(May 16 and 23, 2017)
I thought that this course was about functions. We did a great exploration of them for the first four weeks. But in the last two weeks, something magical happened. Our math circle transcended a single topic to arrive at an exploration of the essence of mathematics, a near-visceral experience of true mathematical thinking. I’m so excited about it that I’m writing a lot. I don’t want to lose any of this story. And writing this gives me the pleasure of reliving it a little bit. This list is a little preview of this report:
• Inverse, composite, and random functions
• Applying logic
• Divination versus math
• Attending to precision
• A quick proof
• The point of all this
• Is this demonstration a proof?
• Is a set of a million pieces of data sufficient for proof?
• Is certainty attainable?
INVERSE, COMPOSITE, and RANDOM FUNCTIONS
S put a function machine (named Richard) on the board and worked with the group to determine that the rule was y = 10x+1. “What if you run the numbers through backward?” I asked. “What’s the rule then?” They figured out that the inverse (the backward, undoing rule) was (x-1)/10. Then F put up a rule called Triangle with a very limited domain. This rule turned out to be 9x with an inverse of x/9.It got really interesting when we made composite functions of these two. What happens if you run a number first through Triangle and then through Richard? What if you run it through both but start with Richard? How does Triangle’s very limited domain affect what you can do when you bring in another function? How do you find the inverse of a composite function? About half of the students could have gone on and on and on with this discussion. But the other half were starting to tune out. (You may want to find out which camp your child fell into to know whether to go further with this topic at home.) It was time for another function machine.
It got really interesting when we made composite functions of these two. What happens if you run a number first through Triangle and then through Richard? What if you run it through both but start with Richard? How does Triangle’s very limited domain affect what you can do when you bring in another function? How do you find the inverse of a composite function? About half of the students could have gone on and on and on with this discussion. But the other half were starting to tune out. (You may want to find out which camp your child fell into to know whether to go further with this topic at home.) It was time for another function machine.R came up to the board with the machine named Bill. The rule turned out to be x^2 +x. It involved a lot of tedious calculations when the inputs were big. So R limited the domain to numbers 3,000. It still involved a lot of tedious calculations! But he coached them on picking strategic in numbers so that they were able to figure out this less-than-obvious rule.
R came up to the board with the machine named Bill. The rule turned out to be x^2 +x. It involved a lot of tedious calculations when the inputs were big. So R limited the domain to numbers 3,000. It still involved a lot of tedious calculations! But he coached them on picking strategic in numbers so that they were able to figure out this less-than-obvious rule.
Finally, T came up with a machine that really did to generate random numbers. “Is it a random number generator?” someone asked. “Yes, it is.” Someone protested. The group debated whether a random number generator fits the definition of a function, as T insisted it did, and they did have to grudgingly admit that he was right.
“If A is an element of set B, and B is an element of set C, and C is an element of set D, is A an element of set D and how do you know?”
That’s the question I was asking in my mind, but the question that came out of my mouth was different. I didn’t want to overwhelm these middle schoolers with abstract language, so I gave them names of things and didn’t use the language of set theory. A few of the students engaged in the discussion, but many of them got distracted by the caterpillars that seemed to be all over the place. I asked about the difference in the kind of thinking required to solve the function machines versus the thinking required for this set/categorization problem. Only one student could feel a difference. She said that this problem used logic and the first didn’t so much. I realized that I need better problems for next week to get everyone on board.
DIVINATION VERSUS MATH
“What is divination?” I asked.
“Isn’t it finding water?” said someone. I had to clarify that I meant the other divination – the kind they teach at Hogwarts. We briefly discussed modes of divination they had heard of. Then I read them this passage:
“Struck argues for a cognitive basis to divination. Moments of insight, like the ones Malcolm Gladwell wrote about in Blink, might be linked to some of the same processes that guided divination in the ancient world. It’s the art of the inexplicable hunch, as sophisticated as it is primal. “There’s a very limited set of things that all humans do,” Struck says. “There’s eating, walking, and there’s divination.”
Struck sees similarities between the early tools of literary interpretation he wrote about in The Birth of the Symbol, with their emphasis on extracting a hidden message, and classical models of divination—attempts to make meaning out of the world.
That same impulse also gives rise to one of the darker aspects of our current political climate—the proliferation of conspiracy theories, interpretation gone disastrously awry. From a classical perspective, the dividing line between sending away to the oracle at Delphi and believing that the Clintons are running a child-sex-slave-powered pizza parlor is improbably thin.
“As humans,” Struck says, “we’re meaning-making machines. Like a bird is hard-wired to build a nest. It’s a way of making the world habitable for ourselves.” Paranoia, he says, is “one of the prices we pay for this amazing capacity for thought.”Struck also sees a connection between the modern voting process and divination. “In these presidential elections, we’ve got to figure out how to treat every single individual in our society. That’s a huge calculation. And every four years we boil that complexity down to A and B. It’s a bit like drawing rocks,” a popular divination practice.
Struck also sees a connection between the modern voting process and divination. “In these presidential elections, we’ve got to figure out how to treat every single individual in our society. That’s a huge calculation. And every four years we boil that complexity down to A and B. It’s a bit like drawing rocks,” a popular divination practice.
“Life is confusing,” Struck continues. “And it’s much more complex than we’ll ever figure out. There’s magnificent beauty and terror too. Sometimes there’s a benefit to stepping back and limiting the variables. Divination is a way of limiting the variables. As complex as the situation is, let’s step back and look at this sheep’s liver right now.”*
What in the world did that have to do with mathematics, I asked the students. We spent the rest of session 5 comparing and contrasting math and divination.
ATTENDING TO PRECISION
“I’m going to do my own function machine,” I announced at the start of our final session. “Your job is to discern the rule. I’ll put it on the board as a mapping diagram. The domain is the first name of anyone I have seen.”
“Seen as in you’ve been dating, or seen as in your saw with your eyes?” asked S.
“Ah, I see the ambiguity here. I mean anyone my eyes have beheld. Someone name a person I have seen.” The students looked at each other with confused looks. “Name one person you know I have seen.”
“C?” suggested someone, naming one of our group members.
“Yes! I have definitely seen C. When you put C into the machine out comes M.” (I named another participant in the group.) Confusion gave way to curiosity.
“What if you put in S?” asked someone excitedly.
“When you put in S, out comes C.”
“Huh?” said several students, since C was whom we first put in, and now he was coming out. Several posited that the rule/output/range only applied to people in our math circle.
“How would you test that?” I asked, unwilling to reveal anything about their conjectures. I was working on fostering their own methods for testing conjectures. They realized they could test by putting in someone outside of math circle to see what happened. But who could they say to put in, considering it had to be someone I had seen. How would they know who I had seen? Then our adult helper walked past.“Put in Meryl!” said someone.
“Put in Meryl!” said someone.
“When you put in Meryl, out comes Robert Wadlow?” This generated a universal “huh?” Who is Robert Wadlow, they all wanted to know? “I can’t say right now, or that would give away the rule. How can you further your testing?”
“Put in Harry Potter!”
“When you put in Harry Potter, out comes Professor McGonagall.”
“You’re seen Harry Potter?” asked someone suspiciously.
“I’ve seen him in the movies. That counts.” People seemed excited, confused, suspicious. Everyone was participating.
“Put in yourself!” someone ordered.
“When you put Rodi in, out comes my husband Sam. Some of you have seen him but most of you have not.”
“I know,” said F. “Put in Bob!” F pointed at the whiteboard that still had a diagram of our very first function machine from six weeks ago. The participants had never let me erase this over the weeks, even though we were short on boards.
“When you put in Bob, out comes Harry Potter.”
“But Bob is just a drawing, not a real person!” argued the students. “You can’t put him in!”
“Bob cannot be excluded from the domain of this function. The test for inclusion in the domain is that I have seen him, not that he’s alive or real. So Bob stays.”
“In that case,” argued someone else, “Harry Potter isn’t even real. He’s just a figment of someone’s imagination!”
“You are right. I didn’t think of that,” I acknowledged. “I’m imagining the movies, so I’ll have to change Harry Potter to the actor who plays him in the movies, Daniel Radcliffe.”
“You’ll have to change Professor McGonagall too!” ordered the students.
“Maggie Smith!” someone yelled out. I crossed out both character names and replaced them with the actor names on the board. “But I think Maggie Smith might be dead,” argued someone else.
“Doesn’t matter if she’s alive or dead. I have seen her in the movies so she stays.” (Meryl chimed in here that Maggie Smith is indeed alive.)
The students resumed testing people in our group. When S goes in, M comes out. “Wait a minute!” objected someone. “M came out when you put C in.” Others corrected this objection with the definition of a function – that different inputs can have the same output, the rule is that a single input can’t have multiple outputs. Back to testing. When T goes in, Rodi comes out. “Put in F” requested someone. Hmmm…. I wasn’t sure who comes out when F goes in.
“You’ll all need to stand up,” I told them. Everyone stood up. “Ah,” I nodded, “when F goes in R comes out.”
“Does this have something to do with height?” asked the other F. I wasn’t going to yield any information, and told the group that they have to figure out how to test her conjecture. They put in someone else and then posited that everyone who came out was taller than their corresponding input. If you’re right, I told them, then you should be able to predict who will come out when you put in the next person. They put in F. But there are several people in the class taller than F. How could this work?
The range (output) must be defined by naming just one person taller than the person put in, the class concluded. I congratulated them for figuring out what I had been thinking, but in the spirit of true mathematicians, the arguing and assumption attacking wasn’t over. (This may be my favorite part of math!) “How do you know that Daniel Radcliffe is taller than Maggie Smith?” I had to justify that one with describing a scene from the movie. “How do you know that as he grew he didn’t become taller than her? He was short in the first movie but not by the end.” I conceded that I should have been more precise when I input him, maybe specifying something like “in the first movie.” “How do you know that Robert Wadlow is taller than Meryl?” I defended this with the statement from the Guinness book that he was 9 feet tall. We went on like this until no one could find any more assumptions to attack or concepts/ideas/objects that required more precision.**
A QUICK PROOF
If B is taller than A and B is shorter than C and A is taller than D, prove that C is taller than D.
I gave this problem to the students but asked them to supply names. We did it with Lucille, Parker, Josh, and Steve. The group collaboratively proved it, with S up at the board and the students dictating to her. I was temporarily out of the picture, a place I’d like to be more often during math circle. I love it when the students take over and no longer need me!
THE POINT OF ALL THIS
“We just did 2 problems that revolved around height. Reflect for a moment about the kind of thinking you were doing during the function machine. And now for the proof. How did each feel? What was different about them? Which was easier for you?”
The students discussed this for a little while. They concluded that in the second height problem we started with rules and then reached a conclusion, whereas in first height problem we started with examples to work up to rules. We discussed inductive versus deductive reasoning.*** Which do you think is better? Which is more important? Which comes first, the chicken or the egg? That question isn’t in jest; it’s relevant here as we discussed how some of the storied ancient Greek philosphers declared that deductive reasoning is the be-all and end-all, while some others said that you can’t start with making deductions if you don’t have anything to start with. Where do you get starting info? The students had widely varied opinions about which type of reasoning they preferred and why, but all agreed that you do need both.
IS THIS DEMONSTRATION A PROOF?
Cut a triangle out of a piece of construction paper. Tear off each of the four vertices (“corners” the students called them). Take each of these torn-off vertices and set them on the same point with the adjacent edges touching the next one. **** You can see that the three angles together make a line, or 180 degrees. Ta-da, 180 degrees in a triangle! (The students all did this with their own triangles that they cut out.)
But is this a proof? I was so happy to see the kids vehemently shaking their heads “no!” I didn’t even need to tell them that this is a demonstration but not a proof. They told me that there may be some other way to make a triangle where you don’t get a line. Go kids!
IS A MILLION PIECES OF DATA SUFFICIENT FOR PROOF?
Once a mathematician named George played a little game: Start with 2. How many prime factors does it have? Just one. One is odd. So the Odds score 1. How many factors does 3 have? Just one. Score another for the Odds. How many does 4 have? Two. Score one for the Evens! The Odds are ahead, 2-1.
I worked through scoring this game with the students for a bit. Once you get to 11, the Evens catch up and the score is 5-5. This means that 5 of the numbers from 2 through 11 have an even number of prime factors and 5 have an odd number of prime factors. When you try 12, the odds move ahead again because 12 has an odd number of prime factors. So far, the evens have never been ahead.
Will they ever get ahead? The students posited some conjectures.
I told them that George tested many numbers and formed a conjecture (known as Polya’s Conjecture because George’s last name was Polya) that the evens will never get ahead. When every number up through 1,000,000 was tried, most people thought the conjecture was true. “What do you think?”
“That must have taken a loooong time,” said F. *****
Even though no one had thought that the triangle demonstration was a proof, many of our math circle students thought that evidence from a million trials was convincing enough. I was more than a little surprised. I had to stop mentally patting myself on the back for how astute the group had been about the triangle demonstration. “Then why do you think I’ve chosen to share this problem with you?” I asked.
“Oh no, there’s a larger counterexample, isn’t there?” asked M. A few people groaned. M was right. Polya formed his conjecture in 1919, but in 1962 Lehman discovered that the evens pull ahead at 906,180,359.
“So when is enough enough?” I asked. “Is there a number of pieces of data you need for absolute certainty?”
“You can never be sure,” posited several students. “You’d have to try every possible input and that would be an infinite number in most cases and trying an infinite number of things is impossible.”
IS CERTAINTY ATTAINABLE?
“Let’s look at a function machine again. If I put in 1 and out comes 2, and 2 in 3 out, and 3 in 4 out, what do you think the rule is?” (You’re adding one.) “So the rule is y = x + 1, right?” (Yep) “So I’m going to posit my own conjecture. I’ll call it Rodi’s conjecture.” (Brief digression when students objected to naming my conjecture after my first name and not my last.) “Rodi’s conjecture states that in the function y = x + 1, y will always be greater than x. Is there a way to prove that without trying every number into infinity?”
Yes, yes, yes, yes, yes! This was the real aha moment (I think/hope). The students saw that this was a statement that could be tackled with Proof. They could think of several ways to do it. But we didn’t have time. I ended the session/course with a quick recap of our big ideas:
• Inductive reasoning differs from deductive, and mathematics primarily (but not exclusively) uses deductive reasoning.
• No matter how many pieces of date you have, data points alone do not constitute proof.
• Mathematics requires proof.
Now goodbye, and have a great summer! Then something that hasn’t happened in one of my math circles ever: the students applauded. I don’t know whether they were applauding me, or themselves, or the ideas in the bullet points, or some combination, but it was great! (My conjecture is that they were applauding a sense of group accomplishment.) Thank you all for allowing this wonderful course to happen.
*Jamie Fisher, “Peter Stuck’s Odyssey,” The Pennsylvania Gazette, May/June 2017, http://thepenngazette.com/peter-strucks-odyssey/
** Please email me if you know a single word verb in the English language that means “to make more precise.” There definitely should be a word like this. Precisify?
***More info on deductive vs. inductive reasoning here: http://mathforum.org/library/drmath/view/55695.html
****Here’s a nice visual to show how to do the triangle tearing: https://www.quora.com/What-should-a-triangles-angles-add-up-to (scroll down to diagram at bottom of page)
*****Her comment made us wonder whether there were calculators back in 1919. I suspected that there were but wasn’t sure. I’ve since looked it up: https://en.wikipedia.org/wiki/Calculator