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\"This book shares ideas about integrating mathematical problem posing with the use of computing technology in the context of K-12 mathematics teacher preparation. Problem posing has been on the mathematics education agenda for a long time. Over the centuries, it appeared in different didactic forms as a way of enriching one's learning experience through investigating mathematical ideas, exploring conjectures, and solving worthwhile problems. In the digital era, mathematics teacher training may include learning the skill of formulating different mathematical problems that the appropriate use of technology affords. As shown in the book, problem posing skills can be supported by two major theoretical positions that stem from technology integration: didactical coherence of a posed problem (Chapter 2) and technology-immune/technology-enabled (TITE) problem posing (Chapter 3). In order to connect theory and practice of mathematical problem posing in the digital era, the book includes examples of problems posed by teacher candidates enrolled in different technology-rich mathematics education courses taught by the author over the years. These examples are analyzed through the lenses of the proposed theory. In addition, the book shows how technology can be used to reformulate rather advanced problems from the traditional (pre-digital era) problem-solving curriculum. The goal of reformulation of such problems is at least two-fold: to make them compatible with the modern-day emphasis on democratizing mathematics education, and to find the right balance between positive and negative affordances of technology. In particular, an argument can be made that through achieving such a balance one uses technology appropriately\"-- Provided by publisher.

The material of this book stems from the idea of integrating a classic concept of Fibonacci numbers with commonly available digital tools including a computer spreadsheet, Maple, Wolfram Alpha, and the graphing calculator. This integration made it possible to introduce a number of new concepts such as: Generalized golden ratios in the form of cycles represented by the strings of real numbers; Fibonacci-like polynomials the roots that define those cycles' dependence on a parameter; the directions of the cycles described in combinatorial terms of permutations with rises, as the parameter changes on the number line; Fibonacci sieves of order k; (r, k)-sections of Fibonacci numbers; and polynomial generalizations of Cassini's, Catalan's, and other identities for Fibonacci numbers. The development of these concepts was motivated by considering the difference equation f_(n+1)=af_n+bf_(n-1),f_0=f_1=1, and, by taking advantage of capabilities of the modern-day digital tools, exploring the behavior of the ratios f_(n+1)/f_n as n increases. The initial use of a spreadsheet can demonstrate that, depending on the values of a and b, the ratios can either be attracted by a number (known as the Golden Ratio in the case a = b = 1) or by the strings of numbers (cycles) of different lengths. In general, difference equations, both linear and non-linear ones serve as mathematical models in radio engineering, communication, and computer architecture research. In mathematics education, commonly available digital tools enable the introduction of mathematical complexity of the behavior of these models to different groups of students through the modern-day combination of argument and computation. The book promotes experimental mathematics techniques which, in the digital age, integrate intuition, insight, the development of mathematical models, conjecturing, and various ways of justification of conjectures. The notion of technology-immune/technology-enabled problem solving is introduced as an educational analogue of the notion of experimental mathematics. In the spirit of John Dewey, the book provides many collateral learning opportunities enabled by experimental mathematics techniques. Likewise, in the spirit of George Plya, the book champions carrying out computer experimentation with mathematical concepts before offering their formal demonstration. The book can be used in secondary mathematics teacher education programs, in undergraduate mathematics courses for students majoring in mathematics, computer science, electrical and mechanical engineering, as well as in other mathematical programs that study difference equations in the broad context of discrete mathematics.