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This paper describes changes in students’ geometrical knowledge and their creativity associated with implementation of Multiple Solution Tasks (MSTs) in school geometry courses. Three hundred and three students from 14 geometry classes participated in the study, of whom 229 students from 11 classes learned in an experimental environment that employed MSTs while the rest learned without any special intervention in the course of one school year. This longitudinal study compares the development of knowledge and creativity between the experimental and control groups as reflected in students’ written tests. Geometry knowledge was measured by the correctness and connectedness of the solutions presented. The criteria for creativity were: fluency, flexibility, and originality. The findings show that students’ connectedness as well as their fluency and flexibility benefited from implementation of MSTs. The study supports the idea that originality is a more internal characteristic than fluency and flexibility, and therefore more related with creativity and less dynamic. Nevertheless, the MSTs approach provides greater opportunity for potentially creative students to present their creative products than conventional learning environment. Cluster analysis of the experimental group identified three clusters that correspond to three levels of student performance, according to the five measured criteria in pre- and post–tests, and showed that, with the exception of originality, performance in all three clusters generally improved on the various criteria.

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... Since 1980, attention has been on investigating tasks that encourage the development of higher forms of thinking, conceptual understanding, a functional connection of different contents within or between different areas of mathematics, and the development of specific skills and strategies. Among these tasks, a special place is occupied by open tasks that have several solutions and can be solved in several different ways (Foong, 2002;Foster, 2013;Klavir & Hershkovitz, 2008;Levav-Waynberg & Leikin, 2012;Schoenfeld, 1992). Tasks that have multiple correct solutions are represented in research to a lesser extent than tasks with one solution that can be solved in several different ways. ...

... Different types of tasks can be effective both as a didactic and a research tool, especially open tasks. In addition, with open tasks, the process of solving is more important than the correctness of students' solutions because it provides a deeper insight into their knowledge and skills (Levav-Waynberg & Leikin, 2012). The multiple-solution task As the term 'solution' can have several different meanings (Polya, 1954), consequently the term 'multiple solution tasks' is used for different types of tasks: both for tasks with multiple solutions strategies (or methods) and for tasks with multiple correct solutions (true outcomes) or 'open-ended tasks' (Levav-Waynberg & Leikin, 2012;Schoenfeld, 1992;Sullivan et al., 2013). ...

... In addition, with open tasks, the process of solving is more important than the correctness of students' solutions because it provides a deeper insight into their knowledge and skills (Levav-Waynberg & Leikin, 2012). The multiple-solution task As the term 'solution' can have several different meanings (Polya, 1954), consequently the term 'multiple solution tasks' is used for different types of tasks: both for tasks with multiple solutions strategies (or methods) and for tasks with multiple correct solutions (true outcomes) or 'open-ended tasks' (Levav-Waynberg & Leikin, 2012;Schoenfeld, 1992;Sullivan et al., 2013). Therefore, before using a term, it is important to highlight what is meant by it. ...

Open-ended tasks in mathematics offer significant pedagogical value for teaching and learning due to their flexibility in solution approaches. Problems with multiple solutions, especially those whose solution variability is not initially evident, have attracted increased attention in mathematics education research and classrooms. This study delves into the abilities of 5th-grade elementary school students to tackle such problems. The objectives encompass evaluating their success in solving open-ended tasks, scrutinizing their solution methodologies, and pinpointing factors hindering their ability to identify all possible solutions. To achieve the study's objectives, a set of 12 content-specific open-ended tasks spanning various domains was developed. The empirical investigation involved 245 11-year-old students from diverse urban and suburban schools in Croatia. This paper provides a detailed analysis of student achievements and solution approaches for two tasks from the instrument. The findings reveal that students possess limited experience and awareness regarding the potential for multiple correct solutions to the tasks. Furthermore, they exhibit a restricted range of strategies and demonstrate limited conceptual understanding, which may impede or prevent them from successfully uncovering all viable solutions.

... Since 1980, attention has been on investigating tasks that encourage the development of higher forms of thinking, conceptual understanding, a functional connection of different contents within or between different areas of mathematics, and the development of specific skills and strategies. Among these tasks, a special place is occupied by open tasks that have several solutions and can be solved in several different ways (Foong, 2002;Foster, 2013;Klavir & Hershkovitz, 2008;Levav-Waynberg & Leikin, 2012;Schoenfeld, 1992). Tasks that have multiple correct solutions are represented in research to a lesser extent than tasks with one solution that can be solved in several different ways. ...

... Different types of tasks can be effective both as a didactic and a research tool, especially open tasks. In addition, with open tasks, the process of solving is more important than the correctness of students' solutions because it provides a deeper insight into their knowledge and skills (Levav-Waynberg & Leikin, 2012). The multiple-solution task As the term 'solution' can have several different meanings (Polya, 1954), consequently the term 'multiple solution tasks' is used for different types of tasks: both for tasks with multiple solutions strategies (or methods) and for tasks with multiple correct solutions (true outcomes) or 'open-ended tasks' (Levav-Waynberg & Leikin, 2012;Schoenfeld, 1992;Sullivan et al., 2013). ...

... In addition, with open tasks, the process of solving is more important than the correctness of students' solutions because it provides a deeper insight into their knowledge and skills (Levav-Waynberg & Leikin, 2012). The multiple-solution task As the term 'solution' can have several different meanings (Polya, 1954), consequently the term 'multiple solution tasks' is used for different types of tasks: both for tasks with multiple solutions strategies (or methods) and for tasks with multiple correct solutions (true outcomes) or 'open-ended tasks' (Levav-Waynberg & Leikin, 2012;Schoenfeld, 1992;Sullivan et al., 2013). Therefore, before using a term, it is important to highlight what is meant by it. ...

Open-ended tasks can usually be solved in different ways, so they have great didactic potential both for teaching and learning mathematics. Amongst open-ended tasks problems that have more than one solution, especially those problems for which it is not possible to know in advance whether they have more than one solution or how many solutions there are, have attracted more attention in the last decades both in mathematics education research and mathematics classrooms. This study aims to investigate 5th-grade elementary school students’ accomplishments in solving open-ended tasks problems that have more than one solution. The objectives are to determine the success of solving open-ended tasks, analyze solution methods, and identify what hinders students from successfully discovering all solutions. For that purpose, an instrument with 12 content-specific open-ended tasks from various fields has been developed. The empirical research included 245 11-year-olds from various urban and suburban schools in Croatia. This paper presents an in-depth analysis of students’ achievements and solution methods on two tasks from the instrument. The results implicate that students have limited experience and awareness that tasks could have more than one correct solution. In addition, the students use a small range of strategies and show limited conceptual knowledge, possibly hindering or preventing them from successfully discovering all correct solutions.

... The solution strategy categorisation presented by Albarracín et al. (2021), based on the productions of secondary school students, was extended for pre-service teachers . A categorisation of all possible solution strategies of these Fermi problems allows us to consider them as multiple solution tasks (Levav-Waynberg & Leikin, 2012), which makes it possible to monitor which solutions are suitable for a particular task and to measure whether the problem solvers know and use more than one solution strategy when they face a sequence of Fermi problems. ...

... Flexibility is an important mathematical skill; indeed, it is necessary for students to acquire the ability to adapt their solution strategies to the characteristics of the task or context (Heinze et al., 2009). In the problem-solving framework, studies on the flexible use of multiple solution strategies found it essential for building deep and connected knowledge (Levav-Waynberg & Leikin, 2012;Star & Rittle-Johnson, 2008). Most studies on the influence of the development of multiple solution strategies and their flexible use have focused on intra-mathematical tasks (Levav-Waynberg & Leikin, 2012;Star & Rittle-Johnson, 2008;Threlfall, 2002). ...

... In the problem-solving framework, studies on the flexible use of multiple solution strategies found it essential for building deep and connected knowledge (Levav-Waynberg & Leikin, 2012;Star & Rittle-Johnson, 2008). Most studies on the influence of the development of multiple solution strategies and their flexible use have focused on intra-mathematical tasks (Levav-Waynberg & Leikin, 2012;Star & Rittle-Johnson, 2008;Threlfall, 2002). Elia et al. (2009) developed a study about flexibility with primary school students who solved a sequence of three non-routine intra-mathematical problems. ...

Fermi problems are real-context estimation tasks that are suitable for introducing open-ended problems in primary school education. To ensure their effective introduction in the classroom, teachers must have adequate proficiency to deal with them. One of the key aspects of problem-solving proficiency is flexibility, but there are few studies on flexibility in solving real-context problems. This study, based on an analysis of the errors made by 224 prospective teachers when solving a Fermi problem sequence, establishes performance levels. In addition, we define levels of flexibility in using multiple solutions across the sequence, which allows us to address the main objective: to study the relationship between performance and flexibility. We found that there are significant relationships between flexibility levels and the number and severity of errors made. Encouraging flexibility in prospective teachers may be an efficient way to improve their performance in solving real-context problems.

... Aizikovitsh-Udi & Amit, 2011;Apino & Retnawati, 2017;Leikin, 2014;Schoevers et al., 2019;Tubb, et al., 2020); the multiple-solution tasks (MST for short) used to investigate the performance or potentials of school children in fluency, flexibility and novelty of creativity (e.g. Handayani et al., 2020;Kwon et al., 2006;Leikin & Lev, 2007;Levav-Waynberg & Leikin, 2012;Leikin, 2014;Sadak et al., 2022); The relationship between students' mathematical creativity and higher-order thinking ability, intelligence, insight and professional knowledge (e.g. Assmus & Fritzlar, 2022;Kahveci & Akgul, 2019;Leikin, 2013;Leikin, 2016); The interaction between cognitive process of problem solving and emotional state on creativity (Cai & Leikin, 2020;DeBellis & Goldin, 2006;Gilat & Amit, 2014;Kozlowski et al., 2019), etc. ...

... Because these three quantitative indicators are concise and clear, they are favored by many quantitative researchers. However, researchers in mathematics education often pay attention to the creativity results of countable problemsolving solutions, they neglect to explore the specific situation of creativity (Schindler et al., 2018), resulting in some inappropriate solutions also being evaluated for creativity (Levav-Waynberg & Leikin, 2012). The statistical conditions of the three indicators have not received due attention. ...

... Correctness means the answer is flawless and accurate, appropriateness can refer not only to correct answers, also the answer may have flaws but the process is reasonable or understandable (Schindler et al., 2018). Appropriateness is particularly suitable for assessing mathematical creativity in complex problem-solving tasks (Leikin, 2013;Levav-Waynberg & Leikin, 2012;Schindler et al., 2018). Therefore, this paper uses appropriateness rather than correctness as a criterion of creativity. ...

Creativity is not only for gifted students, but also for regular ones. This case study was aimed to analyze the appropriateness of tasks and the elaboration of multiple solutions to occasion fourth-graders’ mathematical creative thinking through a documentary multiple-solution counting task in a figurative setting. The data came from the written report of 48 fourth graders in two classes in Taiwan, China. The appropriateness of creativity was reflected in the appropriateness of tasks and solutions, particularly suitable for complex problem solving. Elaboration was detail-dependent, and visualization was beneficial to the analysis of elaboration. The regular students who had just entered the fourth grade could show their creative thinking through different angles (horizontally or vertically) and starting points (holistic or partial), but with slightly more partial and horizontal than holistic and longitudinal, more adaptation than transformation. These fourth-grade students have had the basic mathematical creative thinking capability of adaptation, combination, change, rearrangement, extension or going back by using counting, combining, adding and reducing, overlapping, moving, and diagonal division strategies. Keywords: creative thinking, mathematical creativity, multiple-solution task, primary school students

... The second part included an individual written online test on solving a geometry problem, administered after the two meetings, and lasted for 30 minutes. The problem used for the test has a problem-solving character as it requires students to use their repertoire of knowledge of school geometry and the solution procedure is not straightforward (Levav-Waynberg & Leikin, 2012a;2012b). To prevent cheating activities, we require students to activate their zoom cameras during the test. ...

... The ability to see and show, for instance, that the triangle AFE is congruent to the triangle CDE can be considered that participating students are able to see a different view of the given figure in the problem. This indicates that they are showing flexibility in the problem-solving process (Almeida et al., 2008;Levav-Waynberg & Leikin, 2012a;2012b). ...

... In this case, students should see that two triangles are similar, so the tangent concept can be used (see Table 1). Similar to the previous strategy, the ability of students to see and show, for instance, that the triangle BAE is similar to the triangle BDF can be considered that they can see the figure from a different perspective, and as such show flexibility in the problem-solving process (Levav-Waynberg & Leikin, 2012a;2012b). Either ...

This study aims to investigate the implementation of a problem-solving approach and its corresponding impact on the creative thinking ability of prospective mathematics teachers. A qualitative case study approach was used in this study in the form of observations of learning and teaching processes for geometry topics through the use of a problem-solving approach and of a written test involving 20 prospective mathematics teachers, in one of the state universities in Bandung, Indonesia. The results showed that the implemented problem-solving approach influenced prospective mathematics teachers’ creative thinking in solving problems. The use of different strategies showed prospective teachers’ creative thinking ability in problem-solving processes. The effect of the problem-solving approach on prospective mathematics teachers can be investigated further to obtain a more comprehensive understanding of creative thinking ability.

... A multiple solution task (Leikin, 2009;Levav-Waynberg & Leikin, 2013;Leikin & Lev, 2013) or multiple strategies task (Klein & Leikin, 2020) is a mathematical task with an explicit requirement to be solved with different methods. According to Levav-Waynberg and Leikin (2012), the difference between solution methods can be shown by the utilisation of: (a) different representations of a mathematical concept; (b) different properties (definitions, theorems, auxiliary constructions) of mathematical concepts in a mathematical topic; (c) mathematical tools and theorems from various mathematical domains. Leikin (2018) considers MSTs also as a type of open problem. ...

... There are three types of solution spaces to be considered (Leikin, 2009): individual, collective and expert. The individual solution space represents all sets of solutions generated by a person without the assistance of another person; a collective solution space represents solutions generated by a group of participants; an expert solution space represents solutions produced by expert mathematicians (Leikin, 2014;Levav-Waynberg & Leikin, 2012). The expert solution space represents the fullest sets of solutions known at any given time. ...

... In view of this, appropriate solutions seemed a better option. Levav-Waynberg and Leikin (2012) stress that the evaluation of creativity can be treated independent of the evaluation of correctness as long as solutions are appropriate. ...

Creativity and problem solving are considered to be twenty-first-century competencies, therefore promoting mathematical creativity should be an important part of school mathematics. The study presented in this paper is inspired by the notion of mathematical creativity and the utilization of multiple solution tasks (MSTs) to investigate students’ creativity. Multiple solution tasks are mathematical tasks with an explicit requirement to be solved with different methods. For the purpose of the study, we used a textbook question which allows multiple solution pathways as our research instrument. This MST was administered to first-year mathematics students (18–20 years) along with additional questions related to their experience in school mathematics. The results of the study showed that students had difficulties complying with the demands of the task, but generally have a positive disposition toward MSTs. Moreover, the results underline the necessity of incorporating MSTs in mathematics classroom teaching to create a more coherent corpus of mathematical knowledge. The study also provides an example of how a regular textbook task can be used as an MST.

... In relation to producing more than one solution to a problem (e.g. Levav-Waynberg & Leikin, 2012), we examine the students' ability to construct auxiliary lines, an act that brings to the fore the structure of the task environment. ...

... Previous research regarding student learning skills suggests that the use of multiple solution methods for problem solving develops student creativity and mental flexibility, and increases mathematical understanding, reasoning, and critical thinking (e.g. Elia et al., 2009a, b;Levav-Waynberg & Leikin, 2012). ...

... The new approaches to teaching and learning geometry, such as the development of mathematical creativity, are the modern way of reforming the teaching and learning of geometry based on research findings (e.g. Kell, Lubinski, Benbow & Steiger, 2013;Levav-Waynberg & Leikin, 2012;Singer, Voica & Pelczer, 2017). Among the various areas of mathematics, geometry can be used as a vehicle to develop different ways of thinking in mathematics. ...

... Así, la flexibilidad es un componente importante de la competencia matemática, y en particular, la flexibilidad en resolución de problemas es un ejemplo de conocimiento procedimental profundo y conectado (Baroody, 2003;Star y Seifert, 2006;Rittle-Johnson y Star, 2009a). Levav-Waynberg y Leikin (2012) han estudiado el papel de las multiple solution tasks en el desarrollo de la creatividad matemática, que abordan con tres criterios: fluidez o soltura, flexibilidad y originalidad. Las multiple solution tasks son problemas en los que el investigador dispone de un espacio de soluciones, por lo que puede pedir a los resolutores que resuelvan una tarea de distintas maneras, o usar el espacio de soluciones para analizar si, al variar las tareas, los resolutores utilizan diferentes resoluciones (Leikin y Levav-Waynberg, 2008). ...

... La flexibilidad se considera una competencia matemática importante, ya que es necesaria para que los estudiantes adquieran la capacidad de adaptar sus resoluciones a las características de la tarea o el contexto (Kilpatrick y cols., 2001). En el marco de la resolución de problemas, como se ha dicho, los estudios sobre el uso flexible de múltiples soluciones consideraron que la flexibilidad es esencial para construir un conocimiento profundo y conectado (Silver, Ghousseini, Gosen, Charalambous, y Strawhun, 2005;Rittle-Johnson y Star, 2009b;Rittle-Johnson, Star, y Durkin, 2009;Rittle-Johnson y Star, 2009a;Leikin y Levav-Waynberg, 2008;Levav-Waynberg y Leikin, 2012). En su entrada para la Encyclopedia of the Sciences of Learning, Nistal, Van Dooren y Verschaffel (2012) definen flexibilidad en resolución de problemas de la siguiente manera: ...

... Sabemos que, en los problemas de la secuencia 1, hay cuatro tipos de planes de resolución: Recuento, Linealización, Unidad base y Densidad. Por tanto, son problemas de modelización con un espacio de tipos de resolución conocido de antemano, es decir, podemos considerarlas multiple solution tasks en el sentido de Levav-Waynberg y Leikin (2012). En otros problemas de modelización más complejos, por el contrario, no se puede delimitar de antemano un espacio de soluciones y su variabilidad y dependencia de las hipótesis iniciales del resolutor hace muy difícil estudiar la flexibilidad en el uso de resoluciones (modelos matemáticos). ...

The aim of the thesis is to study the performance of prospective teachers in solving a type of modelling problems involving estimation: Fermi problems, which we will call real-context estimation problems. The use of modelling activities in the classroom is an effective way of connecting Mathematics with the real world. Real-context estimation problems are accessible tasks that allow modelling to be introduced in primary school. However, their implementation is a challenge for primary school teachers, because shortcomings have been detected in their specialised knowledge of mathematical content for teaching, in particular, in their proficiency in problem solving. There is consensus that the flexible use of various types of resolution is a component of problem-solving proficiency. It is therefore of interest to study the flexibility of pre-service teachers in solving real-context estimation problems, and to analyse possible relationships with their performance. In order to address these aspects linked to the flexibility and performance of prospective teachers in solving real-context estimation problems, the research design is complex: two sequences of four problems and two questionnaires are designed, and the research is divided into three parts: the first part is the central one, and is composed of two experiences, in which the N = 224 pre-service teachers involved solve a sequence of real-context estimation problems, first individually and schematically ( resolution plan), and then as a group and performing measurements at the problem site (group and on-site resolution). The second part is based on an alternative sequence of problems in order to validate the results of the previous one with another sample of N= 87 prospective teachers, although it is also proposed to study the effect of syntactic structure on success in solving the problems. The third part deals with the implementation of a questionnaire answered by N = 81 experts in Mathematics and/or its didactics to determine adaptability criteria (what is the best solution) in this type of problems. An analysis of the resolution plans and the group and on-site resolutions, combining qualitative and quantitative techniques, leads to address the research objectives of the thesis: to categorise the productions of prospective teachers and to establish a significant relationship between certain context characteristics and the type of resolution; to categorise and analyse specific errors in real-context estimation problem solving, defining performance levels based on the errors made; to analyse inter-task flexibility (understood as the ability to change the type of resolution from one problem to another in the sequence, depending on the characteristics of the context) and to find relationships between the level of flexibility and performance; to compare individual resolution plans and group and on-site resolutions; to define adaptability criteria for this type of problem and to analyse the adaptability of pre-service teachers. The results offer the opportunity to design problem sequences that promote flexibility and learning from errors, which will contribute to improve the initial training of prospective teachers and enrich their specialised knowledge of mathematical content for teaching. More: https://roderic.uv.es/handle/10550/81850

... The structural evaluation of a completed solution or an attempt at a solution, with a view toward developing alternate solution approaches, is a crucial topic underpinning all the preceding (Mamona-Downs, 2008). Previous researchers have utilized multiple-solution problem-solving to measure and foster students' mathematical creativity, enhancing students' mathematical comprehension, cognitive flexibility, reasoning, and critical thinking Levav-Waynberg & Leikin, 2012). ...

... Multiple methods to a single problem can be seamlessly integrated via geometry. Practically any geometry issue in conventional textbooks may be transformed into a multiple-solution problem (Levav-Waynberg & Leikin, 2012). The following indicators are used to evaluate the mathematical creativity of pupils when solving geometric problems. ...

... The structural evaluation of a completed solution or an attempt at a solution, with a view toward developing alternate solution approaches, is a crucial topic underpinning all the preceding (Mamona-Downs, 2008). Previous researchers have utilized multiple-solution problem-solving to measure and foster students' mathematical creativity, enhancing students' mathematical comprehension, cognitive flexibility, reasoning, and critical thinking Levav-Waynberg & Leikin, 2012). ...

... Multiple methods to a single problem can be seamlessly integrated via geometry. Practically any geometry issue in conventional textbooks may be transformed into a multiple-solution problem (Levav-Waynberg & Leikin, 2012). The following indicators are used to evaluate the mathematical creativity of pupils when solving geometric problems. ...

The definition of creativity among professional mathematicians and the definition of mathematical creativity in the classroom context are significantly different. The purpose of this study was to investigate the relationship between students' mathematical creativity (i.e., cognitive flexibility) and figure apprehension when solving geometric problems with novel auxiliary features such as straight lines and curved lines. In other words, this study determined if geometry knowledge influenced mathematical creativity (cognitive flexibility) in problem-solving. Grade-12 students participated in the intervention. The high school that is the research topic attempts to equip students with academic abilities and is, except for vocational schools, the most popular form of high school among all other types. Such a school was chosen for the study so that a significant proportion of students in Makassar could be represented. In this study, we discovered a relationship between cognitive flexibility and the geometric ability of pupils while solving problems involving auxiliary lines. This indicates that the usage of auxiliary lines as a reference for developing pupils' creative thinking skills must be advocated. In addition, good geometric abilities (e.g., visual thinking, geometrical reasoning) will encourage pupils to generate various problem-solving concepts. This finding contributes significantly to future research by focusing on auxiliary lines.

... A task does not have to have more than one correct answer to be used in the context of mathematical creativity, but it is necessary and sufficient that a task has more than one solution path (Leikin, 2009). Multiple solution tasks explicitly require students to solve mathematical problems in different ways and these tasks can be used both as a psychometric tool measuring students' creative thinking and as an intervention to develop students' creative thinking in mathematics (Leikin, 2009) Researchers who applied multiple solutions tasks to measure students' creative thinking mostly employed geometrical problems as they heavily focused on multiple representations as one technique of multiple solution tasks (Leikin, 2013;Levav-Waynberg & Leikin, 2012;Levenson et al., 2018). Because geometry was heavily emphasized over other subfields of mathematics, Schindler et al. (2018) suggested that future research should investigate whether creativity in mathematics should be considered as a subspecific construct in mathematics (e.g., geometrical creativity, algebraic creativity, numerical creativity). ...

... 2) Different representations: Using different representations is another technique that allows students to apply tools, such as drawings, graphs, tables, and written or verbal symbols to discern connections among several mathematical concepts and create new mathematical knowledge (Ervynck, 1991;Levav-Waynberg & Leikin, 2012;Silver 1997). This can be easily classified under the category of connecting mathematical ideas with other ideas in mathematics, as it is an already common practice in mathematics classrooms. ...

Although teaching mathematics for creativity has been advocated by many researchers, it has not been widely adopted by many teachers because of two reasons: 1) researchers emphasized and investigated mathematical creativity in terms of product dimension by looking at what students have at the end of problem-solving or -posing activities, but they neglected the creative processes students use during mathematics classrooms, and 2) creativity is an abstract construct and it is hard for teachers to interpret what it means for students to be creative in mathematics without further guidance. These can be eliminated by employing techniques of mathematical connections as tools because using mathematical connections can help teachers make sense of how to promote the creative processes of students in mathematics. Because making mathematical connections is a process of linking ideas in mathematics to other ideas and this is a creative act for students to take to achieve creative ideas in mathematics, using the strategies of making mathematical connections has the potential for teachers to understand what it means for students to be creative in mathematics and what it means to teach mathematics for creativity. This paper has two aims to 1) illustrate strategies for making mathematical connections that can also help students’ creative processes in mathematics, and 2) investigate the relationship among general mathematical ability, mathematical creative ability, and mathematical connection ability by reviewing theoretical explanations of these constructs and several predictors (e.g., inductive/deductive ability, quantitative ability) that are important for these constructs. This paper does not only provide examples and techniques of mathematical connection that can be used to foster creative processes of students in mathematics, but also suggests a potential model depicting the relationship among mathematical creativity, mathematical ability, and mathematical connection considering previously suggested theoretical models. It is important to note that the hypothesized model (see Figure 4) suggested in the present paper is not tested through statistical analyses and it is suggested that future research be conducted to show the relationship among the constructs (mathematical connection, mathematical creativity, mathematical ability, and spatial reasoning ability).

... Nonetheless, over the past two decades, researchers from a variety of countries have primarily focused on implementing discipline-specific instructional strategies through the use of a variety of learning strategies. Multiple solution tasks (MSTs) (Leikin, 2009;Levav-Waynberg & Leikin, 2012;Leikin & Lev, 2013), open-ended problems (Kwon et al., 2006), multiple representations and/or visualizations tasks (Bicer, 2021b), model-eliciting activities (Gilat & Amit, 2014), and visualization with technology integration tasks (Idris & Nor, 2010) are some of the learning approaches used to develop mathematical creativity. While several other researchers conducted studies on mathematical creativity on the topic of general instructional practices, such as justice (Luria et al., 2017), making errors (Shriki, 2009), and taking risks (Sriraman, 2017), others examined the relationship between mathematical creativity and specific instructional practices. ...

... According to this analysis, the most frequently occurring keywords are "creativity", "problem-posing", "mathematics", "problem-solving", "teaching", "learning", "cognition", "e-learning", "creative", "education", "task", and "thinking". This set of keywords shows the meaning that creativity in mathematics or mathematics creativity in the learning and teaching process of mathematics can be developed through problem-solving (Moore-Russo & Demler, 2018), problem-posing (Van Harpen & Sriraman, 2013), multiple solution tasks (MSTs) (Leikin, 2009;Leikin & Lev, 2013;Levav-Waynberg & Leikin, 2012), and technology/e-learning (Yuniawati et al., 2020). In addition, when students use creative processes to solve mathematical problems, their cognitive styles, such as spatial, object, and verbal cognitive styles, allow them to comprehend the outcomes . ...

Mathematical creativity is among the most intriguing research fields in the world. This is plausible because research on mathematical creativity, particularly in the field of education, has a positive impact on many dimensions of life. Even though numerous studies have been conducted on this topic, there are still many aspects that have not been examined. Using bibliometric analysis, the authors of this study evaluated scientific articles on mathematical creativity from 2002 to 2022 that were indexed in Scopus using Biblioshiny and VOSviewer. The authors analyzed 162 publications in terms of document distribution patterns and growth trends, contributions and impacts from countries, institutions, authors, and journals, patterns of development and evolution of the theme of mathematical creativity, and future research opportunities. Despite a decrease in the average number of citations per document, the results suggest a significant increase in the number of publications between 2002 and 2022. The United States and the University of Haifa are the nations and institutions with the highest publication output, ZDM-Mathematics Education has the highest impact, and Bicer is a core author who is extremely productive and influential. "Creativity" has been the most popular keyword over the past two decades, but it is not the only one. This study encourages future research on mathematical creativity in mathematics education to not only focus on the theme of discipline-specific instructional practices, but also on the theme of general instructional practices involving more person, process, product, and press/environment creativity.

... In Mathematics, Multiple-Solution Tasks (MSTs) have proved to be a useful teaching and learning tool which can not only foster student creativity but also, measure it (e.g. Leikin, 2014;Levav-Waynberg & Leikin, 2012a;2012b). As Levav-Waynberg & Leikin (2012a) have stated, MSTs in geometry give students the opportunity to investigate numerous solutions by applying knowledge and concepts already taught in school geometry curricula. ...

... Leikin, 2014;Levav-Waynberg & Leikin, 2012a;2012b). As Levav-Waynberg & Leikin (2012a) have stated, MSTs in geometry give students the opportunity to investigate numerous solutions by applying knowledge and concepts already taught in school geometry curricula. ...

This study aims to investigate high school students’ geometry learning by focusing on mathematical creativity and its relationship with visualisation and geometrical figure apprehension. The presentation of a geometrical task and its influence on students’ mathematical creativity is the main topic investigated. The authors combine theory and research in mathematical creativity, considering Roza Leikin’s research work on Multiple-Solution Tasks with theory and research in visualisation and geometrical figure apprehension, mainly considering Raymond Duval’s work. The relations between creativity, visualization and geometrical figure apprehension are examined through four Geometry Multiple-Solution Tasks given to high school students in Greece. The geometrical tasks are divided into two categories depending on whether their wording is accompanied by the relevant figure or not. The results of the study indicate a multidimensional character of relations among creativity, visualization and geometrical figure apprehension. Didactical implications and future research opportunities are discussed.

... Various educational research confirm that the problem tasks create an appropriate environment for students to connect previously acquired and gain new knowledge, develop different skills and competences such as visualisation, choosing a correct and economical problem-solving path and appropriate method, coping with new and unfamiliar situations, etc. (e.g. Levav-Waynberg & Leikin, 2012;Natsheh & Karsenty, 2014;Schoenfeld, 1992). ...

... (npr. Levav-Waynberg i Leikin, 2012;Natsheh i Karsenty, 2014;Schoenfeld, 1992). ...

The paper defines a special type of problem tasks and considers its didactic potential, as well as the success of students in solving the selected problem. The research instrument used is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura). The geometrical task is presented in three versions: as a verbal problem, as a verbal problem with a corresponding image and as a problem in context. The material analysed in the present paper was collected from 182 students in 7 th and 8 th grade of Croatian urban elementary schools. The didactic potential is considered from the aspect of use of mathematical concepts and connections. The success of students in problem-solving is considered from the aspect of implementation of the problem-solving process and producing correct answers, depending on the manner in which the tasks are set up. The results show that the stand-alone problem, as a special type of problem task, has considerable didactic potential. However, the students' skills of discovering and connecting mathematical concepts and their properties are underdeveloped. In addition, the manner in which the tasks are set up considerably affects the process of solving the task and consequently the success of that process. Based on the results of the research, proposals are given for application of stand-alone problems in teaching mathematics.

... Problem solving is a level of intellectual activity that uses existing knowledge to find a solution to a problem (NCTM, 2000). Problem solving and problem posing can be used to develop mathematical creativity (Levav-waynberg & Leikin, 2012). In Silver (1997) research, it is argued that mathematics lessons centered on problemgenerating questions and activities related to problem solving and problem posing can help learners develop more creative thinking in mathematics. ...

Creative thinking and problem solving skills are essential in facing the challenges of the 21st century. This study aims to (1) test the effectiveness of the contextual approach CPS model in improving creative thinking skills in mathematics problem solving and (2) describe creative thinking skills in mathematics problem solving in terms of Self Regulated Learning (SRL). This type of research is a mixed method research that combines quantitative and qualitative research methods sequentially. In this study, 6 research subjects were taken based on the level of SRL in class XI students of SMA N 5 Semarang. The results showed that (1) contextual approach CPS learning is effective in improving creative thinking ability in mathematics problem solving, and (2) research subjects with high SRL category were able to fulfill all indicators of creative thinking ability in mathematics problem solving, namely fluency, flexibility and novelty; research subjects with moderate SRL category only fulfilled two indicators of creative thinking ability in mathematics problem solving, namely fluency, and flexibility; research subjects with low SRL category were only able to fulfill the indicator of creative thinking ability in mathematics problem solving, namely fluency.

... The video recording allowed us to examine their mathematical reasoning within the theoretical frame of Radford [14], that is, from a multimodal approach to mathematics learning in collective solution spaces. This is in line with Levav-Waynberg and Leikin [27], who posited that collective solution spaces can be a helpful "tool for examining the mathematical knowledge and creativity of participating students" (p. 78). ...

This article examines gifted students' (ages 13-16) groupwork on a rich task in mathematics. This study was conducted in Norway, which has an inclusive education system that does not allow fixed-ability grouping. The purpose of this study was to better understand how to cultivate mathematical learning opportunities for gifted learners in inclusive education systems. The analysis was conducted from a multimodal perspective, in which students' coordination of speech, gestures, and artifact use was viewed as part of their learning process. The findings contribute to discussions on gifted students as a heterogeneous group. Moreover, our analysis illustrates how giftedness can be invisible, leading to unrealized potential and low achievement. We suggest that more attention be paid to teaching by adapting to gifted students' individual needs, particularly if the intention is to provide high-quality learning opportunities for gifted students in inclusive settings.

... In choosing not to show students how to produce a solution to a given problem and instead asking them to develop their own solution process, student-centered pedagogical approaches create the context necessary for students to develop rich, conceptual understandings of mathematics beyond procedural knowledge (Mackrell & Pratt, 2017;Papert, 1980) and find connections within and across specific concepts (Noss & Hoyles, 1996). In doing so, student-centered teaching creates the context necessary for students to develop creative problem-solving skills (Ali et al., 2021;Jasien & Horn, 2018;Levav-Waynberg & Leikin, 2012) and promote new ways of thinking about mathematics (Bland, 2019). ...

While research has shown that students benefit from student-centered pedagogies, few studies have considered the benefits of this pedagogical approach for educators as they learn through teaching. In response to this need, we analyzed interviews, lesson plans, and video observations from five teachers in elementary schools across the United States who varyingly engaged student-centered and teacher-centered pedagogies. Our analyses revealed that the participating teachers developed a wide breadth of teacher knowledge regardless of their pedagogical approach. However, the teachers who employed student-centered teaching reported more pedagogical content knowledge gains for themselves than the teachers who used direct teaching.

... Menurut [11], hampir semua masalah geometri pada buku teks dapat diubah menjadi multiplesolution task. Hal ini disebabkan masalah geometri dapat diselesaikan melalui beragam solusi menggunakan berbagai konsep dan sifat-sifat pada geometri itu sendiri [12]. ...

Pengabdian kepada masyarakat ini bertujuan untuk: (a) memperkenalkan bagaimana merancang multiple-solution task dalam pembelajaran matematika bagi guru-guru SMA/SMK Muhammadiyah di Klaten dan sekitarnya; (b) menyelenggarakan lokakarya dan pelatihan guru dalam merancang multiple-solution task dalam topic geometri analitik. Pelatihan dilaksanakan di SMA Muhammadiyah 1 Klaten, Jawa Tengah. Pesertanya adalah para guru SMA/SMK Muhammadiyah di Klaten dan sekitarnya. Hal itu dilakukan melalui tahapan sebagai berikut: persiapan, terdiri dari koordinasi internal dan eksternal; tahap implementasi yang meliputi pengenalan dan pelatihan desain tugas solusi ganda dalam pembelajaran matematika. Kegiatan bakti sosial ini diikuti 24 guru dari 17 sekolah Muhammadiyah di Klaten. Peserta telah dilatih secara teknis dengan pengalaman baru merancang multiple-solution task dan mengembangkan multiple-solution task dalam topik geometri analitis; para peserta terlibat penuh selama pelatihan. Ada umpan balik dan kebutuhan untuk peningkatan dalam mengintegrasikan multiple-solution task dalam pembelajaran matematika.

... The purpose of video-recording the collaborative work was to gain insight into collective solution spaces. According to Levav-Waynberg and Leikin (2012), collective solution spaces can be a helpful "tool for examining the mathematical knowledge and creativity of participating students" (p. 78). ...

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.

... To help overcome this challenge, the use of tasks with multiple solutions and the use of more open-ended problems have been recommended strongly by mathematics educators and reformers as a way of helping students build connections between procedural and conceptual knowledge. These recommendations are based on research that has shown use of open-ended problems can increase students' flexibility in problem solving (Kwon et al., 2006;Levav-Waynberg & Leikin, 2012;Rittle-Johnson & Star, 2007, 2009) and conceptual understanding of the material (Rittle- Johnson, 2009) while encouraging collaboration and mathematical discourse (Chan & Clarke, 2017;Cohen, 1994) . ...

... For instance, it has been reported as one of the ways to foster connectedness of students' mathematical knowledge and positively impact students' creativity and flexibility, contributing to developing students' mathematical understanding and problem-solving skills. It has also been found to be a valuable tool for examining students' mathematical knowledge and creativity (see, e.g., Levav-Waynberg & Leikin, 2012a, 2012b. ...

Task design is an important element of effective mathematics teaching and learning. Past research in mathematics education has investigated task design in mathematics education from different perspectives (e.g., cognitive and cultural) and offered a number of (theoretical) frameworks and sets of principles. In this study, through a narrative research in the form of autoethnography, I reflected on my past teaching and research experience and proposed a (theoretical) framework for task design in mathematics education. It contains four main principles: (a) inclusion, (b) cognitive demand, (c) affective and social aspects of learning mathematics, and (d) theoretical perspective(s) toward learning mathematics. This framework could be used as a tool for critically reflecting on current practices in terms of task design in teaching mathematics and research in mathematics education. It may also contribute to ongoing research in mathematics education about task design and enable or enhance opportunities for dialogue between lecturers, teachers, and researchers about how to design rich mathematical tasks for teaching and research purposes.

... While for many decades creativity in mathematics teaching and learning was largely overlooked (Haylock, 1987;Leikin, 2009a), luckily, in the last decade, we observed an exponential development of research publications related to creativity in mathematics (Leikin & Sriraman, 2022). We see the development of knowledge and skills and the development of creativity in a circular manner: more advanced knowledge and skills allow better creative processing, while creativity can serve as a mechanism for the development of knowledge and skills (Guberman & Leikin, 2013;Levav-Waynberg & Leikin, 2012;Pitta-Pantazi et al., 2022). Thus, we argue that creativity-directed activities are an effective instructional tool. ...

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 and problem-solving insight. Insightful and divergent thinking are at the base of mathematical creativity. This chapter analyzes the mathematical challenge embedded in problem-solving tasks from the point of view of evoked mathematical insight and the use of multiple solution strategies. While a variety of variables (such as conceptual density, level of concepts, length of solution or use of different presentations) determine the complexity of mathematical problems, the insight component and the requirement to solve problems in multiple ways increase the mathematical challenge of the task. Researchers distinguish between different types of mathematical insight as they relate to the distinction between mathematical expertise and mathematical creativity. In this chapter, we introduce a distinction between mathematical tasks that allow insight-based solutions and tasks that require mathematical insight. We provide empirical evidence for our argument that tasks that require mathematical insight are of a higher level of complexity.KeywordsInsightful solutionMultiple solution strategiesInsight-requiring tasksInsight-allowing tasks

... The inhibition process requires mental effort on the part of the solvers. Thus, high-level students usually display higher creativity capabilities than lower-level students (Kattou et al., 2013;Levav-Waynberg & Leikin, 2012). Usually, teachers state that MOTs and ITs are more difficult to solve than MSTs since higher cognitive skills are required. ...

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 both in MOTs and MSTs increases both the complexity of the task and the mathematical curiosity of school students, making Math-Key tasks inherently mathematically challenging. In addition, Math-Key tasks require a change in socio-mathematical norms, and thus, the program is didactically challenging. To provide scaffolds for teaching and learning processes Math-Key tasks are accompanied by exploratory and task-directed dynamic applets (DA). The exploratory nature of the DA enables solvers of Math-Key tasks to understand the problem structure and to support the teachers’ orchestration of classroom teachers. We characterize Math-Key tasks using several examples and explain the task directness of the DA. Integration of the Math-Key program within the regular curricular sequence is a part of the recommended curricular change suggested in this chapter.KeywordsMath-Key TasksDynamic appletsMultiple solution strategiesMultiple solution outcomesSolution spacesOpen-start problemsOpen-end problemsSolution completeness

... For example, counterexample task reports, analyzed and classified by their obviousness and how they conformed to the premises and conclusion of a proposition, could improve students' understanding of the logical status of examples and statements (Buchbinder & Zaslavsky, 2009). Seeing many different correct examples helps enrich the class example space and encourages development of knowledge and creativity (Levav-Waynberg & Leikin, 2012). It also provides an opportunity to challenge students' common belief that only one correct answer exists and that it resides in the head of the assessor (Bennett, 1993). ...

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students’ exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic activity format for if–then propositions and implemented the activity on a platform that collects and analyzes students’ work. Finally, we searched for ways to use variation theory to analyze ninth-grade students’ recorded work. We scored and classified data and found correlation between patterns in exploration stages and the conjectures students generated. We demonstrate how automatic identification of mistakes in the early stages is later reflected in the quality of conjectures.

... Using multiple mathematical representations is a way for enhancing mathematical creativity (Boaler et al., 2016;Bicer, 2021a,b) as it enables students to flexibly shift between various representations and alternate their solutions when they encounter new problems (San Giovanni et al., 2020). It has been suggested that applying multiple mathematical representations can help students figure out connections among various concepts and develop their mathematical knowledge (Ervynck, 1991;Silver, 1997;Levav-Waynberg and Leikin, 2012). ...

This article discusses the cognitive process of transforming one representation of mathematical entities into another representation. This process, which has been called mathematical metaphor, allows us to understand and embody a difficult-to-understand mathematical entity in terms of an easy-to-understand entity. When one representation of a mathematical entity is transformed into another representation, more cognitive resources such as the visual and motor systems can come into play to understand the target entity. Because of their nature, some curves, which are one group of visual representations, may have a great motor strength. It is suggested that directedness, straightness, length, and thinness are some possible features that determine degree of motor strength of a curve. Another possible factor that can determine motor strength of a curve is the strength of association between shape of the curve and past experiences of the observer (and her/his prior knowledge). If an individual has had the repetitive experience of observing objects moving along a certain curve, the shape of the curve may have a great motor strength for her/him. In fact, it can be said that some kind of metonymic relationship may be formed between the shapes of some curves and movement experiences.

... Bicer, A., Chamberlin,S., & Perihan, C, 2020; Haavold, P.O, 2018; Inuusah, dkk, 2019;Piirto, J, 2011; Sebastian, J., & Huang, H, 2016; Tyagi, T.K, 2016; UNESCO, 2006).Kemampuan literasi numerasi diartikan sebagai keterampilan untuk memperoleh, menggunakan, menginterpretasi angka, symbol matematika dalam menyelesaikan masalah, melakukan analisis dan mengambil keputusan. Sedangkan kreativitas merupakan keterampilan yang dapat diukur pada proses pemecahan masalah (problem solving) dan pengajuan masalah (problem posing)(Elgrably, H., & Leikin, R, 2021;Levav, W.A., & Leikin, R, 2012). ...

Integrasi keilmuan menjadi kata kunci yang ada didalam visi misi UIN Syarif
HIdayatullah Jakarta yang kemudian diturunkan menjadi visi misi FITK. Melalui
integrasi keilmuan Inilah, kami ingin menemukan, menelurkan, menciptakan,
mengkreasikan ilmu-ilmu baru serta pengetahuan-pengetahuan baru, penelitianpenelitian baru untuk kemaslahatan umat dan kebaikan bangsa. Integrasi keilmuan
yang dilakukan diharapkan mampu memberikan kontribusi yang sangat sangat
tinggi untuk meningkatkan harkat dan martabat manusia itu sendiri khususnya di
Indonesia.
Proses integrasi keilmuan, keislaman dan keindonesiaan diharapkan akan
menumbuhkan keilmuan-keilmuan baru karena akan mempertemukan berbagai
disiplin keilmuan di samping pengembangan keilmuan itu sendiri. integrasi
keilmuan juga akan diarahkan kembali ke asal mula ilmu itu sendiri yaitu ilmu dari
Allah, ilmu dari Tuhan. memberikan kontribusi untuk peningkatan pendidikan
khususnya yang ada di Indonesia karena melalui integrasi keilmuan Inilah kita
berharap memberikan kontribusi menuju Indonesia emas tahun 2045. Selain pada
proses Integrasi, literasi digital menjadi satu keniscayaan untuk kita kuasai agar kita
mampu mengimplementasikan dan manfaatkan teknologi melalui dunia
Pendidikan. Oleh karena itu integrasi keilmuan, keislaman dan keindonesiaan
melalui berbagai penerapakan kemampuan literasi digital menjadikan pendidikan
yang terbaik untuk Indonesia emas di tahun 2045 nanti. Berbagai pemikiran dan
pengembangan keilmuan yang dilakukan khususnya terkait integrasi keilmuan dan
keislaman difasilitasi melalui forum diskusi yang berlangsung melalui seminar
nasional yang dilaksanakan oleh Fakultas Ilmu Tarbiyah dan Keguruan UIN Syarif
Hidayatullah Jakarta.

... A number of researchers have highlighted the value of generating multiple solutions to a problem for developing problem-solving skills and nurturing mathematical creativity (Leikin, 2009;Levav-Waynberg & Leikin, 2012;Silver et al., 2005). This Fig. 3 The problem that required to explore the mathematics relationships study shows that Taiwanese mathematics teachers provide students the opportunities to develop mathematical creativity by posing questions to facilitate multiple solutions based on flexibly applying the learned knowledge. ...

In this study, we aim to investigate the types of questions that Taiwanese mathematics teachers pose and in which instructional situations they do so during mathematics lessons at the secondary school level. The classroom teaching of six experienced mathematics teachers was analyzed. Quantitative analysis showed that the mathematics teachers tend to give lectures rather than ask questions. When the mathematics teachers posed questions, only about one-fifth of the questions require students to provide high-cognitive responses. We also observed that the mathematics teachers differed in the number and type of questions they asked in different instructional situations. A cross-examination of the types of questions and the lesson structures revealed that two-thirds of the mathematics teachers asked high-cognitive questions when practicing or reviewing the content with the students. The qualitative analysis further identified three instructional purposes for high-cognitive questions: connecting the meaning of mathematical concepts, stimulating multiple solutions to a problem, and exploring mathematical relationships across different problem contexts. The results imply that mathematics teaching at the secondary school level in Taiwan is more teacher-centered, and the mathematics teachers do not often ask questions during classroom teaching. However, the teachers tend to ask high-cognitive questions for assessment purposes to ensure that the students have understood the concepts and can proceed to advanced mathematics.

... In the curriculum, the classroom culture is also stated as an environment where learners "approach problems through different ways", "share their thoughts, strategies and solutions" with their friends and "value different ways of solutions" (MoNE, 2018, p. 13). These statements support mathematical creativity and flexible thinking because the students are encouraged to solve the problem situations in more than one way (Levav-Waynberg & Leikin, 2012;Silver, 1997). Moreover, the curriculum expects teachers to use instructional practices that allow students' creative thinking processes in the classroom (e.g., …in a way that allows the development of creative thinking skills…") (MoNE, 2018). ...

The majority of existing research have repeatedly embedded problem solving and problem posing in the assessment of students’ mathematical creativity, but there is a lack of studies focusing on the relationship between these two regarding mathematical creativity. In this study, we aimed to examine whether there is a relationship between the constructs of creative ability in mathematical problem posing (CAMPP) and creative ability in mathematical problem solving (CAMPS) and to examine the structure of this relationship through confirmatory factor analysis. The participants were 187 sixth-grade students in Turkey. Data were collected by two creative ability tests, namely CAMPP and CAMPS. We used a rubric to characterize mathematical creativity by interpreting scores of in the dimensions of creative ability (fluency, flexibility, and originality) in the context of problem solving and problem posing. The findings showed that mathematical problem posing and mathematical problem solving both constituted the constructs of CAMPP and CAMPS respectively, based on the dimensions of creative ability. Moreover, the structure of the relationship between the constructs of CAMPP and CAMPS can be explained better with a constituted higher-order factor of Creative Ability in Mathematics (CAM) rather than placing one of these factors as a sub-construct under the other one.

... Students are asked to solve MSTs in multiple ways, using different properties, theorems, representations, or relationships (Leikin, 2009). Often, students' written products are then evaluated (e.g., Levav-Waynberg & Leikin, 2012) since the ability to produce different solutions is considered to reflect personal mathematical creativity (Leikin & Lev, 2013). ...

In the age of artificial intelligence where standard problems are increasingly processed by computers, creative problem solving, the ability to think outside the box is in high demand. Collaboration is also increasingly significant, which makes creative collaboration an important twenty-first-century skill. In the research described in this paper, we investigated students’ collaborative creative process in mathematics and explored the collaborative creative process in its phases. Since little is known about the collaborative creative process, we conducted an explorative case study, where two students jointly worked on a multiple solution task. For in-depth insight into the dyad’s collaborative creative process, we used a novel research design in mathematics education, DUET SRI: both students wore eye-tracking glasses during their collaborative work for dual eye-tracking (DUET) and they each participated in a subsequent stimulated recall interview (SRI) where eye-tracking videos from their joint work served as stimulus. Using an inductive data analysis method, we then identified the phases of the students’ collaborative creative process. We found that the collaborative creative process and its phases had similarities to those previously found for solo creative work, yet the process was more complex and volatile and involved different branches. Based on our findings, we present a tentative model of the dyad’s collaborative process in its phases, which can help researchers and educators trace and foster the collaborative creative process more effectively.

... The use of the decimal scoring scheme and the edges of 10% and 40% were introduced and justified by Leikin (2009Leikin ( , 2013. Further validation of the scoring scheme was performed in multiple studies since 2009 (e.g., Leikin et al., 2017;Levav-Waynberg & Leikin, 2012). Within the space limitations of this paper, we do not repeat here justification of the edges. ...

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 strategies and the creative product to PPI outcomes (i.e., strategy creativity and outcome creativity). Furthermore, we draw a connection between the openness of tasks and their power for the evaluation of strategy creativity and outcome creativity, demonstrate the aptness of PPI tasks for the evaluation of both types of creativity, and examine the connections between them. The model for the evaluation of creativity that we used in this study, was initially designed and validated using analysis of problem-solving strategies when solving multiple solution tasks. We previously extended the model to evaluation of PPI outcomes, and we here demonstrate its implementation to evaluation of creativity of PPI strategies. To examine connections between creativity of PPI strategies and creativity of PPI outcomes, we focused on PPI by eight experts in mathematical problem solving who were members or candidates of the Israeli IMO team. We present empirical evidence for the distinctions between strategy creativity and outcome creativity, and for the connections between them. We analyzed strategy creativity as a unique characteristic of problem-solving experts. We found that higher strategy creativity does not necessarily lead to higher outcome creativity, and that a high level of strategy originality correlates with outcome flexibility. We conclude that creative product and creative process are two distinct characteristics of cognitive processing linked to creativity-directed problem solving.

... McMullen and colleagues (McMullen et al., 2017) found that students' strategies for working with rational numbers predicted later pre-algebra skills. Similarly, Levav-Waynberg and Leikin (2012) found a relationship between geometrical knowledge and the strategies used by students who engaged with tasks for which there were multiple possible solution methods. In addition, Lemaire and Siegler (1995) found that improved adaptivity in strategy use was one explanation for increased speed and accuracy in multiplication tasks for French 2nd graders. ...

Background
In this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies.
Results
Our results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees.
Conclusions
Although it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning.

... The conceptual understanding problems proposed by teachers, even if they look simple, are problems that require an understanding of mathematical concepts that are relational in nature, which relate one mathematical concept to other mathematical concepts, and require complex mathematical thinking skills that are not just asking the procedure for solving a problem (Budhi & Kartasasmita, 2015;Posamentier & Stepelman, 1990;Skemp, 1976). The problem-solving problems proposed by the teachers can often be categorized as non-routine problem-solving problems, often coming from mathematics competition, which requires the teachers to do proving processes, and require critical as well as creative thinking Koichu & Leron, 2015;Levav-Waynberg & Leikin, 2012;Weber, 2005). In addition, several problems proposed by the teachers having the type of problem-solving are open-ended problems, namely mathematics problems that require creativity, divergent thinking, both in terms of the problemsolving processes and of finding a variety of answers that meet the problems (Ho & Hedberg, 2005;Koichu & Leron, 2015;Kwon et al., 2006). ...

One of the competencies for mathematics teachers that needs to be developed continuously is professional competence. However, even if efforts for developing teachers’ competencies have been made formally by the government, it seems still lacking. This study, therefore, aims to develop mathematics teacher professional competencies through an informal development model using social media. This research used a qualitative method, a case study design, involving 19 mathematics teachers from various regions in Indonesia in the informal development process in the range of 2019-2021. The informal approach was carried out using question-and-answer techniques and guided discussions on mathematical problems. From the teacher development processes, 30 mathematics problems and their solutions were collected. As an illustration of this development process, this article presents five problems and their solutions, including solutions for two mathematics problems on conceptual understanding and three mathematics problems on problem-solving. We conclude that this informal approach is fruitful in helping mathematics teachers solve mathematics problems. This study implies that the teacher development process carried out in this study can be used as a model for informal teacher development by other higher education academics in their respective places.

Pentingnya kemampuan berpikir kreatif dalam pembelajaran dicanangkan pada kurikulum 2013. Salah satu tipe soal yang dapat digunakan untuk mengukur kemampuan berpikir kreatif siswa yaitu Multiple Solutions Task (MST). Soal-soal berbasis etnomatematika dapat diujicobakan karena memadukan kebudayaan lokal dengan materi matematika sehingga memotivasi dan meningkatkan kemampuan berpikir kreatif siswa. Penelitian bertujuan mendeskripsikan kemampuan berpikir kreatif siswa dalam menyelesaikan soal MST yang berbasis etnomatematika. Subjek penelitian yaitu 18 siswa kelas IX A pada salah satu SMPN di Panekan. Metode pengumpulan data yaitu dokumentasi, tes tulis dan wawancara. Data dianalisis dengan teknik interaktif dari yang spesifik hingga yang umum, dan melibatkan berbagai tingkat analisis. Hasil penelitian yaitu Tingkat Kemampuan Berpikir Kreatif (TKBK) menunjukkan terdapat 8 siswa (44%) pada TKBK 0 (tidak kreatif), 7 siswa (39%) pada TKBK 2 (cukup kreatif), 2 siswa (11%) pada TKBK 3 (kreatif) dan 1 siswa (6%) pada TKBK 4 (sangat kreatif). TKBK 0 (tidak kreatif) lebih mendominasi daripada TKBK yang lain. Tingkat kemampuan berpikir kreatif siswa dalam menyelesaikan masalah Multiple Solution Task (MST) berbasis etnomatematika dalam penelitian ini belum mendapatkan hasil yang cukup baik. The importance of creative thinking skills in learning was proclaimed in the 2013 curriculum. One type of question that can be used to measure students' creative thinking skills is the Multiple Solutions Task (MST). Ethnomathematical-based questions can be tested because they combine local culture with mathematics material to motivate and improve students' creative thinking skills. This study aims to describe students' creative thinking skills in solving ethnomathematical-based MST questions. The research subjects were 18 students of class IX A at one of the SMPN in Panekan. Data collection methods are documentation, written tests, and interviews. The data are analyzed by interactive techniques from the specific to the general and involve various levels of analysis. The results on the Creative Thinking Ability Level (TKBK) showed that there were 8 students (44%) at TKBK 0 (not creative), 7 students (39%) at TKBK 2 (creative enough), 2 students (11%) at TKBK 3 (creative) and 1 student (6%) in TKBK 4 (very creative). TKBK 0 (not creative) is more dominant than the other TKBK. The level of students' creative thinking skills in solving ethnomathematical-based Multiple Solution Task (MST) problems in this study has not gotten good enough for results.

TD is inherently a cultural phenomenon, shaped by the intricate interplay of genetic predispositions, sociocultural experiences, and environmental influences. Within the context of individual development, cultural provisions and interventions constitute an integral component of TD. Cultural provisions and interventions are viewed as developmentally responsive when they address developmental needs and goals in a timely, proactive fashion, strategically positioned within specific TD contexts to accommodate diverse needs and challenges encountered by individuals during distinct developmental processes and phases. Recent research spanning the decade from 2010 to 2020 shows some degrees of alignment with the imperative of developmental responsive characteristic of the provisions/interventions research. It emphasizes the role of these provisions and interventions in initiating and sustaining TD, fostering positive talent growth trajectories. Nevertheless, there is a compelling call for a more systematic and programmatic research approach, one that pursues a specific line of inquiry on provisions and interventions across time, to comprehensively address this multifaceted category of TD research.

Los problemas de Fermi, adecuados para primaria, plantean una situación real y abierta que permite desarrollar y comparar múltiples estrategias, lo que requiere que los maestros sean adaptables (capaces de escoger la más apropiada). El objetivo de este trabajo es caracterizar y analizar la adaptabilidad de futuros maestros cuando resuelven estos problemas. Para ello, la investigación se divide en dos estudios. El Estudio 1 presenta una encuesta dirigida a expertos en educación matemática; el análisis de sus respuestas permite vincular las características contextuales de los problemas con estrategias, y estas, con criterios de adecuación (precisión, rapidez y rigor). Estos resultados conducen a una caracterización de adaptabilidad que nos permite abordar el Estudio 2 con futuros maestros, y se concluye que la mayoría de los resolutores adaptables usan estrategias de manera no sistemática.

Η έννοια της αναλογίας είναι μια από τις πιο σημαντικές έννοιες των
μαθηματικών, ενώ παράλληλα ο αναλογικός συλλογισμός αποτελεί έναν από
τους πιο σπουδαίους μηχανισμούς της γνωστικής ανάπτυξης του ατόμου.
Ωστόσο, η παγκόσμια ευρεία χρήση της αναλογίας στα προγράμματα
σπουδών έχει ως άμεσο επακόλουθο την δημιουργία παρανοήσεων και
συγκεκριμένα ότι το αναλογικό μοντέλο μπορεί να εφαρμοστεί παντού,
ακόμη και σε μη αναλογικές καταστάσεις. Το φαινόμενο αυτό αναγράφεται
στην βιβλιογραφία ως ψευδαίσθηση της αναλογίας και συναντάται σε
διάφορους τομείς των μαθηματικών όπως στην άλγεβρα, στις πιθανότητες
και στην γεωμετρία. Ειδικότερα, στον τομέα της γεωμετρίας έχει
παρατηρηθεί από ένα ευρύ πλήθος ερευνών ότι οι μαθητές τείνουν συνεχώς
να αντιμετωπίζουν τις σχέσεις μεταξύ μήκους και εμβαδού ή μεταξύ μήκους
και όγκου ως γραμμικές αντί ως τετραγωνικές ή κυβικές αντίστοιχα.
Στην παρούσα ερευνητική εργασία γίνεται μια προσπάθεια μελέτης
αυτού του φαινομένου σε 10 μαθητές διαφορετικών τάξεων από το Δημοτικό
μέχρι και το Λύκειο, με κύριο σκοπό να εξακριβωθεί ανάλογα με τις
επιδόσεις των μαθητών σε αναλογικά και μη έργα μέσα από έξι διαφορετικές
φάσεις αν το φαινόμενο της ψευδαίσθησης της αναλογίας είναι ανεξάρτητο
της ηλικίας των μαθητών και να διαπιστωθούν πιθανοί τρόποι
καταπολέμησης αυτού του επίπονου φαινομένου. Τα αποτελέσματα
καταδεικνύουν και επιβεβαιώνουν ότι η ηλικία των μαθητών δεν παίζει
κάποιο ουσιαστικό ρόλο, καθώς αυτή η παρανόηση εντοπίζεται τόσο σε
μαθητές Δημοτικού όσο και σε μαθητές Γυμνασίου και Λυκείου.
Παράλληλα, διαπιστώνεται ότι η χρήση αναπαραστάσεων, όπως
διαγραμμάτων σε τετραγωνισμένο χαρτί, καθώς και η μέθοδος των
πολλαπλών τρόπων λύσεων και η συμπλήρωση κενών αποτελούν μια πιθανή
διέξοδο και αντιμετώπισης του φαινομένου.

A frequent concern about constructivist instruction is that it works well, mainly for students with higher domain knowledge. We present findings from a set of two quasi-experimental pretest-intervention-posttest studies investigating the relationship between prior math achievement and learning in the context of a specific type of constructivist instruction, Productive Failure. Students from two Singapore public schools with significantly different prior math achievement profiles were asked to design solutions to complex problems prior to receiving instruction on the targeted concepts. Process results revealed that students who were significantly dissimilar in prior math achievement seemed to be strikingly similar in terms of their inventive production, that is, the variety of solutions they were able to design. Interestingly, it was inventive production that had a stronger association with learning from PF than pre-existing differences in math achievement. These findings, consistent across both topics, demonstrate the value of engaging students in opportunities for inventive production while learning math, regardless of prior math achievement.

The purpose of this study was to quantify the relationship between pre-service teachers’ spatial visualisation skills and their mathematical creativity through problem-posing tasks. A group of 62 pre-service teachers completed the Purdue Spatial Visualisation test and took the mathematical creativity test through problem-posing tasks. Pearson’s product-moment correlation coefficients in SPSS-28 were employed to find the correlation between pre-service teachers’ spatial visualisation skills and their mathematical creativity (i.e. fluency, flexibility, and originality) through problem-posing tasks. Results showed a moderate to high correlation (r = .523) between pre-service teachers’ fluency scores and their spatial visualisation scores, and this correlation was statistically significant (p < .05). Similarly, the results showed a high correlation (r = .619, p < .05) between pre-service teachers’ flexibility scores and their spatial visualisation scores. However, the results also revealed a weak correlation (r = .218) between pre-service teachers’ originality scores and their spatial visualisation scores (p = .240). Possible explanations for these correlations in relation to existing studies about mathematical creativity and the implication for future studies are provided.

The purpose of this systematic review is to reveal the research findings that suggest instructional practices to foster the creativity of students in mathematics. Although several studies have investigated the effects of various instructional practices influencing the mathematical creativity of students, little is known about how the findings of this collective body of research contribute to the understanding of what instructional practices should be integrated into a mathematic classroom to further foster the mathematical creativity of students. In this systematic review, the knowledge of instructional practices that foster the mathematical creativity of students were categorized under two main factors including: 1) discipline-specific instructional practices and 2) general instructional practices. The discipline-specific instructional practices were problem-solving, problem-posing, open-ended questions, multiple solution tasks, tasks with more than one correct answer, modeling/model-eliciting activities, technology integration, extendable tasks, and emphasizing abstractness of mathematics. The general instructional practices were providing students with ample time to think creatively about real-world related mathematical problems in a judgment free and collaborative classroom environment so that they take risks to share their mathematical ideas and use informal words. Integrating all of these instructional practices into mathematics classrooms can provide opportunities for students to discover their potential creative abilities in mathematics.

Mathematical creativity is an important topic both in research and in practice. Especially in research and practice on secondary school level, it is gaining increasing significance with a growing body of tasks, practices, and empirical studies being developed and conducted. At the same time, the growing field of research on mathematical creativity on secondary school level goes along with an increasing variety of perspectives on mathematical creativity. The aim of this chapter is to provide a systematic literature review to characterize the landscape of the current state of empirical research on mathematical creativity on secondary school level. After a scanning process, we analyzed 22 research articles with respect to their perspectives on mathematical creativity and inductively identified five perspectives that are predominant in current research on mathematical creativity on secondary school level. In the chapter, we present these perspectives and we identify and discuss research trends as well as research gaps relevant for future work.

Pentingnya kemampuan berpikir kreatif dalam pembelajaran dicanangkan pada kurikulum 2013. Salah satu tipe soal yang dapat digunakan untuk mengukur kemampuan berpikir kreatif siswa yaitu Multiple Solutions Task (MST). Soal-soal berbasis etnomatematika dapat diujicobakan karena memadukan kebudayaan lokal dengan materi matematika sehingga memotivasi dan meningkatkan kemampuan berpikir kreatif siswa. Penelitian bertujuan mendeskripsikan kemampuan berpikir kreatif siswa dalam menyelesaikan soal MST yang berbasis etnomatematika. Subjek penelitian yaitu 18 siswa kelas IX A pada salah satu SMPN di Panekan. Metode pengumpulan data yaitu dokumentasi, tes tulis dan wawancara. Data dianalisis dengan teknik interaktif dari yang spesifik hingga yang umum, dan melibatkan berbagai tingkat analisis. Hasil penelitian yaitu Tingkat Kemampuan Berpikir Kreatif (TKBK) menunjukkan terdapat 8 siswa (44%) pada TKBK 0 (tidak kreatif), 7 siswa (39%) pada TKBK 2 (cukup kreatif), 2 siswa (11%) pada TKBK 3 (kreatif) dan 1 siswa (6%) pada TKBK 4 (sangat kreatif). TKBK 0 (tidak kreatif) lebih mendominasi daripada TKBK yang lain. Tingkat kemampuan berpikir kreatif siswa dalam menyelesaikan masalah Multiple Solution Task (MST) berbasis etnomatematika dalam penelitian ini belum mendapatkan hasil yang cukup baik.The importance of creative thinking skills in learning was proclaimed in the 2013 curriculum. One type of question that can be used to measure students' creative thinking skills is the Multiple Solutions Task (MST). Ethnomathematical-based questions can be tested because they combine local culture with mathematics material to motivate and improve students' creative thinking skills. This study aims to describe students' creative thinking skills in solving ethnomathematical-based MST questions. The research subjects were 18 students of class IX A at one of the SMPN in Panekan. Data collection methods are documentation, written tests, and interviews. The data are analyzed by interactive techniques from the specific to the general and involve various levels of analysis. The results on the Creative Thinking Ability Level (TKBK) showed that there were 8 students (44%) at TKBK 0 (not creative), 7 students (39%) at TKBK 2 (creative enough), 2 students (11%) at TKBK 3 (creative) and 1 student (6%) in TKBK 4 (very creative). TKBK 0 (not creative) is more dominant than the other TKBK. The level of students' creative thinking skills in solving ethnomathematical-based Multiple Solution Task (MST) problems in this study has not gotten good enough for results.

Many studies in the past decades have focused on low and typical mathematics achievers, yet little is known about children with high mathematics achievement, particularly at a young age. The current study aimed to fill this gap and started from the early work of Krutetskii (1976) as a theoretical lens to study the characteristics of high mathematics achievers in primary school. Krutetskii’s work was extended with more recent research on mathematical cognition. A simultaneous investigation of mathematics-specific abilities and general motivational and cognitive factors allowed us to examine their unique contributions to high mathematics achievement. Participants were 162 children, that is, 81 high mathematics achievers and 81 average achievers (8- to 10-year-olds). Children completed measures assessing their mathematical cast of mind (attention to number), flexible mental calculation (adaptive number knowledge, strategy variety), and striving for elegance (use of varying strategies for multidigit arithmetic). We also measured children’s need for cognition, spatial visualization ability, and working memory. There were significant group differences on all tasks, except for the attention to number task. A logistic regression analysis revealed that strategy variety, need for cognition, and spatial visualization ability were significant predictors of group membership. These data suggest that strategy variety, need for cognition, and spatial visualization ability might be critical characteristics of high mathematics achievers in primary school. The identification of such characteristics might be an important first step in creating supportive educational environments for these children.

Most research has focused solely on understanding high school or college students’ mathematical creative thinking abilities while understanding younger students’ creative thinking in mathematics was ignored. These studies of older students have focused mainly on students’ creative products rather than creative processes. The authors of the present study investigated four young elementary school students’ creative processes through eye-tracking (ET) and stimulated recall interview (SRI) techniques while they engaged in multiple representations (MRs) of mathematical problems. Our qualitative case study revealed what phases four elementary school students’ creative processes in MRs involve and how they achieve original ideas. The results revealed that neither Wallas’ (1926) creative process of mathematicians nor Schindler and Lilienthal’s (2020) creative process of a high school student could fully explain the creative processes of four young elementary school students in the present study. The findings from the present study emphasize that four young students’ creative processes are difficult to predict as it is non-linear compared to professional mathematicians’ creative process, but young students’ creative processes can be demystified through ET and SRI techniques. The present study also emphasizes the importance of external factors (e.g., teachers, peers, environment) for the four elementary school students to get different perspectives to achieve creative ideas in mathematics.

Bu araştırmanın amacı Suriye uyruklu öğrencilerin problem çözme sürecinde kullandıkları bilişsel ve üst bilişsel stratejileri incelemektir. Çalışma, 2020-2021 Eğitim-Öğretim yılı Güz döneminde, Hatay ilinde tamamen Suriye uyruklu öğrencilerin eğitim gördüğü bir ortaokulda 8. Sınıf öğrencileri ile gerçekleştirilmiştir. Nitel araştırma olarak desenlenen çalışmada katılımcılar, amaçlı örnekleme yöntemlerinden kolay ulaşılabilir durum örnekleme yöntemiyle seçilmiştir. Veriler sesli düşünme protokolü uygulanarak çevrimiçi bir uygulama aracılığıyla klinik görüşmeler aracılığıyla toplanmıştır. Veri analiz sürecinde öğrencilerin problem çözme sürecinde transkript edilen sesli düşünceleri incelenerek hangi bilişsel ve üst bilişsel stratejileri kullandıkları incelenmiştir. Elde edilen bulgular öğrencilerin bilişsel stratejileri üst bilişsel stratejilerden daha fazla kullandıklarını göstermiştir. Öğrenciler okuma, kendi cümleleri ile ifade etme, hipotez oluşturma, hesaplama yapma bilişsel stratejilerini kullanırken görselleştirme ve tahmin etme bilişsel stratejilerini kullanmamışlardır. Üst bilişsel stratejilerden ise yorum yapma ve problemi anlamaya yönelik soru sorma gibi üretici olmayan stratejileri kullanırken, problemi analiz etme, çözümü kontrol etme, çözümü savunma gibi üretici stratejileri pek fazla kullanmadıkları görülmüştür. Öğrenciler en fazla problemi anlama sürecinde zorlanmış, plan yapma sürecini göz ardı etmiş, şekil, şema, tablo vb. temsiller kullanmamışlardır. Araştırmanın sonuçlarına göre Suriye uyruklu öğrencilerle çalışan öğretmenlerin problemi anlama, tahmin ve görselleştirme aşamalarına önem vermeleri önerilmiştir.

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 development of attention paid to mathematical creativity in schools. A systematic search conducted to examine research on creativity in mathematics education journals and research in the field of mathematics in creativity-related journals, yielded 49 papers that described empirical studies on creativity in mathematics education. The retrospective analysis examined the following: (a) the main goals of the studies; (b) conceptual frameworks used by the researchers; (c) research methodologies; and (d) tools and tasks. Of these, 18 studies examined the relationships between creativity in mathematics and other characteristics, 25 were on creativity related to instructional practices and mathematical tasks, and 6 studies investigated teachers’ creativity-related conceptions and competencies. Of these studies, 29 used quantitative methodologies, 10 of which integrated analysis for problem solving or problem posing strategies or outcomes. Ten additional studies employed qualitative methodology (mainly content analysis of lesson observations). We conclude the paper with a short review of papers in this special issue, and outline directions for future research on creativity in mathematics (education).

Background and Aims: The overall purpose of the present study is to study the
extent of efficiency of education on improving high school students` attitude towards
math with an emphasis on multiple solutions.
Methods: The research methodology is action research which has more affinity
with a qualitative approach, but it has quantitative aspects as well, in terms of its
experimental methods and statistical tests. In this study, 47 students participated in the
experimental group and 54 students in the control group, all of whom were studying in
Tehran`s public schools for girls in both mathematics and empirical sciences in the
academic year 2015-2016. The research tools are Palacios, Arias & Arias (2014)
questionnaire which assesses students` attitude towards math in Likert Scale in the
form of four components, namely, the perception of utility, mathematical self-concept,
perception of mathematical incompetence and enjoyment of mathematics.
Results: The findings reveal that all the attitude components have a positive
meaningful effect on math and empirical sciences groups. The difference is that the
math group has improved more in the enjoyment of mathematics. Also, results of
Tukey’s posthoc test show that the mean of the intervention group in math and
empirical sciences are higher than the control group in all components of attitude.
Conclusions: The positive growth of all attitude components in both groups and the
significant distance between the intervention and control groups in all components of attitude,
suggest that this approach has been successful in improving that students` attitude towards math.

In this study we utilize the notion of learner-generated examples, suggesting that examples generated by students mirror their understanding of particular mathematical concepts. In particular, we explore examples generated by a group of prospective secondary school teachers for a definition of a square. Our framework for analysis includes the categories of accessibility and correctness, richness, and generality. Results shed light on participants’ understanding of what a mathematical definition should entail and, moreover, contrast their pedagogical preferences with mathematical considerations.

School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathematized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum. Usiskin (Z. Usiskin, 1987), for instance, has described four dimensions of geometry: (a) visualization, drawing, and construction of figures; (b) study of the spatial aspects of the physical world; (c) use as a vehicle for representing nonvisual mathematical concepts and relationships; and (d) representation as a formal mathematical system. The first three of these dimensions require the use of spatial reasoning.

This article describes a case study in mathematics instruction, focusing on the development of mathematical understandings that took place in a 10-grade geometry class. Two pictures of the instruction and its results emerged from the study. On the one hand, almost everything that took place in the classroom went as intended—both in terms of the curriculum and in terms of the quality of the instruction. The class was well managed and well taught, and the students did well on standard performance measures. Seen from this perspective, the class was quite successful. Yet from another perspective, the class was an important and illustrative failure. There were significant ways in which, from the mathematician's point of view, having taken the course may have done the students as much harm as good. Despite gaining proficiency at certain kinds of procedures, the students gained at best a fragmented sense of the subject matter and understood few if any of the connections that tie together the procedures that they had studied. More importantly, the students developed perspectives regarding the nature of mathematics that were not only inaccurate, but were likely to impede their acquisition and use of other mathematical knowledge. The implications of these findings for reseach on teaching and learning are discussed.

For the gifted mathematics student, early mastery of concepts and skills in the mathematics curriculum usually results in getting more of the same work and/or moving through the curriculum at a faster pace. Testing, grades, and pacing overshadow the essential role of creativity involved in doing mathematics. Talent development requires creative applications in the exploration of mathematics problems. Traditional teaching methods involving demonstration and practice using closed problems with predetermined answers insufficiently prepare students in mathematics. Students leave school with adequate computational skills but lack the ability to apply these skills in meaningful ways. Teaching mathematics without providing for creativity denies all students, especially gifted and talented students, the opportunity to appreciate the beauty of mathematics and fails to provide the gifted student an opportunity to fully develop his or her talents. In this article, a review of literature defines mathematical creativity, develops an understanding of the creative student of mathematics, and discusses the issues and implications for the teaching of mathematics.

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers' understanding of other functions of proof than just the traditional function of 'verification'. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called 'reconstructive' approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry.

The recent development of powerful new technologies such as dynamic geometry software (DGS) with drag capability has made possible the continuous variation of geometric configurations and allows one to quickly and easily investigate whether particular conjectures are true or not. Because of the inductive nature of the DGS, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical concern. In this article we discuss the implications of the development of this new software for the teaching of proof and making proof meaningful to students. We describe how three prospective primary school teachers explored problems in geometry and how their constructions and conjectures led them see proofs in DGS.

the goal of many research and implementation efforts in mathematics education has been to promote learning with understanding / drawing from old and new work in the psychology of learning, we present a framework for examining issues of understanding / the questions of interest are those related to learning with understanding and teaching with understanding / what can be learned from students' efforts to understand that might inform researchers' efforts to understand understanding
the framework we propose for reconsidering understanding is based on the assumption that knowledge is represented internally, and that these internal representations are structured / point to some alternative ways of characterizing understanding but argue that the structure of represented knowledge provides an especially coherent framework for analyzing a range of issues related to understanding mathematics (PsycINFO Database Record (c) 2012 APA, all rights reserved)

The concept of understanding in mathematics with regard to mathematics education is considered in this volume. The main problem for mathematics teachers being how to facilitate their students' understanding of the mathematics being taught. In combining elements of maths, philosophy, logic, linguistics and the psychology of maths education from her own and European research, Dr Sierpinska considers the contributions of the social and cultural contexts to understanding. The outcome is an insight into both mathematics and understanding.

Während der letzten dreißig Jahre haben sich viele nordamerikanische Mathematikdidaktiker dafür eingesetzt, dem Beweisen im Mathematikunterricht der Sekundarstufen eine geringere Rolle als bisher zuzuweisen. Diese Auffassung wurde großenteils mit Entwicklungen in der Mathematik selbst, mit den Ideen von Lakatos und mit geänderten sozialen Werten begründet. Der folgende Artikel versucht zu zeigen, daß keines dieser Argumente einen solchen Auffassung swandel rechtfertigt und daß das Beweisen weiterhin seinen Wert für den Schulunterricht behält, wegen seiner zentralen Bedeutung für das mathematische Arbeiten und als ein wichtiges Mittel zur Beförderung von Verständnis.
Abstract
Over the past thirty years, many mathematics educators in North America have suggested that proof be relegated to a lesser role in the secondary mathematics curriculum. This view has been shaped in large part by developments in mathematics itself, by the thinking of Lakatos and by changing social values. This paper argues that none of these factors justifies such a move, and that proof continues to have value in the classroom, both as a reflection of its central role in mathematical practice and as an important tool for the promotion of understanding.

Our concern in this study was to examine the relationship between problem-solving performance and the quality of the organization of students' knowledge. We report findings on the extent to which content and connectedness indicators differentiated between groups of high-achieving (HA) and low-achieving (LA) Year 10 students undertaking geometry tasks. The HA students' performance on the indicators of knowledge connectedness showed that, compared with the LA group, they could retrieve more knowledge spontaneously and could activate more links among given knowledge schemas and related information. Connectedness indicators were more influential than content indicators in differentiating the groups on the basis of their success in problem solving. The tasks used in the study provide straightforward ways for teachers to gain information about the organizational quality of students' knowledge.

Abilities are always abilities for a definite kind of activity ; they exist only in a person's specific activity. Therefore they can show up only on the basis of an analysis of a specific activity. Accordingly, a<span style=background-color: #ffff00;> mathematical ability exists only in a mathematical activity and should be manifested in it. (S. 66)

This paper discusses multiple way problem solving as a habit of mind and claims that it is an effective tool for the development and diagnostics of advanced mathematical thinking. I introduce a notion of solution spaces of a mathematical problem and exemplify the ideas using several problems. The concepts of symmetry and continuity are presented here as important ideas that should be a part of habitual mathematical behavior in the context of advanced mathematical thinking.

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 originality of solutions. In this article, the MST research instrument is introduced in connection with the theoretical analysis of the research literature and then explained and analyzed using geometry students' performance results on one MST. The analysis shows that the research instrument differentiates between students belonging to high- and regular-level instruction groups and sheds light on the interrelations between components of geometry knowledge and creativity.

In this article we examine students' perspectives on the customary, public work of proving in American high school geometry classes. We analyze transcripts from 29 interviews in which 16 students commented on various problems and the likelihood that their teachers would use those problems to engage students in proving. We use their responses to map the boundaries between activities that (from the students' perspective) constitute normal (vs. marginal) occasions for them to engage in proving. We propose a model of how the public work of proving is shared by teacher and students. This division of labor both creates conditions for students to take responsibility for doing proofs and places boundaries on what sorts of tasks can engage students in proving. Furthermore we show how the activity of proving is a site in which complementarity as well as contradiction can be observed between what makes sense for students to do for particular mathematical tasks and what they think they are supposed to do in instructional situations.

At the K-12 level one assumes that mathematically gifted students identified by out-of-level testing are also creative in their work. In professional mathematics, “creative” mathematicians constitute a very small subset within the field. At this level, mathematical giftedness does not necessarily imply mathematical creativity but the converse is certainly true. In the domain of mathematics, are the words creativity and giftedness synonyms? In this article, the constructs of mathematical creativity and mathematical giftedness are developed via a synthesis and analysis of the general literature on creativity and giftedness. The notions of creativity and giftedness at the K-12 and professional levels are compared and contrasted to develop principles and models that theoretically “maximize” the compatibility of these constructs. The relevance of these models is discussed with practical considerations for the classroom. The paper also significantly extends ideas presented by Usiskin (2000).

This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of didactical contractand double bindto uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching—what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition—are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.

This report, prepared for and published by the Mathematical Association of America's Committee on the Teaching of Undergraduate Mathematics, includes a description of the state of the art on problem solving, lists available resources, and makes recommendations regarding the place of problem solving in the college curriculum and ways to teach it. The report recommends (1) an approach to teaching mathematics that fosters an alert and questioning attitude in students and that actively engages them in the process of doing mathematics, (2) a series of problem-solving courses at various levels of sophistication as regular offerings in the standard college curriculu, and (3) a series of texts for problem-solving courses at all levels to be developed and disseminated. Specific suggestions are given on how to teach problem solving, especially pertaining to the role of the teacher and ways of organizing the class. Some typical problems and class discussions are provided. Then follows an extensive annotated bibliography of problem-solving resources, with characterizations of the type of course for which each appears most appropriate, its focus or subject matter, and its level. Journals, books, and articles are listed separately. Finally, the problem-solving questionnaire and responses are briefly presented. (MNS)

One of the four cornerstones of the National Council of Teachers of Mathematics (NCTM) "Curriculum and Evaluation Standards for School Mathematics" asserts that connecting mathematics to other subjects in the curriculum and to the everyday world is an important goal of school mathematics. This yearbook is designed to help classroom teachers, teacher educators, supervisors, and curriculum developers broaden their views of mathematics and suggests practical strategies for engaging students in exploring the connectedness of mathematics. Following are the section and chapter titles. Part 1: General Issues: (1) "The Case for Connections" (A. F. Coxford); (2) "Connections as Problem-Solving Tools" (T. R. Hodgson); (3) "Connecting School Science and Mathematics" (D. F. Berlin & A. L. White); and (4) "Using Ethnomathematics To Find Multicultural Mathematical Connections" (L. Shirley). Part 2: Connections within Mathematics: (5) "Connecting Number and Geometry" (L. Leake); (6) "Using Functions To Make Mathematical Connections" (R. P. Day); (7) "Making Connections with Transformations in Grades K-8" (R. N. Rubenstein & D. R. Thompson); (8) "Transformations: Making Connections in High School Mathematics" (M. L. Crowley); (9) "Using Transformations To Foster Connections" (D. B. Hirschhorn & S. S. Viktora); and (10) "Connecting Mathematics with Its History: A Powerful, Practical Linkage" (L. Reimer & W. Reimer). Part 3: Connections across the Elementary School Curriculum: (11) "Learning Mathematics in Meaningful Contexts: An Action-Based Approach in the Primary Grades" (S. L. Schwartz & F. R. Curcio); (12) "Measurement in a Primary-Grade Integrated Curriculum" (L. Rhone); (13) "Connecting Literature and Mathematics" (D. J. Whitin); (14) "Connecting Reasoning and Writing in Student 'How to' Manuals" (N. F. Grandgenett, J. W. Hill, & C. V. Lloyd); and (15) "Connecting Mathematics and Physical Education through Spatial Awareness" (D. V. Lambdin & D. Lambdin). Part 4: Connections across the Middle School Curriculum: (16) "Seeing and Thinking Mathematically in the Middle School" (G. M. Kleiman); (17) "Projects in the Middle School Mathematics Curriculum" (S. Krulik & J. Rudnick); (18) "Carpet Laying: An Illustration of Everyday Mathematics" (J. O. Masingila); (19) "Mathematics and Quilting" (K. T. Ernie); and (20) "Randomness: A Connection to Reality" (D. J. Dessart). Part 5: Connections across the High School Curriculum: (21) "Connecting Geometry with the Rest of Mathematics" (A. A. Cuoco, E. P. Goldenberg, & J. Mark); (22) "Forging Links with Projects in Mathematics" (J. W. McConnell); (23) "Baseball Cards, Collecting, and Mathematics" (V. P. Schielack, Jr.); (24) "Experiencing Functional Relationships with a Viewing Tube" (M. R. Wilson & B. E. Shealy); (25) "Breathing Life into Mathematics" (K. M. Johnson & C. L. Litynski); and (26) "Students' Reasoning and Mathematical Connections in the Japanese Classroom" (K. Ito-Hino). (MKR)

Investigates the extent to which visual considerations in calculus can be taught and be a natural part of college students' mathematical thinking. Recommends that the legitimacy of the visual approach in proofs and problem solving should be emphasized and that the visual interpretations of algebraic notions should be taught. (YP)

This study analyzes development of teachers’ mathematical and pedagogical conceptions in systematic (through learning) and craft (through teaching) modes and the relationships between them. We focus on teachers’ conceptions of the meaning and potential of multiple-solution connecting tasks in school mathematics. We found that in systematic mode teachers increase primarily their shared conceptions and that the development of mathematical conceptions precedes that of pedagogical conceptions. Only in craft mode do they develop new understandings while their mathematical and pedagogical conceptions become integrated and advance each other mutually.

This study aims to deepen our understanding of the development of teacher knowledge in systematic (through learning) and craft (through teaching) modes. The main research tools of this study were multiple-solution connecting tasks. We used the notion of solution spaces to analyze the data and demonstrated that modifications of the teachers’ solution spaces were situated in their practices of varying types. We found that the implementation of multiple-solution connecting tasks in systematic mode meaningfully developed teachers’ problem-solving performance due not only to the reproduction of solutions offered during the course but to the production of new solutions. Furthermore, in creating opportunities for students to solve tasks with multiple solutions, teachers expanded their personal solution spaces. We conclude that the combining of systematic and craft modes is an optimal condition for the development of teachers’ knowledge.

Reflects on a year spent as an observer and presents observations of the individual as an understander of number and mathematics in general. Theoretical ideas, observations, and curriculum materials are discussed that provide ways of enhancing cognitively guided instruction. A rationale is given for the concept of understanding understanding, which in an extended form implies a process that occurs linearly over time with many rounds. Thus, understanding a child's understanding brings about teaching that builds understanding. Examples are given of how this process works in children's understanding of mathematical principles. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

The subject of this essay is the mathematics curriculum: What should we be teaching in mathematics, and in what ways? This issue has been a focus of my problem solving work for nearly two decades, and I have written about it at length from that perspective. However, I am going to take a different point of view in this essay. Here I shall take a distanced perspective, in order to reflect on some difficult issues. Mathematics education is at a turning point. Some radically new programs are being proposed, and the abolition of some familiar programs is being proposed as well. This is a good time to ask, What do we really know? How much of what we think we know is based on a firm knowledge base, how much on informed guesswork, how much is really just opinion? How much of what we plan to do reflects cultural biases, rather than established fact? These are thorny questions. I shall explore the following four major issues related to curriculum: questions of content, tracking, problem-based curricula, and the role of proof. My goal is to be as honest about what I know, and what I don't know, as I can be.

It is reasonable to assume that a subject of presumably universal appeal must rely on just one style. In spite of its universality, mathematics employs many styles. In particular, there are many styles of proof. In this paper we present and analyse a number of proofs of a property of the area under an hyperbola due to Gregory of Saint-Vincent, a mathematician of the first half of the seventeenth century. There is a baroque and prolific quality to the architecture of his proofs, and this quality points to a connection between a culture and the discovery of a mathematical theory. An historical perspective shows that, in addition to many styles, the universality of mathematics implies a variety of procedures.

In this paper, we describe a one-day professional development activity for mathematics teachers that promoted the use of comparison
as an instructional tool to develop students’ flexibility in algebra. Effective use of comparison in mathematics instruction
involves using side-by-side presentation of problems and solution methods and subsequent student discussion of these multiple
solution methods to highlight the similarities and differences among problem-solving techniques. The goals of the professional
development activity were to make teachers aware of how to use comparison effectively in their instruction, as well as to
impact teachers’ own flexibility in algebra by using comparison instructionally during the professional development. Our analysis
of teachers’ experiences in the professional development activity suggests that when teachers were presented with techniques
for effective use of comparison, their own understanding of multiple solution methods was reinforced. In addition, teachers
began to question why they relied exclusively on one familiar method over others that are equally effective and perhaps more
efficient and started to draw new connections between problem-solving methods. Finally, as a result of experiencing instructional
use of comparison, teachers began to see value in teaching for flexibility and reported changing their own teaching practices.
KeywordsComparison–Flexibility–Mathematics teacher professional development–Multiple solution methods–Algebra