Attractive mathematical induction
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Другие материалы по предмету Математика и статистика
S(m) S(m+1) then for every n ? k, the statement S(n) is true. This can be depicted as follows:
For example, Task 4: Prove that an = 5 . 2n - 3n+1, if a1 = 1, a2 = -7 and an+2 = 5an+1 - 6an for all n ? 1.another induction scheme is Downward Mathematical Induction: Let S(n) be a statement involving n. If S(n) is true for infinitely many n, and for each m ? 2, S(m) S(m-1) then for every n ? 1, the statement S(n) is true. Its graphical depiction is:
For example, Task 5: Prove that the statement "the geometric mean of n positive numbers is not larger than the arithmetic mean of the same numbers" is true, i.e.,
At schools, teaching the method of mathematical induction, usually the simplest schemes are covered however more complicated schemes can describe parallel mathematical induction and structural or two-dimensional mathematical induction. (Andzans, Zarins, 1983, p. 70-99)
The Value of Multimedia in Learning
Multimedia learning is the process of learning, usually in a classroom or wordsly structured environment, through the use of multimedia presentations and teaching methods. This can typically be applied to any subject and generally any sort of learning process can either be achieved or enhanced through a careful application of multimedia materials. Multimedia learning is often closely connected to the use of technology in the classroom, as advances in technology have often made incorporation of multimedia easier and more complete.general, the term "multimedia" is used to refer to any type of application or activity that utilizes different types of media or formats in the presentation of ideas.is the combination of various digital media types, such as text, images, sound, and video, into an integrated multisensory interactive application or presentation to convey a message or information to an audience. (Shank, 2005, p. 2).helps people learn more easily because it appeals more readily to diverse learning preferences.connection between multimedia learning and technology is usually made because advances in technology often make the use of different media easier and less expensive for schools and teachers. (Wiesen, 2003).
Multimedia Learning Object "Mathematical Induction"
In view of the above suggestions, I used the options offered by the e-learning software Lectora (
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).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.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,
2)gifted students who study for mathematics competitions and olympiads,
)teachers wishing to present the nature and potential of the mathematical induction method in an attractive manner,
)anyone who wants to find out the link between the method of mathematical induction, growth and life processes.
Conclusions
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.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 (
List of References
1.Andzans, A., Zarins, P. (1983). Matematiskas indukcijas metode un varbutibu teorijas elementi. Riga: Zvaigzne
.France, I., France, I., Slokenberga, E. (2011). Komplektizdevums „Matematika 10. klasei". Riga: Izdevnieciba LIELVARDS.
3.Grunschlag, Z. (2002). Induction. Retrieved April 7, 2011, from
.Gunderson, D. S. (2011). Handbook of mathematical induction. Theory and applications. NewYork: Taylor and Francis Group
.Pierce, R. (2008). Maths Fun: Tower of Hanoi. Retrieved April 7, 2011, from
.Seg Research. (2008). Understanding Multimedia Learning: Integrating multimedia in the K-12 classroom. Retrieved April 7, 2011, from
7.Shank, P. (2005). The Value of Multimedia in Learning. USA: Adobe Systems. Retrieved April 7, 2011, from
8.Spector, L. (2011). The Math Page. Topics in Precalculus. Retrieved April 7, 2011, from
.Steinhaus, H. (1983). Mathematical Snapshots. Canada: General Publishing Company, Ltd
.Шульман, T., Ворожцов, A. B. (2011). Знакомство с методом математической индукции. Retrieved April 7, 2011, from
.Wiesen, G. (2003). What Is Multimedia Learning? Retrieved April 7, 2011, from