2. Fibonacci Series


a. The Fibonacci Series for Elementary School Students

Children learn by pattern recognition and by applying previous learning to the extension of problems they already know how to solve.  Given the pattern 1, 2, ,what is the next number in the sequence?  Because there are many possibilities we need to be given more information.  If we were told that it was an arithmetic sequence, the next number would be 3, while if we knew that it was a geometric sequence, the next number would be 4, because 2 x 1 = 2 and 2 x 2 = 4.  Yet, if we  were told that the sequence is 1, 2, 3, 5, , you would know that it does not fit the pattern of either an arithmetic or geometric sequence.  Based upon previous knowledge, you might guess it to be a prime number sequence and the next number would be 7.  If, though, we were told the sequence was 1, 2, 3, 5, 8, , you would have to try another strategy.  At this stage even some first grade students will recognize that the next number is obtained by adding the last two, making the sequence  1, 2, 3, 5, 8, 13, ....

Somewhere around this stage all the students would begin to see the pattern and guess that the next number is 21.  You can see it takes a number of trials with this method of delivering the lesson.  It is part of the process to teach students to think beyond what they know.  Educators often shortchange this process by using very simple repetitive sequences or playing simple games that may be fun but do not effectively give the proper incentive to truly develop the thought processes.  Many will terminate the session here with no further follow up.

Some very creative instructors (and even some children) will ask to continue the sequence to the left instead of the right.  This process can be tried with some second graders.  I have even seen adults who have had trouble with this simple but novel look at the problem.  Now instead of adding the last two numbers, we subtract the leftmost number from the number to its right.  Let us look as the sequence develops:

2 - 1   =   1                1, 2, 3, 5, 8, 13, ...

                                   1 - 1   =   0                1, 1 ,2, 3, 5, 8, 13, ...
                                   1 - 0   = 
1                 0, 1, 1, 2, 3, 5, 8, 13,

The sequence eventually becomes 13, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, , .

The same problem as seen by a middle school student