**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.

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.

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.

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. hducharme@gmail.com , 510 417 5736.

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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, more frequently higher. So I politely emailed Clark asking him if I could try changing some of octaves in rehearsal. He agreed, and was in fact thinking along the same lines. He’s left the particulars to my “good musical taste”.

With all but my highest piccolo reeds working, I’ve transposed this passage up a couple of octaves. The bass player also doubles the part, six octaves below, and together I think the high and low sound pretty cool together. Clark likes it.

January 2 — Usually when I fail at something musical, it seems that I didn’t work hard enough or practice long enough. But I’ve just stumbled badly in my first rehearsal since break, missing several cues that I thought I had nailed before. I think this was due to a strategic mistake rather than lack of practice: the last few days I had gotten too fancy for my own good, deciding upon tricky reed changes and unnecessary articulations without really knowing my part down cold — a mistake of over-enthusiasm — and so after devoting a few hours to interpretations I can’t sustain, the fundamentals have suffered. It’s now time for triage.

January 8 — Overall, the twelve rehearsals have had a predictable progression: core instrumentalists, then adding our seven percussionists, then singers, so far all off-site at the Paul Dresher Studios. Our playing area is accurately spiked so that from the beginning we can practice what it’s like not to be able to see each other, and, at many times, not to be able to hear each other very well. Then rehearsals move to the Crucible, where the singers work on staging and we get used to local acoustics and working with Brendan Aanes, the sound technician. Some performers can follow Barnaby’s conducting only from television monitors. A direct sight line is not always possible.

It is a tech-heavy production. Whenever the performers take a break, the Crucible tech crew swarms upon the stage to weld, screw, rivet, staple, paint, wire, and duct tape stuff. Two days ago, Friday, we started rehearsing with the fire effects, after the lecture demo of what to do if we catch on fire. I get used to a column of fire in front of me and other blasts of fire every time we play certain passages. I imagine the fire will be more intense if I play sour notes. Barnaby gets a smoke machine that is intended to intensify the mood at his podium, although he might prefer to be able to see the score cleanly.

Today’s rehearsal, from make-up to final notes: 8 1/2 hours. Tomorrow, Monday, is our public run-through, Wednesday, opening night.

]]>— 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

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]]>- 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.

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• 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.

]]>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.”

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!*

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**

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.

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