Review of concepts, forms and different ways of representing the methods of mathematical induction, characterization of its ideas and principles. Features of a multimedia learning object students and teachers on the example of the University of Latvia.
Аннотация к работе
Wilf, Professor of Mathematics from the University of Pennsylvania has said: "Induction makes you feel guilty for getting something out of nothing, and it is artificial, but it is one of the greatest ideas of civilization." (Gunderson, 2011, p. Mathematical induction is like real life when a little sprout grows and blossoms into a magnificent flower, when a small acorn transforms into a huge oak tree, when two cohabiting people develop a family, when substantial aims are born of a simple thought, when a single drop of water creates a puddle, when great love thrives from a single sight, and when a large house is built by putting together brick by brick. That’s the basic idea behind what is called "the principle of mathematical induction": in order to show that one can get to any rung on a ladder, it suffices to first show that one can get on the first rung, and then show that one can climb from any rung to the next. To understand the method of mathematical induction, several teachers of mathematics both in Latvia and abroad, make students solve the task about the Towers of Hanoi, invented by the French mathematician Edouard Lucas in 1883. The figure demonstrates that the number of handshakes for one person equals to 0, two persons have one handshake, three persons - 3 handshakes, four persons - 6 handshakes, five persons - 10 handshakes and six persons - 15 handshakes.Mathematical induction teaches students not only mathematics but also life - in order to develop we need to start with the minimum, take the first rung, the first step. The story of mathematical induction coincides with several verities of life, for example, the famous French author Antoine de Saint-Exupery said: "To be a man is to be aware, when setting one stone, that you are building a world." Students accept, understand and love things that are related to life and reality. Therefore it is important that students have practical work: use domino, build towers of Hanoi, make visual models of tasks, calculate statement values in Excel spreadsheets for n = 1, 2, 3, 4, 5, 6... and only then they can move to the general and complicated cases when n = k and n = k 1. The Internet is also rich in materials, for example, the search engine Google listed 1 310 000 results for the searched phrase "mathematical induction" on 18 April 2011. Gunderson, Professor of Mathematics at the University of Manitoba have described in their books the possibility of using schemes to depict methods of mathematical induction.
Вывод
Mathematical induction teaches students not only mathematics but also life - in order to develop we need to start with the minimum, take the first rung, the first step. The story of mathematical induction coincides with several verities of life, for example, the famous French author Antoine de Saint-Exupery said: "To be a man is to be aware, when setting one stone, that you are building a world." Students accept, understand and love things that are related to life and reality. Therefore it is important that students have practical work: use domino, build towers of Hanoi, make visual models of tasks, calculate statement values in Excel spreadsheets for n = 1, 2, 3, 4, 5, 6... and only then they can move to the general and complicated cases when n = k and n = k 1.
Lots of books have been written about the method of mathematical induction. The Internet is also rich in materials, for example, the search engine Google listed 1 310 000 results for the searched phrase "mathematical induction" on 18 April 2011. Whereas signs of interactivity were present only in two search results: 1) interactive test (http://www.themathpage.com/aprecalc/ precalculus.htm) and 2) the POWERPOINT presentation (http://www.slidefinder.net/ 2/202_20 induction/19762525). Only two authors: Agnis And?ans and Pcteris Zario?, Professors at the University of Latvia and David S. Gunderson, Professor of Mathematics at the University of Manitoba have described in their books the possibility of using schemes to depict methods of mathematical induction. These schemes are easier understood by students if placed into interactive environment, for example, Excel spreadsheets or Multimedia learning object.
This work has been supported by the European Social Fund within the project "Support for Doctoral Studies at University of Latvia".
List of References
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