# Attractive mathematical induction

## 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.

 Рубрика Математика Вид реферат Язык английский Дата добавления 11.02.2012 Размер файла 1,1 M

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University of Latvia

# The inductive method plays a significant role in understanding the principle of mathematics. Although, the range of the problems concerning the usage of the mathematical induction method has grown, in school syllabi very little attention is paid to the issue. If mathematical induction teaching methods are improved, more and more students would become interested in it. This is a powerful and sophisticated enough method to be acceptable for the majority. For students the learning process sometimes may seem boring, therefore we can attract their attention with the help of information technologies. It can be done by creating multimedia learning objects. In that way teachers can work easier and faster, paying more attention to practical assignments. The created multimedia learning object "Mathematical induction" serves as successful evidence to that statement.

## Another analogy for mathematical induction is given by Hugo Steinhaus in Mathematical Snapshots in the 1983 (Steinhaus, 1983, p. 299). Consider a pile of envelopes, as high as one likes. Suppose that each envelope except the bottom one contains the same message "open the next envelope on the pile and follow the instructions contained therein." If someone opens the first (top) envelope, reads the message, and follows its instructions, then that person is compelled to open envelope number two of the pile. If the person decides to follow each instruction, that person then opens all the envelopes in the pile. The last envelope might contain a message "Done". This is the principle of mathematical induction applied to a finite set, perhaps called "finite induction". Of course, if the pile is infinite and each envelope is numbered with consecutive positive integers, anyone following the instructions would (if there were enough time) open all of them; such a situation is analogous to mathematical induction as it is most often used.

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. Task 1: three rods and a number of disks of different sizes are given. Only smaller disks may be placed on larger disks. All disks from the first rod have to be moved to the third rod by employing minimum moves, as shown in Figure 3. Several mathematicians have invented programs for visual solution of this task. For example, Figure 4 shows that applet is based on the Tower of Hanoi. Applet created by David Herzog (Pierce, 2008).

Figure 3. Tower of Hanoi

Figure 4. Interactive solution of the task

Many teachers ask their students to create visual models in order to understand mathematical induction. For example, Task 2: At a party, everybody shakes hands with all attendees. If there are n people at the party and each person shakes the hand of each other person exactly once, how many handshakes take place? Handshakes may be described visually, where persons are marked as circles, but handshakes as line segments, as shown in Figure 5:

Figure 5. Visual interpretation of Task 2

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. Students can further make their own conclusions that for n number of persons the number of handshakes will be. This can be easily checked for several n values by using the options in MS Excel, as shown in Figure 6. The n values n = 1, 2, 3 ... are entered in the first row. But the values of expression (n-1) . n : 2 are calculated in the second row. Besides, the values in Excel spreadsheet can be calculated very quickly by using the sensitive point and dragging it with cursor as far as you wish.

## Multimedia Learning Object "Mathematical Induction"

In view of the above suggestions, I used the options offered by the e-learning software Lectora (http://www.trivantis.com) and in 2010 created the multimedia learning object "Mathematical Induction" which starts with the quote: "Mathematical induction is the mode of thinking which makes us think generally and prove that the statement is true for all values."

Figure 7. Basic page of multimedia learning object

Figure 8. Task in multimedia learning object

It includes the following parts:

- Introduction;

- Description of general and separate statements;

- Interactive examples for general statements;

- Description: What is mathematical induction?

- How to graphically depict the method of mathematical induction?

- Seven tasks with solutions and visual depiction of each task, graphical schemes, value calculation in Excel tables and the proof with the help of mathematical induction method;

- Tasks for independent solution (themes: equalities, inequalities, divisibility etc).

Multimedia learning object is attractive, richly illustrated and interactive. For example, by clicking Excel icons you can open electronic spreadsheets and calculate values of the given tasks. Also, the multimedia learning object offers to view videos about the domino effect in operation, about the seed which grows into a beautiful flower and about the erection of the Towers of Hanoi. While the task graphic interpretations or squared lines provide the possibility to view what is hidden behind each tinted square.

The aim of multimedia learning object is to provide learners with the possibility to understand and learn the method of mathematical induction in a user-friendly manner and speed. It is available for students and teachers in Latvia by attending the classes at Extramural Mathematics School of the University of Latvia. It can be used by

1) students learning the method of mathematical induction in accordance with the requirements of mathematics curriculum standards,

3) teachers wishing to present the nature and potential of the mathematical induction method in an attractive manner,

4) anyone who wants to find out the link between the method of mathematical induction, growth and life processes.

## 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 Andювns and Pзteris Zariтр, 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".

## 4. Gunderson, D. S. (2011). Handbook of mathematical induction. Theory and applications. NewYork: Taylor and Francis Group

5. Pierce, R. (2008). Maths Fun: Tower of Hanoi. Retrieved April 7, 2011, from http://www.mathsisfun.com/ games/towerofhanoi.html

6. Seg Research. (2008). Understanding Multimedia Learning: Integrating multimedia in the K-12 classroom. Retrieved April 7, 2011, from http://s4.brainpop.com/new_common_images/files/76/76426_BrainPOP_ White_Paper-20090426.pdf

7. Shank, P. (2005). The Value of Multimedia in Learning. USA: Adobe Systems. Retrieved April 7, 2011, from http://www.adobe.com/designcenter/thinktank/valuemedia/The_Value_of_Multimedia.pdf

8. Spector, L. (2011). The Math Page. Topics in Precalculus. Retrieved April 7, 2011, from http://www. themathpage.com/aprecalc/precalculus.htm

9. Steinhaus, H. (1983). Mathematical Snapshots. Canada: General Publishing Company, Ltd

10. Шульман, T., Ворожцов, A. B. (2011). Знакомство с методом математической индукции. Retrieved April 7, 2011, from http://ru.wikibooks.org/wiki

11. Wiesen, G. (2003). What Is Multimedia Learning? Retrieved April 7, 2011, from http://www.wisegeek.com/ what-is-multimedia-learning.htm

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