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

Technique 1 Gears • My son drew arrows on gears to indicate direction. • As problems got more complicated, we suggested he could substitute symbols for arrows; he chose to use 1 and 2. • When the most complicated problem was solved, he re-substituted incorrectly and started to get very confused and frustrated. Book pages • When we moved on to the kid/young teen level, my son became very frustrated talking about the problem. My husband made a book with him and they numbered pages and then talked about tearing out different pages. • 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? Once he understood the upper and lower limits, he loved the challenge of manipulating the figures to see if all outcomes between 5 and 10 were possible.

I'm not sure what the problem numbers are, but we worked on the Wheels with Cogs problem, and the 2^100 problem, yesterday. It was really fun! I worked with Alyssa (going into 8th grade) and Taryn (going into 6th grade). Both attended the Rio Rancho Math Camp last month. **What we learned about Wheels with Cogs** If one wheel is going clockwise, any wheel that touches it is going counterclockwise. If the wheels are different sizes, then they go around at different speeds. For example, if a wheel with 5 cogs is the 'driver', and it is touching a wheel with 9 cogs, then for every revolution of the 5 cogged wheel, the 9 cogged wheel goes around 5/9ths. It would be interesting to do some measurements and do the math to figure out if the distance travelled along the circumference of both circles is the same or not. If you make a circle of cogged wheels, and you have an odd number, then you end up with a conflict of which way one of them should turn—counting one way from the start, the "last" one should be clockwise, but counting the other way, the "last" one should be counterclockwise. If you actually try to build this, when you put the last wheel in, all the wheels will stop turning. We did most of this exercise on pencil and paper, but we also used some lego pieces that were cogged wheels for the circle example. We also looked at some cool animations of cogged wheels spinning online. **What we learned about 2^100** Both Alyssa and Taryn started writing down all the powers of two starting with 4, 8, 16, 32, etc. They thought the brute force method would work. They started slowing down at about 1024, and then they decided to go as far as 2^20. When they got there, Taryn thought he could multiply that answer by 5 to get 2^100. I tried to convince him that was nothing close to doubling 2^20 another 80 times. I think he understood that. Then Alyssa started noticing the pattern of 2, 4, 8, 6 in the last digit. She then wrote down 2, 4, 6, 8 as a group of four numbers, 25 times. Checked by counting the digits up to 100, and said that 6 was the last digit. I tried to point out that she could have divided 100 by 4 and figured out that since the remainder was zero, the answer was the last digit in the group of four. It might have been better to draw a circle with the numbers 2, 4, 8, 6 around the outside of the circle, and then count to 100, and see if she noticed that this has something to do with division and remainders. We then checked the answer on the internet and found that 6 was the correct answer--if you can believe anything you read on the internet. From a teaching standpoint, I noticed that it is tricky to offer the kids guidance and support without leading them directly to the answer I wanted them to find. Asking open ended questions is not as easy as it sounds! These were both great exercises--lots of fun to watch the kids think. They were both really engaged.

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

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.

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

Not a lot of insight to report about the first part of problem 3, which I again did with about 10 people from 5 to 65. Some got it, some didn't, at least within the time frame. My husband and my son, both of whom have good 3-D visual minds, figured it out first. Then, once some people had figured it out, younger children just got discouraged and gave up (so I did the "what would be the easiest way to place them for this problem" thing, which they enjoyed much more). Eventually one other adult and one other teen figured it out, but after that point even the older students gave up and just wanted to see the answer from the ones who figured it out. So sustaining interest in figuring it out yourself after some peers have done it was an issue for me. We also did the MAA-AMC #3 problem with a different group of two moms and four children. One family considered themselves good at math; the other family didn't. I gave them the problem "What is the correct ordering of the three numbers 8^10 , 12^5 and 2^24 ? I first asked them what they just intuitively thought was the answer, and almost everyone chose D giving highest exponent highest value. Then the math family started just powering through the calculations (each working separately without using calculators or anything) while the non-math family just kind of floundered around. So I started to work with them about writing out the equations and could they do anything to simplify or make the equations more comparable. They wrote them out OK, but they still didn't get anything they could do. Eventually all three of the math family finished their calculations, apparently correctly, because each one handed in a written answer that was the correct choice. Then they started fiddling around with the re-formulating the equation to make them more comparable. With a few hints or prompts from me, they figured out the alternative approach as well. I think the non-math trio understood what we did once we showed them and explained it to them, but I don't know that they would have ever figured it out themselves. So, in general, I'm having a problem supporting or encouraging the people who can't figure it out fairly easily to stick with it or to be more creative in their thinking about how to figure it out.

Oh, I forgot this part of problem 1, the one about the book pages. I made a sheet and had them write about their thinking process in solving the problem. Out of five people, two adults, two teens, and one child, the adults and one teen figured it out, and the others gave up. Here are how the ones who completed the problem explained their thinking: **Adult 1**: There are 20 pages. On each page, there is one odd number and one even number. I separate the odds and the evens. If you add evens to evens, they will be even. If you add odds to odds an even number of times, the result will be even. Therefore, the answer has to be even. **Adult 2**: I visualize the numbers as blocks. Even numbers create whole squares or rectangles. Odd numbers have a missing piece. But if you can match two odd numbers together, they can connect to make a whole square or rectangle--in other words, they make an even number. As long as every odd number has a mate, there are no missing pieces. Adding in the even numbers doesn't change that, so I visualize all of the numbers coming together to make a big, even rectangle. No matter what pages are ripped out, there is an even and an odd number on each. Since 20 is an even number, we know each odd number has a mate. I tested it out on 1-20 and 2-21. **Teen**: Even. There are two pages on each paper. One is odd and one is even, so it is like 1 + 2 = 3. 20 x 3 = 60. So, I got three very different successful approaches to the problem. Carol

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!

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!

PROBLEM 1 ![alt text][1] I did a simplified version of the AMC problem from Essay 1: “A farmer wants to maximize the amount of grass she can plant in her sheep pen with a limited amount of fencing material. How should she build her fence?” My students were R, age 13, and J, age 9, both homeschoolers with strong conceptual knowledge but few algorithms under their belts. R is eager and confident in math, J is the opposite. To make it even more challenging, they are sisters. From the start, R attempted to distill the essence of question while J tried to change it. This battle played out over the entire hour. J didn’t like any of the premises, including the fact that I called it a “problem.” I read her the explanation of the word “problem” from the prelude to assignment 1 (who wrote this?). That calmed her a bit. Throughout, each girl alternated between enthusiasm and discouragement. As we worked, I talked about “flailing” as a problem-solving technique. J said “I don’t want to flail.” I suggested we do “organized flailing,” and we sallied forth with that as our approach. Each time discouragement nearly prevailed, I read one of Dr. Tanton’s steps to problem solving from Essay 1. Each of these steps rejuvenated our problem solvers. I read the whole paragraph that goes along with Step 1 (“Take a deep breath and relax”) and both girls breathed a sigh of relief. At one point later J stormed out of the room, but was listening when I read Step 3. She bounced back in saying, “I think I have an idea!” R was ready to give up near the end but was energized by the un-numbered step “identify the penultimate step.” We attacked the problem by using props (2 pieces of yarn), asking questions, making assumptions, changing assumptions, forming conjectures, and rejecting conjectures. By the end, we hadn’t a definitive answer, but our flailing had been successful, albeit exhausting, so far. Both girls want to finish the problem at a future date. Some of the **questions** that arose from this problem (few were answered today): • Which piece of yarn is bigger? • What is the question? • Can grass grow everywhere in the pen? • How can we know for sure which shape has the biggest area? • What is area? (“How much space something takes up in 2 dimensions,” says R.) • Do we want grass coverage in 2 dimensions or 3? • What is a dimension? (“Counting how big things get,” says J.) • How many dimensions does a point have? • Can we trust our eyes if we make the shapes with yarn and eyeball them? **Conjectures**: • Shape doesn’t matter • Shape does matter • A square fence would be best • A circle would be best (This became the girls’ working conjecture before problem-solving began.) • The farmer could replant seeds indefinitely • The farmer could create a system to replenish the grass (“A circle might be bigger but replenishing is better,” argued J, since the pen could then be built more comfortably/humanely for our sheep Justin when maximum area was not the only consideration.) • The fence could allow the sheep to get grass from outside the pen • We could use various methods to answer this (eyeballing, area formulas, measuring, guessing, doodling…) **Assumptions**: • The farmer can replant/replenish (We did the question with this assumption, with J inventing numerous engineering solutions.) • The farmer cannot replant/replenish (We spent most time operating under this assumption.) • The farmer is using the most efficient planting methods At this point, the girls worked together with a ruler to make measurements of their yarn shapes. R began applying area formulas, while J began to draw shapes to make sense of it. She had finally let go of her desire to change the problem from a math question to an engineering challenge. “I can think much better when I’m doodling,” she explained. R calculated areas for a rectangle, square, and circle. The circle was the biggest so far. But more **questions** arose: • Are our results significant, since we just measured rough yarn shapes? • How could we calculate this without tedious arithmetic? (R did not want to use a calculator, but the fractions in the measurements were ugly – so common when using the English system!) • How do you multiply fractions on a simple calculator? • What other shapes should we try? While this problem was very challenging to address with such different ages, I loved that they each benefited from the other’s contribution. The older child gained from the “thinking outside the box.” The younger child heard a lot of math terms for the first time, and is very very curious to learn what they mean. I am building her math studies for 4th grade based upon her questions from these problems. I was surprised that no one thought of super-imposing the shapes, or of getting some actual grass seed and counting how many fill the space. I also was surprised by the pretty interesting tangential discussions of dimensions and pi. I enjoyed seeing the power of verbal acknowledgements of the emotional side of mathematics. I wish we had time for R to do this problem in an abstract manner (no props, no numbers); I asked her to think about this on her own later. I also would have like to have a few more kids (we will next week). Then kids can have closer-aged peers to bounce ideas around. I’ll send separate reports about problems 2 and 3. I had hoped, unrealistically, to do all 3 in one sitting. [1]: /storage/temp/90-img_0849+(640x480)+(2).jpg

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.

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!”

Kids liked the problem with gears very much. Older kids also. I guess they liked it because they knew what to do immediately. And it is funny to look at their rotation, too. We've also played the problem of 20 (or 4) pages, matches and triangles. I'd love to try it with integer powers also but we haven't had enough time for them this week. One of the kids has had a moment of enlightenment with 20 pages, and it was great to see his face and hear him cry "Aha!!" For others it was difficult to get the idea - why are we questioning about odd or even sum at all? They were staring at the book without any idea, and after some time of talking about sums the've got bored. Pins or matches were also difficult and that amased me a lot because kids of 9-10 years were not able to solve the problem for younger kids about 6 matches and 4 pilows. They were trying to put 6 or 10 matches in the right way and then they told me it was not possible, but they didn't explain why. With the triangles I've drawn a picture and ask my son what similar figures did he see there. He've found a lot, but not the pair of triangles that helps to solve the bigger problem. The good thing is that I saw him thinking about the problem without looking at the picture and he confirmed that he can see it in his mind. Next time I'll talk to them more before giving a problem - about what they know, their ideas, what they want etc.

**(1) Warm-up** Gears was easy for both of them and tried to increase speed! For mix-pins Parsa tried to approach algorithmically and find a solution which works for making mix for random number of pins! Amir tried to approach visually. But he started from Ten-Pins! Parsa has simplifier role for Amir. He tried to show Amir that starting from smaller number of pins works better. Amir reached the right solution which Parsa did. but in a wrong way! This was good thing for both of them. Amir revised his steps and found his mistake. **Problem1(Area of Triangle):** Parsa knows Thales Theorem. So he solved it easily! Amir knows how to solve areas of circles and right triangles. Amir tried to start from what he know and redrew it. But he stuck! **Parsa drew a line and asked: "Amir what is it? What are possibilities?" Then Parsa told that it can be a triangle! a triangle with one 180 degree angle and two zero angles!** He could not continue exactly from what he thought. But finding heights from shadows with a ruler could be a great experience for them! **Problem2 (finding max value):** This problem was challenge for both of them! Amir only knows that we can use variable x for representing different values but he does not know about algebraic operations or equations and Abs function. But Parsa know Abs function and quadratic equation but he had no visual idea about equation of circle or square. We tried to remove what we don't know and turned it to simpler form: x+y + x-y = 2 and x - 6x + y Amir learned simple equations from Parsa and they could struggle with solution. **Problem3(comparison):** Parsa had no problem to solve problem. But his challenge was to teach square root to Amir! His strategy was similar to folding and unfolding! Amir learned and solved it!

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 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/).

![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

![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

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."

We have played with the first two problems so far. My older boy is definitely more confident with the math and the patterns. At least he has the language to recognize and explain what he is seeing. http://youtu.be/b0Jub6-Bcms Both boys loved creating the gears and could tell easily which direction each should spin. I had just started the pin problem and you can see my old boy caught the even odd question right away, but then a reminder went off on my phone and I had to leave for a meeting. We did not get a chance to return to the math. http://youtu.be/dM4YnCDI2CQ Edit Finally got around to problem 3. I laid out three strings and crossed them as in the problem. I told the boys that they couldn't move the ends, but had to make the lines so they didn't cross. At first the 8 year old picked up a string and said it was too hard. He then dropped the string but it landed behind one of the ends. I could see understanding click and he quickly picked up strings and rearranged them in a correct order. It took about 10 seconds. The younger boy had been distrcted and came back to see the problem solved and was disappointed. I quickly made the problem a bit more difficult by saying they could not go behind the middle end point. The problem was still dolved in seconds.