1. MATHEMATICAL VISUALISATION Mathematical Association of America
THE POWER OF
MATHEMATICAL VISUALISATION
James Tanton
Mathematical Association of America

2. What made me a mathematician:
Designed by Macgregor Campbell @mainsequence

3. EPIPHANY …
Purple = 13
White = 12

4. Visualization in the Curriculum
My culture shock: Drawing pictures in algebra and other high-school math
classes very much eschewed. (Looked down upon even!)
* “Visual” or “Visualization” appears 34 times in the ninety-three
pages of the U.S. Common Cores State Standards
- 22 times in reference to grade 2-6 students using visual models for fractions
- 1 time in grade 2 re comparing shapes
- 5 times re representing data in statistics and modeling
- 4 times re graphing functions and interpreting features of graphs
- 2 times in geometry re visualizing relationships between two- and three-dimensional objects.
* Alberta curriculum: Recognised as core HS mathematical process:
[V] Visualization “involves thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the world” (Armstrong, 1993, p. 10). The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them.

5. One historical change away from the visual:

6. The vinculum makes complicated expressions straightforward to unravel.
Compare this with
etc.
Why the change?

7. radix The vinculum still makes an appearance today.
Is this 10 or is it 4?
radix
Line segment
Line segment AB.

8. Obviate student mistakes!
This vinculum meaning “group.”

10. Two examples of using the power of Mathematical Visualisation
in the curriculum.
Algebra: Piles and Holes
Probability: Garden paths and Area
Other examples appear in other sessions:
Area Model: Using it throughout the curriculum
Exploding Dots: a K-12 (and beyond!) story
Brilliant Thinking: Summations, Quadratics, and loads more

11. PILES AND HOLES
A single visual says it all.

12. A story that isn’t true.
When I was a child my parents sat me in a sandbox.

13. But then one day I had the most astounding flash of insight of all!
YOUR TURN:
What’s opp 2 ?
What’s opp 5?

14. I realized too that the opposite of a hole is a pile.
YOUR TURN:
What is the opposite of the opposite of a hole?
b) What is the opposite of the opposite of the opposite of three piles?
c) What is the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of seven holes?
In this untrue story I had discovered negative numbers.

15. One problem …
Society doesn’t use the term “opp.” It uses a little dash to denote opposites.
= 1
Also, society doesn’t like this.
It invented this notion of “subtraction” for the addition of the opposite: 3 – 2 = 1.
ANNOYING!

16. Read
10 – 5 as (Ten piles and five holes)
-5 – 7 as (Five holes and seven holes)
6 – 4 +1 as (Six piles and four holes and 1 pile)
SUBTRACTION IS JUST THE ADDITION OF THE OPPOSITE.

17. Distributing the Negative Sign
This is the opposite of … 3 piles and 2 holes.
- (3 + -2) =
Clearly that is … 3 holes and 2 piles.
- ( ) + 5 =
6 – (5 – 2) = 6 –
2 – (20 – x) = 2 – x = x - 18

19. PROBABILITY via GARDEN PATHS
Send a large number of people down a forked garden path.

21. Roll a die and then toss a coin. What is P(EVEN and HEAD) ?

22. I toss a small coin and then a big coin and then roll a die.
What is:
P( HEAD and HEAD and {5,6}) ?

23. Infinite geometric series formula.

24. A tough real-world problem handled well with the area model.

25. THANKS! JAMES TANTON tanton.math@gmail.com www.gdaymath.com
OTHER WEBSITES:
All my EARCOS 2017 PowerPoints appear here:
Problem-Solving in the Classroom:
MAA’s CURRICULUM INSPIRATIONS