The goal of the case study presented in this paper was to examine a student's perspective on crea... more The goal of the case study presented in this paper was to examine a student's perspective on creative products in project-based learning. In this paper we dismantle, by means of the theory of shifts of attention, a two-month long sequence of events that preceded an unexpected invention made by a ninth-grade student: the student invented a new mathematical symbol, and valued this invention higher than his solution to a complex mathematical problem. Key Words: Sense of creativity, shifts of attention, unusual sign, PBL, sequences
The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insi... more The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insight solution to a challenging problem by a ninth-grade student. A three-week long solution process was analysed by means of the theory of shifts of attention. We argue that concurrent focusing on what, how and why the student attends to when working on the problem can adequately explain his insight.
EXPLORING INSIGHT: FOCUS ON SHIFTS OF ATTENTION, Jul 2015
There is a famous tale about the schoolboy Gauss, who was able to compute the sum of the first 10... more There is a famous tale about the schoolboy Gauss, who was able to compute the sum of the first 100 integers with great rapidity. Mathematics teachers and educators frequently use the tale to demonstrate to their students what insight in mathematical problem solving may look like. Hayes (2006) collected and analysed more than a hundred versions of the tale. In all the versions, a pivotal part of the young Gauss's insight is described as noticing the pattern 1 + 100 = 2 + 99 = 3 + 98 and so on. However, there is no agreement on what " the method of Gauss " could be. Hayes (2006) found that the most widespread interpretations of the method are folding, double row and average, which correspond to the formulas respectively (see Hayes, 2006, and Tall et al., 2012, for additional approaches to the problem). The three formulas are algebraically equivalent, but the ways of attending to the string of the first n integers leading to each of them are different. We will never know which method young Gauss used and how he noticed the pattern. In this article, however, we make empirically-based suggestions about how the pattern was found by a 15-year-old student Ron, when he sought, with his classmate Arik, a formula for the sum of the first n integers. Ron and Arik needed to find a formula in order to accomplish a solution to a challenging problem that they had chosen for their research project in mathematics. In an attempt to account for the three-week long sequence of events that preceded Ron's (seemingly) serendipitous invention of the Gauss formula [1], we use the lenses provided by the Mason's (1989, 2008, 2010) theory of shifts of attention. Accordingly, our goal for this article is to reveal the potential of this theory as an analytical tool that can (at least partially) explain the course of the exploration towards the insight solution.
The goal of the case study presented in this paper was to examine a student's perspective on ... more The goal of the case study presented in this paper was to examine a student's perspective on creative products in project-based learning. In this paper we dismantle, by means of the theory of shifts of attention, a two-month long sequence of events that preceded an unexpected invention made by a ninth-grade student: the student invented a new mathematical symbol, and valued this invention higher than his solution to a complex mathematical problem. INTRODUCTION This article is part of a series of reports, in progress, on the results of a research project " Open-ended problems in mathematics " (Palatnik, in progress). At the beginning of a yearly cycle of the project, a 9 th grade class of one of schools in Israel is exposed to a set of about 10 challenging problems. The students choose a particular problem to pursue and then work on it in teams of two or three for several weeks. The initial problem serves as a basis for follow-up inquiries, which last for additional 2-3...
The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insi... more The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insight solution to a challenging problem by a ninth-grade student. A three-week long solution process was analysed by means of the theory of shifts of attention. We argue that concurrent focusing on what, how and why the student attends to when working on the problem can adequately explain his insight.
The goal of the case study presented in this paper was to examine a student's perspective on crea... more The goal of the case study presented in this paper was to examine a student's perspective on creative products in project-based learning. In this paper we dismantle, by means of the theory of shifts of attention, a two-month long sequence of events that preceded an unexpected invention made by a ninth-grade student: the student invented a new mathematical symbol, and valued this invention higher than his solution to a complex mathematical problem. Key Words: Sense of creativity, shifts of attention, unusual sign, PBL, sequences
The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insi... more The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insight solution to a challenging problem by a ninth-grade student. A three-week long solution process was analysed by means of the theory of shifts of attention. We argue that concurrent focusing on what, how and why the student attends to when working on the problem can adequately explain his insight.
EXPLORING INSIGHT: FOCUS ON SHIFTS OF ATTENTION, Jul 2015
There is a famous tale about the schoolboy Gauss, who was able to compute the sum of the first 10... more There is a famous tale about the schoolboy Gauss, who was able to compute the sum of the first 100 integers with great rapidity. Mathematics teachers and educators frequently use the tale to demonstrate to their students what insight in mathematical problem solving may look like. Hayes (2006) collected and analysed more than a hundred versions of the tale. In all the versions, a pivotal part of the young Gauss's insight is described as noticing the pattern 1 + 100 = 2 + 99 = 3 + 98 and so on. However, there is no agreement on what " the method of Gauss " could be. Hayes (2006) found that the most widespread interpretations of the method are folding, double row and average, which correspond to the formulas respectively (see Hayes, 2006, and Tall et al., 2012, for additional approaches to the problem). The three formulas are algebraically equivalent, but the ways of attending to the string of the first n integers leading to each of them are different. We will never know which method young Gauss used and how he noticed the pattern. In this article, however, we make empirically-based suggestions about how the pattern was found by a 15-year-old student Ron, when he sought, with his classmate Arik, a formula for the sum of the first n integers. Ron and Arik needed to find a formula in order to accomplish a solution to a challenging problem that they had chosen for their research project in mathematics. In an attempt to account for the three-week long sequence of events that preceded Ron's (seemingly) serendipitous invention of the Gauss formula [1], we use the lenses provided by the Mason's (1989, 2008, 2010) theory of shifts of attention. Accordingly, our goal for this article is to reveal the potential of this theory as an analytical tool that can (at least partially) explain the course of the exploration towards the insight solution.
The goal of the case study presented in this paper was to examine a student's perspective on ... more The goal of the case study presented in this paper was to examine a student's perspective on creative products in project-based learning. In this paper we dismantle, by means of the theory of shifts of attention, a two-month long sequence of events that preceded an unexpected invention made by a ninth-grade student: the student invented a new mathematical symbol, and valued this invention higher than his solution to a complex mathematical problem. INTRODUCTION This article is part of a series of reports, in progress, on the results of a research project " Open-ended problems in mathematics " (Palatnik, in progress). At the beginning of a yearly cycle of the project, a 9 th grade class of one of schools in Israel is exposed to a set of about 10 challenging problems. The students choose a particular problem to pursue and then work on it in teams of two or three for several weeks. The initial problem serves as a basis for follow-up inquiries, which last for additional 2-3...
The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insi... more The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insight solution to a challenging problem by a ninth-grade student. A three-week long solution process was analysed by means of the theory of shifts of attention. We argue that concurrent focusing on what, how and why the student attends to when working on the problem can adequately explain his insight.
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Papers by Boris Koichu
Key Words: Sense of creativity, shifts of attention, unusual sign, PBL, sequences
Key Words: Sense of creativity, shifts of attention, unusual sign, PBL, sequences