Proceedings of the 11th International Conference on Mathematical Creativity and Giftedness (MCG 11)
22.08.2019 – 24.08.2019, Universität Hamburg, Germany
Band 5 der Reihe Conference Proceedings in Mathematics Education
Münster 2019, ca. 390 S., 17 cm x 24 cm
Print: ISBN 978-3-95987-131-0, 38,90 €
Ebook: ISBN 978-3-95987-132-7, 35,90 €
https://doi.org/10.37626/GA9783959871327.0
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Documentation of Review-Process
- What – What is being reviewed? All papers in the book
- Who – Who conducts the peer review? 2 external peer reviewers
- How – What is the level of anonymity? double anonymised
- When – At what stage is the peer review being conducted? Pre-publication
- Peer review is overseen by: member of the editorial board of the edited book
This volume contains the papers presented at the 11th international Conference on Mathematical Creativity and Giftedness (MCG 11) Including the Highly Gifted and Creative Students – Current Ideas and Future Directions, held from 22.08.2019 – 24.08.2019 in Hamburg.
Fritzlar, Torsten; Nolte, Marianne: Research on Mathematical giftedness in Germany – Looking back and ahead. pp 8 – 20
Beginning with William Stern, giftedness research has a long tradition in Germany, whereby in the times of German Partition the developments in East and West were quite different. In addition to psychological studies, didactic research on mathematical giftedness has also been on the rise for about 40 years. The lecture presents important developments and results, especially of mathematics education research in Germany, in their interdisciplinary and pedagogical-practical references. On this basis, further research questions for possible future projects will be put up for discussion.
Key words: mathematical giftedness, developing expertise.
Mellroth, Elisabet; Margrain, Valerie: Teaching highly able Learners in diverse Classrooms: Pedagogical possibilities through collaboration. pp 21 – 31
We argue in this paper that, with appropriate support, teaching of highly able learner can occur in diverse classrooms. We draw on a constructivist theory of learning and a differentiation paradigm (Dai & Chen, 2013). The claim that teachers can orchestrate teaching for highly able students in diverse classrooms is considered with evidence of our own and other data, warrant, backing, qualifier and rebuttal. Results from many studies have given knowledge of learning needs of mathematically highly able learners as well as of successful teaching to meet a diversity of learners. Drawing on our research, and work with school development, we share ideas about possibilities for teachers to support learning for all students, that is, including the highly able, within a diverse classroom. In particular, we advocate the possibilities from professional collaboration and our practice examples illustrate this claim.
Keywords: Gifted education, Differentiated instruction, collaboration
Sriraman, Bharath: Uncertainty as a Catalyst for Mathematical creativity. pp 32 – 51
In mathematics, uncertainty in the form of constraints takes on different forms. Constraints in problems in different domains within mathematics have played a unique role in its development. Paradoxically, constraints are both restrictive as well as liberating. For instance a result that is very generally stated and difficult to prove can be tackled by imposing constraints on the premises, i.e., restricting its scope. Hardy and Wright once noted that the proof or disproof of the twin-prime conjecture“ is at present beyond the resources of mathematics.“ It remains unproven even today. On the other hand, when an impasse is reached on a problem, it is sometimes overcome with the insight of inventing new tools to go beyond the (perceived) constraints of the problem. In this process of liberating the problem from the shackles of uncertainty, a different form of Uncertainty arises in the acceptance/un-acceptance of these „new tools“ or „new methods“ by the field. Both mathematicians (in the history of the subject) and students of mathematics have experienced the frustration when the field or the teacher respectively, do not accept the methods used to overcome the constraints. This dynamic view of insight overcoming constraints and/or insight imposing constraints is also ubiquitous with many heuristic devices used by mathematicians to solve problems. In this plenary, we will survey the role of Uncertainty in the dynamic of insight/constraints as catalysts for creativity. Relevant theories of creativity are used to scaffold the contents of the lecture.
Key words: Arithmetic; “Big C” creativity; Continued Fractions; Euler; Infinite Series; Lambert; Logarithm; Mathematical creativity; Napier; Uncertainty
Brandl, Matthias; Szabo, Attila: Overexcit Abilities, Iconoclasm and Mathematical creativity & giftedness. pp 53 – 58
There are several theoretical psychological concepts in the realm of research on (mathematical) creativity and giftedness, e.g. originality, non-conformism, iconoclasm, overexcitability and high sensitivity. By connecting these aspects to one another we show some concept-immanent interdependencies and congruities. Applying those to the specific area of mathematics we identify a natural relation of the mentioned concepts to the character of performing and dealing with mathematics. Additionally, we derive some consequences for classroom teaching.
Key words: overexcitability, iconoclasm, originality, high sensitivity, creativity, giftedness, mathematics.
Joklitschke, Julia; Baumanns, Lukas; Rott, Benjamin: The Intesection of Problem posing and creativity: A Review. pp 59 – 67
In this article, we take an in-depth look at research on the intersection of problem posing and creativity in order to present its current state of research in a systematic review. A full search in top journals from mathematics education and the Web of Science revealed only 15 articles from different genres, of which 11 were included in the analysis. Those articles were sorted into two clusters, depending on whether the articles focus on the identification or the fostering of creativity.
Key words: Problem Posing, Creativity, Review
Marumo, Jack Mathoga: Providing für gifted Learners in the regular mathematics Classroom: Needs and the way foward. pp 68 – 72
In line with South Africa’s constitution, the National Policy on Education (White Paper 6) clearly states for inclusive education to be a reality, the educational system should accommodate the learning needs of diverse learners, including those that are gifted. Nevertheless, more than a decade following the introduction of Education White Paper 6, most learners with giftedness who attend mathematics in regular classrooms are still having learning problems. There is no consensus regarding who should and should not be classified as a gifted learner in South Africa. The difference in opinion causes confusion in terms of identifying and supporting this group of learners. This paper examines the types of gifted learners and their needs. It further gives recommendations on how to meet the needs of different profiles of gifted learners.
Key words: gifted learners, mathematics, inclusive education, profiles of the gifted and talented.
Mhlolo, Michael Kainose: Setting the Ceiling too low für mathematically gifted Students in South African Schools. pp 73 – 78
In inclusive classrooms empirical evidence shows that half of gifted and talented students do not perform to their best abilities because the assessment tasks were insufficiently difficult to measure students’ true ability or knowledge. This theoretical paper uses the concept of frame to analyse the different aspects of the South African education system that contribute to setting a low ceiling for mathematically gifted students. This is important because setting low ceilings tends to promote gifted underachievement as tasks are too easy and do not encourage challenge or sustained effort. Although many frames could be used to analyse the South African education system, in this paper I argue that (a) the way inclusive education is conceptualised and implemented, (b) the way the curriculum is structured, (c) the level at which the pass mark is pegged as well as (d) the teacher competencies can all lower the ceiling for gifted students thereby inhibiting the maximization of their potential.
Key words: gifted students, underachievement, inclusive education, frame, low-ceiling
Rodríguez-Salazar, Luis Mauricio; Sánchez, Guillermo S. Tovar: What is Epistemology of the Imagination? Theory – Epistemological Bases to Mathematical Reasoning. pp 79 – 85
Given the need to design new strategies for the development of mathematical reasoning to overcome the mathematical reasoning and creativity inhibition, the epistemology of the imagination is presented as a current option that consider the symbolic-imaginative reasoning of the cognitive triad as a fundamental concept. Through the psycho-sociogenic method, the work presents some theoretical and epistemological bases that could help teachers in the development of new learning strategies.
Key words: Epistemology of the imagination, mathematical reasoning, symbolic-imaginative reasoning, psycho-sociogenic method, cognitive triad.
Applebaum, Mark; Leikin, Roza: Girls´Performance in the Kangaroo Contest. pp 87 – 94
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 instance, only insignificant differences were found in Grades 3- 4 for all difficulty levels. Furthermore, on some tasks, the girls‘ performance was better than the boys‘. In this respect, continuous investigation is needed to examine possible factors that make it happen. Furthermore, qualitative data could be collected and analyzed about young students’ thinking when solving different tasks to uncover other possible hidden factors that influence mathematical performance by girls in Kangaroo contest.
Key words: gender, spatial ability, mathematics performance, competitions
Assmus, Daniela; Benölken, Ralf: What do Student Teachers belief about mathematical giftedness? First Insights of an exploratory study. pp 95 – 102
What do teachers or student teachers belief about certain pedagogical and didactic, subject or curricula related facts? Or what might characterize their individual knowledge? Both beliefs and knowledge guide teachers as to their acting in classrooms, for example, regarding the support of different groups of children such as mathematically gifted. This is why professional knowledge bridging objective and individual constructed facets is considered a fundamental base for both teachers’ and student teachers’ training and education. The article presents first insights of an exploratory study focusing on German student teachers’ beliefs about mathematical giftedness.
Key words: Professional knowledge; beliefs; mathematical giftedness.
Aljarrah, Ayman; Towers, Jo: Discerning two creative Acts: Expanding possibilities and divergent Thinking. pp 103 – 108
This study is part of a broader research study exploring collective creative acts in elementary mathematics learning environments. In this paper we use two metaphors—expanding possibilities and divergent thinking—to describe two types of learning acts of a group of sixth grade students while they are working on a mathematical task. We then use examples from the group’s acts to discern some similarities/differences between the two metaphors. We acknowledge that students’ acts under both metaphors are important and desired learning acts that should be promoted and sustained in mathematics learning environments. However, teachers’ awareness of such discernments would promote their abilities and professionality in being alert and responsive in their interventions within the collective to further the evolving structure of mathematical thinking and understanding.
Key words: Creativity, Collective Creativity, Expanding Possibilities, Divergent Thinking.
Benölken, Ralf; Käpnick, Friedhelm; Auhagen, Wiebke; Schreiber, Lea: „Lemas“ – A joint Initiative of Germany´s federal government and Germany´s federal states to foster high-achieving and potentially gifted Pupils. pp 109 – 116
In contrast to other countries, the proportion of high-achieving pupils in international comparative studies in Germany is relatively low. Therefore, Germany’s federal government and Germany’s federal states have launched an initiative in which an interdisciplinary network of scientists together with schools develops guiding principles and adaptive concepts to support achievement. This article gives an initial insight into the structure, aims and projects of the initiative, in particular into a subproject on mathematics.
Key words: support program; high-achieving pupils; capable pupils; gifted pupils.
Cakir, Asli; Akkoc, Hatice: Socio-mathematical Norms related to problem Posing in a gifted Classroom. pp 117 – 123
As a response to the calls for investigating norms and problem posing in gifted classrooms, this study aims to explore socio-mathematical norms related to problem posing in the micro-culture of a mathematics classroom of gifted students. A case study was conducted in a mathematics classroom with twelve students in a secondary school for gifted students. The data collection tools include an observation form, semi-structured interviews, and teacher’s and students‘ notes. The main source of data consists of videos of twelve mathematics lessons. We observed three socio-mathematical norms related to problem posing (re-formulation of problems, generating new problems and sufficiency of the information in the problem) which were not reported in the literature. The paper discusses the implications of SMNs concerning problem posing and giftedness in mathematics.
Key words: Problem posing; socio-mathematical norms; mathematically giftedness; mathematically talented students
De Carvalho, Alexandre T.; Gontijo, Cleyton H.; Fonseca, Mateus G.: Collective Creativity in mathematics: possible Scenarios for shared mathematical Creativity. pp 124 – 129
Our purpose was to analyze the nature of shared creativity in mathematics in three different scenarios: working individually, working in groups without any mediation and working in groups with mediation of power in which the Creative Sharing Methodology was used. We selected 24 students from the 5th graders from a public school in the Brazilian, capital of Brazil. These students responded a test of creativity in mathematics composed of three versions used in each research scenario. It was observed that, in working individually, the teams presented fewer solutions, less varied and common ideas. Already in the collective work in which there was no intervention of the teacher, it was observed an improvement of performance. However, in the collective work with mediation of power, the teams produced less ideas, but with a greater level of originality. We conclude that shared creativity becomes more qualitative when power asymmetry is controlled.
Key words: Mathematics Education. Shared Mathematical Creativity. Creative and Critical Thinking. Mathematical Motivation.
Cilli-Turner, Emily; Savic, Milos; El Turkey, Houssein; Karakok, Gulden: An initial Investigation into teacher Actions that specifically foster mathematical Creativity. pp 130 – 135
While mathematicians and mathematics educators agree that students should be exposed to the creativity inherent in mathematics, there still is a need for further research showing how this can be done at the tertiary level mathematics. This report uses empirical evidence in conjunction with Sriraman’s Five Principles for maximizing creativity framework to explicate teaching practices that can foster mathematical creativity in the classroom. The report provides a practical guide for mathematics teachers who would like to value and nurture creative mindsets in their students.
Key words: teaching practices, mathematical creativity, tertiary-level
Durak, Tugay; Tutak, Fatma Aslan: Comparison of gifted and mainstream 9th grade Students´s statistical reasoning Types. pp 136 – 143
There have been studies investigating gifted students thinking skills while they attempt to solve a challenging problem (Leikin, 2011) but only a few studies specifically focused on gifted students’ statistical reasoning abilities. Garfield and Chance (2000) defined statistical reasoning as “the way people reason with statistical ideas and make sense of statistical information” (p.101). In this study, statistical reasoning types of 9th grade students attending school for gifted (n1=49) or school for mainstream education (n2=42) were compared. Turkish version of Statistical Reasoning Assessment (Karatoprak, 2014) instrument was used to investigate whether there are any differences between these two groups of students’ statistical reasoning types. Descriptive results of 8 subscales of the test and results of Mann-Whitney-U Test for these subscales are reported.
Keywords: gifted, statistics, statistical reasoning, mathematics.
Fonseca, Mateus G.; Gontijo, Cleyton H.; Zanetti, Matheus D. T.; De Carvalho, Alexandre T.: Improving mathematical Motivation from mathematical creativity Workshops. pp 144 – 149
This article presents an analysis about the improving mathematical motivation from mathematical creativity workshops. The research occurred with a group of fifty students of the last year of the Brazilian High School, in a public school of Brasília / DF – Brazil. The students were separated in two groups: control and experimental. The first one took a set of seven mathematical traditional classes (control), while the second one, participated of seven mathematical creativity workshops (experimental). A Mathematical Motivation Scale was used before and after this period. By results, it was verified a significant increase in the level of mathematical motivation of the experimental group.
Key words: Mathematics Education. Mathematical Creativity. Creative and Critical Thinking. Mathematical Motivation.
Freiman, Viktor; Robichaud, Xavier: Fostering young children´s creative Minds: Kindergarten Kids explore school-based stem Lab. pp 150 – 157
The paper presents a study of kindergarten students building bridges using LEGO blocks in a school-based STEM lab. From the theoretical perspective, the study is grounded in the ideas of providing young children with challenging and complex tasks which would enhance fostering their creativity and ingenuity. The initial results make explicit how technology-rich environment stimulates students’ interest in working on constructions using their imagination, creative potential, in collaboration with peers, while being perseverant facing obstacles and dealing with challenges.
Key words: Makerspaces, STEM, complex problem-solving, design, creativity, ingenuity
Gontijo, Cleyton Hércules; Zanetti, Matheus Delaine Teixeira; Fonseca, Mateus Gianni: Creative and critical Thinking in Mathematics: A Workshop for Teachers. pp 158 – 162
Increasingly critical and creative thinking is advocated on the international stage as a necessary capacity for 21st century education. In Brazil, this subject is still something new, especially for the primary education teachers. In order to contribute to the stimulation of critical and creative thinking in mathematics of students between 6 and 10 years old, was organized a workshop for teachers of these school years. The aim of this study was to proceed with a brief study of the validation of this workshop with a group of 27 teachers of primary education. As a result, the unanimity of positive feedback about the workshop and the reports collected from the sample show that it has acceptance and potential for continuing teacher training.
Key words: Mathematics Education. Mathematical Creativity. Creative and Critical Thinking in Mathematics.
Hagelgans, Heike: „But Painting is more fun“ on intervention Possibilities for Underachievers in mathematics Education. pp 163 – 170
Among the group of gifted children there is also the group of underachievers. Underachievement is conceptualized as a considerable discrepancy between high potential and poor school attainment. Underachievement is quite heterogeneous in its appearance and therefore requires individual support strategies in the classroom. In this article, concrete manifestations of underachievement and possible intervention strategies in regular mathematics teaching are presented based on extracts from a case study. This presented case study is part of a larger research project on underachievement in a ninth-grade mathematics class.
Key words: giftedness, expertise, underachievement, interventions in mathematics lesson
Jablonski, Simone; Ludwig, Matthias: „Oh, I do not like that when you have to justify something“ – Difficulties in formulating Arguments as a Basis for the support of mathematical Giftedness. pp 171 – 177
Arguments and argumentations are of high relevance in mathematics education, on the one hand as a learning goal and basis for proofing, on the other hand as a necessary basis for the deeper understanding of number characteristics and relations. The paper focuses on a qualitative analysis of the difficulties in formulating argumentation products of mathematically gifted primary students (age 9-11). With special regard to a structural analysis by means of the Toulmin layout, different categories of argumentation difficulties are specified. The difficulties are of contentual, methodological and motivational nature. After pointing out the difficulties, the results are taken into account for the support of mathematical giftedness with special regard to the support of argumentative abilities in order to exploit the full potential of excellent argumentative abilities.
Key words: giftedness, argument, Toulmin, primary school
Nair, Vinay; Ramasubramanian, Hari: Exploration of unknown: A different approach to foster mathematical Creativity. pp 178 – 184
In India, mathematical talent is nurtured by training students who have apriori shown an aptitude for problem solving techniques in mathematics typically under a time constraint. Many mathematicians argue that mere problem solving does not foster mathematical creativity. This article is about an ongoing experiment spread over 35 weeks (of which 20 weeks have elapsed) involving a group of 40 school students who were given opportunities to foster their creativity by exploring the unknown. The focus was to provide an environment and opportunity to discover mathematical concepts without teaching them formally and observe its role in nurturing creativity. Thus, these children learnt by constructing the mathematical concepts by themselves.
Key words: Creativity, Pattern-observation, Exploration, Critical Thinking, Inquiry, Problem-solving.
Jóelsdóttir, Lóa Björk; Errebo-Hansen, Dorthe: Selection criteria for Students in a danish mathematics talent Program. pp 185 – 190
In this paper, we present the results of an interview study of Danish mathematics teachers’ selection criteria regarding their selection of their most talented math students to participate in a two-year talent program, Talent2 for Grade 5 students. Talent2is a development program for math teachers and talented math students, which focus on developing students’ creativity and Mathematical Mindset. The results indicate that the math teachers have selected their talented students on the basis of either test results or the teachers’ instincts. The teachers have not thought particularly about creativity as relevant criterion. The interview study also reveals that the talent students can be categorized in two groups: students that are generally result-orientated and achieving above average in all school subjects and students with higher level of creativity.
Key words: creativity, mathematics, middle school, talent, selection criteria
Karatas, Fridevs Iclal; Bostan, Mine Isiksal: Mathematically gifted Students´ Reflections on using History of mathematics in mathematics Classroom. pp 191 – 196
The aim of this study is to examine the reflections of mathematically gifted students on using history of mathematics in mathematics classroom. This case study examined 12 fifth grade mathematically gifted students’ opinions about using history of mathematics in mathematics classroom. Students’ opinions were taken with video evaluation forms after they had watched videos on biographies of mathematicians. The findings of the study revealed that mathematically gifted students reflect their opinions about using history of mathematics in a positive way. Students opinions were categorized into three sub-themes: learning more information, demanding further research and motivation to achieve.
Key words: Differentiated instruction, history of mathematics, mathematically gifted students, using videos.
Lee, Yujin; Capraro, Robert M.; Capraro, Mary M.; Vela, Katherine; Bevan, Danielle; Caldwell, Cassidy: Students´ Conceptions of mathematical creative Thinking and critical Thinking in steam PBL activities. pp 197 – 201
Mathematical creative thinking and critical thinking are essential to be successful in STEM-related post-secondary academic and career pathways. Therefore, the aim of the present study was to investigate the development of students’ conceptions of mathematical creative thinking and critical thinking through STEM PBL activities. A group of 39 students in grades 7-12 participated in the intervention (STEM PBL activities) and completed pre- and post-surveys concerning their conceptions of mathematical creative thinking and critical thinking. Results showed that students’ conceptions about their mathematical creative thinking and critical thinking were statistically significantly higher after engaging in STEM PBL activities. The findings from the current study support the importance of STEM PBL for the development of students’ positive conceptions toward creative thinking and critical thinking in mathematics.
Key words: Creative thinking; Critical thinking; Science, Technology, Engineering, and Mathematics Project-Based Learning (STEM PBL)
Simensen, Anita M.; Olsen, Mirjam H.: Tasks that enhance creative Reasoning: Supporting gifted Pupils in inclusive education system. pp 202 – 208
This paper represents the first stage of a wider study on how to support mathematically gifted pupils’ creative reasoning in inclusive education systems. The study focuses on gifted pupils in Norway, which has a one-track education system where pupils are organised in heterogeneous (mixed-ability) classes with few opportunities to meet other gifted pupils. In the present study, we observed gifted pupils’ reasoning when working collaboratively in both heterogeneous and homogeneous learning environments. The purpose of this paper is to discuss the creative potential of one of the tasks used in our study. We present preliminary findings from the study, focusing on pupils’ written products from collaborative work in homogeneous groups. The analysis identified a variety of methods used to solve the tasks, and we used the pupils’ written products from work on one of the tasks to exemplify this variety.
Key words: gifted pupils, learning opportunities, rich tasks, inclusive education
Van Wyk, Motshidisi Gertrude; Mhlolo, Michael Kainose: Examining primary school teacher-support towards mathematically gifted learners in South Africa. pp 209 – 214
This paper reports on teacher preparedness in supporting mathematically gifted learners in mainstream classrooms. Empirical studies in South Africa show that such classes seem not to present conducive learning environments for gifted learners. It is against these observation that this study aims at exploring the support given by foundation phase teachers to mathematically gifted learners in Motheo and Xhariep districts’ primary schools of Free State province. A Hundred and five teachers completed questionnaires and their principals were interviewed from twenty selected schools. The results show that teachers lack training which disadvantages the gifted learners to perform to their full potential. This paper concludes by recommending the continuing emphasis on teacher training in gifted education at higher institutions as well as in-service training at school levels.
Key words: mathematically gifted, teacher preparedness, mainstream, foundation phase, gifted education
Vela, Katherine; Bevan, Danielle; Caldwell, Cassidy; Capraro, Robert M.; Capraro, Mary Margaret; Lee Yujin: Stem project-based learning activities: Opportunities to engage in creative mathematically Thinking? pp 215 – 221
Prior research has shown the use of science, technology, engineering, and mathematics (STEM) project-based learning (PBL) activities in a mathematics classroom increases students’ interest in mathematics and fosters students’ creative thinking abilities. During the summer of 2018, one- and two-week residential STEM camps were held at a university in the southwest region of the United States for 7th – 12th grade students (n=49). Students took STEM PBL courses involving hands-on learning within real-life scenarios. A quasi-experimental design was used to determine how well students’ beliefs about mathematics, interest in applying mathematics, and beliefs about creativity in STEM fields could predict their interest in applying creativity in STEM fields. Results indicated that the variables were strong predictors of students’ interest in applying creativity in STEM fields and ultimately pursuing a career in STEM.
Key words: Beliefs, Creativity, Mathematics, STEM, Mathematics Application, Creative Thinking
Voica, Cristian; Singer, Florence Mihaela: Analogical transfer and cognitive framing in prospective teachers´ problem posing activities. pp 222 – 228
The ability of making analogical transfers is an important skill in mathematics and beyond. The paper reports on the capacity of a group of prospective mathematics teachers to pose analogical problems within a project. It seems that most prospective teachers face difficulties in transferring deeper structural elements of the initial problem to their newly posed problems. The difficulties are related to an obvious lack of experience of communicating mathematical tasks, but their comments reveal also a superficial level of mathematical competence, level made better visible through this type of project activity than through a classical problem-solving task.
Key words: Problem posing, analogical transfer, cognitive framing, teacher training.
Westerhout, Eline; Van Driessel, Isabelle; Van der Helm, Björn: Problem solving in secondary Education: A qualitative Analysis of the differences between highly and mildly gifted student. pp 229 – 235
Recent PISA studies revealed that students’ problem-solving skills are in many countries not as well developed as their mathematical skills. Since highly gifted students are considered better problem solvers than mildly gifted students, the question is in which ways they behave differently. To answer this question, we analysed the problem-solving process of secondary school students at preparatory university level (≥110) using an existing problem-solving model. Both junior (=26, grade 7-9) and senior (=34, grade 10-12) students make most of their errors in the analysis and verification phases. While the highly gifted students (=5, ≥130) encounter fewer problems in the analysis and verification phases, they make more sloppy mistakes in the planning and implementation phases. The findings from this study suggest that highly gifted students are indeed better at problem solving but could benefit from paying more attention to details and mathematical notation.
Key words: giftedness, problem solving, Mathematics, secondary education.
Yazgan-Sag, Gönül: Prospective teachers´ views on mathematical giftedness and on teachers of mathematically gifted students. pp 236 – 241
The aim of the study is to examine prospective secondary mathematics teachers’ views of mathematical giftedness with respect to characteristics of the teachers of the mathematically gifted students. The data were collected through a focus group interview with seven prospective secondary mathematics teachers. Then, the data were analysed by using descriptive analysis. The participants associated mathematical giftedness with genetic factors, social environment, and effort. It is concluded that the participants’ views on mathematically gifted students are limited to their personal experiences with such students and their responses were seen to be under the influence of related films and books they have encountered. They also defined the teachers of mathematically gifted students with regard to personal and professional characteristics.
Key words: Mathematically gifted students, secondary level, prospective mathematics teachers, teacher education
Zioga, Marianthi; Desli, Despina: Improving mathematical creativity in the classroom: A case study of a fourth-grade teacher. pp 242 – 248
This study explores a fourth-grade teacher’s conceptions about mathematical creativity and their development after his attendance of a program concerning the promotion of creativity in mathematics teaching. The teacher completed one questionnaire before and one after the program whereby he was asked to choose a mathematical task that he considers suitable for promoting creativity and to explain the reasons for choosing it. Interviews were also conducted, before and after the program, to highlight his envision of creativity in mathematics and the impact the program had on him. Findings revealed enrichment of his conceptions after having attended the program, mainly with regards to generating and using original tasks as well as to the ways he deals with mathematics teaching for creativity.
Key words: mathematical creativity, primary school teachers, training program
Bulina, Elina; Cibulis, Andrejs: Magic polygons and its usage in work with gifted pupils. pp 250 – 256
One of the gifted, mathematically promising pupils’ needs is to work with recreational, challenging tasks. Very rarely, mathematics teachers themselves create and work with such tasks, so it is useful to encourage them and introduce them to new topics. Magic polygons are little known but very suitable in work with gifted pupils. Such polygons can also be a good source for problems in mathematics Olympiads, competitions, and for pupils‘ and students’ research work. This article provides both simple introductory tasks as well as difficult and unsolved mathematical problems.
Key words: Golygon, magic polygon, polyomino, polyiamond, perfect polygon, tiling.
Heuer, Karl; Sarikaya, Deniz: Group theory via symmetries for enrichment classes for gifted Youth. pp 257 – 263
This paper exemplifies how we design open problem fields for mathematically gifted children. We elaborate the theoretical embedding of this approach and deliver general guidelines before introducing the problem field of ‘group theory of ornaments’. This is on one hand a possible working direction for students who entered the open problem field of designing ornaments, tilings or Escher’s Symmetriezeichnungen, and on the other a motivation to study group theory from the perspective of symmetries. In order to do so, we motivate the abstract definition of groups via the symmetries of regular n-gons and more complex frieze patterns. We then offer an overview of possible inner mathematical follow-up topics and for detours embedding the work into culture and art.
Key words: symmetries, group theory, ornaments, open problem fields, low floor high ceiling, mathematically gifted children, enrichment classes
Jahanshahi, M.; Aliev, N.: Extension and development of different non-newtonian calculus in order to solve different differential and difference Equations based on mathematical education approaches. pp 264 – 269
In this paper, at first we review Newtonian calculus and some Non- Newtonian calculus’s which introduced and extended in recent years. These creations and generalizations have been done by several mathematicians to reach to different goals. Then we centralize on mathematics educational approaches. After that we give some ideas and methods to obtain invariant functions with respect to related derivative and calculus. We will use these invariant functions to solve several and different differential and difference equations
Key words: Invariant function. multiplicative derivative, Discrete multiplicative derivativ
Juhász, Péter; Katona, Dániel: Pósa method: Talent nurturing in weekend math camps. pp 270 – 276
Lajos Pósa has been organizing weekend mathematics camps for highly gifted students to foster their development with a special method since 1988. During these 30 years, he and his disciples led more than 350 camps for more than 1500 students. Students’ work dominantly takes the form of a special team work, and is based on a five-year-long coherent curriculum organized around problem threads, which form a complex web. These threads run parallel, in ‘harmony’, supplementing and assisting each other’s development. Effectiveness of the camps is reflected in the fact that in the past 25 years, almost all members of the Hungarian IMO teams were participated in these camps before.
Key words: mathematically gifted learners, learner autonomy, discovery learning, inquiry based mathematics education, connected task-design, problem thread, teamwork
Kalobo, Lukanda; Mhlolo, Michael Kainose: Mathematics education pre-service teachers awareness of gifted students´characteristics. pp 277 – 283
Gifted students present an array of characteristics in mathematics. However, in South Africa most of the tertiary institutions do not cater for gifted education to provide opportunities for pre-service teachers to be abreast of the mathematically gifted students’ characteristics. This study investigates pre-service teachers’ awareness of mathematically gifted students’ characteristics. This is done to strengthen the claim to introduce a module on gifted education in Mathematics Education programme at the Central University of Technology (CUT), in South Africa. The study followed a qualitative and quantitative approach where sixty-six pre-service teachers’ responses were collected and analysed. The results revealed that pre-service teachers’ have limited acquaintance regarding the characteristics of gifted students in mathematics. Thus, the inclusion of modules on gifted education in the mathematics training program is needed.
Key Words: Gifted education; Mathematically gifted; pre-service teachers, gifted students’ characteristics.
Kasuba, Romualdas; Mazétis, Edmundas: Simple but useful tasks with geometrical content and creative flavour. pp 284 – 289
Nowadays nobody would argue the usefulness of geometrical content. Similarly nobody will say something against the thesis that geometry has always been a difficult art. It is also taken for granted that geometry has its own intuition which makes geometry somehow more complicated to understanding as well as to dealing with. Still the world of geometrical ideas and problems is so challenging that it is worth to start even from really elementary things that are understandable for all.
Key words: Geometrical intuition, challenging tasks, accessibility of problems, attractive formulations, development of concepts.
Koehler, Peter: Visualization of the first steps of number theory for elementary school children – a pythagorean approach. pp 290 – 295
Influenced by the method of the Pythagoreans, who used pebbles to visualize numbers I have developed an approach to elementary math teaching where the students use coloured interlinking blocks and follow a few simple rules to visualize numbers, look for patterns, shapes and sequences, make their own mathematical creations and develop a sense of the more general principles of mathematics. Over the many years that I have been using the method in my math enrichment sessions at the Nueva School, a school for gifted children, I have found that this approach stimulates interest and enthusiasm for math, is a great motivator and can spark mathematical creativity, originality and a joy in the subject.
Keywords: Discovery Learning, Hands-on, Visualization of Number Theory, Figurate numbers.
Phelps, Conny: Incubating mathematical creativity through a molecular gastronomy 101 Saturday enrichment camp. pp 296 – 303
This project report addresses creative applied mathematics used outside the regular education classroom. It examines universal themes and standards of mathematics through the pedagogical framework of the Torrance Incubation Model of Creative Teaching and Learning (TIM) during a university-sponsored Saturday Enrichment Camp for gifted and talented learners. Enrichment camp instructors included experienced gifted facilitators, parents of gifted children, and grown-up gifted children who volunteered their time, energy, culinary expertise, and specialized equipment to explore three modern culinary techniques in a three-hour molecular gastronomy class for gifted and talented learners ages 8-14 years. The hands-on Saturday Enrichment Camp with real world application required participants to apply mathematics and technology to prepare a three-course meal using seasonal locally sourced farm food ingredients from a rural Midwestern community in the United States.
Key words: Creativity, giftedness, mathematics, enrichment, Torrance, molecular gastronomy
Speer, William R.: It´s not alwys simpler to use „make it simpler“. pp 304 – 310
This manuscript describes the problem-solving strategy of “Make It Simpler” in a context that goes beyond using more convenient numbers to exploring metaphor and content change. Challenging problems that do nt appear to have clear or readily identifiable pathways to solution are used to illustrate how “make it simpler” through the use of metaphor and/or context shifts can lead to increased discourse and deeper comprehension of underlying concepts.
Key words: Problem solving; connections; reasoning; quantitative literacy; metaphor; isomorphism
Rodríguez-Salazar, Luis Mauricio; Rosas-Colín, Carmen Patricia; Martínez-García, Ramsés Daniel: Pedagogy of imagination: epistomological foundations to develop mathematical thinking in preschool students. pp 311 – 318
The recent educational reform in Mexico at preschool and elementary level raises the need to propose pedagogical alternatives for the development of mathematical thinking in children in kindergarten, due to the great deficiencies that are detected in later educational levels. Our proposal is based on the theoretical assumptions of the epistemology of imagination, a post-Piagetian approach that explains imagination as a cognitive process in which practical thinking, imaginative symbolic reasoning, and formal reasoning are involved. In agreement with the human cognitive development proposed by Piaget, we propose as a pedagogical alternative to focus on detonating and fostering imaginative symbolic reasoning in preschool children, instead of mathematical thinking that has not yet been formed or instead of teaching mathematical concepts and procedures such as it was imposed by traditional education.
Key words: Epistemology of imagination, mathematical thinking, symbolic-imaginative reasoning, pedagogy of imagination, cognitive triad.
Veilande, Ingrida: Tasks on visual Patterns as the first stage of introducing algebra concepts. pp 319 – 326
The presented paper discusses specific questions of leading mathematical circle for lower grade students at the Correspondence School of Mathematics at the University of Latvia. The paper covers the aspects of solving tasks on visual patterns, the importance of which is recognized by many researchers. Such tasks should be one of the first foundational stages in acquiring the methodology for solving Olympic problems. The paper provides some sample problems and discusses students‘ solutions. Excerpts from teacher’s notes demonstrate active student participation in class where they learn from each other’s ideas.
Key words: Algebraic reasoning, mathematical circles, problem solving, tasks on visual patterns.
Watanabe, Shin: Creativity, technology and „out school“ interesting mathematics with technology during out school. pp 327 – 334
Mathematics should be a creative endeavor and enjoyable pastime throughout one’s life, not just during school and careers. This paper describes the basis for this and gives examples of interesting problems that can be investigated by anyone at home using readily available technology resources. Examples from a mathematics workshop for adults who are out of school and looking for interesting leisure pursuits are also given. The term “out school” is created to describe these activities.
Key words: creativity, mathematics, lifelong learning, technology
Capraro, Mary M.; Capraro, Robert M.; Vela, Katherine N.; Caldwell, Cassidy; Bevan, Danielle; Lee, Yujin: Mathematizing creative stem pbl activities. pp 336 – 339
Prior research has indicated that the use of science, technology, engineering, and mathematics (STEM) project-based learning (PBL) activities within the mathematics classroom can positively impact students’ interest in mathematics and foster their ability to identify the connection between mathematical content knowledge and creative thinking. The purpose of this workshop is to demonstrate how to implement STEM PBL strategies and activities. Discussions include the advantages of using this pedagogical strategy in mathematics classrooms. Participants explore the engineering design process through two mini-STEM PBL activities. At the end of the workshop, participants should feel confident in their ability to implement effective STEM PBL activities that foster mathematical creative thinking.
Heuer, Karl; Sarikaya, Deniz: Workshop: Variations in open problem fields as a tool for mathematical education: from basics to open questions in 90 minutes. pp 340 – 341
This workshop aims to give insight in how to design a research-oriented experience for mathematically gifted pupils in high school. In order to do so we reflect on the creative challenge to ask interesting questions and develop adequate notions for a fruitful research field. The example to which we mainly refer considers tilings of the plain. We offer materials for possibly five days, which only partially build up on each other, for a project week within a high school. We expect the participants of this workshop to work with our work sheets from a student’s and a teacher’s point of view. Furthermore, we want to discuss the general idea behind such material in total. As a key feature, this includes a text-driven approach for variations of mathematical problems.
Key words: Open problem fields, design of work sheets, tilings, Penrose tilings, text-driven variations, creativity, research-oriented experience
Koehler, Peter: An introductory workshop to the visualization oft he first steps of number theory for elementary school children – a pythagorean approach. pp 342 – 344
In this workshop we will create a mathematically and visually stimulating hands-on environment in which participants will discover and create polygonal numbers using interlinking coloured blocks. The approach is inspired by the teachings of Pythagoras and his followers, who used pebbles to turn number sequences into geometrical patterns. The interplay between numbers and geometry will give rise to questions and potential answers which can be transferable to the participants’ own students in a similar setting.
Key words: Discovery Learning, Hands-on, Odd and Even Numbers, Visualization of Number Theory
Sheffield, Linda Jensen: Shape up: Proven spatial activities for elementary students. pp 345 – 346
Spatial ability is a critical, but often overlooked component of success in several STEAM careers that has its beginnings at an early age. In this workshop, we will briefly explore research on the role that spatial thinking plays in educational and occupational STEAM innovation and expertise, including the fact that spatial reasoning is not hard-wired and can improve with practice. Participants will then engage in proven spatial investigations and activities from the National Science Foundation-funded Project M2: Mentoring Young Mathematicians for students in kindergarten through second grade and the US Department of Education Javits-funded Project M3: Mentoring Mathematical Minds for grades three through six students.
Key words: spatial ability, mathematics, geometry, exceptional promise, STEAM
Stender, Peter: Spirograph – A toy as a mathematical problem. pp 347 – 354
The Spirograph is a toy that consists of plastic rings with gear teeth and plastic gear wheels which allow to draw several kinds of hypotrochoid that first of all are just beautiful. From the mathematical point of view occur several questions that can be dealt with by gifted students on different levels – starting at grade six up to university. Selected aspects are shown here.
Key words: Spirograph, Problem Solving, Hypotrochoid .
Bahar, A. Kadir; Kanbir, Sinan: Tapping mathematical creativity through problem solving: Problem- matrix framework for teaching for creativity in the math classroom. pp 356 – 357
Much of academic content and teachers’ daily instruction practices in the math classroom could be considered as an aspect of problem solving (Bahar & Maker, 2016). To support students in developing as creative problem solvers, teachers need a systematic understanding of what a creativity and creative problem solving growth trajectory looks like in the math classroom. For these purposes, presenters offer a ‘Problem Matrix Framework’, which will guide mathematics teachers in cultivating creative potentials through production, fluency, originality, and detail in the solving of complex and realistic problems. Presenters will also share the findings of their study on how different types of problems tap mathematical creativity differently.
Key words: Mathematical creativity, problem solving, problem types, open-ended problem.
Krüger, Nina; Johannsen, Mieke; Smoydzin, Luca F.; Genzel, Henrik; Peritz, Marguerite I.; Meyer, Jakob; Fiedler, Sören; Daseking, Monika: Research within the framework of the hamburger model fort he promotion of particularly mathematically gifted children and adolescents. pp 358 – 365
This symposium is intended to provide insights into research concerning mathematical giftedness conducted within the William-Stern-Gesellschaft für Begabungsforschung und Begabtenförderung e.V. (Association for talent research and fostering; WSG e.V.). More specifically, results on gender differences in mathematical performance, Need for Cognition (NFC), Learning and Achievement motivation (definitions see below), as well as intelligence are presented. Moreover, this symposium aims at establishing new contacts as well as exchanging views with other organizations and programs that are concerned with talent research and fostering to enable a sustainable exchange.
Keywords: Mathematical creativity and giftedness; Formal and informal learning
Auhagen, Wiebke: Affects of mathematically gifted students related to revolving door models. pp 367 – 370
Bruhn, Svenja: Creative processes of first graders working on arithmetic open tasks. pp 371 – 373
Key words: Mathematical creativity, first graders, open tasks, arithmetic.
Fox, Ryan D.: Mathematical interactions with gifted adolescents. pp 374 – 377
Students who demonstrate interest in mathematics pose fascinating, and intellectually challenging, questions to each other and their teachers. As teachers and teacher educators, we make pedagogical decisions based on a quick recall of relevant content knowledge. Applying prior research and interdisciplinary literature, I reflect on my own experiences as a teacher of mathematically interested middle-grade students to determine how content knowledge and gifted education inform decisions to pursue intellectually stimulating conversations.
Key words: Middle Grades education, Specialized Content Knowledge, Statistics
Mellroth, Elisabet; Vinerean-Bernhoff, Mirela; Boström, Mattias; Liljekvist, Yvonne: Differentiated instruction using learning management systems in upper secondary school and university level – A research proposal. pp 378 – 380
There is a need to develop an infrastructure to support and maintain teaching that both challenge all students at their knowledge level and open up the possibility to applying mathematical knowledge in innovative and creative ways. Learning management systems (LMS) are widely used throughout the Swedish school system. However, recent studies shows that few teachers use this resource for teaching development, i.e., using LMS as an instrument to improve forthcoming lessons. In this poster a research proposal is outlined. The aim is to explore how LMS can be used as an instrument for differentiated instruction throughout the intertwined processes of planning, teaching, studying, follow up and assessment.
Keywords: Differentiated instruction, Learning Management System
Nolte, Marianne; Pamperien, Kirsten; Vorhölter, Katrin: Research and development tasks within the framework of the prima-projekt in Hamburg. pp 381 – 384
PriMa is a project aiming amongst others on the promoting of mathematical gifted pupils of elementary schools in Hamburg. Whilst working with the mathematical gifted pupils within the project questions regularly occur. These questions lead to research and development projects within PriMa. Current research projects are presented within this paper.
Key words: mathematical gifted children, research projects, development projects