(September 20 and 27, 2018)
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 17th 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.
*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.