Mathematical Discovery Workshop
Ida Crown Jewish Academy - August 19-20, 2013

Shai Simonson

I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of textbooks…  I should say that on the average, we get about two percent efficiency out of schoolbooks as they are written today. The education of the future, as I see it, will be conducted through the medium of the motion picture...  where it should be possible to achieve one hundred percent efficiency. 

(Guess who?  1922)

Good teaching comes from mastery of a subject, clear communication, patience, empathy, planning, and a sense of humor.  Using technology will not make you a better a teacher except in how it improves these aspects.


What is the mathematical discovery style of teaching?

Is lecture an effective way to teach?  Does a teacher need to be an entertainer as well as an educator?  

To a colleague known for his teaching skills:  What makes a good college CS teacher?  Answer:  They taught high school first.

The "flipped" classroom - a modern popular approach.




  • Height of a Pyramid (Geometry and Algebra)
  • Advanced (Calculus needed) - At what angle must one throw a ball so that it is always moving away from the thrower?

Resources and Links

 Rediscovering Mathematics - MAA

Mathematical Reasoning for Education - Stonehill College


Monday, September 30   -- 

Discovery Lab for Means  (Detailed Notes in Chapter 6 Rediscovering Mathematics)

When is median helpful, and when is mean helpful?  Tricks for fast calculation of median and mean.


74, 64, 82, 93, 90, 76, 78, 81, 85, 76 - test scores

40, 60, 50, 48, 54, 57, 82, 90, 96, 100 - more test scores

1, 2, 3, 3, 3, 4, 5, 9, 10, 10  - survey satisfaction answers

240, 260, 280, 230, 310, 180, 4600, 240, 250, 210 - home prices in thousands of dollars in Skokie

I made 10% a year and then 4% a year on my investments. 
Does this earn the same money as making 7% in each year?
What is the right average in this case?

I ride my bike 15 mph to ICJA and 20 mph on the way home. 
Does this take the same time as riding 17.5 mph the whole way?
What is the right average in this case?

I went bowling and got 130, 150, and 143. 
Is my total the same as if I bowled 141 each game?
What is the right average in this case?

Given two numbers x and y, which average is biggest? smallest?  middle? 
Are the averages ever the same?

Consider this drawing.  Let's investigate the average of the black and light blue distances.

What is a proof?   (Detailed notes in Chapter 1 Rediscovering Mathematics)

Definitions, Axioms, Rules of Inference, Conjectures, Theorems, Proofs


The sum of two even numbers is even.  Prove it!  Easy.

The square of an even number is even and the square of an odd number is odd.  Prove it!  Not so hard.

There are an infinite number of prime numbers. Prove it!  Harder.

The square root of two is not the quotient of any two integers.  Prove it!  Huh?  Much harder.