THE POWER OF MATHEMATICAL VISUALISATION James Tanton Mathematical

THE POWER OF MATHEMATICAL VISUALISATION James Tanton Mathematical Association of America

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

EPIPHANY … Purple = 13 White = 12

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.

One historical change away from the visual:

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

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

Obviate student mistakes! This vinculum meaning “group. ”

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

PILES AND HOLES A single visual says it all.

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

But then one day I had the most astounding flash of insight of all! YOUR TURN: a) What’s 2 + opp 2 ? b) What’s 3 + opp 5?

I realized too that the opposite of a hole is a pile. YOUR TURN: a) What is the opposite of a hole? b) What is the opposite of three piles? c) What is 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.

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

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

Distributing the Negative Sign This is the opposite of … 3 piles and 2 holes. Clearly that is … 3 holes and 2 piles. - (3 + -2) = -3 + 2 - (3 + -4 + 2) + 5 = -3 + 4 + -2 + 5 6 – (5 – 2) = 6 – 5 + 2 2 – (20 – x) = 2 – 20 + x = x - 18

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

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

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

Infinite geometric series formula.

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

THANKS! JAMES TANTON tanton. math@gmail. com www. gdaymath. com OTHER WEBSITES: All my EARCOS 2017 Power. Points appear here: www. jamestanton. com. Problem-Solving in the Classroom: MAA’s CURRICULUM INSPIRATIONS www. maa. org/ci www. theglobalmathproject. org.