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

]]>That got their attention!

I needed to harness their attention because many of the students had come in very excited to see each other. I didn’t want to raise my voice, shush them, or otherwise dampen their spirits. Instead I wanted to quickly channel that enthusiasm into mathematical pursuits. So I ditched my planned discussion of the role of algorithms in computer programming, and instead delved right into something hands-on and interactive, something I had planned for a little later in the session.

After the students flipped their real or imaginary coins 30 times and recorded H or T next to each number on their papers, I asked them to compare lists of outcomes. “Which list appears to be more random, the real coin tosses or the imaginary coin tosses?”

WHAT DOES RANDOM MEAN?

Two groups concluded that the imaginary list was definitely more random. The other two groups agreed that while the imaginary list “looked” more random, the real list was actually more random. This led to a heated debate about what random means, whether streaks can occur at random, whether the outcome of one event affects the outcome of the next, and more. Some of the students had studied probability and some had not, but everyone had something to say. Fortunately, I had to say very little. I did tell them of the gamblers fallacy, and from this discussion they were able to define randomness (not an easy task!).

IS SPOTIFY RANDOM?

I asked their opinions on whether Spotify shuffle is random. Another debate, even more heated. I had spent some time before class today perusing Spotify message boards on just this topic. I shared with the class complaints people had posted about getting too many songs in a row from the same genre. “Yeah, it’s really not random!” said a few students. But the students who knew some probability insisted that this can happen on random lists. Finally, I showed them some graphics about random distributions and Spotify. Finally everyone agreed that the human brain wants things to be more evenly distributed to actually feel random. The coin toss activity, the graphics about random distributions, and the info about the Spotify playlists all come from the same article in the Daily Mail. (I love this article!) Read it for more info about this topic, or better yet, for those of you with children in this class, ask them! They now know for sure whether Spotify is random.

ARE THERE DEGREES OF RANDOMNESS?

We then discussed Random Number Generators (RNGs) – what they are, their purpose, and true RNGs vs. pseudo RNGS. We played with a well-known example of a pseudo-RNG, the Linear Congruential Generator (LCG), which uses an algebraic sequence and modular arithmetic. We talked about remainders, which students often think they’re done with after third grade. “I like remainders better than fractions or decimals,” commented one of the more experienced students. We agreed that when you have a cyclical relationship, remainders might help you with a more intuitive understanding.

“Everything I just told you about RNGs I learned from my favorite youtuber,” I told the class.

“YOU have a favorite youtuber?!” said some, quite surprised.

“Definitely. Eddie Woo.” I encouraged them that any time they want more insight about a high-school math topic to go onto youtube and type in the math topic along with “Eddie Woo” to get a clear and interesting video. They were impressed that he has 70,000 subscribers. “Not bad for a mathematician,” they agreed.

We spent a lot of time on RNGs, but I’m not going into detail here because you can find all the content in Eddie Woo’s videos. One thing that came up in our class that didn’t in the video is curiosity about the precise mechanism for converting space noise to a list of random numbers. I didn’t know precisely how it’s done, but encouraged students to look it up themselves.

The example of the LCG that we did today generates a list with an obvious repeating pattern. Eddie Woo’s second video on this topic shows some graphics of what the LCG produces when you vary the seed number. I would have loved to show this to our group but didn’t have the technology to easily share it. I’d encourage everyone in the group to look at this video, starting at time 7:24, to get a better idea of the kinds of lists the LCG can produce.

]]>Unlike last week, today I came well prepared for these questions, thanks to Ted Alper of the Stanford Math Circle. I had reached out to the 1001 Circles Facebook group for help with this problem, and Ted came to my rescue with another example that better illustrates this characteristic of the algorithm.* The students did the new example, saw what was going on, and then we moved into another discussion of the practical applications of this theoretical model. Last week talked about matching doctors to hospitals. This week we discussed an article (by the Royal Swedish Academy of Sciences) that further explained why this algorithm was Nobel worthy and what others did with the algorithm to extend it.

One thing that we talked about doing last week was continuing the proofs behind this algorithm. We ended up running out of time. I suggested that the students watch Dr. Rhiel's second Numberphile video ("the math bit") to see the proofs, just in case we run out of time in the course. I hope to return to these next week, but I also hope an interested student or two might want to watch the videos and lead a discussion of the proofs with the group.

In our final 15 minutes, we revisited our project of creating individual college-admissions algorithms. I told the students that my goal is to put their algorithms into an excel spreadsheet so that we can run various hypothetical students through it. “Cool!” was the biggest reply.

Finally, I invited students to let me know after class if they wanted to present any of their own algorithms, or those they’re interested in, or proofs of algorithms (see above, hint, hint!) in our final session in a few weeks. I hope some do!

Rodi

*Ted Alper and Benjamin Leis both responded to my post in 1001 Circles and gave me help. I just love that the math circle community is so supportive. Ted recommends that interested teens read the original article by Gale and Shapley, “College Admissions and the Stability of Marriage.” I've read it and would encourage this too. During each class session, I talk with the students about what I've posted in these online reports. Often students want to read more about the things we talk about, so please forward them these reports so they have the links. Thanks, parents!

]]>I was right; the class was captivated by the problem. But it turned out that the problem was much deeper than I was prepared for. (I shouldn’t have been surprised by this, since I knew that this algorithm eventually led to a Nobel prize for one of its creators.)

Does the algorithm give the same result if you start with the men? S said yes, M said no. Several students tried to work it out in their heads with our current example and quickly agreed with S (that the answer is yes). M wasn’t so sure. She was adamant that we actually do it out. So we did. And it turned out that with our specific example, we got the same result. It looked like S was right. Until I told them that the algorithm can favor the person who proposes. “Why did our example work out the same?” The prevailing student conjecture was that it was something specific about the order of our lists. They were all too much alike. Another conjecture was that our sample group was too small. I admitted to the students that I don’t know the math behind this problem well enough to answer (I only learned it a week ago), but told them I’d try to find out. My plan is not to just give them the answer to this question next time, but to give them more scenarios so they can discover it for themselves.

We got into the math (theorems and proofs) behind the algorithm. I got confused in one of the proofs (a proof by contradiction); I wasn’t sure whether we were doing it right. It is a short but confusing proof. Some students were following, some had no interest. Sometimes the younger students just check out mentally when the mathematics gets too abstract. The older students wanted to figure it out for themselves but the younger ones had gotten off the bus so to speak. I was glad to notice that we were out of time. We had been working on this problem intensely for 75 minutes without even a quick bathroom break. Had we more time, I would have given a break right then. But it was time to leave. So I promised everyone that we’d start with it first thing next time. This would give me time to learn the proof better.

On the one hand, it might seem that this was not a successful math circle considering that I couldn’t answer all of the questions and got confused in a proof. But I posit that seeing me struggle with math was very good for the students. Students can expect their leaders to be walking Googles, and that creates a distance. It can make it hard for students to see themselves as mathematicians or even problem solvers. So when they get to see how everyone struggles with some things in math, it can give them hope. No students were annoyed with me for not knowing the things I didn’t know. I think it motivated some of them to try to figure it out for themselves.

The students did enjoy using the puppets. This is my first time using them with teens and I was happy that they enabled the students to go deeper into the mathematics than they may have otherwise. There were a few times when the mathematical struggle was a little intense, and then someone broke the tension by making their puppet say something funny.

By the way, I didn’t write a full recap of session 3, so here’s an overview. We finished work on the Google Page Rank algorithm. Students were exposed to some new mathematics – probability distribution – and had a lot of questions and a lot of fun. Then students then developed their own college admissions algorithms for hypothetical colleges of their own design:

- Kale University (“for really smart people”)
- Collegiate University of Redundancy
- Universal University
- Monster University
- Redundant School of Redundancy
- TSU/Top Secret University (“This is not its name; no one knows its name.”)
- University of the Underworld (“only for people who are dead”)
- Prison College (“an online school for people serving life sentences”)

Just from the list of schools here, students could already see the dangers of using a single algorithm on a large scale.

To be continued in another session.

Oh, one more thing I wanted to tell you: I’ve been showing students pictures of the mathematicians behind the problems we’ve been working on in this course. It is exciting for them to see so many diverse faces (age, race, gender, and even formality – Emily Riehl’s faculty page shows her playing the guitar in a T-shirt!).

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FIBONACCI

Dr. Lawrence used conventional algebraic notation including variables with subscripts and matrices. I wanted to know how comfortable our students would be with this while also keeping their minds wrapped around the definition of an algorithm, so I put a series of numbers on the board: 0,1,1,2,3,5,8,13,…. (This is not something she did.)

Several students quickly identified this list as the Fibonacci series. I asked

- Is this an algorithm?
- Could the number 4 be on the list?
- What is the rule?
- How can you express the rule symbolically?

We discussed, and ended up with the conventional notation for this on the board. We have such a wide range (6 years*) of age and experience in the group that it didn’t surprise me that this algebraic notion was old news to some students and totally unfamiliar to others. We did a quick calculation or two then I reminded everyone of the big picture in this course: algorithms, their applications, and their misapplications.

WHERE DOES THAT PAGE ORDER IN SEARCHES COME FROM?

The class brainstormed what they knew about how Google comes up with the list order. They didn’t know much, but the rest of the class raised their eyebrows at how much one student knew about browsing in incognito mode. Then I gave Dr. Lawrence’s example and we worked through it. We ended up with a graph theory graph on the board. This is not a traditional coordinate-plane, xy-axis type of graph. This is a graph of a network with edges and vertices.

OUR DIVERGENCE FROM THE PRESENTATION

Dr. Lawrence’s presentation used both graph theory and the notation of a system of linear equations with variables and subscripts. Our discussion was juggling the math in the exact same way as Dr. Lawrence. The graph notation was easy for everyone to follow. The equations, though, were not. Once they were on the board, the most experienced students were smiling and nodding but some of the least experienced wore deer-in-the-headlights expressions. Hmmmmm…. what to do?

Since it seemed that everyone understood what was going conceptually, the only issue was the notation. We had to (A) tell the same story without variables, (B) work through some simpler variable scenarios to aid in comprehension for some students, or (C) keep going with this notation with only some people understanding. I thought quickly about the emotional state of the students. Option (A) would work for everyone if only I knew of a way to do it. Option (B) would leave some students bored, and maybe even resentful of being in a class with people who hadn’t seen this notation before. Option (C) would dig a deeper hole of confusion and maybe even anxiety for some other students.

Fortunately, I got very lucky. First of all, M said, “I don’t understand!” relieving some of the tension in the room. Second, I somehow saw a way to do option (A). Phew! I realized that we could use numerical calculations without variables and mark those results directly on the arrows on the graph.

My 20/20 hindsight tells me I should have anticipated this problem before class and have an alternative approach to the problem in my metaphorical back pocket. But I didn’t. I’m feeling grateful that something occurred to me on the spot. I also wished I had talked to the students about the importance of acknowledging their own feelings/reactions in math. We also could have talked about the different ways people react emotionally to math problems. (One student told me later that working with symbols makes her feel good, that they make her feel smart.)

I am happy that the students got to enjoy the delight of an unexpected mathematical result (ask your children, or watch the video!). If you do watch the video, know that we didn’t get through all of it, and will pick up next time at the part where we come up with the probability distribution and test it.

Looking forward to continuing with this problem next week! I do plan to continue to present the material with the algebraic notation, since familiarity will increase comfort and usability for the younger students and will be respectful to the older students. I expect to face the above pedagogical dilemma again and again. This will be fun! (Those of you who know me will know that I am being serious, not ironic.)

Rodi

*Why, you may ask, do we have such a wide range of ages in one group? The answer is that this wide range insures us enough enrollment to have a big enough group for meaningful and energetic mathematical conversation and collaboration. We have 9 students, which allows for many perspectives and insights.

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>>>*What is an algorithm?* The students defined it as an input and output with a certain way of solving something. They brought up and asked some questions about the Pandora algorithm (specifically it’s “randomness”), which we will return to during another session.

>>>*What are some common algorithms and how do they work?* We worked through conversion from Fahrenheit to Celsius and also prime factorization. Then I showed the students an article in today’s Philadelphia Inquirer about new Youth Poet Laureate Husnaa Hassim.

>>>*If you wanted to create a website or app that provides its users with a poem every day, how would you be sure to provide poems that your users would like?* The students said they’d ask users to click if they like the poem, watch for trends (feedback!), and survey users ahead of time for 6 things: what’s going on in your life, how was your day, genres/poets they already like/dislike, your religion, your core values, and your style of humor. A discussion emerged about what the right number of questions would be. “What does this have to do with algorithms,” asked the class. You will see, you will see!

>>>*What would you do if you were in charge of a large, urban, struggling school district and you needed to raise student performance?* The students brainstormed and debated. I asked “who is easy to blame?” for student performance. I asked what variables they would consider if they had to rank teachers to weed out ineffective ones. The students said they’d ask kids, sit in and observe, evaluate teaching performance, and perhaps give teachers a sample lesson to teach. They grudgingly said that if money were an issue, they may have to use student testing data, but that they wouldn’t want to. I then shared the story of teacher Sarah Wysocki’s unfortunate experience with the teacher performance algorithm “Impact” described in detail in Cathy O’Neill’s Weapons of Math Destruction (WMD).

>>>*Do you care how the temperature algorithm works?* Only one student raised her hand.

>>>*Do you care how the prime factorization algorithm works?* That same student raised her hand, and a few put their hands up a few inches, but most kept their hands down very low.

>>>*Do you think people care how the Impact algorithm works?* “They should!” declared D, with many nods of agreement. (I’m hoping that by the end of this course, people will not automatically accept mathematical algorithms – whether pure math or applied – on faith.)

>>>*What is your algorithm for packing your lunch?** The students brainstormed a list (which would, of course, vary by student) of the variables they would consider: food availability, hunger, what they’re doing that day, nutrition, taste, portability, quantity, whether sharing, dietary restrictions, and variety. (Just an aside here: I found it interesting that no one mentioned cost. I wonder how this list would differ had parents written it.)

>>>*Could you write down your algorithm so that someone else could pack your lunch for you?* Students weren’t sure. J began to write her algorithm. Others began to discuss potential difficulties: things that can vary day by day, the skill of the person preparing it, mood, birthdays, availability, seasonality, tiredness. We then discussed the difference between a formal and informal algorithm and also what it means for an algorithm to be trustworthy. “So you’re saying that the algorithm has to be dynamic; it has to react to changing conditions?” Most said yes. J, however, posited that her algorithm didn’t have to be dynamic, that it would work every time. M seemed to agree, and began proposing his own algorithm. J read her algorithm to the group. M and J posited that it might be possible to forsee every variable. People started poking holes in their algorithms. “You’re behaving exactly like mathematicians,” I remarked. “One person posits a proof, and other mathematicians try to find its flaws.” The discussion continued. The debate focused on whether an algorithm must be dynamic in order to be trustworthy. This was a debate I definitely didn’t anticipate when I was planning this session, so it was thrilling to watch.

We were then out of time, to be continued next week.

Rodi

*The lunch packing algorithm example is based upon O’Neill’s dinner prep example in WMD.

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__Our Algorithmic Culture__

Dates: Thursdays, 3:30-4:45pm, 9/7-10/26 (8 weeks, 75-minute sessions, 10 hours total)

Suggested Ages: 13+

What are algorithms and how do they drive our culture? We’ll examine the Google page-rank algorithm, Cathy O’Neill’s National-Book-Award-nominated *Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy*, whether random number generators are really random, the mathematics behind “fake news,” the Euclidian algorithm, and much more. These topics will provide context for a study of the algebra behind algorithms: sequences of instructions that usually involve variables and (algebraic) expressions. These expressions can be organized logically into matrices, programs, flowcharts, etc., to produce solutions to well-defined problems. Then, of course, we will debate the appropriateness of labelling problems “well-defined.” While we’re at it, we’ll delve into some statistics and number theory and then compose some algorithms of our own.

Students should be familiar and comfortable with variables, although the ability to manipulate them is not a prerequisite. While this is a students-only course, interested parents and guardians are invited to participate during the final session.

__Embodied Mathematics__

Dates: Thursdays, 3:30-4:30pm, 11/2-12/7 (5 weeks, no class on Thanksgiving)

Suggested Ages: 5-7

Neuroscience has provided empirical evidence of what we intuitively knew all along: that counting on your fingers enhances learning. The discipline of embodied mathematics employs gesturing and physical interactions with the environment to develop conceptual understanding and to facilitate articulation about mathematical concepts. Year after year, young students come into Math Circle with the idea that mathematics is all about quick computation and nothing else. This course will open students’ minds to the reality that math is about more than numbers and can be explored with more than a computational approach. We’ll use our bodies and surroundings to examine symmetry, 2D and solid geometry, equivalence, measurement, spatial reasoning, and arithmetic computation.

__Invariants__

Dates: Thursdays, 3:30-4:30pm, 1/25-3/1 (6 weeks)

Suggested Ages: 8-10

An invariant is a quantity whose value never changes no matter what you do to the operation under consideration. For example, when you shuffle a deck of cards, the number of cards in the deck remains unchanged. Mathematicians consider invariance one of the most important concepts children need to know as they go through their math educations. In this course, we’ll engage in problem solving, games, flowcharts, storytelling, and a hands-on exploration of the Euler Characteristic to search for and understand invariants.

__The Platonic Solids__

Dates: Thursdays, 3:30-4:30pm, 3/29-5/17 (8 weeks)

Suggested Ages: 10-14

Students will engage in hands-on activities to discover some fundamental principles of geometry. We’ll create and classify the platonic solids as we build with Polydrons. We’ll explore fractals as we attempt to build a 3D Sierpinski Triangle from business cards. We’ll make discoveries about area as we fold paper into squares and triangles. We’ll explore and expand upon Euclidian geometry as we fold more paper. We’ll see math in nature through a look at how ladybugs fold their wings. And in a study of empirical versus logical proofs, we’ll use toilet paper to explore what variables come into play – and how they interact – as we try to figure out the maximum number of times you can fold a piece of paper.

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I thought that this course was about functions. We did a great exploration of them for the first four weeks. But in the last two weeks, something magical happened. Our math circle transcended a single topic to arrive at an exploration of the essence of mathematics, a near-visceral experience of true mathematical thinking. I’m so excited about it that I’m writing a lot. I don’t want to lose any of this story. And writing this gives me the pleasure of reliving it a little bit. This list is a little preview of this report:

• Inverse, composite, and random functions

• Applying logic

• Divination versus math

• Attending to precision

• A quick proof

• The point of all this

• Is this demonstration a proof?

• Is a set of a million pieces of data sufficient for proof?

• Is certainty attainable?

S put a function machine (named Richard) on the board and worked with the group to determine that the rule was y = 10x+1. “What if you run the numbers through backward?” I asked. “What’s the rule then?” They figured out that the inverse (the backward, undoing rule) was (x-1)/10. Then F put up a rule called Triangle with a very limited domain. This rule turned out to be 9x with an inverse of x/9.It got really interesting when we made composite functions of these two. What happens if you run a number first through Triangle and then through Richard? What if you run it through both but start with Richard? How does Triangle’s very limited domain affect what you can do when you bring in another function? How do you find the inverse of a composite function? About half of the students could have gone on and on and on with this discussion. But the other half were starting to tune out. (You may want to find out which camp your child fell into to know whether to go further with this topic at home.) It was time for another function machine.

It got really interesting when we made composite functions of these two. What happens if you run a number first through Triangle and then through Richard? What if you run it through both but start with Richard? How does Triangle’s very limited domain affect what you can do when you bring in another function? How do you find the inverse of a composite function? About half of the students could have gone on and on and on with this discussion. But the other half were starting to tune out. (You may want to find out which camp your child fell into to know whether to go further with this topic at home.) It was time for another function machine.R came up to the board with the machine named Bill. The rule turned out to be x^2 +x. It involved a lot of tedious calculations when the inputs were big. So R limited the domain to numbers 3,000. It still involved a lot of tedious calculations! But he coached them on picking strategic in numbers so that they were able to figure out this less-than-obvious rule.

R came up to the board with the machine named Bill. The rule turned out to be x^2 +x. It involved a lot of tedious calculations when the inputs were big. So R limited the domain to numbers 3,000. It still involved a lot of tedious calculations! But he coached them on picking strategic in numbers so that they were able to figure out this less-than-obvious rule.

Finally, T came up with a machine that really did to generate random numbers. “Is it a random number generator?” someone asked. “Yes, it is.” Someone protested. The group debated whether a random number generator fits the definition of a function, as T insisted it did, and they did have to grudgingly admit that he was right.

“If A is an element of set B, and B is an element of set C, and C is an element of set D, is A an element of set D and how do you know?”

That’s the question I was asking in my mind, but the question that came out of my mouth was different. I didn’t want to overwhelm these middle schoolers with abstract language, so I gave them names of things and didn’t use the language of set theory. A few of the students engaged in the discussion, but many of them got distracted by the caterpillars that seemed to be all over the place. I asked about the difference in the kind of thinking required to solve the function machines versus the thinking required for this set/categorization problem. Only one student could feel a difference. She said that this problem used logic and the first didn’t so much. I realized that I need better problems for next week to get everyone on board.

“What is divination?” I asked.

“Isn’t it finding water?” said someone. I had to clarify that I meant the other divination – the kind they teach at Hogwarts. We briefly discussed modes of divination they had heard of. Then I read them this passage:

“Struck argues for a cognitive basis to divination. Moments of insight, like the ones Malcolm Gladwell wrote about in Blink, might be linked to some of the same processes that guided divination in the ancient world. It’s the art of the inexplicable hunch, as sophisticated as it is primal. “There’s a very limited set of things that all humans do,” Struck says. “There’s eating, walking, and there’s divination.”

Struck sees similarities between the early tools of literary interpretation he wrote about in The Birth of the Symbol, with their emphasis on extracting a hidden message, and classical models of divination—attempts to make meaning out of the world.

That same impulse also gives rise to one of the darker aspects of our current political climate—the proliferation of conspiracy theories, interpretation gone disastrously awry. From a classical perspective, the dividing line between sending away to the oracle at Delphi and believing that the Clintons are running a child-sex-slave-powered pizza parlor is improbably thin.

“As humans,” Struck says, “we’re meaning-making machines. Like a bird is hard-wired to build a nest. It’s a way of making the world habitable for ourselves.” Paranoia, he says, is “one of the prices we pay for this amazing capacity for thought.”Struck also sees a connection between the modern voting process and divination. “In these presidential elections, we’ve got to figure out how to treat every single individual in our society. That’s a huge calculation. And every four years we boil that complexity down to A and B. It’s a bit like drawing rocks,” a popular divination practice.

Struck also sees a connection between the modern voting process and divination. “In these presidential elections, we’ve got to figure out how to treat every single individual in our society. That’s a huge calculation. And every four years we boil that complexity down to A and B. It’s a bit like drawing rocks,” a popular divination practice.

“Life is confusing,” Struck continues. “And it’s much more complex than we’ll ever figure out. There’s magnificent beauty and terror too. Sometimes there’s a benefit to stepping back and limiting the variables. Divination is a way of limiting the variables. As complex as the situation is, let’s step back and look at this sheep’s liver right now.”*

What in the world did that have to do with mathematics, I asked the students. We spent the rest of session 5 comparing and contrasting math and divination.

“I’m going to do my own function machine,” I announced at the start of our final session. “Your job is to discern the rule. I’ll put it on the board as a mapping diagram. The domain is the first name of anyone I have seen.”

“Seen as in you’ve been dating, or seen as in your saw with your eyes?” asked S.

“Ah, I see the ambiguity here. I mean anyone my eyes have beheld. Someone name a person I have seen.” The students looked at each other with confused looks. “Name one person you know I have seen.”

“C?” suggested someone, naming one of our group members.

“Yes! I have definitely seen C. When you put C into the machine out comes M.” (I named another participant in the group.) Confusion gave way to curiosity.

“What if you put in S?” asked someone excitedly.

“When you put in S, out comes C.”

“Huh?” said several students, since C was whom we first put in, and now he was coming out. Several posited that the rule/output/range only applied to people in our math circle.

“How would you test that?” I asked, unwilling to reveal anything about their conjectures. I was working on fostering their own methods for testing conjectures. They realized they could test by putting in someone outside of math circle to see what happened. But who could they say to put in, considering it had to be someone I had seen. How would they know who I had seen? Then our adult helper walked past.“Put in Meryl!” said someone.

“Put in Meryl!” said someone.

“When you put in Meryl, out comes Robert Wadlow?” This generated a universal “huh?” Who is Robert Wadlow, they all wanted to know? “I can’t say right now, or that would give away the rule. How can you further your testing?”

“Put in Harry Potter!”

“When you put in Harry Potter, out comes Professor McGonagall.”

“You’re seen Harry Potter?” asked someone suspiciously.

“I’ve seen him in the movies. That counts.” People seemed excited, confused, suspicious. Everyone was participating.

“Put in yourself!” someone ordered.

“When you put Rodi in, out comes my husband Sam. Some of you have seen him but most of you have not.”

“I know,” said F. “Put in Bob!” F pointed at the whiteboard that still had a diagram of our very first function machine from six weeks ago. The participants had never let me erase this over the weeks, even though we were short on boards.

“When you put in Bob, out comes Harry Potter.”

“But Bob is just a drawing, not a real person!” argued the students. “You can’t put him in!”

“Bob cannot be excluded from the domain of this function. The test for inclusion in the domain is that I have seen him, not that he’s alive or real. So Bob stays.”

“In that case,” argued someone else, “Harry Potter isn’t even real. He’s just a figment of someone’s imagination!”

“You are right. I didn’t think of that,” I acknowledged. “I’m imagining the movies, so I’ll have to change Harry Potter to the actor who plays him in the movies, Daniel Radcliffe.”

“You’ll have to change Professor McGonagall too!” ordered the students.

“Maggie Smith!” someone yelled out. I crossed out both character names and replaced them with the actor names on the board. “But I think Maggie Smith might be dead,” argued someone else.

“Doesn’t matter if she’s alive or dead. I have seen her in the movies so she stays.” (Meryl chimed in here that Maggie Smith is indeed alive.)

The students resumed testing people in our group. When S goes in, M comes out. “Wait a minute!” objected someone. “M came out when you put C in.” Others corrected this objection with the definition of a function – that different inputs can have the same output, the rule is that a single input can’t have multiple outputs. Back to testing. When T goes in, Rodi comes out. “Put in F” requested someone. Hmmm…. I wasn’t sure who comes out when F goes in.

“You’ll all need to stand up,” I told them. Everyone stood up. “Ah,” I nodded, “when F goes in R comes out.”

“Does this have something to do with height?” asked the other F. I wasn’t going to yield any information, and told the group that they have to figure out how to test her conjecture. They put in someone else and then posited that everyone who came out was taller than their corresponding input. If you’re right, I told them, then you should be able to predict who will come out when you put in the next person. They put in F. But there are several people in the class taller than F. How could this work?

The range (output) must be defined by naming just one person taller than the person put in, the class concluded. I congratulated them for figuring out what I had been thinking, but in the spirit of true mathematicians, the arguing and assumption attacking wasn’t over. (This may be my favorite part of math!) “How do you know that Daniel Radcliffe is taller than Maggie Smith?” I had to justify that one with describing a scene from the movie. “How do you know that as he grew he didn’t become taller than her? He was short in the first movie but not by the end.” I conceded that I should have been more precise when I input him, maybe specifying something like “in the first movie.” “How do you know that Robert Wadlow is taller than Meryl?” I defended this with the statement from the Guinness book that he was 9 feet tall. We went on like this until no one could find any more assumptions to attack or concepts/ideas/objects that required more precision.**

If B is taller than A and B is shorter than C and A is taller than D, prove that C is taller than D.

I gave this problem to the students but asked them to supply names. We did it with Lucille, Parker, Josh, and Steve. The group collaboratively proved it, with S up at the board and the students dictating to her. I was temporarily out of the picture, a place I’d like to be more often during math circle. I love it when the students take over and no longer need me!

“We just did 2 problems that revolved around height. Reflect for a moment about the kind of thinking you were doing during the function machine. And now for the proof. How did each feel? What was different about them? Which was easier for you?”

The students discussed this for a little while. They concluded that in the second height problem we started with rules and then reached a conclusion, whereas in first height problem we started with examples to work up to rules. We discussed inductive versus deductive reasoning.*** Which do you think is better? Which is more important? Which comes first, the chicken or the egg? That question isn’t in jest; it’s relevant here as we discussed how some of the storied ancient Greek philosphers declared that deductive reasoning is the be-all and end-all, while some others said that you can’t start with making deductions if you don’t have anything to start with. Where do you get starting info? The students had widely varied opinions about which type of reasoning they preferred and why, but all agreed that you do need both.

Cut a triangle out of a piece of construction paper. Tear off each of the four vertices (“corners” the students called them). Take each of these torn-off vertices and set them on the same point with the adjacent edges touching the next one. **** You can see that the three angles together make a line, or 180 degrees. Ta-da, 180 degrees in a triangle! (The students all did this with their own triangles that they cut out.)

But is this a proof? I was so happy to see the kids vehemently shaking their heads “no!” I didn’t even need to tell them that this is a demonstration but not a proof. They told me that there may be some other way to make a triangle where you don’t get a line. Go kids!

Once a mathematician named George played a little game: Start with 2. How many prime factors does it have? Just one. One is odd. So the Odds score 1. How many factors does 3 have? Just one. Score another for the Odds. How many does 4 have? Two. Score one for the Evens! The Odds are ahead, 2-1.

I worked through scoring this game with the students for a bit. Once you get to 11, the Evens catch up and the score is 5-5. This means that 5 of the numbers from 2 through 11 have an even number of prime factors and 5 have an odd number of prime factors. When you try 12, the odds move ahead again because 12 has an odd number of prime factors. So far, the evens have never been ahead.

Will they ever get ahead? The students posited some conjectures.

I told them that George tested many numbers and formed a conjecture (known as Polya’s Conjecture because George’s last name was Polya) that the evens will never get ahead. When every number up through 1,000,000 was tried, most people thought the conjecture was true. “What do you think?”

“That must have taken a loooong time,” said F. *****

Even though no one had thought that the triangle demonstration was a proof, many of our math circle students thought that evidence from a million trials was convincing enough. I was more than a little surprised. I had to stop mentally patting myself on the back for how astute the group had been about the triangle demonstration. “Then why do you think I’ve chosen to share this problem with you?” I asked.

“Oh no, there’s a larger counterexample, isn’t there?” asked M. A few people groaned. M was right. Polya formed his conjecture in 1919, but in 1962 Lehman discovered that the evens pull ahead at 906,180,359.

“So when is enough enough?” I asked. “Is there a number of pieces of data you need for absolute certainty?”

“You can never be sure,” posited several students. “You’d have to try every possible input and that would be an infinite number in most cases and trying an infinite number of things is impossible.”

“Let’s look at a function machine again. If I put in 1 and out comes 2, and 2 in 3 out, and 3 in 4 out, what do you think the rule is?” (You’re adding one.) “So the rule is y = x + 1, right?” (Yep) “So I’m going to posit my own conjecture. I’ll call it Rodi’s conjecture.” (Brief digression when students objected to naming my conjecture after my first name and not my last.) “Rodi’s conjecture states that in the function y = x + 1, y will always be greater than x. Is there a way to prove that without trying every number into infinity?”

Yes, yes, yes, yes, yes! This was the real aha moment (I think/hope). The students saw that this was a statement that could be tackled with Proof. They could think of several ways to do it. But we didn’t have time. I ended the session/course with a quick recap of our big ideas:

• Inductive reasoning differs from deductive, and mathematics primarily (but not exclusively) uses deductive reasoning.

• No matter how many pieces of date you have, data points alone do not constitute proof.

• Mathematics requires proof.

Now goodbye, and have a great summer! Then something that hasn’t happened in one of my math circles ever: the students applauded. I don’t know whether they were applauding me, or themselves, or the ideas in the bullet points, or some combination, but it was great! (My conjecture is that they were applauding a sense of group accomplishment.) Thank you all for allowing this wonderful course to happen.

**-- Rodi**

*Jamie Fisher, “Peter Stuck’s Odyssey,” The Pennsylvania Gazette, May/June 2017, http://thepenngazette.com/peter-strucks-odyssey/

** Please email me if you know a single word verb in the English language that means “to make more precise.” There definitely should be a word like this. Precisify?

***More info on deductive vs. inductive reasoning here: http://mathforum.org/library/drmath/view/55695.html

****Here’s a nice visual to show how to do the triangle tearing: https://www.quora.com/What-should-a-triangles-angles-add-up-to (scroll down to diagram at bottom of page)

*****Her comment made us wonder whether there were calculators back in 1919. I suspected that there were but wasn’t sure. I’ve since looked it up: https://en.wikipedia.org/wiki/Calculator

We spent most of our third session creating mapping diagrams, which are a visual way to present and evaluate functions. The students were surprised, after several weeks of numerical function machines, that functions do not have to involve words. They can involve words and more. We mapped students to their drivers, students to their ages, and students to their pets. “Two of these are functions and one is not. Which one is not a function and why?” The students only needed this question to come up with the mathematical definition of a function.

Then we mapped the set of possible statements to true or false. The student-provided statement “ghosts are real” “blew up the machine.”

“What can we do to make this statement not blow up the machine?” I asked. The students suggested limiting what you put in the machine to things that are definitely statements, since the mathematical definition of a statement is something that can be proven true or false. “Possible statements” was too vague. In other words, the students limited the domain of the function. They also suggested adding another possible “out” value: “maybe.” In other words, they expanded the range of the function. Then we went back to the earlier mapping that turned out not actually be a function (students to pets, since some students had multiple pets). “How could we turn this into a function? The students did it by changing the rule.

We played around with more functions when we mapped students to things they like. I posited that for everything they like, they could come up with a function within that realm. I challenged the group to come up with functions for each thing, and they did with gusto.

Almost half the kids not there – it was a very different atmosphere because the small group of 7 allowed for intimate conversation. We will review mapping diagrams next week so those absent can get up to speed. I plan to get the kids who were in attendance to lead that activity.

We did some other activities last week (session 2). Here’s a quick overview:

• Function machines – we revisited the Bob series (from week 1) and came up with rules to “undo” what the machines had done. In other words, found inverses.

• Modelling of functions – if you fold a paper strip in half, how many layers of paper do you have? If you fold it again, how many layers? Again? Turns out 1 fold produces 2 layers, 2 folds 4 layers, 3 in 8 out, 4 in 16 out, 5 in 32 out. (The kids counted to be sure.) Is there a pattern? Is this a function? Can you make predictions for bigger numbers? How can you be sure?

• Cutting a cake: If you make a cut in a round birthday cake that connects 2 points on the circumference, how many pieces do you get? If you connect 3 points, how many pieces? It seemed like the same pattern was emerging (consecutive “in” numbers corresponded to “out” numbers of 2, 4, 8, 16. You’d think the next would be 32 . But... when you draw it and count, you get 31. Hmmm…. What could this mean? This generated a conversation about the need for proof, including a visual proof of the commutative property of multiplication.

• Patternicity: I read aloud an article on the human need for order then we gleefully discussed this. Actually we debated it, not just discussed.

• Function machines again: I actually began session 3 with the whiteboards that had our work from paper folding and cake cutting. Then I did a function machine where you put in 1 and out comes 2, in 2 out 4, in 3 out 8, and so on up to out 256. The kids spent some time at the beginning and end trying to figure out a mathematical way to describe the rule. Their current conjecture is that it has something to do with multiplying numbers by themselves, but no one is sure what numbers or how. https://www.flickr.com/gp/talkingsticklearningcenter/YA4BA0

“When will we go back to the Recaman Sequence?” asked M at the end of session 3. Next time, first thing, I promised. (Last week it rained, this week I didn’t get there early enough to set it up.)

-- Rodi

REFERENCES

FUNCTIONS WITHOUT NUMBERS: Thinking Mathematics #4 Functions and Their Graphs, James Tanton http://www.lulu.com/us/en/shop/james-tanton/thinking-mathematics-4-functions-and-their-graphs/paperback/product-5508096.html

PAPER FOLDING ACTIVITY: Tanton’s Take on Models of Mathematics http://www.jamestanton.com/wp-content/uploads/2012/03/Curriculum-Essay_April-2017_Models.pdf

NEED FOR ORDER: https://psychcentral.com/lib/patterns-the-need-for-order/

THINGS YOU LIKE FUNCTIONS: 1001 Circles Facebook group (a group where people joyfully share ideas for playfully exploring mathematics)

The goal of today’s session was to get everyone familiar with what typical functions look like. Over the weeks we’ll be ramping this up. Today, though, was the first session of a six-week “Picnie Math” course on functions designed for students approximately age 10-13. We had a circle of 13 students ages 9-14 picnicking on the grounds of Awbury Arboretum. I hope we have such lovely weather every Tuesday!

We started with a series of Function Machines. The students had a lot of fun designing the machine and coming up with fun sounds it makes. I’ve never done function machines with students this old and didn’t know how it would go over with them in a group. I know tweens and teens are playful individually and in small groups, but here they were in a larger group, and no one knew everyone in the group. But within minutes, they were joyfully playing. No is ever too old for function machines!

They named the first one Bob. With a slight revision and new rule, the old Bob became Bob 1.0 and the new Bob became Bob 2.0. Now that the students had the hang of how to do this, they were clamoring for the addition to Bill Cipher to Bob to create Bob 3.0. I had no idea who Bill Chipher is and wasn’t doing well drawing according to their instructions, so R came up to the board, took the marker out of my hand, and took over. Bob 3.0 was renamed Bob & Bill. This machine took numbers, doubled them, and added 1. (Remember that for later.)

Bob 4.0, which was renamed Robert, led to some interesting debate. No matter what number the students put in, the “out” number was 1. Most agreed that the machines makes every number become 1. M argued that the rule could be that it divides each number by itself. Several students then said it could be any number of rules. Then came the real mathematical thinking: Can we be certain? How could we test? Could we put a certain number in to determine whether it’s turning things into one or dividing them by themselves? “What if you put in infinity?” Said M. Out comes 1. “What if you put in zero?” Still you get one. Then everyone agreed that the rule was not dividing a number by itself. I stressed that we need proof, so no one thought we had proven what the rule is for sure, just that we had disproven one conjecture about it.

“Fold a piece of string in half. While it is folded, make 1 cut. How many pieces of string do you have? Continue with another piece of string folded in half, making 2, 3, 4, and 5 cuts.” We didn’t have any string so I passed out long, thin strips of newspaper instead. It turned out to matter greatly. You can do things with paper strips that you can’t do with string. What if you cut the strip in half lengthwise, wondered the students? That changes the problem. (Fun!) We had to make sure that we only did with the strips what you can actually do with string to do the problem as intended. “What is the point of this activity?” I asked.

“To get us bored so we fall asleep?” joked J.

“No,” said S, “it’s so you can see a pattern – that the number of pieces is always 1 more than double the number of cuts, since you have that folded section on the end.” This from the youngest in the group; most of the older kids didn’t have a clue what he was talking about until they played with the strips and scissors some more.

“You’re absolutely right,” I said, “although I was thinking about something else. Can you tell that you’re creating a function here?”

“It’s Bob and Bill!” exclaimed M. I also find it exciting to realize how many things in math are telling an analogous story.

“This is a function machine,” I said. “You can express functions in lots of ways. What would be the input and output numbers here?” We wrote them on the board in a table, function-machine style.

“This next problem,” I said, “was created by a Columbian mathematician named Bernardo Recamán Santos, who is still working in Columbia.” Many eyes opened widely in class, as S had just returned from Columbia and many of the students knew it. “You mean the District of Columbia, not Columbia-Columbia, right?” asked someone doubtfully.

“I do mean the country of Columbia. S, you may have seen Bernardo Recaman walking down the street.” Eyes opened even more widely. I brought up Recamán’s home and that he’s still alive walking down the street intentionally. Most people don’t imagine mathematicians being alive, working, and walking down the street. And certainly not in South America.

Here’s the problem: Start at 0 on a number line. On move 1, move 1 space. On move 2, move 2 spaces. And so on. Move to the left if possible, but don’t go into negatives and don’t repeat. Can you hit every number on the number line?

All 13 kids stood in a line and moved together on a sidewalk chalk number line. Keeping track of moves was tricky. “Can we start over?” asked F after they had lost track. Wise question. They started over several times. Finally they started marking off their moves with chalk to keep track. It was fun to hear the conjectures change and evolve. R extended the number line as I only had set it up to go to 40. When they ran out of room on sidewalk, we debated extending the line into the (active) parking lot. Then F suggested “why not just curve it around to run alongside the first part of the line.” Great idea. But then R announced that we were out of chalk. Enthusiasm was waning among some participants. The question was compelling, especially since it’s an unanswered question in mathematics, but the work was tedious. We discussed what would happen if we were to wait till next week to continue. The line and its markings would definitely be gone. There was too much sun glare to make an accurate photographic record. The consensus was that we write down on paper what number the group was standing on and what move number they were on, and that I would try to quickly recreate the line next time. Then we went back to our picnic area for one last problem.

“So we did a bunch of functions today that everyone here knows the answer to, then one that no one knows the answer to. Now here’s one that at least someone knows the answer to but I sure don’t.” The group spent the final 10 minutes looking for patterns in this problem that Gordon Hamilton of Math Pickle had suggested for our group (thank you!). The problem is a function machine. Lots of mini-patterns emerged, but the group was not able to come up with an over-arching rule. “Does the same thing have to happen to every number?” asked F. No, but there does need to be a pattern. “If you don’t even know what the pattern is here, why do you expect that we could figure it out?” asked someone.

“There are plenty of things I don’t know that students can figure out.” They looked at me dubiously. Then we were out of time so I sent them home, although some lingered to keep working on the problem.

Rodi

*A NOTE ON NAMES:* In past years, when I wrote these reports I used different letters for students’ names than their actual ones when they shared a first initial. This year I’ve stopped doing that. So when you see M here, you don’t know which M said what. We have 2 Ms, 2 Fs, and 3 Ss. Just email me if you want me to tell you which things your own child said – I’m happy to fill you in.

Function Machines – the students design a “machine,” the facilitator states the input and output (often but not always numbers), and the students must discern the rule.

Bill Cipher info: http://gravityfalls.wikia.com/wiki/Bill_Cipher

Teenagers and play: I had a conversation with Natural Math author Yelena McManaman about my experience with the tweens and teens doing function machines. She told me of a study of teens on playgrounds: playgrounds are geared to kids 12 and under, but when teens were given access and time, they ended up playing joyfully too. I couldn’t find the study online, but if you search “the importance of playgrounds to teenagers,” a lot of interesting articles come up.

String cutting problem: http://www.cehd.umn.edu/ci/rationalnumberproject/01_1.html

Primitive lightning: http://mathpickle.com/project/primitivelightning/

Racamán sequence: http://mathpickle.com/unsolved-k-12/ (click on “grade 5,” go about halfway into video), https://mathlesstraveled.com/2016/06/12/the-recaman-sequence/, https://es.wikipedia.org/wiki/Bernardo_Recam%C3%A1n_Santos (this one is in Spanish). Even though the “Unsolved K12” committee recommends this problem for grade 5, that’s really just a minimum recommended age, as no one has ever solved this.

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