What I want to see more than anything in my classroom is focused, joyful work. Within limits, I don’t care all that much what the children choose to work on; I just care that the work on it with attention, effort, and enthusiasm. And yet, I find it nearly impossible not to meddle, and the longer the children focus on a particular project, the more I feel the urge to interrupt them, sometimes to try to guide them to something else and sometimes in a weirdly backwards effort to inspire them to keep working on the project longer.

I’ve suspected all year that I might be interfering at less than helpful moments, but in a full classroom with lots of children who really *do* need redirection fairly often, it’s easy not to notice. On the other hand, with only eight children, most of whom (being older) have fairly involved projects to work on, there really *isn’t* that much for me to do a lot of the time. Yes, there are lessons to give, but if I gave lessons with the sort of frequency that I’d give them in a class of 30, they’d do nothing but sit in lessons all day and get very, *very* annoyed with me.

So what the heck’s going on? Here’s a few things I’ve thought of.

- It’s really, really hard to accept that sometimes my job means sitting back and
*not*“teaching”. As soon as I realize the best thing to do is get out of the way and prepare materials, observe, update records, or even read a book, the next thought to cross my mind is “but what if someone sees me? They’ll think I’m not doing my job!” - It’s very hard to ignore the (perceived) societal/parental/administrative pressure to push the children to focus on certain things. I find myself thinking: “okay, you’ve been working on those origami polyhedra all day. It’s time to stop unless you’re going to write a report on platonic solids too. Besides, you haven’t done any writing, or reading, or math, at all today.” So what? They’re concentrating. Hard. And they’re developing spacial skills. And fine motor control. And curiosity about polyhedra, which just might evolve into that report on platonic solids, at least if it’s not too obvious that that’s what their guide is secretly hoping will happen. And anyway it’s a chance to sit down with some children that I rarely have a chance to talk with at any length and hear what they have to say. Verbal expression is a local curriculum requirement, too (not that it’s one that Montessori children have any trouble with!).
- Despite much of a lifetime spent in Montessori (and other progressive) schools, I’m still influenced by the disturbingly common belief that children who are enjoying themselves must not be learning, or working hard. Yes, the enjoyment gets out of hand and spills over into just plain distracted silliness now and then (and is that really the end of the world?), but most of the time, the giggling means the creative juices are flowing. I
*want*that to happen. And yet I interrupt it. - (This is the not-beating-myself-up reason.) There
*are*legitimate reasons to interrupt the children. They do need to be offered new lessons (even if not too many), and sometimes they do get lost in their projects and need help to refocus. And yes, society has given us (the children and me together) an obligation to make sure they learn certain things. As a first year teacher, I’m not always sure what’s what. Is now a good time to give this lesson? Should I insist they come? Is that giggling a sign of distraction or of focused work? Should I let the distraction go and see what happens, or should I head it off now before it gets really out of hand? Is the work they’re choosing right now a way of avoiding other things? Does it matter? Do we need to have a conversation about responsible choices? I don’t think there are any universal answers to these questions, though I continue to hope that someday I’ll develop an instinct for when to intervene and when to leave well enough alone. In the meantime, all I can do is fumble around trying things and seeing what happens.

Surely there’s some valuable life lesson to be learned from that. Someone else can find it. My urgent priority is a nap.

]]>From the perspective of a Montessori guide, this is my favourite bit:

Did he answer you the way you expected to be answered?Yes. He was very honest to me and very, very nice. I think all adults should be honest to kids with their answers and take them seriously. They’re living people, too. I especially hate when adults dumb it down for me.

Should you be treated like an adult?I’m not sure if that’s a good thing or a bad thing. I like being taken seriously, but I’m just not ready to be an adult. I don’t want to pay taxes. I never want to do that.

I could quote this interview further, but I’d just end up quoting the whole thing. Just read it. It’s amazing.

]]>I bring this up because I made some interesting observations today. It was pouring and I’m not feeling well, so we decided it was best to have indoor recess today. We let the kids get legos, k’nex, board games, plastic dinosaur models, etc. from the after school care room and just let them have at it. Yes, it was a bit loud and a bit of a mess (next time, we will work on cleaning up before going on to the next thing, just like the children are expected to do the rest of the day), but here’s the interesting part: I saw more concentration in that room during that recess than I’ve seen all year. Yes, they were building with legos and playing Monopoly, but they were doing it *intensely*. After my partner in crime took a few students off for a Going Out, I just couldn’t bring myself to put an end to recess. It was so much easier to observe and let them keep doing what they were doing than to try to push them all to do “real” work (and I just didn’t have the energy for the fight), and anyway, I was curious and we had French so I wasn’t planning to give any lessons anyway.

After about an hour of legos and whatnot, a few students started choosing other work. Two sat down and started writing, a few more started making Zentangles (and giving each other lessons on how to make them), some others practiced calligraphy, and a few more found some stencils in the after school care room and decided to use them to make felt pillows. I had a few kids who I knew would be pretty restless by mid-afternoon since they had no chance to run around, so I wrote a note to one of them that said something like “go into the hall, do ten jumping jacks then hop down the hall and back on one foot.” I spent the rest of the afternoon writing command cards for various children, which eventually involved things like “pick up all the k’nex on the floor using tongs” (they had a lot of fun with that one, though the floor didn’t get clean), and “do a chequerboard problem, stopping after each row and walking down the hall and back with a plastic dinosaur balanced on your head.” One command got out of hand (they did not come up with the interpretation I expected them to) and there were a few out-of-control children, but it didn’t seem any worse than normal (if a bit louder). It did take us half an hour to clean up (something we’re going to work on next time), but on the whole, it was a seriously fun afternoon.

So all this made me wonder: should my classroom be more like this more of the time? Dr. Montessori wrote about putting toys in her first Casa. She eventually took them out because she noticed the children were more interested in other materials in the environment, not because she thought playing with toys was a waste of time. I don’t think she ever did the same thing in an elementary classroom (she didn’t move on to elementary children until many of her ideas about education were already formed), but I have to wonder if I’m unfairly expecting the kids to gravitate towards the “real” work without ever giving them the opportunity to saturate themselves with “playing” (and that implies that drawing, building with legos, and playing Monopoly are a waste of time, which doesn’t seem likely). In my training, I was taught that the single most important measure of success in the classroom is that children are *concentrating*. By that measure, this was the most productive and learning-filled afternoon they’ve had all year.

I’m not sure I’ll want to or be able to continue running my classroom this way, but I’m going to remember this afternoon. If nothing else, I think it’s the first time I did what I was really taught to do. Since it was recess, I had no agenda beyond safety and a reasonable noise level, so I observed first and **then** decided where to give input.

I have a boy in my class who we’ll call Ryan. Ryan is six and has been obsessed with the bead squares since the beginning of the school year. He’s been building pyramids out of them, stacking them on top of each other in different ways to see what will balance, and sometimes just being silly with them. So I decided to show him something that they’re intended to be used for and I gave him a lesson on squaring. I showed him how to fold up the short 5 chain (5 connected wires, each wire containing five beads) to make a square, showed him the notation for squares, and let him work out how many beads were in the square. After we did one together, he decided to do **all** the bead chains. Here’s the result:

That was a full morning’s work for a very proud six year old. Next up: cubing.

]]>- The students follow their own interests. The children often announce that “according to my research…”
- Nearly every class activity is a Going Out. To be fair, she organizes many of the trips and often the whole class goes along, but usually around the children’s interest. And given their adventures, who can blame all the kids for wanting to go (well, except for Arnold)? (Of course, in the real world, most guides won’t send their children off to the inside of a volcano and then show up to rescue them in the nick of time, but we do this with our imaginations every day.)
- No homework.
- No one has ever seen a child in her class complete a worksheet or take a test.
- She has a wild and outrageous dress sense. (-:
- They keep a lizard in the class. (We had an iguana in my upper el class, so I buy this one.)
- She explains only what the children need to continue their learning. The children have to find their own answers.
- The children work on their own schedules, not all on the same thing at the same time.
- No grades.
- The children obviously *love* their work. None of them have lost the “spark”.

For more on the “spark” that Montessori encourages, watch Trevor Eissler’s new video.

Can you think of any more reasons?

]]>Here’s my theory: Some students struggle with economics because they do not fully understand the mathematical tools economists use. Profs do not know how their students were taught mathematics, what their students know, what their students don’t know – and have no idea how to help their students bridge those gaps.

The biggest difference is in the use of calculators. Older professors may never have had access to them, whereas younger students may have started using them at age 6 or 7.

I’ve written before about the problems with over-reliance on calculators, but I’ve never thought of the problem in quite these terms. As someone who grew up in the calculator age but with teachers who generally didn’t allow calculator use (or, in college, wrote problems that made calculators useless), I always thought of the issue as a matter of all-too-common bad teaching, but not as something that could lead to serious misunderstanding between professor and student.

The part I find most interesting is her claim that there is more than just an “arithmetic gap,” there is a “mental math gap” because all this calculator use has led to different ways of thinking about mathematics:

But the mental arithmetic gap has more subtle implications. Mental calculations often require intuition about, and comfort with, the use of fractions. Pre-calculator: 1/3+1/3=2/3. Calculator era: 0.3333….+0.3333….=0.6666…. Pre-calculator: “To multiply by twenty-five, divide by four and add two zeros (25*Y=1/4*100*Y)” Calculator: Multiply by twenty-five. Back in the day, fractions were easier than – or at least not much more difficult than – decimals. Calculators make fractions obsolete.

I’m particularly intrigued (though not surprised) by Dr. Woolley’s point about the tendency to gravitate towards decimals vs. towards fractions. It has been a long-standing frustration for me that my students *insist* on converting their final answers into messy decimals, even when the fractional answer they had was quite elegant. For many of my students, it seems that *fractions just aren’t meaningful answers*. I’d always assumed that this was just a strange cultural difference between high schools in the United States and most of the professional mathematics world: teachers and textbooks insist on answers in decimals, but most mathematicians find fractional answers more elegant. (Of course, most mathematicians rarely come up with answers that have enough actual numbers in them to convert to a decimal). It has never occurred to me before that this different might be the result of calculator use, but now that it has been pointed out to me, it makes perfect sense.

But (putting my Montessorian hat on), I think there is a deeper problem with this trend to prefer decimals over fractions: fractions are more concrete. This probably sounds crazy to those who struggle with math, most of whom find fractions especially challenging. I suspect this is because the *algorithms* for computing with decimals are indeed easier to learn than those for fractions, and with a calculator, they are completely trivial. There’s no need to remember when or how to find a common denominator, etc, etc.

The trouble is that it’s harder (though not that much harder) to fully grasp what those decimals mean. Yes, you can make a concrete “manipulative” representation of decimals, but since they operate on an exponential scale, in order to represent millionths, you’d have to represent units with something like a half meter cube, and even then your millionths cubes would only be half a centimeter on each side. That can give a nice visual impression, but it’s not so good for actually moving pieces around to solve a problem (especially if you also want to include whole number categories; your thousands would take up most of a room).

Moreover, once you get this concept (and it is a crucial one) that a tenth is ten times bigger than a hundredth and a hundred times bigger than a thousandth, I think that’s the end of the serious mental reasoning that you can do with these visual impressions. Hundredths, thousandths, etc. are just too small for any realistic mental estimation. If I ask you to picture 0.33333 can you do it without looking at the number and saying “oh, that’s about a third” and then thinking of one third? I can manage to picture three tenths and three hundredths and three thousands, etc, but this is an absolutely meaningless pile of blocks, not a useful “amount.”

On the other hand, fractions, which are trickier to manipulate on paper, can be beautifully represented. Take a circle, chop it into two equal pieces. Now you have halves. Take the same circle, chop it into three equal pieces: thirds. Now chop the circle into four equal pieces: fourths. With these pieces, we can not only get a visual impression of how big different fractions are and how they relate to each other; we can physically see why it is that adding 1/3 to 1/4 without finding a common denominator is meaningless, and we can see that 7/12 is the same size as 1/3 and 1/4 put together, even though the numbers are all different. We can use our hands to find equivalent fractions and then learn how to find common denominators. And most importantly, we can learn to estimate these amounts. Where 0.33333 converts to a meaningless pile of blocks in my head, when I think of one third, I see a third of a circle. (It’s also red and metal, but I think this may be an artifact of many years as a Montessorian!).

Perhaps this view is the result of being a Montessori child, since I was exposed to fractions for much longer and in a far more concrete form. If the order had been reversed and I’d spent years working with very, very concrete decimal materials, would I think better in decimals? I doubt it. Many of my students seem to think that the decimals are more concrete and easier to “think in,” but I wonder, is this because they can relate to those decimals as old friends, seeing where they fit in the continuum of numbers and estimating how big they are compared to other, more common numbers, or is it because the decimals “look right” but fractions look scary and strange?

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At any rate, last week I had a chance to go birdwatching with about 25 children, ages 6-9. The children had their own teacher with them, and we had a naturalist guide to help us find and identify the birds. As far as I could tell, the children had a wonderful time. They were certainly enthusiastic. However, the entire walk had a bit of a harried feel as the guide rushed us along, often dragging the children away from watching a bird they found interesting, only to drag the children back when she wanted to stop and watch something.

The more I watched this interaction, the more it became apparent to me that our guide was birding like an adult and that *this is not how children approach watching birds.* For many adult birders, a big part of bird watching is finding as many different species as possible, looking especially for rare birds and “life birds” (ones the watcher has never seen before). For these children though, the excitement lay in finding and *successfully identifying* familiar birds and watching what those birds are doing. Add to this a lack of basic birding skills, such as using binoculars and navigating a field guide, and I think the children missed out on a lot of potential fun during the trip, even though they had a great time.

Since then, I’ve been thinking about how to introduce children to the pleasures of bird watching. This is just my brainstormed list and many of the ideas are no-brainers. I’m sure there are experienced birders and experienced teachers who will have more ideas. Please share!

*Practice using binoculars* before going out in the field. This is definitely a skill, and not a trivial one for a six year old. Learning to find and focus on an object that isn’t flitting around among the trees is probably a good starting point.

*Put up a bird feeder near the classroom/house. *Learn the regular visitors before going out and trying to identify more birds in the field. At our feeder here, we regularly have cardinals, blue jays, grackles, cowbirds, blackbirds, house finches, nuthatches, chickadees, several kinds of woodpecker, titmice, and at least five kinds of sparrows. That’s a lot of birds to learn without even going outside, and a feeder gives lots of opportunity to really practice studying the birds closely.

*Learn how to use a field guide.* To some extent, I think this is a skill that has to develop over time. In my experience, to really use a guide successfully, I need to already have a fairly good idea of what I’m looking for. Still, if nothing else, knowing something about the organization and parts of a bird guide would be very helpful. I brought two guides with me on our trip, and they were in use for most of the time we were out. I noticed that the guides were very helpful when the children came and asked me for help looking for something. Either they wanted to see a particular bird, and I helped them find it, or we had a pretty good idea of what we’d seen, and I could direct them to the right section. But when I just gave the children one of the books, they didn’t need to know how to use them. One child told me she’d seen a swallow-tailed kite, which she almost certainly did not see here in Ohio. (I didn’t see the process, so I don’t know if she chose the kite because it looked similar to something she’d seen or if she chose it at random.) Something as simple as instruction on how to use read a range map would have been helpful.

*Choose a local bird guide. *I brought *Birds of Ohio* and* The Sibley Field Guide to Birds* (Eastern Region) with me. Both books were used a great deal, but it was quite apparent that the children had an easier time with the Ohio book, simply because it had fewer birds in it. The Sibley guide may be the bird bible, but it is difficult to use.

*Learn just a few birds at a time, learn how to recognize them and learn something about them. *I have long had a sort of disgusted fascination with turkey vultures because when I was 10 or so someone told me that they had evolved to eat carrion and so they can eat rotting food that would kill an adult human and they pee on their heads to keep clean. (I have since learned that the second part doesn’t seem to be true: they have bald heads to stay relatively clean after sticking their heads inside animal carcasses and they pee on their *legs* to increase evaporation and keep cool). Sometime later, someone told me how to tell a flying turkey vulture apart from a flying hawk. The result is that a) these birds are interesting to me because I know something about how they live and how they’re different from other birds and b) I am capable of recognizing them *on my own *(and c) turkey vultures are gratifyingly gross, a fact that counted for a lot more when I was ten)*.* This is something that seemed to be lacking on our trip. Our guide pointed out many types of birds to the children, but without added information on how to recognize them and how they live, the birds were just temporary curiosities.

*Go birding in a small group or have enough experts that each can work with a small group. *A lot of learning about birds happens on the fly from listening to a more experienced birder (at least, this is how I’ve learned). With one expert and 25 children, it’s impossible for the guide to know what all the children are seeing or for all the children to see the birds the guide points out.

*Conduct a bird survey.* Instead of going out and looking for as many species as possible, conduct a count of common species the children know well. On our trip, the children pointed out every robin, blue jay, cardinal, and chickadee they saw, because they knew how to recognize these birds and the birds are easy to find. (They especially like the chickadees, which, in this park, would eat birdseed out of your hand.) Alternatively, go out with the intent to look for a few particular new but likely birds. I suppose this could be frustrating if those birds don’t cooperate by showing up.

*Learn about bird families.* Knowing something about the different families of birds and how they behave seems to me to be a crucial start for learning about birds. It helps narrow down possibilities enough that a field guide can be used successfully, and besides, just recognizing what family a bird belongs to can be a major success when first learning about birds.

I’ve been thinking about this issue a lot lately, because I love birds and, although I’m a beginner when it comes to recognizing birds, I’d love to share this interest with children. Still, my goal is to help the children learn what they want to know and to explore independently, so teaching them about birds is no good if they are totally dependent on me to tell them what they’ve seen.

]]>So how’s the training? Let’s just say that Montessori training is *not* Montessori. There is so much to cover in so little time, that training basically consists of four to six hours of lectures every day, during which the trainees frantically write down/type what the trainer is saying. Homework consists of proofreading, formatting, and illustrating the lectures, which amount to several hundred pages of writing/editing every week. Since there’s so much to cover, there’s not a lot of time for questions, discussion, or reflection, though that should get better when the real training starts, since we’ll have two hours every day for supervised practice time, which should help us absorb everything we’re learning.

I was warned that the training wouldn’t be like a Montessori classroom this by many, many Montessori teachers before I started the training, but the reality has been shocking and, honestly, a bit distressing. During this long weekend (we’re off until Wednesday!), I’ve been thinking a lot about how to keep myself motivated while I slog through the year, since I know I have a very, very good reason for doing this. Today, I remembered something I came up with during my last meditation retreat, and I tried it.

During retreats, we spend some of our time practicing a technique called “metta” meditation. Metta is a Pali word that describes a feeling of kindness and generosity towards all beings. It’s often translated as “lovingkindness.” The idea of metta meditation is to strengthen your capacity for metta, and it’s usually done by repeating a series of four or so phrases silently to yourself. The phrases are wishes of well being, and traditionally, you start by doing metta for yourself, and then for a dear friend or a “benefactor” (someone who has taught or supported you). From there, you move on to a neutral person (someone you have no strong feelings about, like your mailman, say), and then an “enemy.” After that, you offer metta to larger and larger circles of people (and animals, and spirits, if that’s your thing), until eventually you offer metta to all beings everywhere.

At any rate, on my last retreat, the usual progression got rather messed up for me when I found myself offering metta for all children, everywhere. I’ve since realized that those phrases represent my deepest wish as a Montessori teacher, and they will be my mantra this year as I try to keep myself focused on my goal. They are my prayer for the future of our world.

*May all children be safe from harm*.

*May all children be safe from fear.
May all children be loved and cared for.
May all children be free to grow.
May all children live in peace.*

I’ll be blogging about the elementary (1st to 6th grade) training, but if you’re interested in other ages, my friend Emily will be blogging about the Primary (ages 3-6) training that she’s taking this year, and her friend Zoe is currently blogging about taking the Assistants to Infancy (ages 0-3) and Primary trainings.

Okay, back to trying to find places to put things, cleaning up the piles of cardboard strewn all over the living room, and endeavoring not to keel over in the heat. We Oregonians aren’t so good at humidity.

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