1. Teaching Critical Thinking in the Age of Big Data
Ryan g. mclaughlin
Berkeley preparatory school, tampa, Fl

2. The P-Beauty Contest
On your slip of paper, write a number between 1 and 100. The winner will be the person who guesses closest to 2/3 of the average (going over is ok).
Although it might seem like a completely random game, there most definitely are good strategies available, so choose your number wisely!

3. Strategies, Strategies, Strategies
What numbers were wise choices?
Were there any numbers that could never win?
Assuming that everyone’s choices were a bit all over the place, we might expect the average to fall right around 50.
When played in a classroom setting, round 1 usually goes to a student who picks a number in the low 30’s
We know, before we begin, that numbers above 66 can’t win.
In general, extremely low numbers and extremely high numbers will probably end up being outliers.

4. What happens in H Block…
Raw Data
Line Plot
Round 1: 23, 35, 45, 5, 44, 87, 75, 77, 78, 43, 47, 45, 72, 57, 47, 19, 35, 75, 41
Round 2: 17, 19, 24, 19, 33, 34, 42, 8, 25, 28, 13, 22, 11, 23, 27, 31, 10, 22, 29
Round 3: 4, 11, 3, 6, 5, 4, 59, 10, 10, 7, 9, 7, 15, 9, 3, 10, 10, 10, 14

5. …doesn’t always stay in H Block
Classroom Discussion
“I figured the average would be right in the middle” ~Winner of Round 1
“Well, I thought everyone would pick around the same number that won last time, so I took 2/3 of that and rounded.” ~Winner of Round 2
“I think the trick is to stay one step ahead of everyone else” ~Winner of Round 3
“So, do we have any homework tonight?” ~The Class Pragmatist

6. So what can we accomplish with this?
1) Teach students to solve problems that don’t have predetermined answers

7. Alternative Epistemologies
Deductive epistemology:
“I started with these definite assumptions. Through syllogistic reasoning, I proceeded to reach these conclusions.”
“All men are mortal. Socrates is a man, therefor Socrates is mortal.”
Bayesian epistemology:
“I started with these assumptions about the likelihood of my hypothesis. By acquiring new data, I was able to adjust my belief about the probability of the hypothesis being correct.”

8. Moving Beyond Algorithms
Algorithmic thinking:
School is about learning procedures. If the procedure is followed to the letter, I will obtain the correct answer.
Problem Solving:
School is about applying principles and concepts to unfamiliar situations in meaningful ways.

9. The Prisoner’s Dilemma
Review the following situation carefully…and try to put aside concerns about morality, loyalty, and friendship for just a moment. Go at this from a purely mathematical standpoint…

10. You and a friend are suspects in a bank robbery case
You and a friend are suspects in a bank robbery case. The police believe you both did it, but they don’t have enough evidence to convict either of you: they really need someone to confess or snitch. So they bring you both to the police station, and put you into separate rooms. The rooms are on opposite sides of the station, and you have absolutely no way of communicating with your friend.
The police offer you both the same deal: if you snitch on your friend, but he doesn’t snitch on you, you will go home free, and he will get 10 years in jail. Likewise, if he snitches on you, but you stay silent, he will go home free and you will be the one with 10 years in jail. If you both snitch, then each of you will get 6 years in jail. If neither of you snitches, you will both get 2 years in jail (the police have found some minor weapons charge they can stick you with if the bank robbery charge fails).
You only have a minute to decide whether you will snitch or not. What will you decide?

11. But wait, before you decide…
You should know something: a study published last year in the Journal of Economic Behavior and Organization found that university students playing a similar game only chose to stay silent about 37% of the time.

12. How should the statistic I quoted have affected your decision?
Reflection Question
How should the statistic I quoted have affected your decision?

13. Answer: Not at all

14. Another Reflection Question
Is the mathematically correct answer the same thing as the ethically correct answer?

15. So what can we accomplish with this?
2) Teach students to be careful consumers of statistics

16. “I was forced to confront the ugly truth: people had deliberately wielded formulas to impress rather than clarify.”
~Cathy O’Neill, “Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy”

17. Like a Bridge Over Troubled Data…
From German Bridges to Facebook

18. Leonhard Euler

19. Is there an Euler Path?
An Euler Path goes over every “edge” exactly one time
It is ok to hit the same “vertex” or “node” more than once… remember, we care about the bridges, not the land masses!
Keep a close eye on the “degree” of the “vertex”—that is to say, try to find some patterns with how many edges connect with each vertex and whether or not there is an Euler path.

20. Is there an Euler Path?

21. Is there an Euler Path?

22. Is there an Euler Path?

23. Is there an Euler Path?

24. Is there an Euler Path?

25. Is there an Euler Path?

26. Is there an Euler Path?

27. So, when is there an Euler Path?
Eventually we arrive at the following pattern:
Euler Paths can happen when there are exactly two vertices of odd degree
Euler Paths can happen when there are zero vertices of odd degree
And that’s it!!!!!

28. Can we solve the Konigsberg Bridge Problem Now?

29. So how does this relate to Big Data?
Eulerian paths are now used to:
Help reconstruct DNA sequences from fragments
Rank search results on Google
Find your friends on Facebook and other social networking sites
Much, much more!

30. So what can we accomplish with this?
3) Teach students to make deeper connections between data points

31. Thank you so much! Please keep in touch!
Ryan McLaughlin Upper Division Mathematics Instructor Berkeley Preparatory School Tampa, Fl