Abstract
Abstract In this paper, we present several different approaches to formula for the sum of integer powers of two in accordance with different representations of this sum or different algebraic methods for its computation. Our long-term experience shows the effectiveness of discussion on this theme for enhancing interest and creative thinking of the students about solutions of various problems, not only in mathematics but also in others fields of knowledge.
Highlights
The purpose of this article is to draw the attention of high school, college and university teachers to didactic possibilities of different representations of one wellknown problem that in most tutorials is presented in the same key
Different approaches to one task, especially to such a basic, as the sum 1 + 2 + 22 + ... + 2n let the student see a lot of mathematical ideas in simple terms, improving understanding and getting interested in learning mathematics
Mathematics teachers are always looking for new challenges for their students to attract interest in learning mathematics and provoke creative thinking
Summary
The purpose of this article is to draw the attention of high school, college and university teachers to didactic possibilities of different representations of one wellknown problem that in most tutorials is presented (and taught) in the same key. Mathematics teachers are always looking for new challenges for their students to attract interest in learning mathematics and provoke creative thinking Such trials may be connected to different mathematical problems, but they may integrate different themes by applying different methods to solving a single problem. We will show several different approaches to discover and prove the formula for the sum of non-negative integer powers of two: n These approaches are based on simple arithmetic and algebraic operations, or on finding interesting connections (analogies) of this sum with some real or abstract processes. We believe that such a diversity of approaches to the single problem stimulates creative thinking and increases interest in the study of mathematics. We only quote one of them: “Creativity is a process of becoming sensitive to problems, deficiencies, gaps in knowledge, missing elements, disharmonies, and so on; identifying the difficult; searching for solutions, making guesses or formulating hypotheses about the deficiencies; testing and re-testing these hypotheses and possibly modifying and re-testing them; and communicating the results.” (Torrance, 1966, p. 8)
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