Reading Mathematical Texts as a Problem-Solving Activity: The Case of the Principle of Mathematical Induction

Reading mathematical texts is closely related to the effort of the reader to understand its content; therefore, it is reasonable to consider such reading as a problem-solving activity. In this paper, the Principle of Mathematical Induction was given to secondary education students, and their effort to comprehend the text was examined in order to identify whether significant elements of problem solving are involved. The findings give evidence that while negotiating the content of the text, the students went through Polya’s four phases of problem solving. Moreover, this approach of reading the Principle of Mathematical Induction in the sense of a problem that must be solved seems a promising idea for the conceptual understanding of the notion of mathematical induction.

Arithmetic, philosophical issues in

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.

UMA SOLUÇÃO PARA O PROBLEMA DE FLÁVIO JOSEFO

http://dx.doi.org/10.5902/2179460X14613In this paper, we present a didactic sequence of activities designed for a group of high school students, aged 15-18 years. Our main objective is to study a problem proposed by the mathematician Josephus, around the year 64. Initially, we review some content like numerical sequences, including the special cases of the Arithmetic and Geometric Progressions. Then, we introduce some notions about Recurrence Relations and the Principle of Mathematical Induction, allowing the generalization of some concepts and results which are already known intuitively by the group of students. This article is part of the dissertation entitled A Proposed Approach to Josephus Problem Applied to High School prepared by the student Erondina Marcia D. S. da Silva under the guidance of the Professor Dr. Gobbi Luciane Tonet.

Improving awareness about the meaning of the principle of mathematical induction

This work is based on our conviction that it is possible to minimize difficulties students face in learning the principle of mathematical induction by means of clarifying its logical aspects. Based on previous research and theory, we designed a method of fostering students’ understanding of the principle. We present results that support the effectiveness of our method with teachers in training who are not specializing in mathematics. Fomentar la conciencia sobre el significado del principio de inducción matemática Este trabajo está basado en nuestra convicción de que es posible minimizar las dificultades de los alumnos cuando se enfrentan al aprendizaje del principio de inducción matemática mediante la clarificación de sus aspectos lógicos. Basándonos en la investigación y teoría previas, diseñamos un método para fomentar la comprensión del principio por los alumnos. Presentamos resultados que respaldan la efectividad de nuestro método con profesores en formación no especializados en matemáticas.Handle: http://hdl.handle.net/10481/3507

m-Dimensional Schlömilch Series

By using the principle of mathematical induction a simple algebraic formula is derived for an m-dimensional Schlömilch series. The result yields a countably infinite number of representations for null-functions on increasingly larger open intervals.