ASSIGNMENT 2: Share your stories about problem groups 1, 2 and 3

I attempted the problems with a group of kids aged roughly 5-9, weighted to the younger end. There was a couple of younger ones in the mix but I think they were a bit shy and didn't manage to really get involved. {I have not yet really worked on Problem 1 but hope to do so with my own kids} **Problem 3** We did this first. The two older kids (8 and 9) worked on the diagram on a whiteboard and solved it pretty quickly. Other younger kids managed it afterward on the board too - one good strategy was to join one diagonal and then it clicked that you had to go *round* the other 'homes' to match the pairs. I also set up chairs with baby and adult stuffed animals to join up. I attached the baby animals with yarn to their chairs so the yarn would trace the paths. I was a little worried they'd get very tangled but this wasn't too much of a problem. The younger kids really enjoyed moving the animals, they quickly caught on to the idea of changing the arrangement of the parent animals. In fact they were more interested in finding different arrangements than in solving the problem! We did manage it eventually although I think the fact that they couldn't see the arrangement from above hampered understanding to some extent. There was also some disagreement about who would get to move each animal... It definitely piqued the kids interest, but I had one group who got it very quickly and another who wanted to play for a while. This meant I lost some of my students early on as they went off to play outside (the timing of this first meeting is not ideal as it is right after another class for most of the kids). **Problem 2** My eldest then wanted to move on the 'paper dolls' she had made. I explained the principle and the kids were very excited by the idea of getting into loops themselves. They had a bit of trouble arranging themselves but quickly caught onto the idea of one 'pillow' per 2 heads. One child very quickly said that there would be half the number of pillows as children for an even number of children. We began to test the theory but got rather distracted by tangential explorations into different body arrangements. They tried linking different parts of their bodies and making strange kinds of loops. I think given some more time and focus we might have tried it again with the paper dolls and noticed some more patterns. In fact I would like to try this again with my own children. Unfortunately in our math circle the kids then got involved in playing musical chairs and that was kind of the end... I think what I found hardest was keeping the attention of more kids than about 4 at a time. I think a couple of the older ones might have benefitted from more involved problems as well, which I had not prepared. The strategy of getting the kids physically involved in acting out the problems worked great in terms of engaging them - they loved it. But it also gave them ample opportunity for distraction and play - they were not vastly invested in solving the problem, and needed frequent nudges from me to get very far in reaching an answer. I think it worked for open ended problems, but not so much for getting from a to b. A bit like trying to walk somewhere with young children in fact! It would be interesting to see how far they would have got without my prodding, although I suspect it would have ended up in piles of giggling children! I also feel like a larger group need time to cohere. This was the first time we had done this, and the kids were already a bit stir-crazy from their day. After the novelty of play dynamics with a given group wears off I think (hope?) I might have more luck engaging them in specific problems and their possible solutions. Next time I think I will be more clear about starting and explaining the problem; there was a fair bit of peripheral activity which i think was distracting. It might also be worth splitting the kids into groups to attack the problems. Better get planning!

I did the first gear problem with about 10 people, ranging from 5 to 65 in age. Everyone I worked with had a pre-existing understanding of how gears worked, so everyone did the problem the exact same way. I gave them a sheet with the problem image on it and asked what way the gears with the arrows would turn. First, everyone just turned their fingers in the direction of the gear. But then it got too complicated (where the gears branch out), and so then they took a pencil and started drawing arrows on each gear. Although they understood what they were doing, all the children made at least one error about which way the gears would turn. The biggest mistake, especially among younger children, was that they didn't always draw the arrow in the same place, and that led them astray. So, for example, if the gear was going clockwise, they drew an arrow at the top pointing right. Then, for the counter-clockwise gear, they would draw an arrow at the top pointing left. So they went on, alternating directions in which the arrow was pointing. However, at some points, for a clockwise gear, they would draw the arrow pointing right, but at the bottom. But that would actually move the gear in a counter-clockwise direction. The younger ones got frustrated when I pointed out their errors, because they thought they were right because the arrows were always alternating. I then asked them if they could figure out an easier way to do this than mechanically drawing an arrow on each gear. All the adults and a couple of the teens eventually figured it out they could just count and know odd is clockwise and even numbers were counterclockwise (or however it is in the puzzle), but some of the teens and none of the younger ones couldn't figure it out without some pretty direct instruction. THEREFORE, if I were to do this again, what I would do, especially with younger students, is have them choose a color for clockwise and a color for counterclockwise and have them color the gears instead of drawing arrows. Not only would this eliminate errors, but it would be easier for them to see the pattern. The frustration level of being "wrong" when they knew they were "right" because the arrow was pointing in the opposite direction from the one before prevented a couple of the youngest students from completing the exercise while we were together, although the moms took some sheets home to try again when they were in a better frame of mind.

Well we've tried yesterday with Problem #3. I was the best student among us )) My babies had a fun with modeling the balls and sausages, and enjoyed the story about Mother Ball, Father Ball and Baby Ball, which are needed to find each other. The abstract mind of my 3y.o. daughter (Maya) is enough to let her agree that Mother and Father could look like coloured balls, but 2y.o. son (Vic) was sceptical about this ))) So Maya tried to connect balls each other but she didn't keep in mind balls of similar coloures working with the nearest balls. And idea of non-crossing way seems totally not clear for her. And Vic had a fun copied our movement and activity with modelling, drawing the way between the balls but only as he understood it. So he draws something here and sticks something - he doesn't mind if it is fun. alt text

After our first meetings, I have a better idea what levels my groups are working at: **K-1st group =** lots of intuition, counting, minimal addition **teen group =** rules-oriented, some intuition, mostly at pre- or early-algebra level (ie, not yet comfortable with exponents) I wrote a thorough report of our K-1 meeting: [Math in the Park K-1][1] The teen group only got through a couple of the puzzles. They are not going to be able to work through the MAA AMC puzzles as written, so I either need to focus on one problem per session or choose easier versions for them. And I need more practice at scaffolding while "being invisible." That's hard! We started with the isosceles triangle with inscribed circles, to see what they would notice about the drawing. They had memorized (or partially memorized) some area formulas, which they tried to apply (using a ruler) with varying success. One of the boys came up with this question: "Can circles really be perfectly round?" Then we moved to the 2^100 problem, which I thought would be a relatively quick one. Not so---it took up the rest of our hour. I gave them the stack of paper (cut in half 100 times) question to take home and wonder about. [1]: /storage/temp/106-math+in+the+park+ k-1.pdf

Problem 1 – I introduced a set of gears to my daughters, 2 ½ & 5, along with the help of my husband, who demonstrated some ways to make them work together. As Rodi mentioned above, the movement of the gears was better visualized as a wave, rather than gears moving in opposite directions. The girls immediately caught on to how the gears fit together to move, and enjoyed building different shapes and “machines”. My 5 yr old and I discussed what kinds of things are powered by gears...clocks, bikes, etc. and did a mini-scavenger hunt to find some around the house. We also spoke about what could stop a gear from working properly – misalignment, something caught in the gears, etc. She then built a “clock” and played a game with her sister in which the gears broke due to one of these and they had to fix it. It was great to see the interaction between the two as they looked for the “broken” gear and either taught the other how to fix it, or had each other find the problem on their own. Problem 2 – I used counting bears to illustrate the concept of 2^100, borrowing another member's idea of cats and kittens. We began with a mama bear who had 2 baby bears, who then each had 2 baby bears of their own. My 5 yr old continued up to 32 bears, and then we examined the patterns together. Problem 3 – For this problem, we utilized the iPhone app, Flow Free. The game encourages the girls to find the most efficient path to the color's mate without crossing any lines. The difficulty increases with each level. Both girls caught on right away, and their problem solving skills improved as they continued. My 5 yr old is currently on level 28, and the 2 ½ year old made it to level 5.

QUESTION/COMMENT ABOUT THE GEAR PROBLEM: When I read and did the gear question myself, I never thought about "changing direction." Most writing I have seen about this problem talks about the change of direction in terms of the gears rotating either clockwise or counterclockwise - total reversal of direction. When I did the problem, OTOH, I visualized, and even drew, a flow of motion that never reversed itself; it was basically a curve or wave that pushed the next gear along in that same direction (north/south/east/west, perhaps). In this model, nothing alternates. I'd like to find out from people doing this problem with kids how children conceptualize the movement, and would like to ask people to mention it in their write-ups if anything notable comes up about this. Thanks!

Math Circles 2013-07-13 We were 7 kids and 8 adults (!), we worked about 80 min. on a Saturday morning at Dor's place. We got to work on two of the three problems, with the first completed satisfactorily and the second partially completed. We ended off with a hand game that Dor had learnt in a German biergarten many years ago. All parents were thoroughly pleased as, I think, were the kids. The atmosphere was relaxed and playful. Dor did most of the facilitation, with sporadic contributions from other parents. Then almost everyone stayed on for another couple of hours for games and lunch. And iPad. And the parents who were not making lunch sat around to discuss subversive mathematics pedagogy... We made new friends. There was a sense of something very positive, and there was general interest in doing this again next week, hosted by Silvia & Russ. For Problem 1, a useful model was a sequence of domino blocks that were either face-up for clockwise or face-down for counterclockwise. ![alt text][1] For Problem 2, kids enjoyed working with matches, ![alt text][2] and some clued in that they could go 3D... Here are some implementation issues that came up in our postmortem, I mean postvivo: Parents' role during the session. Are they spectators or facilitators? If they are to help, how might they do so? Do all parents need to be on board with the lesson plan and goals? For the most during our session, the parents sat back and watched. We would like to change that. One of the motivations to involve more parents is to create opportunities for these parents to practice, and possibly get feedback, on how to work with children on these problems. Forms of representation, and in particular symbolizing. We recognize that some forms of representation are uniquely powerful models, for example algebraic symbolic notation supports reasoning, inference, and generalization in ways that transcend figural and diagrammatic models. In that sense, moving toward symbols appears to be a positive decision. And yet we noted that the concrete models children built with the substantive materials were often adequate for thinking through the (gears) problem. The models appeared to bear all the information relevant to solving the problem - there was no need for another level or phase of representing. Other times, though, multiple media can be very useful for problem solving, such as in the case of the Pins problem, where it might have helped to keep a record of aggregated findings from the group and then detect patterns within that aggregation. The question here is how to introduce paper and pencils (or markers) in a way that does not appear contrived. Embracing group variability. Our kids were K-3, so that they had had quite a variable exposure to mathematics content. Also, we had one ADHD child, and one child on the Autistic spectrum, so that we witnessed a variety of engagement forms. Finally, a couple of kids (boys) tended to dominate group conversations. We thought it might be useful to develop strategies for embracing this diversity. We would rather not divide and conquer, and yet sometimes that seemed to be the only way of maximizing engagement. Keeping kids on the theme. The kids were creative in taking the problems in innovative directions. Sometimes this was an opportunity to make explicit the problem's premises, such as when kids started building vertically the pins (matches), holding them up with clay. That is when we realized we had never stated that we are working only flat on the surface. Other times, we were not quite sure how or even whether to get kids back on track. [1]: /storage/temp/104-mathcircle2013-07-13-berkeleyproblem1d.jpg [2]: /storage/temp/105-mathcircle2013-07-13-berkeleyproblem2a.jpg

The biggest take-away for me from the first set of problems is to not over-plan with my kids. The 6-year-old can smell "official math problem" a mile away and becomes immediately resistant. The key for me to engage him is to be playful. Problem #1 For the book-page numbers, he shut down quickly when we started adding page numbers together. Later, I tried the card game on him and it was a hit. He decided quickly that knowing the sum of the numbers on the two cards was sufficient for determining the parity and became uninterested in the product of the numbers. When I asked him how he was able to determine the parity of the mystery card, he said (something like): "When you add two evens together, the answer is even because there are no extras to start with so there are no extras when you add them. When you add an odd and an even, the odd has an extra but the even doesn't. So the answer will have an extra and will be odd. When you add an odd and an odd there are two extras that go together. So the answer is even because there aren't any extras left." Problem #2 My 6-year-old was not all that interested in the powers of two until I found an online hundred board where he could click to color them. He was able to guess at a pattern with the last digits, but at that point he was "done" and did not want to pursue any further questions on the matter. Problem #3 I tried the A-B-C problem with my 4-year-old. I started with "oh, look at this cool rock. Let's pretend it's a gray squirrel. Can you help me find ones to be a red squirrel and a black squirrel?" We also easily found their favorite foods (acorns, pinecones, maple seeds). I ended up using string for the paths and he had no trouble finding ways to connect the critters to their foods without crossing paths. At this point I asked him if two squirrels could share each food - could we add a second path for each squirrel so that still no paths would cross? He said "No, I think we should find more acorns and pinecones." Shortly thereafter we had another place to get to and I was unable to get the same momentum going for the problem. In the future I will need to make sure that there is adequate time to "play" and also "play math."

![alt text][1] **PROBLEM 3** I had chosen the classic math problem “Gas Water Electricity” to practice the strategy “Engage in wishful thinking.” I had done this in a math circle several years ago with the instruction “How can this be done?” It turns out that in the classic version, it can’t be done. One child in that circle had gotten very frustrated, nearly in tears. I was hoping that by rephrasing the instruction to “Can this be done?” and using the Wishful Thinking strategy, the girls would reach this conclusion in a less painful way. Immediately J said, “I like this problem.” She is a visual/spatial learner, a style that this problem as great for. Also, there is no arithmetic. As is her style, she immediately started changing the problem by moving the houses and declaring that lines can overlap without crossing and still be legal. In other words, she was engaging in wishful thinking on her own. I named this strategy, and identified it each time she used it to worked the problem. Unfortunately, R had seen this problem before and knew that there was no solution in the classic sense. I tried to get her to engage in wishful thinking anyway to notice and attack assumptions, and just see if there’s a solution if you change the problem. “I don’t like this strategy of wishful thinking,” she pouted. She just couldn’t get past her preconceived notions of the question. She also stated “I don’t want to cheat.” To her, changing the problem is cheating. To J, changing the problem is opportunity. There are a lot of things you can change and test in this problem. But my girls didn’t do them. R wasn’t willing to try. J was stuck on her overlapping idea – she still thinks it will work but is having trouble figuring out how to draw it. I hope to return to it with both of them in the future. If I were to do this problem again, I’d do it the same way, but make sure that no one had seen it before. One other observation I’d like to share about these 3 problems is the idea of sharing the board. When I lead math circles, I do it in a room with multiple boards and children are welcome to come up and take over the class at any time. At home, we only have a teeny whiteboard, making sharing virtually impossible. For the third problem I did let them use the board since we didn’t have long lists of numbers that couldn’t be erased. I think this contributed to J’s enjoyment. I hope to get a bigger board for home, or even just use sidewalk chalk outside to get everyone more involved. [1]: /storage/temp/102-img_0861+(800x600).jpg

![alt text][1] ![alt text][2] **PROBLEM 2** The strategies of “reread the question” and “do something” were of immense help in this problem. I asked R and J how many total animals would you need to hang them from a crane pyramid-style for a circus act, and let them define the terms of the problem. J really got into this information-gathering stage: she lined up and measured the heights of her stuffed animals and researched typical heights of cranes. She chose the Flat Top Tower Crane (344’ tall) for this problem, and determined at an average height of 1’ tall per animal that we would need 344 rows of animals. There were other questions, answers, and assumptions that went into clarifying the problem, prompting R to complain “I don’t like this stage. I want to get to the problem solving.” Her eyes quickly lit up, though, when she found a flaw in J’s calculations (they are sisters, after all). Since the girls had decided that the animals had to hang 1-animal length above the ground for dramatic effect, only 343 rows were needed. “Is it going to get harder?” she then asked. “I’m confused,” said one girl and “I don’t understand this,” said the other, once we got into problem solving. We repeated the “reread” and “do something” again and again. I worked with each girl separately for parts, as J probed the problem with stuffed animals and rubber bands, and R did with a chart of data. They periodically helped each other. J got frustrated when she ran out of animals long before row #343. She doubted getting a solution. R got frustrated trying to generalize (Is this a function of number of animals, or number of legs? How do you express this with exponents?). Interestingly, they both got stuck at similar points. They both realized that this problem is tedious and boring without a generalization. Once again, we reread and did something. J hung the animals from a hook on the ceiling and we tried to spread them out to see the row-to-row relationship better. I did ask some leading questions (something I don’t like to do, but will when a kid is getting too discouraged). Finally, she had an epiphany: “Each row is one Leg Number as big as the row before!” (The decided-upon Leg Number was 2, and since J hasn’t worked much with multiplication before, this was her conceptualization of that concept.) “Yes, it is – you’re right!” I replied. She jumped up and down and whooped and hollered. This may have been her first math epiphany ever. In the meantime, R had written out her answer on the board as a sum of a list of exponential numbers. She was still discouraged that she didn’t have a way to simplify this. She was at a conceptual impasse, having never done much with exponents. I showed her how to represent this symbolically as a summation sign, but required her to provide the variables. That thrilled her. Using the sigma was powerful. What a relief to show something as a single instruction. Her task for the future will be to figure out a way to actually simplify this. If I were to do this problem again, I would do it in a multi-week math circle. We could explore how the answer might (or might not) be different if we used 4-legged animals. Middle-school kids could learn so much about exponents from it. Younger kids could have more time to dramatize it, gain more conceptual understanding, and write out number sentences. I did like the idea of stating the steps (reread, do something) again and again as almost a mantra. This is something that we do in math, but don’t always remember to verbalize. [1]: /storage/temp/100-img_0855+(300x225).jpg [2]: /storage/temp/101-img_0856+(300x225).jpg

I tried first two activities with one member of my family (female, adult, not trained for maths except of regular high school courses). On the first problem (wit 20 pages out of the book), her initial question was if they took out 20 pages or 20 sheets of paper. After precision, she said that it must be even since 20 is an even number, but then she said we should start with page number 1, so the back will be 2, therefore, 1+2=3 and 3x20=60, so the sum should be even. And she added that this holds for any number of page we begin with. From my point of view, as educator, I see this problem as a very interesting to engage people, including very young one in a game of questioning – like what would be if 20 pages counted as 1 sheet=2 pages? What if we take out pages not as sequence but separately from different parts of the book? What would be with any n number taken out? What can we say about divisibility of the sum by a different number (like 3, 4, 5, etc.). Regarding the second task, my participant said first that it may be 0 as last digit since 100 finishes with 0 – interesting that for the second time she also thought that some hints are in a given number (like 20 in the first task and 100 in the second) but after she realized that it cannot be 0. Then she said it should be either one of 2, 4, 6 or 8 but could not figure out which one. It is interesting to see with this problem how it is important to develop (as early as possible) a culture of ‘seeing mathematics’ through different patterns (thus developing ‘mathematical cast of mind’ (term used by Krutetskii, 1976). The task can also be an excellent way to introduce math that is overlooked by curricula, like modular arithmetic, and do so using another context as well, like circular clock (see for example http://www.shodor.org/interactivate/discussions/ClocksAndModular/).

We were working today with Problem #1 with a ladder, a rope and a bucket. Throwing the rope over an odd crossbar we have the bucket moves up while the rope is pull down. And when the rope goes over an even crossbar than the bucket moves up pulling the rope up too. Children didn't ask anything, just enjoed the pulling the rope. I tried to pay their attention to the fact of different ways of the rope to rise up the bucket but they were busy enough just pulling the rope up, the higher is the better. ![![alt text][1]][1] [1]: /storage/temp/99-14072013-sg208523-2.jpg

I'm working on a report about the K-1st grade group, but I need to wait for permission from one of the parents for photos. Meanwhile, here are some highlights: We had five children playing with the math problems and five adults (counting my teen daughter, who helped when she wasn’t taking pictures), so after I introduced a puzzle, we were usually able to work one-on-one with the kids: to respond directly to what each child was doing, reflect their thinking, and offer little suggestions to nudge them along. We adults probably talked more than we should have—listening without offering advice has always been hard for me. To my surprise, the children found the gear pictures easier to work with than the physical gear set, perhaps because the lack of motion let them focus on the teeth where two gears met: this gear is turning this way, so which direction will it push the next gear? All five children worked intently for several minutes. The three older children successfully solved even the challenge puzzle (the long, branching line of gears)—well, sort of: I think I saw final answers that went both directions, so one of them must have been wrong, but the kids clearly understood how each gear turned the next one, even if they lost track of which way the arrows were pointing as they went along. The younger siblings had the idea of spinning, but drawing arrows to keep track of direction was beyond them: one girl drew circles inside most of the gears, though she skipped some of the smaller ones, and a boy made several long chains of circles on his paper. I adapted the exponents problem as an addition puzzle, to put in order the sums 10+8, 5+12, and 2+24. I used a three-bears coloring picture (Papa Bear likes big numbers, etc.) for sorting out the answers. The kids immediately said that 24 was a Papa Bear number and 2 belonged to Baby Bear. Two boys insisted that 1 should be on the page somewhere because it was wrong to let 2 be the smallest number, and then one of them pointed out that zero would be even smaller. None of the children recognized the sums as representing individual things to be sorted. In their opinion, “10+8” wasn’t a number. The kids went to the playground while the parents helped pick stuff up. One girl told my daughter, who was acting as playground guard: “I’m done with math. It was really fun!”

Technique 1 Gears • First my son drew arrows on gears to indicate direction • As the problems got more complicated, we suggested he could substitute symbols for arrows. He chose to substitute 1 and 2. • When the most complicated problem was solved, he re-substituted incorrectly and started to get really confused and frustrated. Book pages • My husband and son made a book and they numbered the pages and then started tearing out different pages to see the result. • We found it useful to let my son discover the alternating pattern: o If you tear out one (an ODD number) page, the sum of page numbers is ODD o If you tear out 2(EVEN) pages, the sum is EVEN • After doing a few more, my son noticed the pattern and he extrapolated to: 20 is EVEN therefore the sum is EVEN Technique 2 Pins • We gathered up 10 Lego minifigures and made 10 pillows out of paper. My husband said that 10 people were going camping but they all forgot their pillows. For safety from bears, they have to sleep in a circle. What is the minimum number of pillows that they need to make (if they each share a pillow with one other person-H:H)? What is the maximum number of pillows that they could make (H: P)? Is it possible to arrange the sleepers such that they require any number of pillows between 5 and 10? My son loved thinking about these imaginary campers and manipulating the pillows and minifigs.

Not a lot to say about the second problem, which I just did with my son, except that it linked up to some number pattern charts we had done earlier this year as part of our exploration of vedic math. Oh, and the resource I mentioned earlier, http://freemars.org/jeff/2exp100/question.htm , was a great addition to the 2^100 problem. Carol