```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,
We think functionally when we
use the scientific method…
And when we use
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).
In the 19th and 20th
centuries functions
became transformations
M. C. Escher
In 1979 Bob Frankston
and Dan Bricklin
Enabled by a new technology
Replacing algorithmic
thinking with functional
Yet in classrooms we
rarely if ever learn to
think functionally
We still concentrate on algorithmic thinking to solve exercises.
What
if…
students
learned to
use…
...
functional thinking organizes
problem solving into:
• Inputs
• Rules
• Outputs
Functional Thinking
is analytic
Functional Thinking
is dynamic
functional thinking:
Where we focus on
patterns:
Science April ‘88
Tables
Visual
Graphs and
Charts
Formulas
Place Value
Place Value
9
8
7
6
5
4
3
2
Concrete
1
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?
0
1
=======>
0 0 6 9 8
8
9
6
0
0
9
Ones
Tens
=
=
Hundreds
=
Thousands
=
Ten Thousands
=
8
0
9
0
6
0
0
0
0
0
0
0
0
0
0
Guess and Check Lesson 1: Introducing Spreadsheets
1
How do you think you could get the spreadsheet
Do you have a guess on how. Write it down here.
2
Flexible
Input 1
+
Input 2
=
Sum
1
+
5
+
6
2
+
5
+
7
3
+
5
+
8
4
+
5
+
9
4 How many did you get right?
5
+
5
+
10
numbers on your
6
+
5
+
11
7
+
5
+
12
Step
1
2
Did I bring your table from
Input to this Check sheet?
to add 2+3 in the check
column, click in the cell and
type =2+3 Enter.
Check
|WHAT|IF|MATH|
Work Problems: House Painting
1 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?
2 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.
3 Setup the Problem
Comment
29
Represent the function
with…
...a table with inputs and
outputs
3 Build a table for each painter.
Picasso
Vermeer
Input Output
Input Output
How much of the house did
Vermeer paint between the end of
hour 3 and the end of hour 7?
What is Vermeer's painting rate?
30
```