Information for the Concord and Alameda Classes

Parents, here is an overview of the curriculum for next year’s math classes for children around early second grade — Thursday morning in Alameda and Tuesday morning in Concord. CURRICULUM — MIQUON The Concord class will begin in the fall at about halfway through the Miquon Red Book, and we will choose topics as needed for the children to finish up the Red Book, then begin the Blue.  The Alameda section, having stared in January of this year, is ready to begin the Blue Book in September. To help you understand the Miquon series, I’ve reproduced three representative pages from the Red Book, with comments. Because Miquon organizes their workbooks into more or less parallel topics, more advanced students can work with similar topics in the Blue book while others work in the Red Book.  This works because what I teach in the lessons is problem-solving, which isn’t taught in any of the Miquon books,  nor any other standard curriculum.  You need live lessons for that.  But the parts complement each other:  problem-solving in our weekly lessons together plus skill reinforcement during the week as homework. red book fractions This is the last page of Miquon’s 24 page introduction to fractions.  Miquon (and the Waldorf curriculum)  is unusual in teaching all four operations in first grade, plus fractions, but I think this is correct.  Children should get this broad conceptual understanding from the beginning. red book division The last page of the division unit.  Children can solve these problems with Cuisenaire Rods or a novel abacus invented by the RightStart Math.  Additionally, now that they know the meaning of all the operations, the children can solve word problems.  With my Alameda class recently, for example:  The baker baked sixteen pies, but after she set them out to cool, a rascally boy took two of the pies and ate them.  Half of the remaining pies were lemon, half were cherry.  How many lemon pies were there?  Children solve this three ways: arranging manipulatives to tell the story, with a drawing, and finally with a math sentence. red book multiplication This page demonstrates one of the reasons I like the Miquon series.  These two sets of questions are quietly about the distributive property and about average, without cluttering things by prematurely mentioning those terms.  This is math the way a mathematician sees it, and it is intellectually beautiful.  In class, the children translate the ideas behind these exercises into words and then into hand gestures. CURRICULUM – RIGHT START I encourage parents to use a second curriculum alongside the Miquon books.  RightStart Math is a curriculum of 300 math games, designed by a mathematician, and strongly influenced by Montessori methods.  They bundle a kit with several decks of cards, game instructions, and an innovative abacus.  The games appeal to young children and they offer a good complement to the Miquon workbooks. As good as Miquon is, it is even better for the child to get their nose out of a workbook now and then and do math in a more visceral game format. TIME In September, I suspect the children would only be up for lessons that are 1 1/2 hours, but I would like to ramp them up to being able, by January, to lessons that are two hours long.  I’ve been successful at keeping childrens’ enthusiasm for 1 1/2 hours without a break, but it all depends on the group.  For a longer class, they would need a small break. SCOPE/SEGUE The Miquon series is paced at one workbook per semester.  We will stay with Miquon through Red, Blue, Green, Yellow, and at least part of Purple, and then segue in about two years to the Beast Academy series, published by The Art of Problem-Solving.  This is the deepest and best-written math curriculum that I’ve found for grades 3-5.  I can talk at length about it and show you some of their books, but that math is two years away! In the mean time, if you have more immediate questions about next year’s math, please email or phone me. , 510 417 5736.  

Machine: Fire Opera

Preparing the Accordion Part December 2 — The day after the first rehearsal, composer Clark Suprynowicz emailed to ask if I was available to play on the Machine opera. (The accordionist originally hired was unable to play the music.)  So I missed the first rehearsal, and then misunderstood when the music would arrive and I’ve now received it three days before my first rehearsal. December 5 —First rehearsal.  Invariably, some passages I thought would be easy turned out to be tricky, and vice versa.  Here’s one I didn’t suspect. The notes seem easy enough, but to play them isn’t. The bass and tympani are playing offbeats, but nobody is playing any onbeats for miles around.  In my part, Clark had thoughtfully written the offbeat bass line, but I missed that, so that in rehearsal I kept sliding the beginning of each measure over so that it coincided, incorrectly, with the offbeats.  You can get a feel for the problem by listening to the metronome at our tempo of 212 clicks per minute. It is enough to imagine each click as the onbeat, but try imagining the onbeat falling halfway between each of the metronome clicks.  Now tap your foot to that imaginary onbeat.  Conductor Barnaby Palmer wasn’t too happy when I kept getting this wrong, and I was incapable of correcting it in the flurry of rehearsal. But I’ve been ruminating since on how to lock into that beat.  My solution is now to tap the onbeat with my left foot, and the strong offbeat with my right. That seems to work. (This note isn’t accordion-specific, only insofar as most accordionists have weak rhythm.) December 19 — There are a few places where I am supposed to play synthesizer instead of accordion, since the synth has a wider range of timbres, especially sounds that don’t sound like an ordinary accordion.  I tried this out in the first rehearsal, but it sounded sterile.  After some discussion, I ended up borrowing an effects box from Joel Davel, the marimba player, and running the accordion sound through it as synth substitue.  For example, here are the opening measures with synth, then accordion through the effects box. What it lacks in timbre it makes up in emotion, I hope. True story: several years ago at a party I found myself bragging expansively to someone about how the accordion is superior to the synth because of the nuance available through the bellows; a foot pedal on a synth is a crude comparison.  After I drifted on for a while, I asked him what he did for a living.  “I’m a synth player,” he replied. December 20 — Barnaby asked the guitars to play a passage more quietly. Guitarist John Schott quipped that, “Well, when we come across a passage that is actually playable, we kind of like to play it louder.” December 22 —  Even after the composer has written the music, there are, invariably, interpretations that are left to the player.  This same gap is understood between architect and carpenter, designer and craftsman, as here, between composer and musician.  How to make those small-scale interpretations depends how my part fits in with the whole, but so far, I have little idea what the opera is about.  I can’t hear much of the singer’s words, and I haven’t seen any of the action on stage.  So I emailed director Mark Streshisky to get a copy of the libretto and the short story upon which Machine is based. I’ve now read these, and have a better feel for the tone of each passage, so I can start to figure out how to interpret my part appropriately. One accordion player’s interpretation is direction of bellows, and I’ve begun writing in bellows direction, just as a string player would write in where to play with an up-bow and where with a down-bow. For example, in this line I prefer to change bellows direction with each note rather than multiple notes in the same direction.  It seems more in keeping with all the drama going on. There’s no time for musicians to discuss every small interpretation, so I was happy to hear in rehearsal last night that the cellist came to the same decision in her interpretation of the same line. December 23 — Fooling around some more:  I have several trills sprinkled throughout my part, and I try out different effects. From the composer’s pen, they all look the same, but by varying speed, bellows, and even treble and bass notes, I can get different trills. I start marking my part.  (Contemporary accordion notation uses curvy lines, like a sine waves, to suggest movement of the bellows.) December 30 — We are on break from rehearsals, but I spent Christmas week playing less accordion than I had hoped.  I went about a few overdue repairs: a new, tighter bass strap, futzing with my shoulder strap adjustments, building a chair riser to get me to the exact right height, an access case for the effects box, and fixing a few reeds that had gotten dust in them.  Over time, dust, lint, and other air impurities get sucked into the accordion and lodge themselves between the reeds and their metal housing, and I had a few smaller, higher reeds which weren’t sounding properly.  So I cleaned those out and vacuumed all the innards to forestall this happening again elsewhere.  I’ve  discovered, however, my tiniest piccolo reed doesn’t just have dust, but is broken.  It’s on the high C, above the highest note of standard accordions, and I don’t have a ready substitute around. So I order the reed from Italy and learn to avoid that note for this show. December 30 — With dead composers, it’s easy, you can just decide however you want to play their music and they can’t say anything about it.  But Clark is quite alive.  In many places, now that I’ve heard how the accordion part fits in, I am starting to prefer a different octave, sometimes lower,

Teaching Biography

— Fifteen years teaching math, chess, and design classes after school, mostly at public schools in Berkeley — Four years tutoring math, teaching SAT and GRE for Kaplan, and guest teaching mathematics at Waldorf schools in Northern California — Four years teaching at math, music, and main lessons at two Waldorf schools in Marin — Four years teaching music and mathematics at two schools in Oakland — B.A., Mathematics, Oberlin College, 1979 I was the first of my family to go to college, and studied Mathematics at Oberlin because that’s what my father wanted me to study, although he didn’t live to see me graduate. And in high school, in a rush of all-night creation before a calculus exam, I invented, or re-invented, four very different ways to calculate the area of a circle. I didn’t have the terminology of problem-solving that I do now, but looking back, but looking back, that was my first times of discovery and creation.  In college, the same happened again with a discovery, published by my professor, of a way to greatly reduce the computations required on huge matrices. After school, I worked as a carpenter, then accordionist, and I never thought that I would work with children. I became music director at a church in Oakland, and was thinking of becoming a priest. I began teaching music only because the principal at the adjoining school asked me to. That summer, a parent asked me to tutor her seventh grade boy in math, although she had no idea that I had majored in it; she  only knew me as a music teacher. I was instantly captivated by the challenge. I tutored Curtis once a week while spending days and evenings holed up at the Ed/Psych Library at Cal, reading all I could of the literature on how to teach math. It was not an especially lucrative endeavor. Factoring in study time, I earned .25/hour from my tutoring that summer. After four years of teaching in Oakland, I taught at two Waldorf schools in Marin as a music specialist, math specialist, and class teacher for eighth grade. There I learned the most I ever have about the art of teaching. Though I disagree with some of the specifics of the Waldorf approach, its pedagogy is the most profound and most spiritual I have encountered thus far, and it still informs my thinking. After four years in Marin, I decided to leave elementary schools to teach independently. For four years I taught SAT/GRE for Kaplan, traveled regionally as a guest teacher at Waldorf schools, and tutored math privately. Three years later I decided to leave Kaplan during Lent when I was brought to the conclusion that my work there was spiritually empty, so I didn’t go on the Master Teacher tack they were developing. At this time I also tutored math to high school and middle school students.. Individual tutoring is a wonderful complement to classroom teaching; ever since, I have thought of this tutoring period as my research and development time, when I could closely observe what goes on in a student’s mind, something that’s a lot harder to do when teaching a group. I assumed I was taking a very different path when I decided to spend less time teaching math and more time playing accordion, but the two have proven more similar than I once thought. Although I studied classical accordion from age five, I’ve always been attracted more to teaching it than to performing. All my music students have been adults, and I’ve been privileged to participate in journeys of great intimacy as we encounter music in their lives. I’ve also done many shows, which have taught me that teaching math to a group of children is a form of performance. It is caring for an audience, it is improvisation with an audience.  For the past fifteen years I have taught after-school classes to children in grades K-6 in Berkeley/Oakland, 200+ students per semester.,. I’ve been surprised at how much I have enjoyed developing new Kindergarten curricula, as I had thought my strongest teaching was around eighth grade. in  It’s been particularly important to me to encourage girls in math and chess., which has led to SuperGirl Math and Chess for Girls, and led me to hire a gender consultant to critique my teaching and discuss the academic research on how girls learn. I founded and currently direct an annual chess tournament, now in its sixth year. And at lunchtime Math Circle last year, with 32 positions available, we had 76 sign-ups — 76 children who preferred doing math to playing on the playground. Henri Ducharme August, 2013  

K. Building Imagination: A Curriculum of Blocks

After teaching Building Imagination for three years now, I’ve sorted out four sorts of learnings going on: Cognitive, expressive/therapeutic, artistic and social. Cognitive: This is the mathematical and engineering side of things: how would you design a staircase, how could you make this road ramp up at an angle, can we imagine where the foundation would go before we build it? In the past, we’ve built the Mayflower from simple plan and elevation drawings, we’ve built Gulliver after first imagining what postures were do-able within the constraints of our blocks. Often the students just think they’re playing. Or as one student said, “This class is really hard but really fun.” Expressive/therapeutic: The children have strong desires to build particular things. More than once students have vetoed what I thought would be the perfect challenge for them, and I have had no choice but to follow along and lead from behind. Sometimes there is a quality of “playing house” or “playing race cars” in their building. Or happy discoveries, as when last year we built a dragon, and a child jumped over the dragon, but subversively because he presumed it was wrong, but then I invited all to jump over the dragon, as that had been the plan all along. Artistic: We play with balance, stability, symmetry, representational and non-representational, etc. At the end of class, I often have them look at each others work, “like you were walking in an art museum.” Although they can’t yet articulate why, the children can tell when something is beautiful — there is often a gasp, and then silence, or comments of “That is so cool.” Social: Sometimes students work well together, sometimes it is a challenge. There are many exercises I use to teach them to cooperate, for example: “Work in pairs and in silence. Take turns, you add one block, then your partner adds one block, then you add a block. Your goal is to make a wall that has a pattern.” I have been teaching this class for several years now. The last several weeks of the Fall of 2012 offers a great illustration of the kinds of tasks your students will be performing in this class: Building Gulliver — teacher-led, the tallest and most impressive structure we built. The tallest student had to climb on top of a chair, on top of a table, in order to complete the project Decide what you want to build — it was their call, the opposite of Gulliver. They ended up building a city complete with zoo, runaway dinosaurs, garbage dump, police and ranger station Build the school — our most representational project, in which we began by taking a “field trip” through parts of the school. I built them a miniature of a chair in the classroom, which they thought was very cool. Build an abstract sculpture  — or as I said it to the children, “build something that isn’t something and that looks cool.” I then demonstrated two aesthetic criteria, balance and pattern, which many incorporated into their creations. and finally, Build something that doesn’t exist — abstract or representational, their choice, but hopefully using balance and pattern. Initially I will try to nudge them towards working together, but that may be beyond them. Most of their work will be without my assistance. By the end of the course their building skills would have improved in terms of mathematics, architecture and engineering, their self-expression and in their abilities to work with one another and represent that which they observe.

SuperGirl Math 5: Fractions are Boring

When I mentioned yesterday to the SuperGirl 5 Math Group that the next unit coming up in their school curriculum will be adding fractions, there was a huge chorus of groans, and they took a minute to vent. They all complained that “Adding fractions is soooo boring.” and they seemed to use ‘boring’ in three different senses: • This is not challenging us. We’ve already done this before. • This is pointless. There is no good reason to be adding fractions anyway. • This is treating us like little kids, insulting our intelligence. We’ve been doing fractions since second grade. Of course, whether these perceptions are true or not matters less than that they are indeed the girls’ perceptions. There’s also a paradox here, that they may be bored with adding fractions but I know from SGM Summer Camp that they are shaky on even understanding what fractions mean, and in a well-sequenced curriculum, understanding what fractions mean comes before adding them up. But maybe it would be better to not to have a well-sequenced curriculum. As luck would have it, I had planned two activities to teach the meaning of fractions, but mixed up were a bunch of other topics they won’t see till sixth grade or so. The first was a poker-like card game. One round, the winner was the person who had the best cards to fulfill 1 / (x + 2y) = 0 (Winner is she who comes closest to 0) This led to the tricky strategy that if you want the fraction to be as small as possible, then the denominator, (x + 2y), should be as big as possible — which I could only explain by reaching for those boring pizza slices they shared back in second grade to demonstrate the meaning of fractions. In another round, three girls dropped out of the bidding because they claimed they had poor hands for (x + y) / 2 (x + y) = 1/2 (Winner is she who comes closest to 1/2) What they didn’t realize is that any two cards substituting for x and y will always give 1/2 for the whole expression. Everybody is a winner! I connected this to the boring fourth grade idea of equivalent fractions. In the second activity, each girl had a number taped to her back and had to deduce what that number was. The only clue was that if each added their hidden number to the number on the back of some other girl, the two would add up to 1, so they had to figure out who was paired with whom. In one round, Laila deduced, for example, that 200% must equal 2, since that person had to be paired with someone who had −1 on her back, and 2 + −1 = 1. Although none of them could have solved any whole round individually, they could solve each round together in about ten minutes. At this stage in their curriculum, they haven’t been officially taught percentages, but they’ve certainly seen then around, and maybe the girls are right, maybe it would insult their intelligence to pretend that they can’t do any problem-solving with percentages. And the same with the several other topics that I had jumbled into the mix, including all of the algebra topics that we discovered in our poker game. The fun of problem-solving isn’t just for advanced students, and this group is more or less average, “at grade level”. We still ran into such misconceptions as 1/8 = .8, or 1/0 = 1, each of which had to be carefully debunked, again. But there appeared to be a lot of excitement and a lot of messy learning yesterday, and perhaps that was more important than the sometimes bumpy ride. Students have taught me something similar in chess class. There is a months-long stretch after learning how the pieces move but before learning to play a whole game, and years ago I designed a curriculum to cover that stretch. But students rebelled (boys especially), saying in effect, “I don’t want you to teach me all these clever ideas. I would rather have the excitement of playing many games, losing many games, and slowly bumping into those same edeas. Then will be the right time to teach me.” The style is more ad hoc and opportunistic, like yesterday when we figured out that Naomi would have edged Meliya by .1 in a poker round if she had only played her cards right. After that motivating loss, I know she’ll nail it the next time around. But in chess, it took me about two years to ease up on my carefully planned curriculum, because I can be quite stubborn when I know I am right.

SuperGirl Math 3: Meditative Doubling

We began with a continuation from our last lesson: “What is the highest number that you can name, can write, and would know how to count up to?” When we left off that highest number was 12, 999. After another fifteen minutes of new answers and discussion, in which I kept responding, “…And what would be one higher than that number?” they rested at a new summit, 99,999. After that, the abyss. We also reviewed number circles from last lesson. We used the number circle to chant the 2’s times tables. Then we moved to today’s lesson. The scroll suspended from the ceiling shows the pattern of doubled numbers: 1 , 2 2 , 4 3 , 6 … 19 , 38 20 , 40 . We noticed various patterns in the chart, and talked about how you could use those patterns to try to predict, “What would be double 14?” etc. You may be wondering what is an accordion doing there. I wanted to give them a calm, even serene experience of doubling. So I played a calming Phillip Glass piece as I questioned and they answered on beat, “18?” (beat 2 .. beat 3 .. beat 4..) “36” …. I had asked the girls to bring meditation cushions. Some took it quite seriously and chanted their doublings with upright posture. Then break. After break, we observed doubling through folding newspapers, 1, 2, 4, 8, 16. When they returned to our room, the scroll had been changed. It had been taken down and a new one put in its place, but this one was blank on both sides. They were surprised. We ended with a final doubling practice with only the blank scroll to focus their attention. The very last problem, accompanied by the Glass, was “0?” (beat 2 .. beat 3 .. beat 4..) “0.”

SuperGirl Math, info

The Second Grade SuperGirl Math Class is currently full. Who: 8 Girls When: alternate Mondays, 4:00 – 5:30 Where: parent houses in Berkeley Why: challenge + review, to help girls to be excited about math, and to be strong in math How much: $150 for five meetings through December A second session will begin in January. Registration will open in early December. From the SuperGirl Math Camp over the summer: Well first off let me say it’s  been great, thank you, and Shira went in one day from saying “I don’t like math” to saying “camp was great”. What can I say, you’re a gem! … I’d love to have it continue during the school year. — Tamar, parent I am very impressed with your commitment to this camp, and Naima is having a great time! … I would like to continue something with you during the school year …. Thanks again for making this so engaging for the girls!

SuperGirl Math Camp, August 5-9

Some daily highlights from our Camp In measuring the body proportions of some cartoon characters using rubber bands, we had to fold some paper. (It’s a long story, but your girls can explain.) In folding our papers, Laila had an interesting question. “The top paper is folded in fourths, and the bottom paper is folded in fourths, but since the top is way smaller than the bottom, maybe I should call it eights, or something else?” They discussed this for several minutes, and I then summarized the discussion by saying, “Maybe there is no such thing as ‘one fourth’, but maybe it always has to be one fourth of something.” Using Cuisenaire Rods, we made models of various halves, thirds, and fourths, and I asked the girls to arrange their answers in a way that was both beautiful and showed the logic of the math. This team was obsessed with the idea of the fractions portraying a fireworks, and I went along with that. We discussed how that idea might dictate the placement of the various fractions, and they came up with an imaginative answer. Enough photos of papers and gizmos. This is a candid of Meliya in pure thought. I showed the girls a way of visualizing multiplication, then we practiced on some problems such as 13 x 14 = ? Although we started solving these using wooden gizmos, the room became eerily silent when they solved using nothing but their imaginations. The point of this was not to learn yet another way of multiplying, but to demonstrate that multiplication can be pictured. That is a valuable lesson that they will use this year, although these particular problems are harder than what they will see in school. That picture of multiplication is directly related to the “lattice method” that they’ve already learned in school, although the students don’t realize that yet. DAY TWO: Math Field Trip I can’t believe that I hadn’t noticed this before, despite several previous visits to the exhibit. It looks at first glance as if each of the smaller rectangles in this work is 1/20 of the whole, but I was startled to notice that the rectangles are subtly different. Why? What’s going on? Wooden rulers appeared, and 6/9 of the students rushed to measure the rectangles. Figuring out ratio of length:width by subdividing. Sophie was the first to start doing this intuitively, but when I started referring to this as “Sophie’s trick”, there were several protestations of “I did it too!” DAY THREE 2 cups make a pint 2 pints make a quart 2 quarts make a half-gallon 2 half-gallons make a gallon Although this list is conventional for us, if you think about it, there is something odd with it, something that breaks the pattern. Our language has lost the older term for “half-gallon”, which used to be called “pottle”. Our early folk measurement system was elegantly based on the powers of 2, and the units continued far less than a cup and far greater than a gallon: …. 2 jills make a cup, …. 2 gallons make a peck, 2 pecks make a pail, …. I used this today as an introduction to thinking multiplicatively instead of additively. In other words, what is the next number — 2, 4,… ? Fourth graders will say that the pattern continues “2, 4, 6, 8…” and while that is true enough, fifth graders should start to see that there is another possibility, “2, 4, 8, 16,…” In the adult word, the second answer is far more important. In the photo above, students are solving, “How many gallons in a hogshead?” Ten minutes later, the answer is 256. Along the way, we again bumped into reciprocals. “If 16 cups make one gallon, how many gallons make one cup?” We had had a variation of this problem on Monday, when Stella complained, “This is making my head explode.” Although I haven’t formally taught reciprocals since then, her head is doing a lot better now. She raised her hand to answer the question, and then matter-of-factly said the correct answer. By Friday, I predict she would answer the question with some adolescent impatience, as in how could I be so dumb as to even ask her the question? Amanda and Leila solved the problem in a surprising way, although the fact that I call it “surprising” may just mean that I don’t understand how they think. It also meant that they were very happy. When Hannah and Nico (not pictured, sorry) worked on the problem, they realized the answer would be 16 x 16, and Hannah asked for paper and pencil to figure that out. But I reminded them, “Wait a second — we just did something like that the other day. Can’t you just solve this by imagining the picture instead of always running to paper and pencil?” And so they did. We ended the morning by developing a visual understanding of the four basic operations with whole numbers, and then extending that visual understanding to fractions. FIN

Episodes from SuperGirl Math Camp, July 15-19

This inaugural demonstration camp was the “proof of concept”, and the concept seems to work: a girls-only math camp, focusing on what they’ll see come the new school year. For the next SuperGirl Math Camp coming up August 5-9 (also for fifth graders), we have nine girls signed up, one slot remaining. I promised the girls that we would begin and end end each day with “Weird”, an unusual application of fractions. On our first morning, we studied the proportions of Bart Simpson and other characters using marked-up rubber bands. You can see that Bart’ head measures a little more than a third of his overall height. We did two experiments applying ratios to cooking. One with orange juice: can the girls taste the difference between a 1:3 concentrate:water and a 2:5 ratio? Our second ratio experiment, trying different ratios of butter:sugar:flour for cookies. We also added various toppings. At first they referred to the cookies by their toppings (“the one with almonds and vanilla”) but they came to realize that toppings were superficial, and that the structure of the cookie was determined by the butter:sugar:flour ratio. Missing, for example, is a photograph of the inedible greasy cookie blob because the ratio of butter to flour was way too high. In the middle of each morning, we had two sessions of “Practice.” Here, the girls took 45 minutes to make representations of halves, thirds, and fourths. They are using Cuisenaire Rods, an elegant math manipulative that I’ve used to teach everything from kindergarten math to algebra. In the course of their work, we cleared up several misunderstandings they had about the nature of fractions. One of our two field trips was to the Hazel Wolf in the Brower Center, Berkeley, where there is an interesting exhibit on relationships and sociability — a relevant topic for fifth grade girls. In the photo are two maps of race by Census tract in Alameda County, the usual way you or I would probably represent it on the paper, and a very different way of visualizing it in the artwork. Both are true; both are very accurate. The girls stayed with this over a half hour to puzzle through the differences. The two sketches on the right are of a landscape in the exhibit, showing the proportions of rock, sand, and sky. But in her first drawing, the student saw the proportions inaccurately — that happens to most of us when we encounter extreme proportions; our mind doesn’t believe what our eyes see and we tend to see proportions as less extreme. The second is her corrected version. [And I made a mistake by not insisting that her second sketch be the same overall size.] We are fortunate to have camp in a home rather than in an institution, and we used the moods of three contrasting rooms for our daily Weird, Practice, and Talking Time (plus a fourth, the kitchen for our experiments.) During Talking Time, I asked questions that they answered in their math diaries and then shared aloud: what are your past experiences with math? when (in any field) have you taught yourself something new? And we continued the theme of “relationships.” Since we’ll be meeting twice a month during the school year, we discussed how we can support each other and work together as a team. During our Tuesday Math Walk, I posed a problem: “How much money does the City of Berkeley collect on this block in parking meter fees each day?” It took them 2 minutes, 51 seconds to solve, and I saw so many good morals from the process that I decided to go deeper into it. On Wednesday, after much discussion and several rehearsals, we shot a video reenactment of solving the parking meter problem. Then on Thursday, an unexpected payoff: overnight, one student spontaneously made up her own problem of a similar sort, “how many times does the gate in front of my house swing back and forth in a year?” She is starting to see her world differently! In the end, more important than giving the teacher the right answer is asking yourself interesting questions.

K. Building Imagination: The Castle, the Challenge, and the Secret Hideout

Parent Krista (mother of Riley) observed class yesterday and has generously shared her observations. I was able to observe my daughter’s class this week and found it fascinating. The kids were building a castle with blocks. I watched them negotiate their thoughts about the construction project…., “no, I think this round piece should go here in the corner because it looks more like a castle that way”. I noticed that the kids naturally broke into smaller groups of 2-4 to conquer a section of the castle. There was a side project going on in which the kids were trying to fit different shapes together to match the shape of a rectangle. Some really got into this and praised other kids when they thought of a creative way to do it different. What a fun class! I see my daughter learning team building skills and early geometry lessons all in one! For another view, our castle-building in pictures.  I wasn’t able to get a good shot of everyone, but this will give an idea of the overall circus, with its three simultaneous rings.   First Ring: The Castle Second Ring: The Challenge Third Ring: The Secret Hideout Epilogue:

Math Circle: Problem Solving for Ten-Year-Olds

In this post, a description of our first three problems in compass constructions, there are no pictures of happy children playing with math, but I will try to imply their happy minds playing with math by describing the challenges I gave to the children and how they responded. You may have seen constructions in high school geometry, but instead of being taught by rote — as I was — the topic is presented as problem-solving challenges for fourth and fifth graders. The chance to dig in and pursue problem-solving, through mistakes and detours and dead ends and byways, makes all the difference, turning the topic from rote learning to an adventure. First challenge: Can you make this? Here is a student solution. Most of the students were able to get this within a half hour; a few figured it out instantly. For some, they first needed practice in manipulating the compasses before they could draw this. A few children drew something like the drawing below, which I praised in class as “a wonderfully wrong answer.” I’m not referring to the four petals on this flower rather than the six I drew for them. More important is that the student isn’t playing the construction game the way Euclid would want. In order to draw this, the student had to eyeball where to put the compass point, and that’s not allowed. Next class, I will challenge the students to figure out how to do this alternate construction, which is beautiful in its own way, but according to Euclid’s rules. Second challenge: draw an equilateral triangle. Most students tried to do something like this, a little measuring, a little eyeballing, both of which aren’t allowed and won’t get the job done. After a student showed me this solution, I asked her if she could use her compass to measure if all three sides are truly equal. She did — you can see the little arcs she drew — and found that it’s close, but not quite correct. Since they had done the first six-petal challenge correctly, I asked the students to study that and see if that could help them with the triangle problem. Could they see the implicit equilateral triangle? When some students excitedly told me they could see the triangle, even though it was not actually drawn, I told them that the are either 1) crazy or 2) wonderful mathematicians. Third challenge: find the midpoint of a line segment. Here is a clever incorrect answer, in which the student essentially started with the answer, a circle plus a diameter drawn through the center, already dividing it perfectly. A few other students got an iterative solution: they did trial bisections by eye, and kept adjusting the compass more and more finely until it appeared to be correct. Very clever, very advanced, but not allowed by Euclid. A couple students got the correct answer (that I was looking for) pretty quickly, but others want to figure it out for themselves, and aren’t interested in merely copying answers. The challenge for me as teacher is how to frame this problem in a way that can help the students to solve it. Next lesson, I will try to elicit from them that the solution is symmetrical, and even though the drawing looks very simple, being symmetrical is a big deal. It might suggest to some that we should make our constructions symmetrical if we hope get the same solution.. I think that should be enough to get many of them over the hump, but I really don’t know. We’ll all find out next class!

Math Circle: Venn Diagrams and Students of Color

I met recently with half a dozen students of color at Jefferson Elementary, whose teachers had encouraged them to try out the Math Circle class.  The idea was for me to lead a demonstration class in which they could meet me and sample the math. Besides the math that we tried out, compass constructions, I also wanted to explain to the students why I was there. Since we’ll be studying sets in the upcoming Math Circle, I decided to explain using the sets of a Venn diagram. I began by asking the students to fill in the blank:  “All of us [the students] are _________, but Mr. Ducharme is not ___________.”  After a few more obvious answers, a student gave the answer I was looking for: “We’re all brown and black and yellow and different colors, and you’re not!” I introduced the term “people of color.” I then put a 2’x3’ piece of paper before the students and asked them to grab about 150 little plastic bears I had borrowed from kindergarten, to represent all the third, fourth, and fifth graders at the school.  They had nostalgic memories of playing with the bears in kindergarten, but I warned them that this lesson would go from being funny and fun to surprisingly serious.  I drew the first circle of the Venn diagram, and asked them to divide the bears into two groups: students of color and not students of color (i.e., white students.) They came up with about 110 students of color and about 40 white students.  I asked Ms. Reed, one of their teachers who was working with us, to correct their division.  The correct answer is just about the complement of what the students had: over two-thirds to three-fourths of the students are white, only a minority are students of color.  I introduced the term “bias” and we discussed possible reasons that their answers were so biased. The students were already starting to get quiet, and now serious as I added the second intersecting circle:  students who are strong enough to do Math Circle vs. students who are not. I defined the the terms, we went over the possible sets, now four of them, and again I asked the students to distribute the tokens accurately. This is their answer before Ms. Reed made a few smaller corrections. Finally I added the third circle, students who would actually sign up for Math Circle class at Jefferson vs. those who would not sign up.  We reviewed the eight resultant sets, I asked students to distribute the tokens one more time, and we discussed their answers.  Then I told them, “I don’t know for sure what will happen here, I can’t predict the future, but I can tell you the past, what has happened in fact in my last three Math Circle classes at three different schools.” Using numbers instead of tokens and only looking at the set of student who signed up for class — for I won’t guess about students I didn’t teach — this is the sobering result. The optimistic students had placed about six tokens for the intersection of {students of color + strong enough to do Math Circle + actually sign up}, for their school alone. The reality, as of my last three schools, was nearly zero.  That, as I explained, was why I had volunteered to come for an hour to introduce myself and Math Circle to them. We discussed these results and then moved on to more innocent math of compass and straightedge constructions. When I left, four of the students, who appear strong enough to do challenging math, wanted very much to sign up for Math Circle. Addendum, four months later: Those four did sign up and stayed with it, as did thirteen others to make a class of seventeen. We had majority girls and about half students of color. It was a great class.

Chess for Girls + Chess for Boys: why?

I began teaching chess nine years ago. As I recall, that first class had about sixteen boys and two girls. I have taught separate Chess for Girls and Chess for Boys for about six years at Malcolm X, two at Berkeley Arts Magnet, and many more girls are playing (roughly equal numbers at MX, but parity may take a few more years at BAM). Each year I ask the girls if they would rather have one class with boys and each year they give that idea a strong thumbs down. There are advantages for both girls and boys in separating the classes. For girls: The literature on teaching girls-only math classes is mixed on whether girls end up scoring higher on standardized tests, but unambiguous in finding that they do gain in self-confidence, they ask more questions, they like math more. My observations in chess are similar: the girls, on the whole, aren’t winning as much as the boys in chess, but all the behavioral indicators are much more positive than when girls and boys had class together. “Culture” can seem like an amorphous force to deal with, but girls at both schools have heard stories of how girls are not supposed to play chess. A fifth grader at BAM told me that when she was in first grade, there was some sort of sign-up sheet posted on the wall for a chess class just for girls.  She remembers that somebody scribbled, “Chess is for boys” on the sheet, and her recollection is that no one signed up except for her.  The first year of Girls’ Chess here at Malcolm X, a parent I didn’t know once passed by our classes.  He stopped in to laugh condescendingly, shake his head and say, “Girls can’t play chess. Why are you trying to teach them?”   For boys: There is a different atmosphere in the boys’ classes: typically more boisterous, more physical, more competitive. I think that is fine, and I’m glad they can have a hour together doing intellectual labor (not sports) in which “boys can be boys,” without the mediation of girls’ presence. And together: This school year I invited the top eight students in the chess tournament to come to the chess class of the other gender. This gives the stronger students more stronger opponents to play with, and it has worked out fine so far.

Math Circles: info

In General: Math Circles began a generation ago in Eastern Europe, originally for high school students. Their purpose isn’t to cover topics in the usual algebra through calculus sequence, but instead to instill passion for math by presenting advanced math topics that a student would typically not see, topics that hint at what math looks like to a mathematician. Or more radically, as I have told some of my students, “What you think math is, is not math.” In recent years, Math Circles have been attempted for younger children in elementary school. But at each level, the structure is the same: a carefully chosen problem is presented to the students, and they take it from there. They come up with ideas, they discuss, they argue, they bump forward. They are guided by the teacher, but they make their own discoveries, they are free to make their own mistakes. It may take us an hour or two or three more to work through a problem. One solution may suggest other problems, perhaps a conjecture, a generalization. We begin again. Done well, the process gives students a sense of excitement and wonderment for the beauty of higher mathematics. This Session: I teach three Math Circle classes at Jefferson, Berkeley Arts Magnet, and Malcolm X schools. We will cover four new topics: – constructions with compass and straightedge, hopefully rediscovering the beginning of Euclid’s geometry – some math behind Braille + graphing our learning – making and solving Soma Cube puzzles – playing and understanding the game of Set I’m excited to be doing two topics that get at spatial reasoning, as research indicates that girls especially can use more experience with this — and we’ll even have a tactile dimension with our Braille.

Chess: improvement

Dear Parents of Boys’ Chess Students: Last week I posed a simple question to your boys, “What do you need to change for you to become a better chess player?” and was delighted by their surprisingly thoughtful answers. Two_suggestions are now part of our routine. One was about focusing — several wanted to be able to “focus better.”  I told them that was a smart goal and described my high school chess tournaments, where there would be hundreds of students in a gymnasium together, all absolutely silent and focused. I asked them how long they thought they could play and focus in silence.  Their predictions tended to be long:  six hours, nine hours, etc. I suggested we start with twelve minutes of silent focusing and take it from there. It was a challenge, but they did it, and we have practiced this each week since.  On Tuesday, we stayed silently focused for fifteen minutes straight. The second goal that came up was what is technically called “board vision,” the ability to see the perils and possibilities of the whole board. This is the opposite of “tunnel vision,” narrowly focusing on just one piece or area of the board.  To address this, I had the boys switch to a new board after each move, so that they would see a new position each time and have to reassess the whole board. This had the added benefit of making several boys simultaneously share responsibility for each board, which meant a lot less ego invested in each game, freeing the boys up to think more abstractly about the positions. They liked both of these exercises, and rose to the occasion admirably.  One student commented that the class was “very peaceful.” 

Chess: Tournament

[From a parent letter describing the second year of the Malcolm X Chess Tournament, two years ago.] — My vote for the best quote of the tournament, so far, is a fourth grade student, speaking of her last round of play:  “My heart was pounding so much that I could feel my glasses moving up and down.” — I overheard one student telling a friend how big a deal it would mean to be second grade champion.  Overall, people played much more seriously, especially third and fourth rounds.  A few faculty told me about how the tournament was all the buzz in their classrooms, and we had more children spectators than last year. [It has kept growing, from 32 students the first year to 62 last year, the third.] — There were several notable upsets of a student winning who wasn’t “expected” to win.  For better or worse, the students do come with their own hierarchies in how they see each other — “I am better than him, and he just beat that person, so I will be able to beat that person.”  Sometimes it was hard on people who used this logic and still lost.  The children are more even in ability than they realize.  For example, none of the students who were grade level champions last year could hold on and repeat as champions this year. [From last year’s tournament.] There will be a couple new awards in the tournament this year, one for the most beautiful checkmate and one for a pair of students who together create a great game.  Current contender for #1 is [a student] who had a very spare mate using bishop, knight, and pawn.  Often students prefer to get two queens and bludgeon their opponent to death.  His solution is much more elegant. Current contenders for #2 are a pair of students.  When one didn’t understand the rule about pawn promotion, the other, instead of being frustrated or condescending, gave her a series of hints and made a game out of the situation.  That attracted a few other students who also tried to help out; it was a very sweet scene.

MathGames: The Seven Ravens

[From a class last year at Berkeley Arts Magnet.] I’d like to describe our classes by telling you about Wednesday’s class. We began The Seven Ravens, Grimm’s fairy tale number 25.  In my retelling, the hero, now a girl, faces three challenges before she can turn her brothers and sisters from ravens back into human form.  The first challenge is to count the stars. I threw out over a hundred small plastic balls for them to catch.  Then their challenge was to count them all.  At first they looked like the chase scene of a Buster Keaton film, with everyone walking by themselves, not working together, maniacally counting “One, two, three …” but they were inadvertently kicking the balls around so their counting wasn’t getting anywhere.  I waited them out for a couple of minutes and someone saw that it was all madness, and commented, “We’ll never be able to count the stars. They keep moving around!”  Next another student got the idea of grouping the balls by color — which makes sense enough, they are skilled at grouping. But when we counted one pile (27) and then the next (16), they realized that since they don’t know how to add, and so that wasn’t going to work. I then decided to help them out by suggesting groups of ten, and showed them that a triangle formation of 4 + 3 + 2 + 1 (like bowling pins) yields 10.  So they set about arranging the balls into triangles, but since the rug and floor made for a hard surface,  in order to not have the balls roll around, the students needed to place them very carefully and to work together to get all balls into triangles of ten. Finally it was time to count.  But even with our neat triangles of ten, most of the students still didn’t “believe in ten”.  They started counting individually within each triangle: “One, two, three … “  I let them do this for several minutes.  But at one point [a student] looked at the orange balls and at a glance proclaimed, “There are 41.”  I stopped the class. She had understood the four groups of ten + one extra.  I pointed this out to the children, and now they counted all the balls by tens: “Ten, twenty, thirty, …” There were 121 in all.  That is a lot of adventure to count the stars, and the hero of The Seven Ravens still has two more challenges to go before she can change her brothers and sisters back into human form. The next challenge involves a pesky old dwarf who only talks in negatives:   It’s not the tallest, not the smallest, It’s not the round one, have you found one? The final challenge will involve a path with steps numbered 1, 2, 3, … 30. They children will have to figure out a hidden step by guessing: for example someone guesses “12” and my thumb indicates that the hidden step is “higher” than 12; perhaps the next clue indicates that the next step is “lower than 25.” They have to keep these clues straight as we hone in on the number. We have done five fairy tales so far, and like The Seven Ravens, they turn into puzzles that challenge them to think creatively, to work cooperatively, and to use their bodies, either in small motor movements or full-out games on the playground.  We may also use drama by having them act out the story. The story: your children scared me last week, every one of them, by how attentive they were in listening to the introduction to The Seven Ravens.  We had been doing games from the previous stories for three full weeks, and that was my mistake, that was too long without a new story.  So they prodded me, “Can you tell us a story?” and when I said yes, they immediately assumed their positions on the rug, on the couch, crowding around me.  And they waited with such a sense of anticipation.  Teachers always want students to “listen better”, but I had better be careful what I ask for.  Right before I began, I had an odd memory.  My high school calculus teacher was fond of saying that “mathematics is the language of science.”.  But your children have taught me that fairy tales and stories are the language of mathematics. 

Math Circle: gender consultant, oddness, and Santa Claus

[A parent letter from November, 2012 at Emerson Elementary.] Dear Parents, A few notes from our recent and upcoming Math Circles: 1. In an effort to figure out how to teach girls better, I have hired a gender consultant, someone with an M.A. in Women and Gender Studies, to observe and critique my teaching, and to discuss with me some of the academic literature on gender and math.  Sarah has visited class once so far, but she’s been great at spotting some of my teaching errors and challenging assumptions I have about working with girls.  I would welcome more people to talk with about these issues. If any of you have a formal background in gender or simply an interest from having observed yourself and your children, please tell me and let’s talk. This is a moment when I am unusually teachable. One of the things that Sarah noticed last week was that three of the girls had independently written above their scratch work a sentence stating that the problem we were working on was hard. (It was, in fact it was impossible.)  Reflecting on the fact that this dovetails with some of the literature on girl’s self-perception, I decided to begin class yesterday differently than  I usually do.  I started with three short anecdotes about my confronting, and sometimes failing to solve, problems in my own life that were hard.  I then asked the children to reflect on when they’ve had problems that were really hard — in school or otherwise — and how they approached them. They became quiet and earnest.  I told them that my teaching is not really about math; Math Circle could just as well be titled, “This is a Hard Problem.  What are You Going to Do About It?”  That’s as far as we got yesterday, but I’ll follow up in future lessons. 2. Yesterday we finished our penultimate lesson on parity (the mathematics of odd and even numbers).  They’ve worked on five short problems that are each about parity , but which have had such varied settings that the children haven’t realized that the problems are all related.  Next lesson will be a summation, or as I told them at the end yesterday, “Next lesson, all will be revealed — or not.” Sometimes in class we treat math as an inductive subject — the students try to find examples that satisfy a certain condition; when they can’t,  they conclude, “It’s impossible.”  This is fine as a heuristic, but in the end math needs to be deductive, depending upon proofs, and not only inductive, depending upon examples.  The next lesson, I expect that they will do their first proof, and to make it more dramatic, we won’t fiddle with examples beforehand, I will invite them to dream up their parity proof out of pure thought. [Note: I did try this the following week, but it flopped miserably. That was probably my fault; most failures seem to happen when I get excited and rush the curriculum.] 3. Next problem due up: How much time would Santa Claus have to visit each family in America that he visits?  How many reindeer would he need to pull his sleigh? This is a Fermi Problem, a back-of-the envelope calculation that uses a little bit of math and a lot of cleverness to estimate something in the real world.  The classic Fermi Problem is, “How many piano tuners are there in Manhattan?” but that’s more of adult-interest.  I made up the Santa Claus problem several years ago and it has been well received.  At first I worried about its Christian reference, but everyone seems to get into the spirit of it. It should take three hours or so to muck through the problem.

Build the World: description and rationale

A year ago, the PTA Afterschool Committee had asked if I could teach a new kindergarten class, an in turn I talked with kindergarten teachers to ask what skills are the students lacking that I could help teach. Their answer: fine motor coordination is something almost all of them need, even children who are doing well in academic subjects. This has evolved to Build the World, first offered in Fall, 2012 at Malcolm X. The initial class description has proven more or less correct: This is a crafts course for kindergarteners in which students build and animate a world.  They will start with a mass of clay and sand which they will sculpt with their hands into land, a mountains, a river, a cave. Weaving with twigs, they will make a bridge, a fence, and stonework to make huts.  Using a wet felt technique, they will create animals and people.  All this the children can do, largely on their own,  with solid materials, no veneers, no tape no glue, just hand manipulation.  And each week I will tell them a creation myth from around the world as we shape our world. We’ll do a couple of things differently in the upcoming Spring class: 1. More fantasy play for the children. We will re-use the stars, rocks, and twig walls from the world we built In the Fall, so that we’ll be able to build a more complete world sooner. My goal is to quickly get to the animate part of creation, so the children can start imagining and acting out stories. We will have a great locale, complete with archetypal cave, mountain, water, plain, and step rocky ascent. Here I have been inspired by “Godly Play”, a Sunday School curriculum started a while back by an Episcopal priest, in which the children do storytelling and fantasy play with figures, usually commercially made, for example, a set of figures to do the Noah’s Arc story.  I’ve seen this in action and am have assisted in teaching Godly Play stories at a church, where it is used for grades 1 and 2. In the Build the World, we play do a secular version of Godly Play, plus making our own figures and dioramas instead of buying commercial products. 2, More stories. The stories affect them deeply, and I would like to do more. The technical difficulty is how to arrange things so that they can work on their craft while listening to the story. There is a tricky balance between how challenging the craft is, how much they need my help to do it, and how well I can tell them, and they can really listen to, the story. Last class, craft and story were separate; this class I will try combining them more.