The Alhambra Banquet

The Touchstone Common Room has been transformed into a palace courtyard, with twinkling lights, a mylar fountain, and carefully painted scenery.

Alhambra new scenery 4Wearing the loose, richly decorated clothing of long-ago centuries and far away places, kids stand tall and proud, representing people whose achievements still shape our lives, even though their names have been forgotten.  Alhambra Caroline and IsyAfter the presentations, gathering around low tables with their families, kids try foods they wouldn’t usually touch with ten foot poles–because in this case, after all, they helped with the cooking.

It’s all magic, magic we made together.

Audrey ShabbasIn designing our version of the Alhambra Banquet, I found both inspiration and practical help in resources assembled by Audrey Shabbas at AWAIR, Arab World and Islamic Resources. As part of a workshop I attended at Clark University in Worcester, Massachusetts, Shabbas described a new approach to the study of medieval Europe, which would include the vital role of Arab Spain, often omitted. To engage students with this hidden history, she had conceived of a banquet in the Alhambra Palace in Granada.

I’m going to quote from a letter I wrote to parents:

[From the Alhambra] we look back to the 700 years of Muslim Spain, in which Christians, Jews and Muslims lived together in relative peace and prosperity; and in which all three religions co-existed comfortably with advances in science, technology, and culture that were largely unknown in the rest of Europe during the same period.

We look out to the world of the Mediterranean, where trade and the common language of Arabic created connections between many cultures; and where the Arab enthusiasm for paper,  an invention brought from China, resulted in an explosion of publishing and translating and an unprecedented exchange of ideas.

Shabbas asked classes like ours to imagine travel through space–the entire Islamic world of the time, around much of the Mediterranean, but also beyond, including the rest of Europe. We also imagined travel through time. Students represented artists, architects, rulers, Sufi mystics, rabbis, Christian nuns, Muslim philosophers, librarians, book collectors, poets, physicians, mathematicians, and astronomers, from across seven centuries of the medieval world.

In our imaginations, in a grand triumph of serious playfulness, all these ancient people came together for a glittering, century-transcending, multicultural, multi-generational dinner party.

Alhambra new wide view of Common Room

When I set out to design the way the AWAIR curriculum would unfold for us, I found inspiration also in the creativity and intellectual ambition of Touchstone students, and in the generosity of their parents’ involvement in the classroom. I knew that we could handle these rich, challenging, unfamiliar worlds and concepts in hands-on ways that would make them real for us.

First, we set our sights high. Literally. Up above the whiteboard at the front of the classroom, I posted Essential Questions for this study:

  • If we could hold an imaginary banquet in about 1400 in Spain, and invite people from past centuries and the whole world known at that time, who could come? What could they talk about?

  • How did life in Al Andalus look and feel and taste?

  • How are we still influenced by the religions and cultures represented at this banquet?

  • What can Al Andalus teach us about the ingredients of successful multicultural societies?

In the light of those questions, we got down to work. During the four or five weeks leading up to the banquet, each student became involved in five different efforts:

Alhambra Nate pointing#1   As a whole group, we learned about the background history and geography. We arranged big file cards into time lines by rearranging our own bodies in a line. We created a giant map of the Mediterranean world in the gym, each person representing an important city. I vividly remember Kate Keller describing the evolution of the mathematical idea of zero, and how that idea found its way from India to northern Europe, by way of Arab Spain. In their classes, our Spanish teachers helped us think about the history of Spain as part of a larger Mediterranean history, and as a hinge between worlds. Taking it slowly and carefully, I found ways to explore Ibn Rushd’s sense of the relationship between revelation and reason–and marveled at the ideas with which these young students could engage.

Alhambra new Don on floor#2   Through a process of choosing from among various professions and roles, each student took on a historical figure to represent. (Adults sometimes filled in gaps. I’m not sure who was being represented by Don Grace in the year of most of these photographs, but the kids did a great job on his clothing!)

Alhambra honored guest listWorking hard to understand challenging sources, students found it exhilarating to take on the identities of people such as Ibn Sinna, known in the West as Avicenna, who brought scientific methods to the study of medicine and healing; or Zubaidah, a queen of Baghdad who set new standards for public works, particularly a series of wells, reservoirs and artificial pools that provided water for Muslim pilgrims along the route from Baghdad to Mecca. It’s driving me crazy not to describe all the people we called back to life. But you can click to enlarge this fairly representative list from 2011.

Alhambra new luke and deanIf a student represented someone well-documented in resources available to us, he or she had to set priorities for what would be included in a two minute presentation. For figures about whom we could find very little, even making careful use of resources on the internet, students branched out to include more information about the person’s areas of work or interest, using resources that would explain monastic life, or Islamic architecture, or the history of mathematics.

Each student checked in with one or two partners, with whom they could share and compare, and stand together in front of the banquet’s assembled audience.

Alhambra new sewing# 3   To help students enter the spirit of these representations, parents worked with them to create special clothing, decorated with magic marker “embroidery” using the motifs and styles of medieval Islamic design.

alhambra matt

#4   Students also prepared scenery and decorations for the Common Room. In our own microcosm of cultural evolution and preservation, we saved some of these murals and columns and window-top decorations from year to year, so any particular class knew that aspects of their work would last, and be built on by future classes.Alhambra new old scenery reused#5  Finally, what’s a banquet without a feast? Some typical foods of Arab Spain could be prepared ahead by the students during projects time, and frozen in home freezers. Other things–vegetarian kufti, chickens roasted with dried fruit, flan–were cooked at home and brought in by parents the night of the banquet–along with pillows to make it easier for all of us to sit on the floor at low tables, and potted plants to help us simulate the lush gardens of the Alhambra.

Alhambra new scenery 2 Alhambra new onion overdoseThrough the weeks before the banquet, kids rotated through projects time groups working with wonderful parent helpers along with teachers and aides, to create clothing, scenery, and food. They read background text related to the clothing, food, and design of medieval Arab Spain. They practiced taking notes, shared what they had understood–and then chopped up tremendous quantities of onion or garlic, or crawled all over the floor collaborating on complex designs.

Alhambra painting sceneryThroughout this entire process, and by the end of the banquet, all the adults involved were captivated and stunned–by the students’ hard work and accomplishments, and by the content they worked to share with us.

What did students get out of it all? I wanted to document this with excerpts from their own reflective writing as we went along, and afterwards–but I’m writing out of a blizzard zone, and can’t find those right now. So I have to try to summarize.

Because these students were so young, almost everything we learned in this study was new. Thinking back, I see faces scrunched up with the effort to grasp strangeness, and glowing with the satisfaction of making sense. Conceptually, they reached far, reached deep, and felt the strength of that reaching.

They also shared my satisfaction in their nitty-gritty skills growth: in note-taking, in handling unfamiliar words and other aspects of challenging texts, in interpreting maps and timelines and other charts, in oral presentation, in giving each other useful feedback.

Alhambra max croppedAbove all, students loved the way most of the learning we shared at the banquet was new to their families–so that their sharing had real focus and purpose. Students felt important, and powerful. I have memories of kids putting on the special clothing they had made (from recycled bed sheets transformed with love and patience)–then straightening up with an amazing light in their eyes.

All of us remember with wonder the physical world of the banquet, so much a product of grand collective effort–but all of us, both students and adults, remember even more the passion in student presentations, the miracles of stepping up, the deeply personal pride.

And, of course, the wild exultation when they were done.

Alhambra cheerI’d be thrilled to hear from past Alhambra Banquet participants, or to help others create their own Alhambra Banquet experience. You can write one of the usual comments, for others to see, or send something directly to me using the contact form below.

Accountability and Projects Time

This post is the second in a series about Projects Time. Here’s a link to the first.

Unless people saw it in action, Projects Time could be hard to explain.

Colleagues at my own school understood, pretty much. They had given my students years of similar experience in younger classes, that helped them be ready to make longer-term choices, and choose on the basis of activity more than work-partner. They sent me students proficient in physical problem solvingprojects tsongas canal adjustment cropped, making things work and getting a kick out of the effort. They sent me students with their natural curiosity and creativity still very much intact, full of the energy and momentum for inquiry. (Museum teachers–here at the Tsongas Center in Lowell–always marveled at the hands-on cleverness and persistence of our students.)

My colleagues also sent me students who knew a lot about working with each other. By the time they reached me, most students already knew how to exercise individual creativity in the service of a group effort—contributing ideas, but not needing to have them adopted by the group every time, and increasingly able to listen to each other.

buildersIn Projects Time, kids could show leadership in many ways. For one thing, they could help to support collaborative working skills for their classmates who needed more time to develop that particular skill set. (After all, nobody has an aptitude for everything.) Year after year, I could count on finding at least one child who would be especially helpful to students brand new to the school–who didn’t know everybody yet, of course, and typically had had fewer opportunities to practice both group skills and hands-on problem-solving.

I loved watching what could happen with minimal adult intervention. I could exercise my care mostly in the background, choosing the groups at the beginning of the year with particular mindfulness, providing appropriately-leveled background reading, giving the most inexperienced students the comfort of a group of two.

In these small interventions, I followed the example of the colleagues from whom I had learned how to teach. Projects Time used and stretched skills the teachers of younger classes had been building for years, in both kids and adults. So they understood it—teachers and students both.

projects out mini buildingsTo the left, a group of students have been exploring construction techniques, ways of supporting weight for example, by creating miniature buildings with natural materials from the slope near our deck.

Still, people who didn’t see it in action found Projects Time perplexing. Parents new to the school sometimes expressed bafflement, until they had an opportunity to join us–or until talkative kids came home bubbling over. If new administrators never came to observe, they might be reassured by the testimonials of other staff, or by reading my parent letters–or they might not.

Most challenging of all, though, for me and for them: teachers from other schools might never have observed projects learning, let alone our situation of groups working parallel on different activities. When I talked with them at conferences and workshops–or family reunions, train rides, all those informal workshops teachers create for themselvesthey said, projects giggling group“Wait– if kids are working away from you, in a small group by themselves or with another adult, how can you be sure what they’re learning? If they’re not all following the same activities, how can you control what’s happening? What do you put on tests?”

projects marble chutes editIn the early years, faced with questions like these, I had to work hard not to get defensive. (It didn’t help that I had no idea how to explain serious playfulness. I just knew it felt right, and worked.) Then, as I developed more and more confidence in my students, I struggled to suppress anger at what I saw as lack of respect for kids.

Eventually, though, I felt sympathetic. As years went by, I increasingly wanted to do one of two things.

  • I wanted to wave a magic wand and give these other teachers my opportunities for knowing students well. My class size, for starters—never more than 18, and more typically 15. Also, my self-contained class, together and with me for most of the day, except for specials and in some cases math. We shared a continuity, richness, and intimacy of group experience increasingly uncommon in the school lives of ten-to-twelve-year-olds. I could know both individuals and the chemistry of a groupknow them really well—and that let me sense the ways I could trust them, and then build on that.

  • projects river model0001Sometimes I just wanted to loan out my inquiry-proficient students to these other teachers, as an example. You have to watch kids who are accustomed to belonging to themselves (like Mister Dog in the wonderful Little Golden Book by Margaret Wise Brown) in order to realize what they can do—what challenges you can give them, and what motivation they will bring to open-ended opportunities.

We did have in place a number of routines to keep me informed about how things were going, for groups as a whole, and for individual kids, and for the other adults in the situation.

  • Whenever possible, I built in freedom for myself to move from group to group, for at least some part of every time block, taking both notes and photographs as I went. (Otherwise, would I have this treasured shot of Lucy Candib and her group digging a hole under the deck, in which to bury various materials, which next year’s class would dig up again, in an ongoing year-after-year investigation called What Rots?)

  • projects Lucy burying rotprojects volunteer notesI asked assistants and volunteers to fill out a quick question sheet at the end of each session, to help keep me informed. (They had time to do this while I led the final wrap-up.)

  • projects Fermi question notes editStudents took notes and made sketches during Projects Time, and then wrote more afterward, summarizing what they’d learned, making additional drawings, listing questions. Sometimes I saw this writing over their shoulders, as groups shared at the beginning of the next session. Sometimes I collected students’ writing, to look for growth and for areas of confusion that needed support. When Sally Kent told me about lab notebooks with carbon sheets that could be torn out, we began using those, and kids could keep the original in their notebook for future reference, and give me the copies.

Still, our ways of observing and guiding students were premised on trust, meant to help us support students in their learning, not grade them on it. (Here’s more about that.)

Looking back, in the light of the current obsession with accountability, I realize that skeptical questioners were asking, ‟How do you keep the kids accountable?”

We worry about holding people accountable when we don’t think they’re likely to stick to their side of a bargain, or approach their part of a task conscientiously, or own some effort. We worry about accountability when we think people are likely to cheat—and people cheat when they don’t feel ownership of the results. Many things can lead to that lack of ownership—some of them outside a teacher’s power to intervene. People cheat when they’re depressed, when they’re overtired, when they lack confidence in what they can really do. There’s a lot to understand in that vicious circle, but demands for accountability don’t change it in any useful way. Or that’s how it looks to me.

Recently, I reread an essay in The Atlantic about why it’s so hard for American educators to understand the success of Finnish education. The article, by Anu Partanen, quotes Pasi Sahlberg, speaking to an audience at Teacher’s College in New York City in 2011. ‟Accountability,” he said, ‟is something that is left when responsibility has been subtracted.” 

I keep mulling that over. I still have questions about Projects Time, and plenty of ideas for changes I’ll make and new things I’ll try if time travel ever becomes available. Still, Sahlberg’s statement helps me understand why Projects Time worked so well almost all the time; why my structures and routines for staying informed were secondary, in fact, to the most important part of the dynamic.

In Projects Time, students were genuinely responsible to each other. They were mutually responsible for the thoroughness and spirit, the seriousness playfulness, of their own groups’ inquiries, obviously. Beyond that, because they so often carried out different inquiries, in parallel, they were learning on each others’ behalf. They put creative thought and boundless energy into the ways theyprojects transportation wheelcha would demonstrate their methods and summarize their outcomes. For example, often they set up stations for other students to try out. To the right, a skit created by a group who’d been researching transportation access issues for people with physical disabilities.

And then there were the puppet shows about water power, or future careers in transportation planning, or… 

transportation puppet shows

So there it is: Irony Alert. The same five-ring circus, the same level of complication that stretched the adults’ ability to be everywhere at once, meant that we didn’t have to.

We didn’t have to hold them all accountable, minute to minute. We held them responsible, instead, and they rose to that, for themselves and for each other.

And, as Robert Frost would say, ‟that has made all the difference.”

For this post, I’ve scanned in some older photographs I found when I went hunting for artifacts from the early years of Projects Time. These kids are now very grown-up grown-ups–finishing med school, having babies, beginning careers in agricultural engineering, as many stories as people. I look at the photographs and grin, and feel–for the millionth time–how incredibly lucky I was to know them when they were prototypes of the energetic, engaged adults they’ve become. So, all of you, thank you again.

projects at the beach

Projects Time

As an incubator for serious playfulness, nothing worked better than Projects Time.

vortex gazersTeaching “big kids”–young adolescents waking up to the world in new ways–I wanted to give them the choices, hands-on experiences, and purposeful collaboration in small groups that would keep them engaged and alert and cooking. Projects Time evolved as a way to frame all that.

It also grew out of adult behavior that can’t ever be taken for granted:

  • Adults made choices about the guidance they offered based on what worked for each particular group of kids, in their individual and group uniqueness–by listening carefully, with a sense of learning targets in our minds, but with the reality of the present always uppermost.
  • projects compost dirt grandmotherAdults dove into hands-on, messy, authentic experience (almost always potentially risky to our dignity.)
  • Adults worked together, as teachers and assistants and committed volunteers, and got a visible kick out of our own collaboration.

Put all together, Projects Time was a bit of a miracle–a twice-weekly, home-grown miracle.

graphing voicesAs we got better and better at running this, we could see the effectiveness of having different small groups working simultaneously on different projects, and then sharing with each other. For example, in the photograph above, a group who’d been investigating sound set up instructions for other students in an end-of-sequence “energy fair”, and two students are trying out the set-up.

Below, in a sharing session at the end of one day’s Projects Time, a group uses their own bodies to demonstrate the arrangement of the states in New England.

bodies as New England states editA little more about logistics

Students and adults came together for Projects Time in two fairly long time blocks—a total of almost three hours every week. Tamara, the teacher who moonlighted as the school’s scheduling wizard, knew that I would accept any other strangeness in my class schedule, in order to preserve those long Tuesday and Thursday afternoon time blocks.

A series of inspired part-time assistants joined us for Projects Time, even when we had no other aide time assigned for the class.  Each year’s volunteer parent coordinator helped me recruit and schedule parents, often well in advance.

Within the nourishing nest of those pre-arranged rich conditions, the students and I could choose our challenges. To begin each sequence, we brainstormed a list of ideas for projects which would make use of various materials and opportunities in and outside of the classroom–and would meet various learning goals.

Some activities, typically, related to our current whole class theme. In the fall and spring, we planned for as many activities as possible to happen outdoors. (For example, in the photograph below, a group discusses a redesign of a water feature in the garden below our deck, taking into account the way water travels downhill.)

projects side gardenA particular week’s list often repeated some of the topics or activities from the previous sequence, because kids wanted to try things they’d seen other students do. “That thing building electric circuits looked like fun–can that be on the list again?”

After we had settled on a menu of possible projects for the next round, each student wrote three or four choices on a sticky note, ranked them, and gave the note to me. (Thinking all this over, it always seems important to me that students were choosing activities, not work partners.) Choosing is hard for some kids, and I let them write down “anything” if they really meant it, but encouraged them to think it through, and predict how different activities would work for them.

Later, I arranged and rearranged the sticky notes to form groups. Usually I started by seeing what would happen if I gave all the students their first choices–and sometimes the groups made themselves immediately, just as easy as that. More often, I needed to give some students their second choices, in order to provide for variety in work-partners and types of activity, both of which felt important to all of us, kids included.

Students’ choices committed them to at least the two blocks of a single week, and sometimes a third block, or even a fourth, in response to popular demand. Longer sequences allowed more time for exploration and follow-through, and students found that rewarding.

projects temperature investigations grinWith very few exceptions, everyone who took part in Projects Time for any length of time felt that it worked, in a unique and exhilarating way.

river group recording some editsStudents experimented and observed and simulated and dramatized, and also had a great time. They took concepts they’d learned from reading and applied them. In the follow-up writing, they speculated about what had happened and why, and what else they might want to try.

There were social benefits, also. Working together in small groups, students got to know each other better. They became deeply involved in inspired arguments. For example, in the photograph below, students conducting a simulation of the effects of transportation argue about a proposed trade.

transportation argument editI’m going to use the next few posts to explore some particularly memorable Projects Time sequences, including the activities Kate Keller designed for our Transportation Choices unit, and some work on A Field Guide to Touchstone.

I also want to share some questions I’m still mulling over. One involves the perennial conflict between coverage of material and effectiveness of student learning experience. Obviously, the Projects Time model isn’t necessarily the best model for covering every detail of content on a long list of state or federal or Common Core standards.

Another persistent and possibly related question involves accountability, a big buzzword in American educational policy right now. Again, it’s obvious that Projects Time wasn’t designed to maximize accountability.

I’ll come back to all that. For now, having given you some snapshots of individual projects, I want to take you on a fantasy helicopter ride, to get a sense of how everything was happening at once.

From our point of view, hovering above the school grounds, we can see a group with a dissecting microscope, at a picnic table behind the main building. (Hooray for extension cords.) The students not currently using the microscope are looking for things in a nearby garden, including creepy crawlers from the compost, to examine when they get a turn. One student sits at the picnic table making a detailed sketch of a flower she found, using a jeweler’s loupe to get a good view of the structure.

Out in front of the school, some kids are measuring the temperatures on top of stones in the wall along the road, comparing with the temperatures they found in the wall spaces underneath those same stones, thinking about the idea of very micro microclimates.

Seth and Ben marble chutes editAnother group, working under the portico to take advantage of a long bench, uses a stopwatch to time their latest marble chute run. They’re trying to maximize the length of the run by maximizing friction, without letting the marble come to a full stop.

Meanwhile, another group is up on the deck outside our classroom, working on a puppet show about water power, in which a dragonfly puppet has become an authority on the differences between overshot and undershot water wheels, and models have been made to demonstrate them.

Somewhere down there, a lucky teacher moves from group to group, carrying her clipboard, with its note-taking sheets about individual students, and its list of stuff to track down for next time. She also carries the camera she wishes she’d used even more.

Although, really, what it needed was video, to capture kids saying, “What if…?” and “Let’s try it again…” and “That is wicked cool…”

Chasing the River, Part 2

In a previous post, I described a student named David who decided to “chase the river”–more accurately the series of brooks and ponds and rivers–that carried his backyard runoff to the sea.

Many students after David found their own watershed pathways, working with the classroom set of topographic maps and other resources. Our distribution varied, from year to year, but we always had a fair number of students who lived in the school’s own watershed, the Blackstone River watershed, along with many from the Sudbury, Assabet, Concord and Nashua sections of the Merrimack watershed to the north, a few from the Charles River watershed, to the east, and a few from the Quinnebaug, to the southwest.

Kids had all sorts of adventures following their watershed pathways, naming nameless brooklets and ponds, asking helpful neighbors what they knew, and playing in the mud.

Once, when we were stumped, I made a house call. Marissa lived on the very top of a hill, typically a watershed boundary. According to the map, her front yard would drain into the Sudbury, and her backyard into the Charles. In actual fact, it all went into the street drains, whose outflow ran into a holding pond, whose outflow ran into the Charles. Not for anything would I have missed that rainy afternoon slogging around in rubber boots, peering down through street drains to be sure which way the water was flowing.

Eventually, with the help of other map freaks, I discovered National Geographic’s Topo software, which let a student use a mouse to trace a pathway on a seamless electronic topographic map, then printed for us the annotated map that resulted. So a student could start with the little trickle that ran across the back of her yard, and keep going, without even needing hip waders or bug repellent. (It’s a shame to miss that entirely, though, and most kids got both experiences: on the map and on the ground.)

Andrew working with Topo cropped

Once a student had done his electronic tracing, he could ask the software for an elevation profile, and see how his pathway was all going downhill, except for a lake or a pond–a long flat stretch–or a place where a hand twitched and the path climbed out of the river briefly. To trace a pathway from brook to pond to river, then flip that path and see it from the side, sliding down the edge of the continent—that was pretty cool.

tracing watershed pathway cropped

If a small group of kids did this together, squeals and screams might erupt from their corner of the classroom, applauding the kid who just kept “sailing” down Narragansett Bay all the way to the open Atlantic. The group above, working with Terry Lunt, parent volunteer extraordinaire, is following the unfamiliar path of a classmate who lives way out in Charlton.

Kids who shared the lower part of their pathways, what they called “the big river,” could meet together and see, for example, how much area was drained, by all their different routes, to become the power of the Blackstone, harnessed for the mills and re-channeled for the canal that gave the Blackstone Valley so much of its identity.

Then what? The maps and their pathways could be put up in the hall for the rest of the school to see.

Topo watershed printouts whole group cropped

Like a dare to everyone passing by: wherever you live, some story like this belongs to you. You can go find out, and you can get started with a map.

Some years, students created posters to share with each other and parents at a special watershed night. Some years, we were even more ambitious, ridiculously ambitious. At the outset of Touchstone’s Older Student Program–supported by a state arts grant and with the help of a wonderful video consultant, Veda Reilly–Katy Aborn and I worked together to create a multi-part video called Voyage to the Sea. Voyage included documentary sections about topics such as the water cycle and water power, as well as story sections about two teams engaged in a regional competition called The Blackstone Watershed Relay.

In the final chapter of the video story, Bill McHenry, then director of Touchstone’s Extended Day Program, posed as the director of the Watershed Protection Association, the fictitious nonprofit that sponsored the equally fictitious relay. Speaking to students who had become fired-up environmentalists as a result of learning about their watersheds, Bill repeated a quote from Bernie McGurl of the Lackawanna River Association, one of thousands of real-life organizations of people who have gathered together to learn about their watersheds and to protect the quality of the water flowing through them.

Water has a voice. It carries a message that tells those downstream who you are and how you care for the land.

Within those words, and within all these connected river-chasing experiences, there are lessons within lessons, like watersheds within watersheds: lessons about the ways learning is grounded and deepened and enriched by attention to place; lessons about everything that flows to us and everything that flows from us, and the responsibilities we share with others and to others; lessons about what we can learn from each others’ small views, flowing together to become a wide view; lessons about the power of the learning community.

If you want to get started on your own watershed adventure, here’s a good place to begin: 

Or you can just put on your boots and head out the door. Don’t forget the map!

Building Average

I’m here to confess: I’ve spent a good portion of my teaching career guiding students in freaking out the cleaning staff.

Each year, in Level 6 math, we built a model of the Average Student, statistically accurate, earnestly assembled, vaguely lifelike. We set it up in a chair toward the back of the room. Usually the students chose a book to balance on its lap. I myself sometimes entered the room, at the end of a long meeting after school, and did a double take.

Traditionally, we took a group photo of the assembled class, with the dummy. Here, for example, is an unusually small class, from the fall of 2010. (Clockwise from the top, Kelly, Ben, Seth, Anna, Lydia, and Gianna,)

average 2010 better

A few weeks post-portrait, when stray arms or eyebrows began to fall off and litter the classroom floor, we held a funeral, usually with dual caskets–since one cardboard box couldn’t hold it all. We paraded more-or-less solemnly to the dumpster, and gave heartfelt testimonials about everything Average had helped us learn–

–which was a lot. If you ask a typical adult what an average is, chances are you’ll get the series of steps followed to find the mean of a set of numbers: add up all the numbers; then divide by the number of numbers.

That’s not wrong, as directions. But what does an average really mean? What can it tell you about a situation or a set of data? What can it not tell?

MathLand­—a wonderful math curriculum no longer in print—gave Level 6 students a chance to explore the idea of ‟average” from the inside. Many years after we had shifted to another curriculum, I kept starting the year with this unit, because it was perfect from so many points of view.

Setting a goal

You could build an average kitten, or an average bookbag–but it worked really well to build an average math class student. Kids took it all more personally, and paid more attention to interesting questions: Is Average identical to any individual in the group? How does the model represent each person’s data?

MathLand provided a data sheet which included a variety of measurable attributes—such as the girth of the neck, or the length of the upper leg from the hip to the knee. The sheet also asked about attributes that had to be described in other ways—such as the color of eyes or hair.

Some questions were yes or no: Do you wear a watch most days? Some questions had been wisely left out. Average was always just Average, neither he nor she. We weren’t asked to measure around the waist, or chest, just shoulder to shoulder.

Some questions deliberately provoked discussion. How do you measure the length of the neck? From the bottom of the ear? From the hairline? The whole class had to stop and decide, together, or the data would be meaningless.

Gathering and recording data

Before we could begin collecting data, we had to choose an appropriate unit of measurement, and an appropriate degree of precision. I did specify metric units, partly because I wanted students to get some practice with decimal numbers. The kids agreed that the measurements had to be at least as precise as the nearest centimeter. Even that could result in very unrealistic hands, though; so we almost always wound up agreeing it should be to the nearest millimeter, which we recorded as a tenth of a centimeter. (Fertile fields, of course, all of this.)

Boys helped boys measure, and girls helped girls. All the data was kept anonymous—and we said that the study subjects were unreachable for clarification of messy handwriting, so the recorded data had to be both readable and reliable.

Working with data

On the other hand, the occasional inscrutable handwriting also offered a relevant opportunity, once we reached the computation stage: If you can only read the data for 11 of the 12 members of the group, what should you use to divide the total? What would happen to the mean if you divided by 12 instead of 11?

Also, once you got your mean, would it tell you anything about the huge variation in sizes of kids this age? No–only if you added information about the range, which wouldn’t actually get built into our model.

Could a very long-legged class member and a very short-legged class member cancel each other out? Yes, in effect. But in a class with several unusually long-legged people, would the mean probably be affected? Yes, again.

Meanwhile, what about the attributes described by words? For those, we found the mode, the most common answer or value, with interesting results. A math class with only 4 out of 13 blue-eyed students could wind up building a blue-eyed Average, if the rest of the kids were divided evenly among brown, green, and hazel. ‟So my brown eyes have disappeared from our Average representation?” a certain kind of kid would ask, even without being paid or prompted.

Representing data:

Ed's arm blueprint croppedAlthough they were working together, every child measured, and recorded measurements. Every child took part in finding the mean or mode for the attributes of his or her team’s assigned body part. Finally, every child drew a “blueprint.” Here’s Ed Pascoe’s blueprint for the arm and fingers.

Julia's face blueprintEach person on the team assigned to manufacture the head and facial features, for example, started out by making a basic sketch of a face, and then labeled the mouth with the mean width of the mouth, the eyes with the color of the mode for eyes, and so on. Here’s Julia Bertolet’s blueprint for the head.

Then, following the suggestion of the curriculum, but apparently against common practice in most places using MathLand, we actually built our model. We were armed:

  • with blueprints, measuring tapes and invaluable partners, for quality control;
  • with brown grocery bags for skin, crumpled newspaper for insides, Sculpey for ears and nose, and miles of masking tape to hold it all together;
  • with paper fasteners for knee and elbow joints and a meter stick taped to the back of the chair to make this character a vertebrate, able to sit up proudly;
  • with the almost invariable blue jeans and t-shirt that fulfilled those modal mandates;
  • and with endless jokes. “Where did you put our torso now?” Etc.

Being mathematicians

All this took time, it’s true. Gobs of time, all of it worthwhile. As teacher, I could observe difficulties with measuring technique, awkwardness with calculators, challenges maintaining focus even with the physical reminder of the unfinished body part. I could identify unusual ability to ask the salient questions, or to solve construction problems, or unusual gracefulness in helping a partner stay on task. The kids could figure out what to expect from, and give to, each other. I could cheer on strengths, provide the necessary re-teaching or skill-building support, and encourage insight—and kids could do all that for each other—within an atmosphere of fun.

We were having fun. We were also thinking about questions central to so many math applications: questions about reliability of data; questions about precision; and questions about whether a calculated answer fits an intuited estimate, given the range of the data. We were doing what many adult users of mathematics do: using that language to explore the world.

And of course, we were united, and found truly memorable group satisfaction, in making life more interesting for the cleaning staff. Or anyone else who wandered by.

average 2010 goofy