides a different view on learning and teaching processes in mathematics education

Solving geometry problems is challenging for students of all ages. The complexity of geometry problems is multidimensional and is linked to visualization, auxiliary constructions required for solutions, computational and proof skills, and deep and robust knowledge of geometry concepts and their properties (definitions, axioms, and theorems). In thi...

The Math-Key program described and characterized in this chapter integrates Multiple Solution-Strategies Tasks (MSTs) and Multiple Outcomes Tasks (MOTs). We demonstrate that MSTs are inherently open tasks while, in contrast, MOTs can either be open or can require attaining completeness of a solution set. We argue that a multiplicity of solutions bo...

In this introduction I explore multiple dimensions of mathematical challenge, including, but not limited to, mathematical, psychological, social, and instructional dimensions, each of which is complex and all of which are interwoven. I discuss the nature and structure of mathematical challenge and argue that it can serve as a magnifying glass when...

Mathematical problem solving is the heart of mathematical activities at all levels. Problem-solving is both the means and the ends of the development of mathematical knowledge and skills as well as of the advancement of mathematics as a science. Researchers distinguish between problem-solving algorithms, problem-solving strategies and heuristics an...

Focusing on geometry proof problems, we describe two types of sets of mathematical problems, each of which includes problems of different levels of complexity. The sets of the first type embrace spaces of problems posed through investigations (PPI) in a Dynamic Geometry environment. These PPI sets can be generated by experts and novices in problem-...

Previous studies on intelligence have demonstrated that higher abilities are associated with lower brain activation, indicating a higher neural efficiency. In other words, more able individuals use fewer brain resources. However, it is unclear whether the neural efficiency phenomenon also appears for mathematical performance that is influenced by b...

This study explored the unique integration of mathematics tutors in teaching high school mathematics in a Virtual mathematics school (VMS). The tutors were three excelling STEM students who did not have any formal preparation for teaching before their work in the VMS. The goal of the study presented in this paper was to design a model of proficienc...

The current study investigated creativity-directed problem-solving processing that explicitly requires solving pattern generalization problems in multiple ways. To examine mathematical creativity, we employed a multiple solution tasks approach, asking participants explicitly to solve pattern generalization problems in multiple ways. The participant...

Mathematical giftedness (MG) is an intriguing phenomenon, the nature of which has yet to be sufficiently explored. This study goes a step further in understanding how MG is related to expertise in mathematics (EM) and general giftedness (G). Cognitive testing was conducted among 197 high school students with different levels of G and of EM. Based o...

We conducted a retrospective analysis of empirical studies on mathematical creativity with special attention to the studies conducted during the last decade. In the paper we present a brief survey of research on mathematical creativity up to 2009 and then a detailed review of empirical studies performed during the past decade. We note an optimistic...

One of the well-known approaches to creativity differentiates between creative person, process, product, and press. In the study presented in this paper we focus on creative process and product associated with Problem Posing through Investigation (PPI) by experts in mathematical problem solving. We link the creative process to creativity of PPI str...

Advancement of self-regulation during complex problem solving and the development of strategical reasoning are among the central educational goals linked to the 21st century skills. In this paper we introduce the notion of “Stepped Tasks”, which are specially designed in Top-Down structure to achieve these goals in mathematics instruction. The pape...

The educational literature often includes debate over the nature and nurture of mathematical giftedness. The varied schools of thought reflect discrepancies between views, which are to a large extent functions of philosophical, political, and economic considerations. To address this debate, in the context of the current ZDM Special Issue, I attempt...

Mathematics teacher educator (MTE) is a complex profession with unique goals and actions. While the realisation and advancement of students’ mathematical potential is a major goal of mathematics education, the major goal of MTEs is promoting mathematics teachers’ (MTs’) professional potential. Here, I introduce the construct of MTs’ professional po...

This study was inspired by the following question: how is mathematical creativity connected to different kinds of expertise in mathematics? Basing our work on arguments about the domain-specific nature of expertise and creativity, we looked at how participants from two groups with two different types of expertise performed in problem-posing-through...

Educational literature indicates that solving open mathematical tasks (OTs) is a powerful creativity-directed activity. However, the use of these tasks with school students on an everyday basis is extremely limited. To promote implementation of OTs in middle school, we manage a large-scale R&D project, Math-Key, which makes open mathematical tasks...

The participants in the study presented in this chapter are high school mathematics teachers teaching high–level mathematics. The teachers were members of a program called Club-5 that adopted communities of practice as a setting for support and encouragement of teachers’ learning and professional development, with the goal of raising the quality of...

The overarching goal of mathematics teacher educators is to facilitate the education and life-long learning and professional development of mathematics teachers. Mathematics teacher educators are usually experts in mathematics teaching. However, this expertise is not sufficient – in their profession mathematics teacher educators require special (ad...

Since each mathematics classroom is heterogeneous with respect to students’ mathematical potential, the quality of mathematical instruction results from matching the level of mathematical activities to different students’ potential. This is also true for classes that study mathematics at high level, in which mathematical challenge is a central elem...

The issue of attracting girls to mathematics has captured our attention when we were analyzing data from the final stage of the Kangaroo mathematics contest in Israel. With general finding showing boys having better results, further analysis of differences across Grades 2-6 indicates that in some grades the gap is smaller than in others. For instan...

In this paper, we describe a study conducted with 68 prospective mathematics teachers (PMTs) who took part in a Euclidian geometry didactical course. During the course, PMTs were required to solve Multiple Solution Tasks (MSTs) and perform mathematical investigations in DGE (Dynamic Geometry Environment) on a systematic basis. This paper describes...

Diese Festschrift ist Frau PROFESSORIN DR. MARIANNE NOLTE zum Eintritt in den Ruhestand gewidmet. Die Autor*innen sind nationale und internationale Kolleg*innen aus ihrer akademischen Schaffenszeit. Neben Beiträgen aus der Mathematikdidaktik, u.a. zur besonderen mathematischen Begabung und zur Rechenschwäche, den Arbeitsschwerpunkten Prof. Noltes,...

There is a strong consensus among mathematics educators, researchers and instructional designers that mathematical problem solving is among the central means—and ends—of school mathematics education. Different ideas, practices and studies in the field of mathematical problem solving are reflected in the volumes, chapters, and papers published over...

We believe that professional mathematicians who teach undergraduate mathematics courses to prospective teachers play an important role in the education of secondary school mathematics teachers. Thus, we explored the views of research mathematicians on the mathematics that should be taught to prospective mathematics teachers, on how the courses they...

This chapter gives an overview of the main directions of the research presented and discussed in the Thematic Working Group (TWG) “Mathematical potential, creativity and talent”. It addresses the issues of nature and nurture of mathematical creativity and mathematical talent as well as of methodological approaches used to study these topics. On the...

In this chapter, we describe different types of programs for students with high mathematical potential in Israel. Nurturing mathematically advanced students requires solving complex problems associated with making decisions about the scope of the programs and the choice of appropriate educational approaches. We focus our attention on classes for st...

In this chapter we describe two National Centers of Mathematics Teachers: The Primary School Teachers' Center and the Secondary School Teachers' Center. The centers are managed by a team from the Department of Mathematics Education in the University of Haifa and are monitored by the Israeli Ministry of Education. In line with their dual affiliation...

This chapter describes a nation-wide project - Club-5 - aimed at improving the quality of mathematics teaching in high school through management of teachers' communities of practice. Club-5 aims to create a comfort zone for project par ticipants and enhance the inquiry components in school mathematics teaching. Mutual support, shared experiences, a...

This paper, which describes neurocognitive studies that focus on mathematical processing, demonstrates the value that both mathematics education research and neuroscience research can derive from the integration of these two areas of research. It includes a brief overview of neuroimaging research related to mathematical processing. I base my claim...

In this response paper, I address the collection of chapters devoted to creativity-directed activities in mathematics. I analyse the chapters in the light of recommendations for the development of twenty-first-century skills that include creativity. I outline the commonalities and differences in the works of different authors through the lens of ac...

In order to achieve the present study’s goal – to understand better the phenomenon of mathematical giftedness – we performed a multidimensional examination of the mental processing in students who exhibited mathematical expertise (EM) at the secondary school level. The study included participants from the three groups: students who excelled in scho...

Creativity and giftedness in mathematics education research are topics of an increased interest in the education community during recent years. This introductory paper to the special issue on Mathematical Creativity and Giftedness in Mathematics Education has a twofold purpose: to offer a brief historical perspective on the study of creativity and...

We urge the reader to solve these problems before reading the paper. Then we suggest that the reader asks himself whether there is a different route or a different analysis of the given situation that leads to a different solution to any of the problems.

This paper presents a part of a larger study, in which we asked “How are learning and teaching of mathematics at high level linked to students’ general giftedness?” We consider asking questions, especially student-generated questions, as indicators of quality of instructional interactions. In the part of the study presented in this paper, we explor...

In this commentary, we analyze the 14 chapters of the book for interdisciplinary themes that unravel and are applicable to mathematics education. In particular, attention is given to interdisciplinary perspectives on the constructs of creativity and giftedness. Dominant themes from clusters of chapters are highlighted.

Observation that the interrelations between mathematical creativity, mathematical expertise and general giftedness are vague is what motivated a large-scale study that explores the relationship between mathematical creativity and mathematical ability. The study employs Multiple Solution Tasks (MSTs) as a tool for the evaluation of mathematical crea...

Invention, innovation, originality, insight, illumination and imagination are core elements of the individual and societal progress along human history from ancient times till the modern society. While these phenomena are often considered as indicators of creativity and talent in science, technology, business, arts, and music; they are also basic m...

This volume provides readers with a broad view on the variety of issues related to the educational research and practices in the field of Creativity in Mathematics and Mathematical Giftedness. The book explores (a) the relationship between creativity and giftedness; (b) empirical work with high ability (or gifted) students in the classroom and its...

Little empirical data are available concerning the cognitive abilities of gifted individuals in general and especially those who excel in mathematics. We examined visual processing abilities distinguishing between general giftedness (G) and excellence in mathematics (EM). The research population consisted of 190 students from four groups of 10th- t...

This paper addresses the neuro-cognitive characterization of super mathematically gifted high school students. The research population consisted of three groups of students excelling in mathematics: super mathematically gifted (S-MG), generally gifted students who excel in school mathematics (G-EM), and students who excel in school mathematics but...

We asked: “What are the similarities and differences in mathematical processing associated with solving learning-based and insight-based problems?” To answer this question, the ERP research procedure was employed with 69 male adolescent subjects who solved specially designed insight-based and learning-based tests. Solutions of insight-based problem...

Mathematical processing associated with solving short geometry problems requiring logical inference was examined among students who differ in their levels of general giftedness (G) and excellence in mathematics (EM) using ERP research methodology. Sixty-seven male adolescents formed four major research groups designed according to various combinati...

The goal of this chapter is to describe recent advances in mathematical problem solving, as they were represented in research reports from the annual international conferences for the Psychology of Mathematics Education. Delimiting the scope of this chapter was a challenge.

This paper analyzes different types of problem posing associated with geometry investigations in school mathematics, namely (I) problem posing through proving; (II) problem posing for investigation; (III) problem posing through investigation. Mathematical investigations and problem posing which are central for activities of professional mathematici...

Co-Chairs: Peter Taylor (Australia), Roza Leikin (Israel); Team members: Viktor Freiman (Canada), Linda Sheffield (USA), Mihaela Singer (Romania), Bo Mi Shin (Korea); Laison IPC Member: Shiqi Lee (China).

This study is based on our belief that mathematics should be challenging in any classroom and that mathematical challenge is among the central factors that determine the quality of mathematics lessons. Choosing challenging mathematical problem for the students is central in teachers’ work while their conception of mathematical challenge can determi...

Super-gifted individuals are considered to be very rare. This paper addresses one part of a larger body of research aimed at characterization of super-giftedness in mathematics. The research population consists of three groups of students who excel in mathematics: super-gifted in mathematics (S-MG), generally gifted students who excel in school mat...

A considerable amount of recent evidence suggests that speed of information processing (SIP) may be related to general giftedness as well as contributing to higher mathematical ability. To date, no study has examined SIP associated with both general giftedness (G) and excellence in mathematics (EM). This paper presents a part of more extensive rese...

In this study we examine the impact and the interplay of general giftedness (G) and excellence in school mathematics (EM) on students' mathematical performance associated with translations from graphical to symbolic representations of functions as reflected in cortical electrical activity (by means of ERP - Event Related Potentials - methodology)....

In this chapter, I analyze multiple solution tasks (MSTs) and mathematical investigations (MIs) and the interplay between them. I argue that MSTs and MIs are effective instructional tools for balancing the level of mathematical challenge in the mathematics classroom and, thus, for realizing students’ mathematical potential at different levels. Addi...

This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analy...

This paper presents an original model for evaluation of mathematical creativity. I describe different stages of the model's development and justify critical decisions taken throughout, based on the analysis of the model's implementation. The model incorporates an integrative theoretical frame-work that was developed based on works devoted to both g...

In this paper we suggest that instruments of neuro-cognitive research enable the evaluation of giftedness in mathematics. We start with a literature review on the related topics presented so as to situate our suggestions within the existing research on giftedness and excellence in mathematics. This literature review allows us later to discuss our f...

This paper presents part of a multidimensional examination of mathematical giftedness. The present study examined the memory mechanisms associated with general giftedness (G) and excellence in mathematics (E) in four groups of 10th -12th grade students (16-18 years old) varying in levels of G and E. The participants first underwent the Raven test f...

This study explores the effects of the presence of external representations of a mathematical object (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric figures. Perfor...

This paper presents part of a multidimensional examination of mathematical giftedness. The present study examined the memory mechanisms associated with general giftedness (G) and excellence in mathematics (E) in four groups of 10th -12th grade students (16-18 years old) varying in levels of G and E. The participants first underwent the Raven test f...

A familiar relationship—the derivative of the area of a circle equals its circumference—is extended to other shapes and solids.

The survey described in this paper was developed in order to gain an understanding of culturally-based aspects of creativity associated with secondary school mathematics across six participating countries. All participating countries acknowledge the importance of creativity in mathematics, yet the data show that they take very different approaches...

We explored transformations that teachers made to modify geometry proof problems into investigation problems and analyzed how these transformations differ in teachers who use a dynamic geometry environment (DGE) in their classes and those who do not. We devised a framework for the analysis of problem transformations and types of teacher generated p...

The study considers mathematical problem solving to be at the heart of mathematics teaching and learning, while mathematical challenge is a core element of any educational process. The study design addresses the complexity of teachers' knowledge. It is aimed at exploring the development of teachers' mathematical and pedagogical conceptions associat...

This paper draws connections between studies in general creativity and studies in mathematics education. Through analysis of the state of the art in the research in creativity as associated with mathematics education we review manuscripts included in this special issue. We consider definitions of creativity and the approaches to studying creativity...

Due to uncertainty regarding the relationship between mathematical creativity, mathematical expertise and general giftedness, we have conducted a large-scale study that explores the relationship between mathematical creativity and mathematical ability. We distinguish between relative and absolute creativity in order to address personal creativity a...

The article demonstrates that multiple solution tasks (MSTs) in the context of geometry can serve as a research instrument for evaluating geometry knowledge and creativity. Geometry knowledge is evaluated based on the correctness and connectedness of solutions, whereas creativity is evaluated based on a combination of fluency, flexibility, and orig...