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]]>(April 5 -19, 2018) In the past, I’ve often made the mistake of getting out “manipulatives”* to help students discover a certain mathematical concept only to find that the students wanted to engage in open-ended exploration. They weren’t interested in my agenda. So, for this course, I put the Polydrons on the table with no guidelines for two weeks. The students just played with them as we worked on other mathematical questions.

Finally, in week three, I said “This week we are only getting out the Polydrons that are regular polygons. Can you sort them so we can put away the irregular Polydrons?” The students quickly learned what regular polygons are. Then I said, “Let’s make some Platonic solids!” What are they, the students wondered. “There are only two rules: they are constructed from regular polygons and all vertices are the same.” The students spent some time asking questions and understanding these rules, playing with the Polydrons with this goal in mind. “Now we can get to the question,” I announced.

“We haven’t even gotten to the question yet?!” exclaimed the students.

“Yep! The question is this: how many different Platonic solids are there?” After some time, the students had discovered three of them (actually four, but they don’t know yet that they discovered a fourth).

THE HANDSHAKE PROBLEM

Since the students in this course ranged far in age (10-14) and didn’t all know each other, in week 1 I gave a classic math problem that easily generates interaction among students:

“If everyone in a room shakes hands with everyone else, how many handshakes will there be?”

The students reasoned that we have 8 people, so it’s 8 times 8. Wait a minute, do we shake our own hand? No. So we each shake 7 hands, 8 times 7=56. So 56 shakes. Done. Confident they had solved it after 3 minutes. “Are you sure?”

“We have to be sure! Let’s try it out!” declared F. They realized soon that shakes were being double counted. 56 divided by 2=28. So 28 shakes. Done, confident they had solved it after 3 more minutes. I insisted they finish gathering evidence (by completing their experiment). They did get 28 after coming up with way to keep track. Confident they had solved it. (F and Z asked clarifying questions – i.e. what if you do two-handed handshakes? etc)

The following week I asked them to generalize their process, which they did. They even they came up with an algebraic formula for it (with a bit of help from me). “How can you be sure that because this works for 8 people, it would work for all numbers of people?” This introduced doubt big time. That’s great news as far as I’m concerned. I am coaching them to doubt conclusions arrived at through induction. I want to move into proof so that they know beyond a doubt that their formula will work for any number of people. In the spirit of true mathematicians, they're asking does is work for multi-digit numbers of people, etc etc etc.

I also challenged the students to explain how this problem relates to the Platonic Solids. No conjectures yet.

FOILED BY MY EXPECTATIONS AGAIN

The handshake problem did turn out to be a great icebreaker. Actually, the students came up with an icebreaker: Go around the table, say your name, one thing you like to do, and name your favorite Youtuber. (Turns out that two of the students “knew” each other from playing Minecraft online, and loved meeting in person.) “Funny you should mention your favorite Youtuber,” I said, since mine is Vi Hart and I brought in one of her videos to show you today. I showed them one of my favorites: Binary Trees.

My mind was aglow with how the students were going to watch this video, become enraptured by the Sierpinski triangles, and demand time to doodle these on their own. Ha! That didn’t happen at all. I was operating under the false assumption that because something happened once before (7 years ago in a math circle) that it would happen again. No one was interested. Even when I told them that you can make a 3D Sierpinski triangle (a Platonic solid!) out of recycled business cards. “Sounds like a lot of work,” several of them muttered. Foiled by my expectations again. Will I ever learn? OTOH many years ago I tried the Platonic solids with Polydrons activity in a course and those students had no interest in that. These kids now are very interested. It’s actually quite wonderful that the same activities turn out differently each time when you let them.

THE BEAUTY/GLORY OF FUNCTION MACHINES

On the first day, S (an experienced math circle participant) asked, “Are we going to do function machines in this course?” I hadn’t planned on it but decided to throw it into the mix as a crowd pleaser. The math that has come out of this so far has been unexpected and delightful.

For those of you unfamiliar with function machines, you play by saying a number that goes in to the machine and the person operating the machine tells you what comes out. Your job is to guess the rule from a series of ordered pairs (in and out numbers).

When J presented a machine, her rule brought up a discussion of **negative numbers** once it became apparent that when the opposite (negative) of a number went in, the same number came out as it did from the original. So what kind of function would generate the same output from its negative? Turns out that squaring a number does this. What is squaring? What happens when you multiply two negatives? And many more questions… The math behind this that I didn’t mention (and wish I had) is that her function, (x^2 + 50)/2 is an **even function**. In mathematical symbols,

f(x) = -f(x).

I do allow the presenting student to use a calculator. This saves time, keeps everyone interested, and opens up its own Pandora’s Box. When S presented a machine, the out numbers didn’t seem to make sense to him. Everyone waited patiently as he input the numbers into different calculators and got different results. (One of the many things I love about this group is their patience.) I mentioned that not all calculators follow the order of operations. This led to a discussion about what the **order of operations** is. Z broke the ice for this discussion with her comment “The order of operations can be confusing.” We also talked a bit about the necessity of knowing how to use parentheses on calculators.

“How would you get the in number for these two functions from the out number? I asked. This led to a discussion about **inverse functions**. I gave two analogies students are often taught for these – (1) undressing and (2) peeling corn. The students seemed to find the undressing analogy (you take of your shoes before you take off your socks even though you put them on in the opposite order) more accessible. F pointed out a flaw in the corn analogy, that there are obvious things smaller and underneath the kernels with corn.

This exploration of function machines looks likes it’s going to converge with the Platonic solids, as both can be looked at through the lens of symmetry. More on that next time.

Also, next time, I’ll tell you more about some of the other things we’ve been talking about – some logic questions, a paper-folding problem, and more.

Rodi

*Manipulatives are physical objects used as teaching tools. In mathematics, they offer concrete experiences with abstract concepts.

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]]>The post Invariants appeared first on Talking Stick Learning Center.

]]>** Piagetian Conservation Tasks** – We did every activity in this article: http://www.cog.brown.edu/courses/cg63/conservation.html. Conservation tasks basically are about invariants. Some cognitive psychologists posit that the ability to do these tasks is not coachable, while others believe it is. Our group had some variety in who could recognize the invariance. All could spot it sometimes. Most had at least one case of not being able to spot it. Interestingly, whether someone had success at these tasks had no bearing on their examination of the Euler Characteristic.

- Line up cubes and count them. If you change the order or the distance between them does it change the count?
- Redistribute blocks into sets – how does that affect the sum? (Note – the students loved using these wood cubes so much that I had to set up before class second time just to give students a chance to play with them
- Pour water in differently-shaped containers. Is there more in a taller thinner container than in a shorter wider one?
- Flatten a play-doh ball. Does this affect how much play doh is there?
- Weigh a play-doh ball in different shapes – does changing the shape affect the weight?

** 1-2-3 Fingers** – kind of like mathematical Rock Paper Scissors – I say “123” and both you and I hold out however many fingers we want.

- Multiply them – if odd, you win, if even, I win (I win every time – tee hee hee!)
- We did this for three weeks, and by the final week most but not all students had figured out the strategy. Lots of fun! (Thanks to Maria Droujkova for this activity.)
- Play this at home!

**Collaboration through NIM**

- We played the game NIM, which you can learn about here: https://mathforlove.com/lesson/1-2-nim/. We played several versions of the game.
- I told the students that the goal of this game is collaboration, since real-world mathematicians get help from each other in solving problems
- What is the best way to collaborate if we play a game that’s me against the team of all of you?
- Students played this for 5 weeks, with their collaboration and mathematical strategies evolving over the weeks. While the game was fun and the thinking got deeper and more sophisticated over the weeks, the collaboration that I demanded was stressful. I didn’t tell them how. Each of them had different ideas. Some people cared more that their ideas (for collaboration methods and for NIM game strategies) got tried. Others cared more that conflict be avoided. I talked about the challenges and benefits of collaboration a lot!

__Cup game__

- You get 7 cups: 5 upside down a 2 rightside up. Your goal is to get them all rightside up by flipping 2 at a time. (Thanks to Maria again.)
- We had very deep math conversations about this game, getting into parity, testing of many cases, changing the rules to see what would happen, and what would proof require if you want to make generalizations.

__Cross-country race__

- We played the game that is “Example #4” on this handout from the Waterloo Math Circles: http://www.cemc.uwaterloo.ca/events/mathcircles/2010-11/Winter/Senior_Mar23.pdf
- Students changed the names of the cities from unfamiliar Canadian locations to things they made up. This made the game more accessible.
- We played it several times, but not enough to be able to make generalizations. The students who did play it most want to play more to discover what happens when you try other starting points, etc. I promised these pictures so that kids can play at home.

__Strings on Cans__

I brought in a bunch of cans of many sizes. I had multiple strings cut to the length of the diameter of each can.

- How many strings does it take to wrap around the can with no overlap or gaps?
- Turns out everyone found that it takes a little more than 3 strings to wrap around the can, no matter what can they used.
- Is that an invariant? The students thought no. I asked how many tire-diameter-length strings it would take to wrap around the circumference of a tire, and everything thought a lot more than three, despite our hands-on results here. Piagetian cognitive psychologists posit that there is a fixed developmental stage at which students can transfer mathematical patterns to other examples. (Of course, not everyone agrees, and I think that with the discovery of brain plasticity and the modern research on mindset, more people are seeing things like transference as something coachable.) I promise you that I am not using your children as my mini-cognitive-psychology lab!
- A parent in the background asked “Are you talking about pi?” It turns out that yes, we were! That number a little more than three, that ratio of circumference to diameter, is pi (my favorite invariant!)

__Euler History__

- I read a little bit about Leonhard Euler in some of the classes so that the students knew that there was a person being the main problem we were exploring.
- I read from Historical Connections in Mathematics – a series that I love.

__Function Machines__

- One goal was to introduce how to play function machines to students who never did. They are super fun. Ask your children how to play and do it at home! (We used rules like x+1, x+20, 100x+1, x-2, but didn’t discuss them in algebraic terms as I am here.)
- Another goal was to do function machines with invariants and have the students figure out what was invariant, so wanted to use rules like subtracting itself (x-x) or 1 if odd and 0 if even. Ran out of time, though.

Thanks to all of you for sharing your wonderful children in the extra-fun course!

Rodi

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]]>The post The Euler Characteristic for Eight-Year-Olds appeared first on Talking Stick Learning Center.

]]>**BEFORE THE COURSE: THINKING ABOUT IT**

I didn’t want to spoon feed the math in worksheet form where I tip my hat to what’s cool about the Euler Characteristic. I spent a long time developing an approach that I hoped would allow students to make some deductions but not be led too much. (See references at the end for my inspirations.) My big question was how much was enough leading but not too much?

**WEEK 1: PRESENTING THE PROBLEM**

The goal was for students to know what the question is. I spent so much time sent on setting up a dramatic narrative because this is a hard problem for 8-year-olds. It’s especially hard because I was hoping that they would come up with the idea that there is a pattern. I did not want to end up telling them that there’s a pattern.

Here’s the setup:

*I need people to play some roles – a farmer, a horse, a carpenter, a secretary, and an accountant. The farmer wants to build pastures for her horse so that there’s a different crop in each for the horse to graze on. Horse, what do you want to eat? Farmer, can you draw some dots to show where you want the fenceposts to be? Carpenter, can you connect the dots with lines to indicate the fences? The rules are that the fences can’t cross and every post has to be connected to every possible other post. Horse, can you count the pastures? (Fun debate here about whether outside the fences counts as a pasture/region.) Secretary, can you keep track on the board everything that we are counting? Farmer, how many dots did you draw? Carpenter, how many fences did you install? Accountant, what do you get when you add the number of dots to the number of regions? *

*The carpenter’s bid depends up on the numbers of dots, lines, and regions. The farmer will hire the carpenter to do the work if the sum of dots and regions is equal to the number of lines. The farmer and horse really want this thing built so the horse can eat that pizza! Will this thing get built?*

Not everyone understood the math. They did get the general gist that the mathematical requirements were not met to get the fence built. “Let’s change it up!” They tried, but the counting got really tedious and confusing.

**WEEK 2 - UNDERSTANDING THE PROBLEM CONCEPTUALLY**

Since I didn’t think the students really understood the problem last week (as mathematicians often don’t at first), we delved into some background. I asked the students how electricians, tile-installers, painters, and carpenters decide how much to charge for a job (“bidding”). What happens if the bid is too high? Too low? How much would you charge to paint the room we’re sitting in right now? The purpose of this discussion was to demonstrate the ideas of formulas/algorithms/rules for bidding on jobs, since our carpenter is putting in a bid to build the fence.

Also, since the diagram the students constructed was pretty complex, I handed out paper and asked them each to draw their own sample pasture, with “any number of dots.” I hoped that if each student had their own example that they created themselves, that they’d understand the problem better. I also hoped that each would create a less-complex example and therefore would have a better shot at coming up with an answer.

Turns out most of the students had a hard time drawing it and sticking to the rules (no lines crossing, connect everywhere possible). Kids did 19, 20, 25 – covered their pages with dots. I thought to myself that I should repeat this in week 3 with an assistant helping the kids draw. I also thought to myself that I could make a handout with our diagram from the whiteboard and dashed lines so that the students could change it. (Alas, I never did either of these things – the assistant or the handout.)

So no progress on the problem this week. (Just like what happens to mathematicians!)

**WEEK 3 – STARTING TO LOOK FOR PATTERNS**

At this point I started to worry about time. I was starting to get nervous won’t have time to connect it to course topic invariants. Unlike Andrew Wiles, we didn’t have a lifetime to make progress on the problem. Only 3 more sessions after today. So I led more than I had originally wanted to. Used the strategy of starting small and gradually building. Kids wanted to jump ahead to larger numbers but I reigned them in a bit. I neglected to tell them that we were using the strategy of starting small – a lost teachable moment. Oh well, can’t get them all, I had to remind myself later when I was beating myself up mentally a bit about this.

**Week 4 – ASKING QUESTIONS**

I wasn’t sure whether all kids are following the record keeping on the board; we needed to make it more clear. I insisted to trying to do this investigation systematically – increasing by 1 the number of points in each trial - but they still wanted to skip 6. Even when they noticed the gap, they didn’t suggest to try 6. I insisted only because we had only 2 sessions left. (Had we more time, I would’ve just let them skip 6.)

C asked *what if you position the dots a different way?*

S asked *why are all the results odd except when there are 7 dots? *

A asked *why are they always going up by 2 except… *

Someone asked *can we do curved lines?*

Someone else asked *can we ever get a different result?*

The students were excited, curious, asking many questions about the problem. Moreover, they were no longer talking about it in context of farmer/carpenter problem. They were saying “dots/lines/regions” not “posts/fences/pastures.” These 8-year-olds had transcended the material world to the abstract! (After class, I asked myself, “Why are we still calling points dots?!” I set an intention to shift terminology to the more accurate term points, which are different from dots. I never explained that difference but did make the shift.)

**WEEK 5 - DOES THE PATTERN HOLD FOR ALL CASES?**

I brought out the students’ original diagram from week 1, the one with 13 points. The students knew exactly how to assess it now. I asked for a conjecture ahead of time: Do you think you’ll end up with points + regions exceeding the number of lines by 2? Most did. Then they counted and discovered that they still got the same result. So is this an invariant? The consensus was… maybe. Most students said we’d need to try more cases, and C argued vigorously for the need to try different arrangements of dots for the completed trials. One student said we’d need to have a proof. (Most students didn’t know what proof meant, so we didn’t get into because of time. Had we more time, we certainly would have.) So some students worked on trying examples with larger numbers of points while C attacked rearranging the points for several cases (4 points, 5 points, and 6 points).

With about 15 minutes left, we shifted gears to a new problem, “Cross-Country Race,” which I’ll explain in a different report - click here for that one.

**WEEK 6 – STUDENT OWNERSHIP**

I had another activity for today that we all started with (in honor of the approaching Pi Day). During the Pi Day activity, students were anxious about returning to our prior problems. Of the only four students in attendance that day, two were desperate (yes, desperate, I mean it!) to get back to what we were calling at that point “The Horse and Carpenter Problem.” The other two were tired of that problem and really wanted to explore the new problem that we started last week. We didn’t have time for both. It seemed that no matter what we chose, half the class would be disappointed.

“You don’t need me for either of those problems. You own them now. How about you two tackle one and you other two tackle the other?”

They looked at me in seeming shock. “We can’t do them without you!” someone exclaimed.

“Yes, you can. You own these problems!” I handed out markers and that was that. They really didn’t need me. I answered a question here and there, checked their progress when they wanted to show me, and that ended our course.

I promised to publish these pictures so that the students can continue to work on the problems at home. Like real mathematicians often find (and we discussed), six sessions just isn’t enough time to tackle really interesting mathematical problems.

**REFERENCES/INSPIRATION**

Joel David Hamkins. Math for Eight-year-olds: Graph Theory for Kids http://jdh.hamkins.org/math-for-eight-year-olds/

Harvey Mudd College Math Department. Mudd Math Fun Facts: Euler Characteristic https://www.math.hmc.edu/funfacts/ffiles/10001.4-7.shtml

Simon Singh. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (book)

Owlcation. Some Practical Applications of Mathematics in Everyday Life https://owlcation.com/stem/Some-Practical-Applications-of-Mathematics-in-Our-Everyday-Life

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]]>The post Embodied Mathematics appeared first on Talking Stick Learning Center.

]]>Here’s a list/description of every activity we did.

** **

**Role-playing the need for math**

In week 1, we acted out scenarios where no numbers were allowed. The students got around this with drawing pictures.

Week 2: no numbers, no pictures

Week 3: no numbers, no pictures, no names of shapes

Week 4: no numbers, no pictures, no names of shapes, no comparison words, and no approximations (at this point we had to use the whiteboard to keep track of all the restrictions)

Week 5: all of the above allowed.

Here were the scenarios:

- Invite me to a party
- Pay me for restoring your sheep’s health
- Resolve a dispute about which army won a battle
- Explain how to cook something (pancakes, cookies, whatever the children knew how to make)
- Explain how to draw a snowman
- Explain how to build a snowman
- Explain how to plant a garden
- Give me directions to your home or your relatives’ home

We had so much fun as students debated and even voted on which words were allowed (Point? Line? Few? Side? Shape? Herd? etc). The students decided each week how the difficulty would be ramped up the following week. They were excited that it would get harder and harder, and it was their idea to make the final week as easy as possible. I didn’t expect this activity to be as popular as it was. The students could have spent the entire 5 weeks doing nothing but this. No one ever got tired of it; they just asked for more and more.

**Simon Says**

We played the game Simon Says but with one twist: with each command, regardless of whether the Simon character said “Simon says,” you have to command the opposite. So if Simon commands “reach your arms to the sky,” the next command has to be “do the opposite of reaching your arms to the sky.” (It’s up to Simon whether to say Simon says, adding in another layer of complexity.) Over the weeks, the students discovered that

- not every command has an opposite
- some commands seem to have multiple opposites (so what does that mean? Does it mean they have no opposite?)
- some commands actually are two commands embedded into one (i.e. stand on your left foot)

If you replace the word opposite with “negate” or “inverse” and replace the word command with “function” the mathematical reasoning involved here may be more apparent. We didn’t use these terms in class, though.

Over the weeks, the game evolved to include the creation of equivalent, not just opposite, expressions. Students could choose to give an opposite or equivalent command and the others had to guess which it was.

**Mirror**

The students stand in a line and the leader strikes a pose. The rest of the group has to mirror it, leading to lots of experimentation with various types of symmetry.

**Ants Go Marching**

The Ants Go Marching is a children’s song that is sung to the tune of “When Johnny Comes Marching Home.” We sang it. “How can we think about or show this idea with our bodies?” I asked, quoting Malke Rosenfeld from the sample chapter of her book Math on the Move. The students first made their bodies into the shapes of the numbers and then wanted to act it out. Problems arose when we didn’t have the right number of people for everyone to stand in the correct formation. In other words, we were playing with divisibility.

**Rhythm Name Patterns**

We clapped the rhythm of every participant’s full name. “How is this mathematical?” I asked, as I asked for every activity during the course. Cyclical patterns, the group came up with after a discussion.

**Sidewalk chalk addition**

I drew a number line from 0 through 8 on the sidewalk. The students jumped to represent operations such as starting on 0 and adding 3, starting on 3 and adding 2, starting on 5 and taking away 4, etc. We did scenarios where the instructions landed them off the line below zero (negative). Then I asked the students to make their bodies face the opposite direction. What happens if you add 2 but you’re facing the other way? What if you take away 3?

In this activity, the wide age spread of the students became apparent. The students ranged from young 5s to a few close to 8. The older students were interested but the younger students wandered away. My original plan for this course had been to do no activities with numbers, but some of the older students begged me to work with numbers right from the start. This activity was to be my compromise. We revisited it a few times for just a few minutes when students needed a break from other activities, but it didn’t become one of our core activities.

**Poi**

I asked my helper Joanna to demonstrate the performance art of poi. “What words would you use to describe what she’s doing?” “How is what she’s doing mathematical?” I was hoping that this would facilitate student’s ability to communicate about math by naming, classifying, and describing poi patterns, and that students would notice the symmetry and periodicity in the motions. They did. Then they wanted to try it. I wasn’t prepared for this, so we couldn’t. (You can do poi with tennis balls in long socks – maybe try it at home.) I was inspired by poi artist Ben Drexler’s article “A Mathematical Approach to Classifying Poi Patterns, Introduction and Basics.”

**Math In Your Feet**

We did an activity from Malke Rosenfeld’s book Math on the Move. (On the book site, click on “Download a Sample” and then find the section “Try it yourself, part 1.”) We invited parents and siblings to do this one too. Students stood in sidewalk-chalk squares and experimented with how many ways they could do certain moves.

**Play Doh Nim**

Play the game Nim with little balls of play doh. In this version, you can smash 1 or 2 on your turn. If you smash the last one you win. This game was another place where the age spread made things difficult so we didn’t return to it on another session, but I plan to do it lots more with other groups. All of the credit for this activity goes to Lucy Ravitch, who described it when she guest-blogged on the Let’s Play Math site.

**Pattern Function Machines**

During the course, one student who had been in math circles before begged to do the activity function machines. (In function machines, students suggest an “in” number, the facilitator reports the “out” number, and the students have to discern the rule after a few examples of ordered pairs.) It took me weeks to figure out how to do this in an embodied way.

We asked the students to stand in a line. The first child was given a red block, the second a blue one, the third a red one. “What color will the next person get?” We did this repeatedly with increasing complexity of patterns and the students creating the patterns. Then we switched and gave the students puppets to hold (and operate, of course). This was tougher since puppets have many more attributes than do wooden cube blocks. The students struggled happily to discern patterns in the line of puppets. The following week we did it with blocks on the table instead of the students carrying them. Had we more time, we would have taken away all props and just had the students stand in certain positions and identify patterns. The eventual mathematical goal would be to move toward abstraction by eventually moving into words then symbols/numbers, but that was not the goal of this course.

Many thanks to helpers Joanna (for facilitating many of the activities) and Maria (for being an extra set of hands). Also to you parents for sharing your wonderful children with us!

Rodi

*Here’s our course description: 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 of 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.

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]]>The post Algorithms, Algebra, and College Admissions appeared first on Talking Stick Learning Center.

]]>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

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]]>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.

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]]>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!

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]]>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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