The journal of secondary gifted education, Aug 1, 2005
aul¹, a 16-year-old member of the team representing Israel in the International Mathematics Olymp... more aul¹, a 16-year-old member of the team representing Israel in the International Mathematics Olympiad (IMO), was interviewed in a study of gifted and average students' strategic behaviors in mathematical problem solving². At the beginning of the interview, the following dialogue took place between Saul and the interviewer: How do you approach difficult geometry problems? Saul:
In this chapter we present our work aimed at interweaving e-learning and face-to-face learning in... more In this chapter we present our work aimed at interweaving e-learning and face-to-face learning in Calculus courses for undergraduate engineering students. This type of blended learning (BL) contains the best properties of e-learning and face-to-face learning and helps overcome many obstacles in traditional teaching. We use our approach in order to improve students' conceptual understanding of theorems. We describe online assignments specifically designed to help students better understand the meaning of a theorem. These assignments are given to students in addition to traditional lectures and tutorials with the objective that they can learn to learn on their own. Students "discover" the theorem and study it independently, by using a "bank" of examples and a lot of theoretical exercises we supply. The assignments are built in such a way that students receive feedback and instructions in response to their Web-based activity.
The length of a (0,l) matrix (a bigraph) is defined and studied *The research was carried out dur... more The length of a (0,l) matrix (a bigraph) is defined and studied *The research was carried out during a visit at the Centre de Recherches Mathkatiques. 'Research supported by Grant DGES-FCAC-FS. 'We use the matrix-theory notation of [3] and the graph-theory notation of [I].
Block-Hankel-matrix inversion and the partial-realization problem, by P. A. Fuhrmann, Ben-Gurion ... more Block-Hankel-matrix inversion and the partial-realization problem, by P. A. Fuhrmann, Ben-Gurion University at the Negev.
This chapter discusses the iterative methods for solving systems of linear equations. The iterati... more This chapter discusses the iterative methods for solving systems of linear equations. The iterative methods are suited for problems involving large sparse matrices, much more so in most cases than direct methods such as Gaussian elimination. The chapter discusses the theory of nonnegative matrices. The theory of M-matrices has played a fundamental role in the development of efficient methods for establishing the convergence and in accelerating the convergence of the iterative methods. The chapter presents the formulas for the classical Jacobi and Gauss–Seidel iterative methods. The convergence results are related to two common classes of situations: (1) those in which A is a nonsingular M-matrix and second and (2) those in which A is hermitian and/or positive definite. A slight modification of the Gauss–Seidel method by the use of a relaxation can be used to produce a method, known as the successive over-relaxation (SOR) method, which can dramatically reduce the number of iterations in certain cases.
International Group for the Psychology of Mathematics Education, 2014
This study focuses on potentially misleading information (PMI) and potentially helpful informatio... more This study focuses on potentially misleading information (PMI) and potentially helpful information (PHI) embedded in the 2-D sketch of a 3-D geometric object. Our quest was to discover whether and how PHI and PMI capture visual obstacles, for high-school students, to desirable comprehension of the sketches used in teaching spatial geometry. We compared the anticipated difficulty of 24 sketches of cubes with different auxiliary constructions, according to their orientation and to the ratio #PHI/#PMI, with the actual difficulty reflected in the scores received by 174 high-school students for comprehending these sketches. The findings suggest: (a) deviations from a normative sketch of a cube affect spatial comprehension; (b) the ratio #PHI/#PMI accounts for a significant part of the students' visual difficulty. Success in learning spatial geometry in high school is frequently attributed to a student's ability to visualize 3-D geometric configurations from 2-D sketches . This attribution is based on the presupposition that human vision and cognition have a high capacity of pattern recognition and synthesis . However, as shown by many researchers, this capacity by itself is not enough to enable an easy completion of missing information in the sketches (
We model the Lights Out game on general simple graphs in the framework of linear algebra over the... more We model the Lights Out game on general simple graphs in the framework of linear algebra over the field $$\mathbb{F}_{2}$$ F 2 . Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring $$\mathbb{Z}_{n}$$ Z n . Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.
Let F p,q be the generalized friendship graph K 1 (pK q ) on pq + 1 vertices obtained by joining ... more Let F p,q be the generalized friendship graph K 1 (pK q ) on pq + 1 vertices obtained by joining a vertex to all vertices of p disjoint copies of the complete graph K q on q vertices. In this paper, we prove that F p,q is determined by its normalized Laplacian spectrum if and only if q ≥ 2, or q = 1 and p ≤ 2.
The program of the conference consisted of thirty six half hour talks, covering a wide range of t... more The program of the conference consisted of thirty six half hour talks, covering a wide range of topics in theoretical and applied Linear Algebra, an open problem session, and informal discussion. The social activities included a reception at the Technion Faculty Club, a banquet, and an Israeli folklore performance.
For each of several S G R", ", those linear transformations 9: R", " -+ R", n which map S onto S ... more For each of several S G R", ", those linear transformations 9: R", " -+ R", n which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M-matrices, the totally nonnegative matrices, the Dstable
and should be submitted in accordance with the instructions given on the inside front cover. An a... more and should be submitted in accordance with the instructions given on the inside front cover. An asterisk placed beside a
It is well known that mathematics students have to be able to understand and prove theorems. From... more It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using ...
The journal of secondary gifted education, Aug 1, 2005
aul¹, a 16-year-old member of the team representing Israel in the International Mathematics Olymp... more aul¹, a 16-year-old member of the team representing Israel in the International Mathematics Olympiad (IMO), was interviewed in a study of gifted and average students' strategic behaviors in mathematical problem solving². At the beginning of the interview, the following dialogue took place between Saul and the interviewer: How do you approach difficult geometry problems? Saul:
In this chapter we present our work aimed at interweaving e-learning and face-to-face learning in... more In this chapter we present our work aimed at interweaving e-learning and face-to-face learning in Calculus courses for undergraduate engineering students. This type of blended learning (BL) contains the best properties of e-learning and face-to-face learning and helps overcome many obstacles in traditional teaching. We use our approach in order to improve students' conceptual understanding of theorems. We describe online assignments specifically designed to help students better understand the meaning of a theorem. These assignments are given to students in addition to traditional lectures and tutorials with the objective that they can learn to learn on their own. Students "discover" the theorem and study it independently, by using a "bank" of examples and a lot of theoretical exercises we supply. The assignments are built in such a way that students receive feedback and instructions in response to their Web-based activity.
The length of a (0,l) matrix (a bigraph) is defined and studied *The research was carried out dur... more The length of a (0,l) matrix (a bigraph) is defined and studied *The research was carried out during a visit at the Centre de Recherches Mathkatiques. 'Research supported by Grant DGES-FCAC-FS. 'We use the matrix-theory notation of [3] and the graph-theory notation of [I].
Block-Hankel-matrix inversion and the partial-realization problem, by P. A. Fuhrmann, Ben-Gurion ... more Block-Hankel-matrix inversion and the partial-realization problem, by P. A. Fuhrmann, Ben-Gurion University at the Negev.
This chapter discusses the iterative methods for solving systems of linear equations. The iterati... more This chapter discusses the iterative methods for solving systems of linear equations. The iterative methods are suited for problems involving large sparse matrices, much more so in most cases than direct methods such as Gaussian elimination. The chapter discusses the theory of nonnegative matrices. The theory of M-matrices has played a fundamental role in the development of efficient methods for establishing the convergence and in accelerating the convergence of the iterative methods. The chapter presents the formulas for the classical Jacobi and Gauss–Seidel iterative methods. The convergence results are related to two common classes of situations: (1) those in which A is a nonsingular M-matrix and second and (2) those in which A is hermitian and/or positive definite. A slight modification of the Gauss–Seidel method by the use of a relaxation can be used to produce a method, known as the successive over-relaxation (SOR) method, which can dramatically reduce the number of iterations in certain cases.
International Group for the Psychology of Mathematics Education, 2014
This study focuses on potentially misleading information (PMI) and potentially helpful informatio... more This study focuses on potentially misleading information (PMI) and potentially helpful information (PHI) embedded in the 2-D sketch of a 3-D geometric object. Our quest was to discover whether and how PHI and PMI capture visual obstacles, for high-school students, to desirable comprehension of the sketches used in teaching spatial geometry. We compared the anticipated difficulty of 24 sketches of cubes with different auxiliary constructions, according to their orientation and to the ratio #PHI/#PMI, with the actual difficulty reflected in the scores received by 174 high-school students for comprehending these sketches. The findings suggest: (a) deviations from a normative sketch of a cube affect spatial comprehension; (b) the ratio #PHI/#PMI accounts for a significant part of the students' visual difficulty. Success in learning spatial geometry in high school is frequently attributed to a student's ability to visualize 3-D geometric configurations from 2-D sketches . This attribution is based on the presupposition that human vision and cognition have a high capacity of pattern recognition and synthesis . However, as shown by many researchers, this capacity by itself is not enough to enable an easy completion of missing information in the sketches (
We model the Lights Out game on general simple graphs in the framework of linear algebra over the... more We model the Lights Out game on general simple graphs in the framework of linear algebra over the field $$\mathbb{F}_{2}$$ F 2 . Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring $$\mathbb{Z}_{n}$$ Z n . Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.
Let F p,q be the generalized friendship graph K 1 (pK q ) on pq + 1 vertices obtained by joining ... more Let F p,q be the generalized friendship graph K 1 (pK q ) on pq + 1 vertices obtained by joining a vertex to all vertices of p disjoint copies of the complete graph K q on q vertices. In this paper, we prove that F p,q is determined by its normalized Laplacian spectrum if and only if q ≥ 2, or q = 1 and p ≤ 2.
The program of the conference consisted of thirty six half hour talks, covering a wide range of t... more The program of the conference consisted of thirty six half hour talks, covering a wide range of topics in theoretical and applied Linear Algebra, an open problem session, and informal discussion. The social activities included a reception at the Technion Faculty Club, a banquet, and an Israeli folklore performance.
For each of several S G R", ", those linear transformations 9: R", " -+ R", n which map S onto S ... more For each of several S G R", ", those linear transformations 9: R", " -+ R", n which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M-matrices, the totally nonnegative matrices, the Dstable
and should be submitted in accordance with the instructions given on the inside front cover. An a... more and should be submitted in accordance with the instructions given on the inside front cover. An asterisk placed beside a
It is well known that mathematics students have to be able to understand and prove theorems. From... more It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using ...
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