(October 19 and 26, 2017) “Today we’re going to do something we’ve never done in a math circle here. We’re going to take a quiz.” Stricken looks on faces. “We’ll do it together, it’s just 2 questions, and I think it will be fun.” Relief. “I’m always on the lookout for things related to algorithms to share with you, and I came across something interesting this week.”

I asked the students to read aloud a few paragraphs on algorithms from the online course “How to Think Like a Computer Scientist,” and then as a group had fun answering the first 2 questions. I didn’t get into any discussion about this at all because I didn’t want kids to think I was pushing programming. But I wanted to plant a seed in their minds.

ALGEBRA

“If one side of a balanced balance scale contains 3 bags of apples and 4 single apples, and the other side contains 1 bag and 5 single apples…”*

I could see students brains already starting to work. “What do you think I’m going to ask?”
“How many apples are in a bag?” said the younger S.

“That’s right. How many?” A number of students gave answers and explanations.

“Now, supposed you live in the time before algebra had been invented. Could you solve this problem?” Now it was much harder. Even when students try to do it without variables, they still were using algebra, just in words. (Hee hee, this is what I had hoped would happen – I was getting ready to talk about the history of algebra and algorithms.) Finally the group came up with a way to solve it by drawing a picture and crossing apples off. This didn’t feel so algebraic. But guess what, it sorta was, and this was the perfect segue into the concepts of “balancing,” “reduction,” and “restoration,” the techniques used by al-Khwārizmī, who some call the “father of algebra.” I told of the origins of the word algebra, al-Khwārizmī’s techniques, and gave a sample problem that al-Khwārizmī solved algebraically using words. We also did another balance-scale algebra problem (from mathisfun.org) to connect the idea of balance to modern algebra.

OUR OWN ALGORITHMS

Then we returned to one of our ongoing questions:

“If you ran a college and had to use an algorithm for student acceptance, what would it be?”

Our work today really focused on the problems of the problems that can arise in developing algorithms.

A new question:

“If our activity is to match our hypothetical students with our hypothetical colleges, would it make sense to finalize our algorithms first or create our students first?”

This seemed a strange question to some. Maybe students thought that this activity was itself some kind of predetermined algorithm. “We’re creating this activity together, as we go along. What should we do?” D’s face lit up with understanding that the group was inventing the activity.

Then W’s face lit up with understanding about why the order could be problematic. “Ah! One could influence the other.” We discussed how people could game the algorithm if they could create student characteristics and the algorithm itself. In the real world the same person would not be essentially applying and accepting (in most cases). What to do? The students thought that in either order there would be a conflict. Maybe they should be done simultaneously, or in a back-and-forth manner. Then one student suggested that each person create their student secretly and then everyone close their eyes and I deal them out. This way, no one would make their own student apply to their own college. Good idea, everyone thought.

I passed out a paper on which I had condensed everyone’s algorithms-in-progress. But how to insure that no one got their own student? We needed an algorithm! Somehow we muddled through this and after one flub on my part everyone ended up with one student applying to one college:

STUDENTS:

Cassy Carlson, Smitty Warben Jagerman Jenson III, Smitty Warben Jagerman Jenson Jr., Anya Reed Woods, Radical Party Dude, Jeff, and A. Neill Human Breen

COLLEGES:

The School, Kale University, The School of Egotism, The First School of Bone Hurting Juice, Collegiate University of Redundancy ,and the Redundant Collegiate University of Redundancy

The math circle participants ran their “students” through their algorithms and had immense fun announcing and posting (on the board) the results. Four “students” got accepted and two “rejected.” We realized that the students were not necessarily applying to the schools that were the best matches, and that using the Gale-Shapley algorithm MIGHT have resulted in a better outcome if the students proposed to schools first, instead of schools proposing to students first.

Class ended with some heated debate when Anya Reed Woods was rejected from The School. “How could not have gotten in?” demanded J. “Our school is too good for her,” replied younger S. J got out of her seat to look at S’s algorithm. They were still debating this as the rest of the students left for the day.

ALGORITHM MACHINES

In our last session, we continued our discussion of algorithms/algebra by playing “Algorithm Machines,” essentially function machines with a new name. I made up (hidden) rules for the students to guess, and the students made up rules for each other to guess. Then we visited the dark side of algorithms when I made up a hard rule but gave one person a slip of paper with a hint. “It’s so obvious,” he said to the others as they posited conjecture after conjecture without figuring out the rule. Frustrations mounted. “But it’s so obvious!” he said – multiple times. Finally one other student was able to piece together the rule from everyone else’s conjectures, but no one else could. I explained that the purpose of this thought experiment was to experience what sometimes happens in real life with algorithms, when they become unfair.

“How did this make you feel?” I asked. Reactions ranged from “It’s not obvious” to “I want to slit his throat!”

EVALUATING ALGORITHMS

We then brainstormed a list of every algorithm we had considered during this course. The students debated which ones were healthy algorithms and which qualify as “weapons of math destruction.” M posited that seemingly harmless algorithms could be used for nefarious purposes, or that there could be unintended consequences. The argument was based on the premise that the Fahrenheit-to-Celsius conversion formula could be used in a context that could disadvantage some people.

One thing we never had time for in the course was discussing the chapter on college admissions in Weapons of Math Destruction. You can get this book at the library. I would highly recommend this chapter!

MORE ALGEBRA

We talked a little more about the etymology of the term algorithm and how it is connected to algebra, and then returned to algorithm machines. We were almost out of time, so I had three students at the board at once creating and demonstrating machines. Debate ensued when the creators disagreed with seemingly correct conjectures about the rule. The students put the rules into conventional algebraic notation and compared them. The students with more algebra experience could see that they were equivalent expressions and equations. Some of the algebra beginners did not see this. For those of you just entering the world of algebra, I’d suggest doing more algorithm/function machines at home to explore the idea of equivalent expressions.

Thank you for these wonderful eight weeks!

Rodi

PS Some of you (both parents and students) were asking when the next math circle will be for this group. We have a spring course on the Platonic Solids for recommended ages 10-14. If it turns out that most of the enrollment comes from students 13-14 we may shift the age range upwards, but sadly as of now we are done with classes for older teens for this year.

*This problem from the book Avoid Hard Work