Attractive mathematical induction
University
of Latvia
ATTRACTIVE MATHEMATICAL
INDUCTION
Aija Cunska
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.
Introduction
Herbert S. 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. 1).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.method of
mathematical induction can be compared with the progress. We start with the
lower degree and, as a result of logical judgments; we come to the general
conclusion (result). The man always tries to advance, tries to develop his
ideas in a logical way, consequently, nature itself makes the man think in an
inductive way.natural beginning of how to prove complicated mathematical things
is to view simple cases. It helps us to visually understand what is required by
the task and gives us essential hints on how to come up to proof.we have to do
is to make the first step towards the result. 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. This is shown in Figure 1.
Figure 1. Rung principle
Figure 2. Domino principle
Different Ways of Presenting Mathematical Induction
Many authors compare mathematical induction to
dominoes toppling in succession. (Gunderson, 2011, p. 4). Suppose that: 1) We
can knock down the first domino; 2) the dominos are so close, that each
previous will knock the following one down when falling. Then all the dominos
will be down, as shown in Figure 2.
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.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.
6.
Task 2 value representation in Excel spreadsheet
. Carefully describe the statement to be proved and any
ranges on certain variables.
. The base step: prove one or more base cases
. The inductive step: show how the truth of one statement
follows from the truth of some previous statement (s).
. State the precise conclusion that follows by mathematical
induction.
Variants
of Finite Mathematical Induction
mathematics induction multimedia training
There are many forms of mathematical induction - weak,
strong, and backward, to name a few. In what follows, n is a variable denoting
an integer (usually nonnegative) and S(n) denotes a mathematical statement with
one or more occurrences of the variable n. (Gunderson, 2011, p. 35).method of
mathematical induction can be successfully illustrated. The assertion S(n) can
be depicted with a line of squares:
If S(1) is veritable, then we can color the first square:
But the condition "from every natural k, if S(k)
assertion is true, follows the verity of the assertion S(k+1), then the
assertion S(n) is true for all the natural n" in geometric way means the
following transition:
In that way we get a belt where the first two squares are
colored:
By repeating the transition one more time, we get a belt
where the first three squares are colored:
Consequently, if we continue in the same way, then gradually
all the infinite belt will be colored and the general assertion n will be
proved. One of the basic schemes
of mathematical induction is Weak Mathematical Induction: Let S(n) be a
statement involving n. If S(1) holds, and for every k ≥ 1, S(k) ® S(k+1), then for every n ≥
1, the statement S(n) holds. This can be depicted as follows:
For example, Task 3: Prove that for n ≥ 2, 4n2
> n+ 11.induction scheme is Strong Mathematical Induction: Let S(n) denote a
statement involving an integer n. If S(k) is true and for every m ≥ k,
S(k) Ù S(k+1) Ù … Ù 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. (Andžāns,
Zariņš, 1983, p. 70-99)
The
Value of Multimedia in Learning
Multimedia learning is the process of learning, usually in a
classroom or similarly 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 (#"535105.files/image017.gif">
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.
List
of References
1. Andžāns,
A., Zariņš, P. (1983). Matemātiskās indukcijas metode un
varbūtību teorijas elementi. Rīga: Zvaigzne
. France,
I., France, I., Slokenberga, E. (2011). Komplektizdevums „Matemātika 10.
klasei".
Rīga: Izdevniecība LIELVĀRDS.
3. Grunschlag, Z. (2002). Induction. Retrieved
April 7, 2011, from http://www.slidefinder.net/2/202_20induction/ 19762525
. 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 http://www.mathsisfun.com/
games/towerofhanoi.html
. 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
. 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
http://ru.wikibooks.org/wiki
. Wiesen, G. (2003). What Is Multimedia
Learning? Retrieved April 7, 2011, from http://www.wisegeek.com/
what-is-multimedia-learning.htm