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f(x)Functional Thinking
11.5.13 Art Bardige
Perhaps the most important concept of mathematics is that of function, which provides us with the means to study dependence and change.
Mathematics Professor Peter Kronheimer, Director of Undergraduate Studies
We think functionally when we use the scientific method…
And when we use spreadsheets for business.
Functional thinking can change the way students learning problem solving.
Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?
To understand this revolutionary pedagogy let us see “from whence we came.”
Copernicus – On the Revolutions 1543
In 1202 Leonardo of Pisa introduced algorithmic thinking as “arithmetic necessary to merchants.”
Enabled by a new technology -- paper
Algorithmic thinking replaced abacist thinking
Algorist vs. abacist 1507
In 1637 Rene Descartes invented a new concept.
In 1673 Gottfried Leibniz named it “function.”
In 1686 Isaac Newton based science on cause and effect -- functional thinking.
In 1734 Leonard Euler gave us the symbol for function -- f(x).
M. C. Escher
In the 19th and 20th centuries functions became transformations
In 1979 Bob Frankston and Dan Bricklin revolutionized business math with spreadsheets
Enabled by a new technology
Replacing algorithmic thinking with functional thinking in business
Yet in classrooms we rarely if ever learn to think functionally
We still concentrate on algorithmic thinking to solve exercises.
...students learned to use…
WhatWhat if…if…
functional thinking organizes problem solving into:
• Inputs• Rules• Outputs
is analytic
Functional Thinking
is dynamic
Functional Thinking
Spreadsheets make functional thinking:
Where we focus on patterns:
Science April ‘88
Visual
Tables
Graphs and Charts
Formulas
Concrete
Spreadsheet Math
Place Value
=======> 0 0 6 9 89
8 Ones = 8
9 = 9 0
6 = 6 0 0
0 = 0 0 0 0
0 = 0 0 0 0 0
I made you a picture to let you play with place value. Type in a number (numeral from 0 to 9 in each place) and watch the bars change. Now look below. How many ones are there? How many tens are there? How many hundreds are there?
Thousands
Ten Thousands
Place Value
Tens
Hundreds
0
1
2
3
4
5
6
7
8
9
1
Flexible
Guess and Check Lesson 1: Introducing Spreadsheets
1
2
Step Input 1 + Input 2 = Sum Check
1 1 + 5 + 6
2 2 + 5 + 7
3 + 5 + 8
3 4 + 5 + 9
4 5 + 5 + 10
5 6 + 5 + 11
7 + 5 + 12
How many did you get right?
Did I bring your table from Input to this Check sheet?
Check every answer.
Adding with a spreadsheet
If you want the spreadsheet to add 2+3 in the check column, click in the cell and type =2+3 Enter.
Now try adding other numbers on your spreadsheet.
Spreadsheet Math
Check your answer
How do you think you could get the spreadsheet to check your answer?
Do you have a guess on how. Write it down here.
29
Start with a problem and a conjecture to answer it.
1
2
3 Comment
What do you guess the answer is. Write your conjecture here--->
Setup the Problem
I think I know how to solve it using the spreadsheet
If you think you can do this already using functions and tables, then go to the
Workspace page and build the table and show us the solution. Be sure that you make the page beautiful and layout the table and headings so that they are clear and anyone can understand them. You can do this anytime.
|WHAT|IF|MATH|
A Problem
Two painters Picasso and Vermeer are going to paint a house. Picasso by himself will take 8 hours to paint the house and Vermeer by himself will tak 12 hours to paint the house. How long will it take them working together to paint the house?
Work Problems: House Painting
30
...a table with inputs and outputs
Represent the function with…
3
Input Output Input Output
What is Vermeer's painting rate? Use it to help you fill in the table?
Build a table for each painter.
Picasso Vermeer How much of the house did Vermeer paint between the end of hour 3 and the end of hour 7?