Olen Dias

Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by means of our new modification of Newtons iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, Chebyshev?Vandermonde-like, and Toeplitz-like matrices, where again the computations are confined to operations with short generators of the involved matrices.

Creativity Research in Mathematics Education Simplified: Using the Concept of Bisociation as Ockham’s Razor

This chapter proposes that bisociation, the Koestler theory of the creativity of the “Aha!” Moment, is the Ockham Razor for creativity research in mathematics education. It shows the power of bisociation in simplifying unnecessary components and in the synthesis of the fragmented ones. The discussion leads through the relationship of bisociation with Piaget’s reflective abstraction, proposes cognitive/affective duality of the “Aha!” Moment and enriches Mason’s theory of attention by the new structure of simultaneous attention necessary for the Eureka experience.

Rate and Proportion Teaching Sequence

Years of teaching remedial mathematics has taught me that traditional, or lecture, method do not always work with our students. Modelling a solution repeatedly and expecting the student to mimic without understanding doesn’t help either. When a question involving rates and proportions is asked, I am confident that a few students will be able to give the correct answer.

Teaching Research Communities

We have spoken of the traditional view of mathematics in the classroom in which the teacher is the authoritarian voice and students – passive recipients of his/her knowledge, and the alternate or reform view exposed by constructivists, as well as Koestler and most prominent mathematical educational institutions in which the focus is on the role of students as active participants in the construction of knowledge.

Tools of Teaching-Research

Vrunda Prabhu remarked at the end of Unit 2 that Teaching-Research, just like mathematics is not a spectator sport. Unit 3 develops that view in an effort to encourage colleagues, teachers of mathematics in schools and colleges to pick up the mantle of this unusually creative bisociative approach to teaching and research. Involvement in teaching-research is not particularly difficult especially taking into account again the comment of William J. Harrington, describing his work of a teacher-as-researcher in Laura R. Van Zoest (2006)

Teacher as the Designer of Instruction

Unit 4 presents the designs of TR investigations and teaching experiments conducted by teacher-researchers in their classrooms. It represents three type of TR activity: daily classroom TR investigations (Chapters 4.2–4.5), construction of learning trajectories through iterated classroom teaching experiments (Chapters 4.6–4.8) and two teaching experiments of opposite types (Chapter 4.9) and (Chapter 4.10).

Creativity Research and Koestler

This section provides an overview of research on creativity in mathematics education: Koestler’s understanding of creativity, the role of creativity with mathematics students, especially those who do not view themselves as gifted and a discussion on the how to support a creative learning environment within the mathematics classroom.

Comparative Study of Three Approaches to Teaching Rates

This chapter has a special aim and organization. It presents the comparative study of three different approaches to the same theme of proportional reasoning, in particular of the concept of the rate. The idea for the study is related to the Chinese Keli lesson study method (Huang & Bao, 2006), whose one of the approaches is the observation of two classes taught by different instructors and presenting different approaches to the same theme (CTRAS 5, 2013) followed by the discussion comparing the instructional approaches.

The Main Themes of the Book

The aim of this introductory unit is to present the foundations of the work, which are based upon two conceptual frameworks, TR/NYCity Model and Koestler theory (1964) of the Act of Creation The coordination between teaching practice and theory (here Koestler theory) – a fundamental component of teaching-research has taken place during the teaching experiment by Vrunda Prabhu and became one of the thematics threads throughout the book.

Underserved Students and Creativity

In this section we discuss the educational issues of immigrant students who frequently make up a sub-cultural often characterized by different language, race and a higher rate of poverty than the predominant culture. In addition to language these students frequently struggle with mathematics both cognitively and in the affect domain, alongside identity issues of what college means to the individual often being the first generation in their family to go to college.