MATH CIRCLE: Questioning Everything (Eye of Horus 2)
EXCITEMENT AND ENTHUSIASM
(September 29, 2013) Many students arrived feeling very enthusiastic about today’s session. Several parents had emailed me about how excited they and their kids were after the last (first) one. K and her mom told me about their own investigations of interesting numbers they had been doing at home. R announced that “I’ve not been able to think about anything other than Math Circle since 1pm today.”
As last week, when people arrived, we played Tens Concentration – this time with a new rule: an option to pick up a third card if desired. Our prior rule was that you can keep your two cards if you can get 10 with them. “I wish we could spend the whole time playing this game,” said someone (again). When it was finally V’s turn, he hesitated with his cards, looked at the ceiling, and asked, “Could you use subtraction?”
On the inside, I was bursting with excitement, as I had been biting my tongue for 2 weeks wondering whether the kids would propose subtraction. On the outside, I calmly replied, “Sure, if you want to. It’s your game. My only instructions were to ‘get ten.’” (For those of you new to Math Circles, one of the leader’s primary tasks is the Biting of the Tongue.) So V figured out a way to ‘get ten” using subtraction. On every subsequent person’s turn, V advised enthusiastically, “You can use subtraction! You can use subtraction!” Some did. Some didn’t.
One student asked the others, “How do you use subtraction?” Several students happily demonstrated, and play continued. A shift in our group had occurred here: the kids had not posed the question to me. A huge goal of Math Circles is collaborative inquiry. The leader is just the secretary, keeping track of things. This Circle is a new mix of kids. About half have been in Math Circles before, although not all in the same group, and about half have never participated. Group cohesion is going to take some time. In my mind, I liken this to my older daughter’s soccer team. In every game, I hear parents yelling from the sidelines, “You girls need to talk to each other!” The team is full of strong players, but the group is new so they have to learn to communicate with each other. I’ve heard Bob and Ellen Kaplan liken a new Math Circle to an immature Quaker Meeting. I suppose that learning how to reach consensus in most settings takes time.
Back to the big math idea here: I explained to the kids that subtraction is allowed in Tens Concentration because I had said “get” in the instructions. In math, making clear what each word in the instructions means is important. In Math Circles, leaders pose questions in a deliberatively vague manner in order to prompt questioning. Mathematicians question everything, right?
Next, I read the part of Ovid’s Metamorphoses about Echo’s relationship with Narcissus. We discussed what each line meant and also described places where we may still “hear” Echo. The students who knew more of the story thoughtfully bit their tongues. I think I am done talking about Echo for this course, but mathematically this would be a good jumping off point for an exploration of reflection. My goals in reading this story are to give the mind a break and also to point out connections between math and everything else. Math, like every “subject,” is not something that exists in isolation.
MOVING FROM THE CONCRETE…
I reached for the blocks. Some students literally dived in and grabbed blocks, while others waited in their seats to receive some. Y took his allotment and moved to a chair and table to immediately begin explorations. Others did this on the floor. Still others waited for instruction.
“Last week you all created your own Narcissistic Numbers. I’d like to clarify some of the math words you used. For instance, some of you suggested that numbers like 1+1, 2+2, 3+3, etc., love themselves and are hence Narcissistic. What did you mean by ‘plus?’”
A few kids looked at me as though I were out of my mind. “Plus means that you add them,” the kids patiently explained.
“What do you mean by ‘add?’” I delved. Silence. Then a few ideas emerged from the group: to put together, to combine, or to count up.
“But what do you mean by ‘count up?’” immediately demanded D of the rest of the group. (Ah, another welcome shift away from the wise-instructor model to the collaboration-among-colleagues model. In a mature Math Circle, each person plays a necessary role. I wonder whether one of D’s roles here will be a Definition Questioner, since he thought to question other terms as the session progressed.) No one was able to define this, so I asked them to show me with blocks. The students were relieved to have an easier task.
… TO THE ABSTRACT…
“Do you count with numbers, numerals, or things?” I asked.
“A numeral is just one thing,” said S to his colleagues.
“Numbers and numerals are the same,” said V. We revisited last week’s discussion of how the terms are technically different but are often used interchangeably. This discussion led to apparent consensus that we typically count with numerals. However, the abstract thinking required here got some kids antsy. A few started climbing under, over, and through the furniture. I refocused their attention with a historical and literary interlude connecting author Rick Riordan and the ancient Greek mathematician Thales. As soon as I mentioned Riordan, O questioned my pronunciation of his last name. I try hard not to use that oft-sanctimonious phrase ‘I heard it on NPR,’ but, well… I heard it on NPR. (Sorry, O.) Then the students began debating with each other (!) whether the names Thales and Thalia (a Riordan character) were similar enough for the name of the person to have influenced the name of the character.
I gave the students three riddles based upon famous statements by Thales:
- What is the only good that is common to all men?
- What is the most active thing in the universe?
- What is the most difficult thing in life?
I’ll emulate the Car Talk guys and not give the answers this week. Ask your kids. They debated these heartily. One person’s conjecture built upon another’s, in true collective inquiry. However, the answer to the riddles were elusive, so I occasionally interjected to say when they were close. (Next time, I probably won’t do this.) The kids did not all agree with Thales, and argued amongst themselves over Thales philosophy. I told the kids how Thales was known for proposing science, as opposed to the gods, as the explanation for natural phenomena. One participant aptly paraphrased this as, “Zeus did not cause that lightning bolt in the sky!” I told them too of how Thales proposed that water was the basis for everything.
“He’s dumb,” said someone. No, I explained, he used science to reach this conclusion. We talk briefly about how deductive reasoning could possibly lead to that conclusion at Thales’ time in history.
…AND THE EVEN MORE ABSTRACT
“Can you count without using numbers or numerals?” I asked.*
“No,” said D with certainty.
“Yes,” said V with certainty. I asked him to demonstrate. He pointed to each chair in the room and said nothing. “I’m doing it in my head,” he explained. This answer required us to precisely define what we mean by “count.”
I asked both O and L for their opinions, and both said “I’m not sure.”
R then announced, “Yes, you can count without numbers. You could put a block on each chair, collect the blocks, count the blocks, and then know how many chairs you had without counting the chairs.” You could feel the excitement in the room. Then I started asking difficult questions: Does it matter how many blocks you use? Does it matter how many are on each chair? Does each chair have to have the same number? Might this method still be using numbers indirectly? With each question (and comment from students), R asked excitedly, “Would you like for me to demonstrate?”
“I am going to require that this conversation continue without blocks,” I said. I explained that one goal of mathematics is to think more abstractly, to work more with ideas in our heads, or on paper, and less with physical objects. Not, I qualified, that we never want to use objects. They are very useful. But you need the ability to think without them too. Some kids appeared shocked by this, but others nodded their heads in understanding. In the continuing conversation, Z had a lot to say, and led the group to a consensus that you need one block per chair for R’s counting method. I threw in the term “one-to-one correspondence,” but didn’t take it further.
Instead, I turned our attention back to Narcissistic Numbers. “Some of you last week suggested 3×3 and 2×2. What did you mean by ‘times?’” Again, a lot of talking erupted at once.
“Three times three is nine,” stated Z. The others agreed. My role here was that of the clueless adult, not understanding: But why is it nine? What does ‘times’ mean? No one could clearly explain, so we returned to the concrete (blocks). Some made numerals with blocks, while others made three rows of three and “squished them together.” For the most part, people paid attention to each other’s work, but several kids had run out of attention and got physical. We were just about out of time now. Kids asked me to photograph their work, so I did. J and L had made a person out of blocks and were writing a list of how many blocks by color they were short to make the person symmetrical. (I don’t even think they realized they were doing subtraction.)
I sent them off with an optional question to think about: Why is 3×3 called ‘3 squared?’
S, Z, R, J, and L stayed after class to have fun placing the blocks back in their boxes in a systematic way. (Again, they probably didn’t know they were doing math.) They wanted to stay longer to use the blocks, but this was not possible. I promised the kids that I’d tell their parents where to buy those blocks .
CONFUSION AND FRUSTRATION
After class, I heard some concerns from one student and from a parent of another student. For these two veteran Math Circle participants (and maybe others?), this course feels different from previous ones. The parent explained that, to her child, it doesn’t seem as focused on one question, nor as playful. The student felt intimidated that some of the other kids seemed more skilled at arithmetic. (“I don’t understand what we were doing; I feel like all those other kids are geniuses.”) This course is different from prior courses for younger children in a number of ways:
- The math skills have changed. This group is now wavering on the cusp between concrete and abstract thinking. That’s a hard place to be. Also, we’re tackling Number Theory, which comes along with a lot of emotional baggage for many kids, teachers, and parents.
- As I mentioned before, we have a new mix of kids. The group needs to form cohesiveness. Also, it’s a larger group with a wider age span than we typically have for this age group. Interestingly, though, it’s not necessarily the oldest kids who demonstrate the most confidence.
- I am requiring the participants to think about terminolgy that most of us learned by rote. What’s your emotional reaction when I ask you to define concepts such as number, numeral, thing, math, add, count, and multiply? I want kids to learn what James Tanton calls “the very important first step to problem solving: Read the question, have an emotional reaction to it, take a deep breath, and reread the question. Have another emotional reaction.”** Struggle and perserverence are necessary precursors to learning just about anything for the long haul.
- We are jumping around the content a lot. Our overriding question is “What is a Narcissistic Number?” The answer involves both multiplication and exponents, skills that no one in our group has mastered (or even heard of). This is intentional. They won’t master these skills by the end of the circle, but should be curiously engaged as they engage in mathematical thinking to strengthen their understanding of addition and its “more powerful” cousins.
I expect the next session to feel more cohesive in terms of group interaction and content. In the first two weeks we were gathering the raw materials for collective problem-solving. I plan to explicitly enumerate these problem-solving skills in class next time. Please keep providing feedback on your children’s Math Circle experiences. That information works to continually improve the course. Also, please email me if you’d like me to write about some specific math activities that you can do at home to reinforce what we are doing here. (I’d love to tell you about a game J invented, “Nerds Concentration,” but I’ve written enough for one day.)
I know this report was long. Since a number of you are new, though, I want to be incredibly clear on what we’re doing. You honor me and your children and math education in general by reading the whole thing, so thank you.
PS I promised the kids that anyone who shows up 10-15 minutes early next time can play some extra Tens Concentration or blocks. Unfortunately, because we have to lock up the building, this cannot happen after class.
*Thanks to Craig Daniels of the Portland Math Circle for this question.
**MAA AMC Inspiration Letter 7, James Tanton http://www.jamestanton.com/wp-content/uploads/2012/09/MAA_AMC_Inspiration_Letter-7_-New-LetterHead1.pdf