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Challenging mathematics and its role in the learning process

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Abstract

Challenge is not only an important component of the learn- ing process but also a vital skill for life. People are confronted with chal- lenging situations each day and need to deal with them. Fortunately the processes in solving mathematics challenges (abstract or otherwise) involve certain types of reasoning which generalise to solving challenges encountered in every day life. Mathematics has a vital role in the class- room not only because of direct application of the syllabus material but because of the reasoning processes the student can develop. ICMI has commissioned a study to investigate the issues of challenge in the learning process. The author is one of the co-Chairs of this study, which is scheduled to have its Study Conference in 2006 and issue its find- ings as a Study Volume in 2008. In this paper he describes some of the attempts to define the concept of challenge itself and discuss various related issues which are being identified in the context of challenge and the learning process.

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... While continuity is a key factor in successful transition, continuity for first year students should not constitute a repetition of prior learning. The importance of student exposure to challenge in mathematics has been acknowledged by international research (Applebaum and Leikin 2007;Taylor 2005;Turner 2010), and yet many of the learning outcomes of the Common Introductory Course are exactly the same as the learning outcomes for the sixth class mathematics syllabus. Therefore, it is hard to see how students can be sufficiently challenged under these circumstances by providing them with sufficient opportunities to develop understanding and reasoning skills. ...
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... The current study has been carried out in the wake of the authors' participation in the ICMI-16 study group, Challenging Mathematics in and Beyond the Classroom. In his presentation of the conference agenda, Peter Taylor (2006) wrote: ...
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This study is based on our belief that mathematics should be challenging in any classroom and that mathematical challenge is among the central factors that determine the quality of mathematics lessons. Choosing challenging mathematical problem for the students is central in teachers’ work while their conception of mathematical challenge can determine the quality of a mathematics lesson. At the same time, little is known about teachers’ views on mathematical challenge. Thus, we explored conceptions of mathematical challenge in two groups of experienced mathematics teachers. The first group (N1 = 9) was asked to define the notion of mathematical challenge and give examples of challenging mathematical tasks. Later the members of the group discussed these examples and definitions. A written response questionnaire was administered to a second group of teachers (N2 = 41) based on answers given by teachers in the first group. We found that the teachers have a broad conception of mathematical challenge and appreciate the relativity of mathematical challenge but are not always convinced that it is possible to incorporate challenging mathematics in everyday teaching in the classroom.
... This study is inspired by our participation in ICMI-16 study group "Challenging mathematics in and beyond the classroom". Peter Taylor (2006) in his presentation of the agenda of the conference wrote: "Challenge is not only an important component of the learning process but also a vital skill for life. People are confronted with challenging situations each day and need to deal with them. ...
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This study arose from our belief that mathematics should be challenging in any mathematics classroom. We analyse conceptions of mathematical challenge of two groups of experienced mathematics teachers. The first group was asked to define the notion of mathematical challenge and give examples of challenging mathematical tasks (N 1 =9). The second group of teachers was presented with a questionnaire based on the replies of the teachers from the first group (N 2 =41). We found that the teachers have a broad conception of mathematical challenge, appreciate relativity of mathematical challenge, but are not always convinced of the possibility of incorporating challenging mathematics in everyday classroom.
... We believe that mathematical challenge is one of the core elements that promote the learning process both in students and in teachers. According to Taylor (2006), ''Challenge is not only an important component of the learning process but also a vital skill for life'' (p. 2). ...
Article
The study considers mathematical problem solving to be at the heart of mathematics teaching and learning, while mathematical challenge is a core element of any educational process. The study design addresses the complexity of teachers' knowledge. It is aimed at exploring the development of teachers' mathematical and pedagogical conceptions associated with systematic employment of Multiple Solution Tasks (MSTs) in a "problem-solving" course for prospective mathematics teachers (PMTs). Our attention to teachers' mathematical conceptions focused on the development of PMTs' problem-solving competences. Our attention to teachers' meta-mathematical and pedagogical conceptions focused on changes in teachers' views concerning the level of interest and level of difficulty of the mathematical tasks. We differentiated between the systematic and craft modes of professional development integrated in the course. Systematic mode involved problem-solving sessions and reflective discussions on collective solution spaces. Craft mode involved interviewing school students. The study demonstrates the effectiveness of MSTs for PMTs professional development.
... We believe that mathematical challenge is one of the core elements that promote the learning process both in students and in teachers. According to Taylor (2006), ''Challenge is not only an important component of the learning process but also a vital skill for life'' (p. 2). ...
Article
The study considers mathematical problem solving to be at the heart of mathematics teaching and learning, while mathematical challenge is a core element of any educational process. The study design addresses the complexity of teachers’ knowledge. It is aimed at exploring the development of teachers’ mathematical and pedagogical conceptions associated with systematic employment of multiple-solution tasks (MSTs) in a “problem-solving” course for prospective mathematics teachers (PMTs). Our attention to teachers’ mathematical conceptions focused on the development of PMTs’ problem-solving competences. Our attention to teachers’ meta-mathematical and pedagogical conceptions focused on changes in teachers’ views concerning the level of interest and level of difficulty of the mathematical tasks. We differentiated between the systematic and craft modes of professional development integrated in the course. Systematic mode involved problem-solving sessions and reflective discussions on collective solution spaces. Craft mode involved interviewing school students. The study demonstrates the effectiveness of MSTs for PMTs’ professional development.
Chapter
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
Chapter
The aim of this section is to address issues related to mathematics challenge as it appears in the curriculum and instructional design. The first part of this section includes two chapters which discuss curriculum developments and how these translate into classroom practices. One chapter refers to early mathematics education while the second to secondary education. The second part of this section includes five chapters and discusses instructional design and more specifically how textbook, tasks, and teaching practices may offer mathematical challenge to students. The first chapter refers to an instructional design and how this may promote mathematical challenge. The second and third chapters explore the way in which mathematical tasks may promote mathematical challenge. More specifically the second chapter investigates Multiple Solution-Strategies Tasks (MSTs) and Multiple Outcome Tasks, while the third chapter the role of problem-posing tasks. The last two chapters investigate how teachers’ actions, the discourse in the classroom, and argumentations may promote the mathematical challenge and offer more possibilities to students to develop deeper mathematical understanding. This section closes with a commentary chapter which reflects on the context of all chapters presented in this section.KeywordsEarly mathematics educationCurriculumMathematical modelingInstructional designTextbookTasksProblem posingExploratory approachDiscourseMathematical argumentations
Chapter
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