COUNTING STRATEGIES
(March 25, 2014) We started out on the floor counting pennies. “I want to buy a piece of candy – a Swedish fish. It costs 10 cents. Do I have enough?” The pennies were arranged in a particular way.1 The students counted, with various results. “Is there more than one way to count them?”
“No!” announced everyone in unison.
“Let’s count them together, since we’re getting different answers.” I put my finger on each penny as we counted. The students did it too. They saw that they were not all using the same counting strategy, and then reluctantly agreed that multiple strategies exist. They debated the monetary value of each coin – some thought that pennies are worth 25 cents each. I quickly gave the correct info6, then rearranged the pennies into two rows: one containing 5 pennies and one containing 6. I put more distance between pennies in the row with only 5, making it appear longer.2
“Which row has more pennies?” I retreated as the students debated amongst themselves. Finally they arrived at consensus after L’s convincing counting demonstration. The shorter row has more pennies. I rearranged the pennies in various ways until the students were experts. After last week’s debate about whether 13 is bigger than 12, I wanted to find out for sure whether each student could understand “conservation of number” tasks. Since not everyone could do so at the beginning of our activity today, but all could with ease by the end, this begs the question: Can conservation of number be taught?3
DEFINING OUR TERMS
Our group was now were ready for the big question of the day: “What does the word more mean?” The kids looked at me incredulously. Why would I ask such an obvious question? I explained that in mathematics, we have to be sure we know what every word means so that we’re all talking about the same thing. What a discussion! But L then insisted that you cannot use the word in the definition5, which effectively shut everyone up.
After some quiet thinking, B arranged each coin in one row directly across from a coin in the other row, then timidly put forth that “This row must be more because this other row is less.” Everyone was quite satisfied with that definition – that somehow the word “less” could be used to define “more.” I accepted this for now because (1) his coin positioning was essentially a visual proof of 6>5, and (2) I know that I’ve asked a group of five-year-olds to do the almost-impossible. They don’t really have the vocabulary and syntax to describe their own reasoning. Since mathematics requires the ability to do so, though, we’ll keep working on it.
“How do we know that 6 is more than 5?” I asked. Despite the positioning of the coins in a visually convincing way, no one could verbalize it. We’ll return to this next week.
OPPOSITES WITH WORDS and SOME LOGIC
“Let’s talk about opposites,” I said, to give their brains a break. “What’s the opposite of up?”
“Down.”
“What’s the opposite of being in a cave?”
“Being out of it.”
Students’ own examples of opposite pairs spilled out. It was clear that they were familiar with this concept. I asked a few more – particularly “What’s the opposite of more?” and “What’s the opposite of less?” – and then we returned to the story of The Very Clever Prince.4
The students had been thinking about it a lot since last week, but felt they had made no independent progress. M reminded us of the various reasons for dismissing prior conjectures. I asked them to discuss their discarded ideas anyway because in math, we can make progress when we examine our mistakes and dead ends.
We talked about why the prince could not do things like steal the cards, kill the guards, etc. I asked “How many guards to you think that king had around his castle?” A’s eyes lit and opened up more and more with each larger number she proposed. Ah, the feelings of excitement of taking mathematical risks! …the power of big numbers! These are real mathematical emotions, which can help us experience the freedom that mathematics is capable of providing.
Our conversation generated a lot of talk about how the prince might be saved if only he could get rid of both Death cards. I then jumped seemingly randomly back into opposites: What’s the opposite of up? (Down!) What’s the opposite of in? (Out!) What’s the opposite of more? (Less!) What’s the opposite of death? (Life!) I had hoped that this line of questioning might trigger some more ideas, but it did not. Oh well. I asked the kids to think more about the idea of getting rid of both Death cards, even though the kids seem to accept that this would be impossible.
A new puzzle:
“A little boy lived on the 15th floor of an apartment building. His mother let him take the elevator alone. But the way he used the elevator was somewhat odd: when he went down, he went all the way to the ground floor, but when he went up, he only took the elevator as far as the 7th floor, and then he took the stairs. How can you explain that?”8
Faces twisted up in perplexed curiosity. All students had been in elevators, and produced numerous conjectures, which served to bring out the limitations and assumptions of the problem. We’ll come back to this another week after it simmers in people’s heads for a while.
CLOCK ARITHMETIC
By then, the kids wanted to jump around our floor clock, so we discussed and jumped counting by threes, asking our usual question: If you jump/count by this number, will you land on every number? They made predictions, jumped around multiple times, and reported back. We diagrammed it on the board and recapped all of our attempts with a list. (This list is hopefully building up to a mathematical realization by the end of the course.)
OPPOSITES WITH PILES OF STONES
Then we moved outside, where I had placed a big pile of stones. For a very long time, I’ve had this notion that very young children can truly understand negative numbers, if only we give them the right activity. I think one such activity is a pared-down physical version of Jim Tanton’s “Piles and Holes” approach. (If only we can get kids to see that negatives are an expression of opposites!) But I ran into a major stumbling block today.
After we did more reciting of opposites, I pointed to my stones and asked “What’s this?”
“A pile of rocks.”
“What’s the opposite of a pile?”
“No pile.”
“Not making a pile”
Hmmm…. not what I was expecting to hear. I asked the kids to make their own single large piles and then make the opposites of their piles. Same reply from most of the kids. I asked each to make 3 piles, and then make the opposites of their 3 piles. Still “No piles.”
Their answers to these questions juxtaposed with my expected responses (something about holes) revealed that the kids saw opposites as negations, while I saw them as vectors (incorporating both magnitude and direction). Only B’s interpretation was closer to the vector approach: “Destroying a pile.”
Most of the students assumed that negation and making opposites are the exact same thing. But they are not always the exact same thing. Is the opposite of, say, finding $3 not finding $3? The difference between those two scenarios is $3. But if the opposite of finding $3 is losing $3, then the difference is $6. (I can’t believe that I’m getting into this subtle logical distinction with five-year-olds! It strains my own brain to think about this. I’m not formally trained in logic so I hope someone who is will straighten me out in case I’m on the wrong track!)
With the kids and the rocks, I was definitely taken aback. It didn’t occur to me that this would come up. I hadn’t thought about it the way most of the kids were thinking about it, and wasn’t prepared to facilitate this conversation. So instead of talking, I got down into the gravel on the ground and started piling it up into a new big pile. “What am I doing?”
“Making a pile.”
“What would I do if I wanted to do the opposite?”
“Not make the pile,” said most, and “Dig a hole,” said B. I was getting nowhere and as so often happens, we were almost out of time. The kids were trying so hard to understand what I was talking about. I could see the intellectual strain on their faces.
I then made things worse by asking “What’s the opposite of nothing?”
“Something.”
“Does it matter what the something is?” I was going somewhere with this, but don’t remember where. Somehow we did, with great difficulty, get to a point where the kids accepted that holes could be the opposite of piles, but that flat ground could also be. (I hope we can resolve this quandary before the course ends in two weeks.) I think many of them were humoring me. They definitely take everything we do here very seriously (as do I). So we spent the last few minutes on something a little more fun: ancient Egyptian numerals.
Last week we had briefly touched upon the idea that there is not only one way to symbolize a certain quantity, so this week I showed them another. I drew some ancient Egyptian numerals and asked for guesses about what each symbol represented, besides quantity. I explained that a scholarly website describes the symbol for a million as “astonished man,” while a newspaper owned by a religious organization describes it as “a prisoner begging for forgiveness.”
“So which is right?” asked the students, still seeing me as the Person Who Knows the Answers (or, Answer). I clearly have my work cut out for me on that score.
“I don’t know.” (Magical words when spoken by a seeming authority.) We briefly discussed how it could be possible that I don’t know something7, and then I chased them out until next week. I can’t wait!
Rodi
PS My reports may get sporadic for the next several weeks due to travel and holidays, but I will definitely report on everything at some point so you can follow up at home.
PPS Thanks to older sibling L for helping me to set up the room and the floor clock.
1 See the photo with 2 children lying on the floor with their fingers on some coins, or click here: http://www.openmiddle.com/dot-card-counting/. Thanks to Sue VanHattum for linking to this site on the Living Math Forum. If you actively engage in mathematical activities with children, I highly recommend this listserve. BTW there was quite an interesting discussion on the listserve this week on the validity of “open middle” problems – if you visit, find thread 27733 labeled “Good Problems.”
2 This activity is described nicely in Alexander Zvonkin’s book Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers (p17). Thanks to Dave Auckly for giving me this book.
3 If you’re curious, try googling this question; some very interesting articles pop up. Much of what we are taught about cognition in young children is based upon the work of Jean Piaget. He observed his own children for years to come up with his groundbreaking theory of developmental stages of childhood, and then later tested his theory on larger sample sizes. Despite various critiques of his research methods, his work is the gold standard upon which almost all early childhood teacher education programs continue to be based. I read an interesting article a few years ago with new questions about the underlying assumptions of his research – particularly the influence that the school environment plays on the emerging of the developmental stages (is it a causal factor?). I don’t know much about this, and would be interested in learning more. If any of you are familiar with scholarly research on how Piaget’s stages play out in children who do not attend school or preschool, would you kindly email me?
4 I first encountered this story at the 2012 Circle on the Road/Julia Robinson Math Festival at the Smithsonian. It was given to me by either/both Dave Auckly (now at the University of Kansas) and Laura Giventhal (Berkeley Math Circle). I got a request to put the story in the report, but don’t think that it is now homework! I have long since lost the written version, so I’m now retelling from hopefully-correct memory: “Once upon a time a princess wanted to marry a prince. Her father the king was opposed. He told the prince, ‘I am going to write Life on a card and Death on a card and turn them face down. Tomorrow you are to randomly choose one. If you choose the Death card you will die, but if you choose the Life card you may marry my daughter.’ That night, the king secretly had the cards changed, so that they both read ‘Death.’ The princess was spying and saw it happen. She alerted the prince, who was clever enough to devise a strategy that succeeded in him marrying the princess. What did he do?”
5 To quote Zvonkin (p80), “Your logic must be mature enough to be aware of the inadmissibility of circular logic.”
6 Zvonkin again (p81) – I guess you can tell what book I’m reading this week! “…one of my guiding principles: never impose your own point of view on a child, even by a hint. But the hierarchy or principles contains another principle, even more important: never follow your principles blindly.” At this point in our circle I could have let the debate on the denominational value of each coin run its course, maybe even with who-knows-what inquiry-type activity to help kids discover the answer, but this topic was so beyond the scope of our circle that I decided to provide the info in order to move the mathematics forward.
7 As you probably realize, am versed in the hierarchy of reliability of sources. But that would have been too big a conversation here, and these kids are probably too young to understand it anyway. (You, though, may want to continue the conversation at home.) The big take-away messages here are (1) multiple interpretations by grownups, and (2) the non-omniscience of teachers.
8 Zvonkin p47
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