The post Category Theory #2: Grappling with Abstraction appeared first on Talking Stick Learning Center.

]]>“No, Penelope is not a pig. She is a pig *puppet*. There’s a big difference,” I replied as we sat down. This seemingly inane comment of mine captured everyone’s attention.

** **

**PIG-PUPPET YEARS**

“You know how it is said that a year for humans is like 7 for dogs and 5 for cats?” I asked? Everyone nodded.

“There are also fox years,” added A.

“Yes, and there are pig-puppet years. But they work in the opposite direction as dog years and cat years. For every 10 years that a human ages, a pig puppet only ages 1 year. So even though I got Penelope almost 30 years ago, her age is really 3.”* We discussed this concept for a moment, and then I announced, “I’m having some trouble with Penelope that you can probably help me with. I’m trying to teach her about numbers, but listen to how she responds to my lessons:”

Me to Penelope the pig puppet: “If Grandma gives you five cookies and Grandpa gives you five cookies, how many cookies will you have?”

Penelope to me: “None, because I’ll eat them all!”**

** **

**DOES CONTEXT MATTER?**

Penelope’s statement set off a huge mathematical conversation. Students had questions and comments about what numbers are, how to explain them, what counting really means, the difference between numbers and numerals, and how numbers first came into being. We talked about all of these things, and then returned to the original scenario. The students tried and tried to teach Penelope the math problem above by changing the context:

Student: “What do you get when you combine five blobs and five blobs?”

Penelope: “One, since blobs squish together when you combine them!”

Student: “What do you get when you combine five pieces of titanium and five pieces of titanium?”

Penelope: “One, since titanium is a metal and metals melt at high temperatures.”

Student: “What do you get when you combine five wooden blocks and five wooden blocks?”

Penelope: “Zero, because I like to play with matches!”

Student: “What do you get when you combine the numeral five and the numeral five?”

Penelope: “Fifty-five, since the fives are right next to each other now!”

Turns out that no matter what context the students came up with (probably 20 examples in all), Penelope had a way to make the problem not work. No matter what, five things plus five things didn’t equal ten.

A: “How can Penelope know so much about other things and not know anything about math?”

Me: “She’s a science prodigy.”

F: “But aren’t math and science related?”

Me: “Yes, but that’s another thing that’s special about pig puppets. We can take some creative liberties.”

By this time, the puppet Penelope had somehow moved from my hand to F’s hand, and the students had taken over both roles – coming up with new contexts and finding ways to contradict the hoped-for result. No one was able to come up with a context that Penelope (in most cases actually M) couldn’t knock down. I was just enjoying the show.

** **

**STRIPPING AWAY CONTEXT**

“What’s the difference between the problem *five plus five equals ten* and the problem *five things plus five things equals ten things*? I asked. The students’ thinking even further intensified. They posited conjectures, debated them, rejected them until S*** said “My brain hurts!” The others agreed.

“What’s the difference between numbers and things?” I asked more directly.

“Well, numbers are something that we made up to talk about things,” answered A.***

“Do numbers exist as things in the natural world?” I asked.

“Yes,” said about half the students.

“No,” said the other half at the same time.

They all looked at each other. Then those that said Yes changed their answers to No.

“Are they ideas?” I asked.

“Yes!” everyone agreed. We talked about ideas versus things. How mathematicians use the word abstract to describe ideas that can then be applied to multiple scenarios.

“Like abstract art,” said S excitedly. Then she quickly reversed herself: “Actually, no, since abstract art is a thing.”

“My brain really hurts now,” said A.

“Do cookies behave logically?” I asked?

“No. People eat them!”

“So would mathematicians rather study things that behave logically or things that do not?” The students all agreed that “logical things” is the answer. I explained that mathematicians like to strip away the context to get at the underlying abstract structure of things. This can reveal similarities, I continued, like in that problem we did last week with the symmetries and arrangements.

But is it always mathematically sound to strip away all context? If a problem is totally abstract, will you arrive at a useful answer?

**RECONSIDERING CONTEXT**

I presented the students a paraphrase of a problem from Eugenia Cheng’s book __How to Bake ____π__:****

*You run a company that takes people on tours. You’re organizing a trip for 100 people.* *You’re renting minibuses and want to maximize your profit. Each minibus holds 15 people. How many do you have to rent?*

The students started out by trying numbers: 10 busses – too many. 9 busses – still too many. Then S suggested dividing 100 by 15, yielding 6.6̅. After some discussion, they concluded that we need to rent 7 busses, so 100 ÷ 15 = 7.

“That’s it? That’s the problem?” said S, a bit disappointed. She was happy to hear that no, that’s just the first part. The problem continues:

*Now you’re shipping some chocolates to a friend. You pre-paid for a stamp that covers the cost of mailing 100 ounces. Each chocolate weighs 15 ounces.* *How many pieces can you send to your friend without paying extra for shipping?*

Immediately the students saw that it’s the same calculation but a different interpretation of 6. 6̅. They all were talking but not so much to each other or me. More like each was thinking aloud, simultaneously. S persevered the longest and gave a solid explanation of why in this case, 100 ÷ 15 = 6.

“So here’s the real problem,” I said to the students. *“Why is the answer 6 when you’re talking about chocolates and stamps but the answer 7 when you’re talking about people and busses?”*

“Context matters,” they all agreed. I quoted Cheng to them: “Be careful not to throw away too much… Category theory brings context to the forefront.”

“What would be the answer,” I asked, “to a person who looked at this problem purely abstractly, with all of the context stripped away?”

“6. 6̅” they all agreed. They definitely were grasping abstraction versus reality and some key points about context. But it was time for a break. People’s brains had started hurting 20 minutes ago.

** **

**FUNCTION MACHINES AND CAKE CUTTING**

I gave everyone an apparent brain break by doing a function machine with them. (The students provide a number that goes “in,” I tell them what number comes “out,” and their job is to discern the rule.)

“Now try this one: If you slice a cake, what’s the function for the maximum number of pieces you can get with a certain number of cuts?” (I also worded it in the language of circles at F’s request: “What is the function for the maximum number of regions you can create with a certain number of chords in a circle?”)

I started sketching this on the board with the students’ verbal instructions (“2 pieces from 1 cut, 4 pieces from 2 cuts,” etc.). But almost immediately, the students were all at the board figuring it out for themselves. Once again, I sat back and enjoyed. Most of the students quickly got 7 pieces from 3 cuts.

“I got 10 pieces from 4 cuts,” announced M.

“Can you get more?” I asked. She tried, without success, and then went on to test 5 cuts and 6 cuts. By then, at least three students had diagrams with 10 pieces from 4 cuts. “You can get more,” I promised. “There’s something that all of you are doing that you could change to get more.” They kept working.

“Can you get more than 7 pieces from 3 cuts?” backtracked S.

“No one ever has,” I said.

“But just because no one ever has, does that mean it can’t be done?” she asked. “Has anyone demonstrated that it definitely can’t be done?”

“That is one of the key questions in mathematics,” I said, so excited by this question. A huge goal of our math circle is to teach kids to be doubters. “In math, it’s not enough that no one has ever done something. There has to be a proof that it can’t (or can) be done for us to believe anything. And yes, there is a proof that you cannot do more than 7.”

“I got 11!” announced A, who had been fervently trying to beat the class record of 10 from 4 cuts.

“Now you’ve reached the number that has been proven to be the maximum.”

My intended point of this activity had been to look for a pattern/function/rule to determine the number of slices. We had a nice sequence of numbers (2,4,7,11), but no one was interested in pattern-seeking. They just wanted to keep testing. So I played the big-number card: “How many pieces could you get from 500 cuts?”

“We would need bigger whiteboards and more markers,” said someone, defeating my attempt to redirect the approach.

I played the ridiculous-number card: “If you had to determine how many pieces you could get from 5,000 cuts, would you rather have a bigger whiteboard or know the rule?” Someone grudgingly said that the rule would be better in that case, but it didn’t detract from anyone’s enthusiasm for drawing.

We were out of time. Had we gotten to the point where students showed interest in determining a rule, I would have burst their bubble anyway with some talk about how patterns don’t mean rules without a proof. (Again, training doubters.) So we ended on a high note with me connecting this activity to the idea of abstraction.

**A FEW OTHER THINGS**

Early in the session, the students brought up the golden goose problem from last week. (*Would you rather have golden eggs, a goose that makes golden eggs, a machine that makes those geese, or a machine that makes those machines?*) Some students had talked about it at home and were reconsidering their answers from last time. We talked about the levels of knowledge you needed for each item in this hierarchy, and how that ties to mathematics. The students wanted to explore whether the answer to the question would be different if we removed the goose as a possibility. I explained that “What would happen if we changed the question a bit?” is exactly something mathematicians ask all the time. This came up later in the class, when S was doing the cake-cutting problem without realizing that the cuts had to be chords, not just random line segments. She made a quick shift from initial disappointment that she had misunderstood the problem to excitement to hear that this might a new way to do this problem that people hadn’t worked on before.

Rodi

*I’m relaying this anecdote so that interested parents can have a jumping-off point to talk more about ratios and proportions at home.

** Eugenia Cheng, __How to Bake ____π__, p19. We also dramatized with Penelope the pig puppet Cheng’s examples of the difference between having memorized the sequence of counting numbers and actually understanding what they mean. (p20) Cheng discusses the cake-cutting problem on pages 33-34. You can find an algebraic explanation of the problem on Wolfram MathWorld and many other places. It’s a classic problem.

***We have two students whose names begin with S, and I’m using S for both. Ditto for A.

****Cheng, p21

The post Category Theory #2: Grappling with Abstraction appeared first on Talking Stick Learning Center.

]]>The post Category Theory 1: Odd One Out appeared first on Talking Stick Learning Center.

]]>ODD ONE OUT

The students found it easy to conclude that the grey duck was the odd one out when all the others were yellow and every other attribute was identical. But what to do when multiple attributes change? After vigorous debate over some pre-determined groups of objects,* the students created their own odd-one-out challenges for each other. They used markers, cubes, pentominoes, playing cards, shape tiles, and rocks.

“Even though this one is the odd one out because of color, you’re wrong. A different one is the odd one out.”

“You’re right that this one is the odd one out, but your reason is wrong.”

Again and again, students stated versions of the above two comments. Serious mathematical discoveries were going on here:

- Changing assumptions changes the answer.
- There can be multiple paths from beginning to end.
- You have to state your reasoning or your conclusion won’t stand up to scrutiny.

Soon, the students changed the wording of their replies:

“You’re right that this one is the odd one out because of color, but that’s not what I had in mind. I was thinking that a different one is the odd one out.”

“You’re right that I was thinking that this one is the odd one out, but your reason is not the reason that I was thinking.”

Do you see the significance of the change in wording?

PROOFS

The students were essentially creating physical proofs, except that here, the conclusion/proposition wasn’t stated at the beginning as it is in a mathematical proof. You could say that the students created the building blocks of mathematical proofs. To paraphrase mathematician Eugenia Cheng,** proofs are like storytelling with a beginning, middle, and end. In the beginning you state your assumptions and definitions. In the middle you state your reasoning. In the end, you state your conclusion (i.e. “ta-da!”). In our group, the beginning, middle, and end of the proof emerged through asking questions, positing conjectures, rejecting or accepting conjectures, and then finally a statement of the author’s reasoning.

The students proved that “this one is the odd one out because it is the only one that”

- is a non-primary color.
- has the white cubes totally enclosed in a boundary.
- doesn’t start with the letter B.
- has no symmetry.
- has no other objects in the group that share a shape.
- has a rough texture.
- has writing of a slightly thicker font.
- etc., etc., etc.

No one set up a group where the obvious attribute (color, size, etc.) was the exception. The exceptional attributes were hidden. Just like in an interesting math problem! I expected this activity to take 5-10 minutes, but it took much longer. I had set an alarm to give us a three-minute break after 50 minutes. When that went off, A said, “Wow, Math Circle goes fast!” It felt like we had just gotten started. We could have done this activity all day. Even when we moved on to other activities, the students took odd-one-out interludes to continue challenging each other.

ABSTRACTION

Another aim of category theory is abstraction – seeking the underlying structure of things in a way that allows you to see a similar structure in seemingly very different things. By “things” I mean mathematical things. By “mathematical things,” I mean things that have a logical structure. I have seen Cheng extend “things” to non-mathematical things. She applies category theory to real life. I hope to do this later in the course, but for now we’re sticking to logical things.

We explored Cheng’s juxtaposition of the symmetries of equilateral triangles to permutations of the numbers 1,2, and 3.*** It may have seemed like this was an activity about properties of triangles, types of symmetries, and one way to calculate a permutation (listing it out). The students did conclude that an equilateral triangle has 6 symmetries and a list of three digits has 6 permutations. But the big question was this: is that just a coincidence? Or (dramatic music here) *could they really be the same problem*?

One thing I love about doing “high-level” math with younger students (here, ages 10-12) is that they are not blinded by too many preconceived notions about math. (Of course, they’re blinded by some. I see five-year-old students convinced that any problem that’s not about number theory is not mathematics and that there’s only one right answer no matter what and that every question has an answer and that the teacher knows it and that there’s only one way to get the answer and and and…. Okay, I’ll get off my soapbox now.)

“Of course, it’s the same problem,” S right away announced. The others quickly agreed with her. We talked briefly about how the underlying structure of things can be the same despite the obvious differences. This idea blew my mind when I first encountered it at an age much older than these students. I suspect that at least some older students might think it’s just a coincidence.

I put “high-level” in quotes because category theory is generally not taught to students before graduate school or possibly third-fourth year college. Cheng is leading a movement to make it accessible to middle- and high-school students. Math Circles in general are hoping to make deep mathematical study accessible to younger and younger students.

GOLDEN EGGS

In our last few minutes, we played an informal round of Would You Rather:

- Would you rather have some golden eggs, or a goose that lays golden eggs?
- Would you rather have a goose that lays golden eggs, or a machine that makes geese that lay golden eggs?
- Would you rather have a machine that makes these geese, or a machine that makes machines that make these machines?

This discussion was rich, going off in many philosophical and mathematical directions. Cheng gives this example**** to make a point about abstraction: “in order to build a machine to do something rather than doing it yourself, you have to understand that thing at a different level.” You have to analyze every step and every implication of those steps. This is some of the thinking we do when our goal is abstraction. We were out of time at this point, so never really did get to a discussion about this perspective. We’ll start there next week.

All of this in 75 minutes – phew! Fortunately, we have five more weeks.

Rodi

*Here are the puzzles on paper that the students debated:

http://www.puzzlesandriddles.com/WordPuzzle15.html

https://twominfun.com/odd-one-out-puzzle/odd-one-out-5/

** __How to Bake ____π__ , Eugenia Cheng, pp66-68

*** Cheng, p17

**** Cheng, p31

The post Category Theory 1: Odd One Out appeared first on Talking Stick Learning Center.

]]>The post “Waggy, Do You Eat Meat?” (Some Basic Tenets of Mathematics) appeared first on Talking Stick Learning Center.

]]>*On a particular island, every inhabitant (puppet) is either a knight, who always tells the truth, or a liar, who always lies. Which puppet is a liar? Which one a knight? You can either listen to their statements, or ask them questions.*

“What’s a statement?” asked A immediately. And our first session was off and running.*

Some deep mathematical thinking beyond the questions of what is a statement, what is the opposite of a statement, and how can you categorize things as statements and their opposites came up:

**Subjective versus objective:** When the students asked the puppets questions, they discovered that some questions did not clarify matters: “Baby Puppy, are your ears floppy?” “Kitty, do you like milk?” “Waggy, is your tail pink?” all resulted in answers that gave different students different conjectures. When the puppet Penelope said “my tail is strong,” the students thought this was a clear indicator of a liar until after the round, when Penelope demonstrated how her skinny short tail could lift an object. The students figured out that words like floppy, like, pink, and strong are subject to interpretation. They learned to use words that leave less room for interpretation to arrive at an answer sooner: “Rooney, do you have two ears?” “Cat, do you have a tail?” While you can certainly argue that there are multiple interpretations of two and ears, we were headed in the direction of precision, one of the basic tenets of mathematics.

**Precision:** When we introduced puppets/characters that sometimes tell the truth and sometimes lie (normals), things got trickier. One puppet told 11 lies then a truth. Another told 8 truths then a lie. The students disagreed on how to categorize them. A identified them as normals. N identified them as a knight and a liar, respectively. They both agreed that a puppet who told half truths and half lies was a normal. A held firm that one exception eliminates a puppet from a category. N argued the definition of the word “sometimes:” a pattern with just one exception does not count as “sometimes.” She felt that “sometimes” was not well-defined. Neither student was able to bring the other around to the other position, but they did both agree that had we been given 5 categories instead of 3 that their answers would then be the same. I forget to mention that these students are just six years old!**

**Functions**: “Waggy, do you eat meat?” Waggy said no. “But he’s a fox and foxes eat meat,” said one of the students,“so he must be a liar.” “But everything else he said was the truth,” said the other.*** They debated this, asked many more clarifying questions, and finally decided that Waggy was a knight despite the meat thing. I explained afterwards that Waggy is really a puppet/actor who was playing the role of a fox but really lives in a bag in my closet and eats nothing. (In my mind, this idea is like nesting dolls or even compound functions, where one function is processed through another before an answer is obtained.) Once the students realized this, it made the game both more complicated and clear at the same time.

**Certainty/Proof**: *If we clarify the word “always” from the original question to mean “with no exceptions,” how many questions do we have to ask the puppets to be certain of their categories? *This question was confusing to the students. (They’re just six years old, after all.) They had various conjectures, all of which were a single number. One student said 16. “So what if that puppet told the truth 16 times, and on the 17^{th} statement or question, told a lie?” I asked. At this point, it was clear that the students’ brains were fried. (Fortunately, I hadn’t gone so far that they got discouraged/frustration.) I had lost track of time. We had been doing math for an hour and twenty minutes. So I sent them home.

**Ownership**: One goal of our Math Circle is for students to own the mathematics, for the facilitator to ideally be a fly on the wall. In the second week of class, A attended, N did not, and a group of new students (ages 5-7) were there. I asked A if she wanted to demonstrate/teach the game of Knights and Liars to the others. She wanted to and she did.

Note for families of new Math Circle participants: I was recently asked what opening activity I like to do for a new course. Here’s what I said. At the Talking Stick Math Circle, we like for the students to have an immediate immersion into mathematical thinking. So whatever problem we take on, we start using the terms "conjecture," "proof," "question," "mathematician," the phrase "I don't know," and for older students the word "assumption" right from the start. This gives many of our students a sharp contrast to some of their other math experiences, and hopefully the beginning of an understanding of what mathematics is. I purposely pose questions that I don't know the answer to. I use the above terms without defining them (until someone asks). Our goal is that eventually (over weeks or months or even years), students will discover the difference between inductive and deductive reasoning. With older students, we talk about that right from the start. Another term important to mention right from the start is "collaboration." I like to give a problem that's pretty impossible for a single students to figure out, but is solvable by a group. Then we talk about how the problem got solved collaboratively.

Rodi

*You can read more about how the game is played from Smullyan’s book “What is the Name of this Book?” or from my reports about this game from another session five years ago: https://talkingsticklearningcenter.org/logic-session-2-knights-liars-percy-jackson/

**I am paraphrasing some of the mathematical language that the students used, but not changing their meaning at all.

***Only 2 students attended the first session. I invited parents and siblings to round out the group, but it turned out that wasn’t necessary.

The post “Waggy, Do You Eat Meat?” (Some Basic Tenets of Mathematics) appeared first on Talking Stick Learning Center.

]]>The post New Math Circle Course Schedules appeared first on Talking Stick Learning Center.

]]>**Unofficial schedule**

__Classic Math Circle Problems__

Dates: Thursdays, 3:30-4:30pm, 9/20-10/18 (5 weeks)

Suggested Ages: 5-7

Knights and Liars, open questions, story problems, pattern making and breaking, explorations of infinity, proofs, and more. We will have fun with these classic math circle activities as students develop the mathematical-thinking skills of asking questions, forming conjectures, testing conjectures, and generally seeking the underlying structure of things.

__ __

__Category Theory__

Dates: Thursdays, 3:30-4:45pm, 10/25-12/6 (6 weeks, 75-minute sessions, 7.5 hours total, no class on Thanksgiving)

Suggested Ages: 10-14

Mathematician Eugenia Cheng describes category theory as “the mathematics of mathematics.” Inspired by Cheng’s book “How to Bake Pi,” we will do activities that use abstract mathematics to see, understand, and generalize the defining structure of things. And by “things” I mean mathematical things, logical things, and social phenomena. Visit her website (eugeniacheng.com) for a preview.

__Queen Dido Problems__

Dates: Thursdays, 3:30-4:45pm, 1/24-3/21 (8 weeks, 75-minute sessions, 10 hours total, no class on 3/7)

Suggested Ages: 13+

In this course, students will explore real mathematics problems from ancient history. These will include Queen Dido problems, Zeno’s Paradox, and ancient inheritance problems. We’ll do the math and put the problems in their historical contexts. We may dabble in a few mythological problems as well. Mathematical concepts will include pre-algebra, algebra, geometry, and some calculus, but pre-requisite knowledge of these topics is not required.

__Polyominoes and Functions__

Dates: Thursdays, 3:30-4:30pm, 4/4-5/16 (6 weeks)

Suggested Ages: 8-10

Polyominoes are a hands-on geometry activity that develop students’ thinking about classification, combinatorics, symmetry, and more. We will also study characteristics of functions via the book Funville Adventures (or via extensions of this book if the students have already used it) and function machines in order to develop algebraic reasoning skills.

(registration information should be posted within a week)

The post New Math Circle Course Schedules appeared first on Talking Stick Learning Center.

]]>The post Platonic Solids: The First Three Weeks appeared first on Talking Stick Learning Center.

]]>(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.

The post Platonic Solids: The First Three Weeks appeared first on Talking Stick Learning Center.

]]>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

The post Invariants appeared first on Talking Stick Learning Center.

]]>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

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

]]>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.

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

]]>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

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

]]>The post Spotify and Random Number Generators appeared first on Talking Stick Learning Center.

]]>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.

The post Spotify and Random Number Generators appeared first on Talking Stick Learning Center.

]]>