The Urge to Interfere

For the past week, I’ve been on loan to part of our upper elementary class while the remainder of the class is on a trip. Even though I brought some of our oldest lower elementary children (the ones who will be moving into the older class next year), I’ve only had between 8 and 10 children each day, and that has given me a chance to notice some quirks of my own behaviour.

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.

  1. 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!”
  2. 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!).
  3. 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.
  4. (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.

Urgent Priorities

From the annals of useful observations for teachers: The things which seem absolutely urgent in one’s first decade of life and the things which seem absolutely urgent at the end of one’s third will generally not be the same.

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

If only we could all be as articulate as this girl

My better half just pointed me towards this interview with a nine-year-old girl named Mason Crumpacker. This young lady made news in the blogosphere when she attended the Texas Freethought meeting and asked Christopher Hitchens about his favourite books. (Why that simple act should be newsworthy, I don’t know, but let’s not quibble.) Anyway, eventually, she (with her parents along for the ride) sat down for an interview with the Dallas News. The transcript is unbelievable. She is calm, articulate, honest, and funny. She also happens to be a Montessorian (though from her description, it sounds like she didn’t think much of her time there).

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.

The puzzle of recess

My understanding of “orthodox” Montessori is that children should a) be working on things they are excited about and having fun with and b) ideally be able to take a stretch/run around/rest break whenever needed. This has always made me wonder why children need to have a designated recess time during which they are allowed to run and to play. I’ve never heard of a school that did away with formal recess time completely. Part of this is the result of the fact that most Montessori schools aren’t perfectly ideal environments built in such a way that the children can go outside whenever they wish. I’m sure part of it is due to legalities (we’re required by law to have organized recess time here), part of it is due to custom and parental expectation, and part of it is due to the fact that recess is often when teachers get their breaks. Even when children don’t get to go outside because of bad weather or something, it seems there is almost inevitably an “indoor recess” where the children are allowed to play games, be a little louder, and generally “not work.”

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.

Six-year-old mathematician discovers squaring

Once again, it has been a ridiculously long time since I last posted. Since I’m now two months into my first year as an actual Montessori guide, I’m not sure that’s going to change much in the near future. I’m just too tired. That said, some moments are too good not to share.

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.

Ten reasons Ms. Frizzle’s class is a Montessori classroom

My better half woke me up this morning by announcing that Ms. Frizzle, the teacher in The Magic School Bus series, is clearly a Montessori Guide.

  1. The students follow their own interests. The children often announce that “according to my research…”
  2. 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.)
  3. No homework.
  4. No one has ever seen a child in her class complete a worksheet or take a test.
  5. She has a wild and outrageous dress sense. (-:
  6. They keep a lizard in the class. (We had an iguana in my upper el class, so I buy this one.)
  7. She explains only what the children need to continue their learning. The children have to find their own answers.
  8. The children work on their own schedules, not all on the same thing at the same time.
  9. No grades.
  10. 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?

Calculator Abuse, Round 3 — Fractions vs. Decimals

Yesterday, a friend sent me this post from Frances Woolley, an economics professor at Carleton University in Ottawa, ON. Her basic point is that there is a “mathematics gap” between older professors and younger students:

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?