Mathematics for Elementary Teachers

Problem Solving

George Polya (1887-1985)

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Mathematician and Teacher of 20th Century

"..'solving a problem means finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable'.." (Billstein 2)

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Quote by George Polya pulled from Mathematical Discovery and re quoted in A Problem Solving Approach to Mathematics for Elementary School Teachers 11th edition by Rick Billstein.

Major contribution was his work in Problem Solving

Steps

1. Identify the problem

Understand the problem

2. Devise a plan

Guess and check

Think of solutions and plug into the problem to check answer

Solve a simpler problem

Use an easier problem based on your current problem

Work backwards

Use a model

Create charts, lists, and use objects

3. Carry out the plan

Be persistant with your current plan

Revise the plan if necessary

If the solution is not visible, rethink the plan

4. Look back (reflect)

Does the answer make sense?

Subtopic

Does it need to be revised?

Is there an easier way to solve it?

Sequences

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Sequence: an ordered list of objects, events, or numbers

Composed of elements (objects found within)

Types of Sequences

Arithmetic Sequences

Sequences or numbers with a common difference

ex. 2, 4, 6, 8, 10

Subtopic

common difference of 2, 4, 6, 8 is 2

Adding 2 to each time will create the next number

Geometric Sequences

Sequence of numbers with a common ratio

ex. 8, -4, 2,-1,..

common ratio -1/2

Recurrence relationship Sequences

Defines a sequence in which the current term is dependent on previous term(s)

ex. 6, 13, 27, 55..

Difference: multiply by 2 then add 1

Product of 1st term 13

13 multiplied by 2 equals 26 then add 1