Finally, a new blog post! What a couple of months it has been. I have been working on this blog post for the past two months, and getting stuck every time I’ve worked on it. This week, I’m reminding myself that this blog is a space for sharing my evolving thinking, not a refereed journal paper or a New Yorker article, and getting this out in public, even though it’s mostly questions.
As much as possible, I try to read scholarly literature and books for teachers on teaching mathematics, nearly all of which comes from a traditional school perspective. Not only is it useful for my teaching, but I find it fascinating to read from the perspective of a Montessorian, because I come in with a different set of givens about what’s normal in a school. The book I’m reading right now, Building thinking Classrooms in Mathematics, by Peter Liljedahl, is especially satisfying in this regard, because he directly tackles those norms.
Liljedahl’s basic premise is that learning doesn’t happen without thinking, but much of what we do in schools promotes non-thinking. This is a big claim, and Liljedahl really takes it in two parts: students spend a lot of time not thinking in school, and much of what is typical in schools promotes non-thinking. (You’ll note that I’m not really getting into the claim that learning doesn’t happen without thinking, and neither does the author. It’s hard to imagine this claim not being the case, although it hasn’t always been taken for granted. I am going to get into this in another post though.)
He gives a really interesting example of how classroom practices can encourage not thinking. He observed students in classrooms doing “now-you-try-one” tasks, where the teacher demonstrates a method for solving a math problem (in Montessori, we could call this the First Period), and then gives another similar problem for the students to try on their own, before going over how to solve the problem. The theory is that students will try to solve the problem, then use the review part to check their own understanding. In fact, in Liljedahl’s observations, only about 20% of students actually do this.
So what were the rest of the students doing? He gives a taxonomy of the different behaviors he observed: slacking (not even pretending to work), stalling (legitimate, but irrelevant, activities like sharpening pencils or going to the bathroom), faking (pretending to work, but actually flipping through the textbook, pretending to write, etc), mimicking (trying to do the work, but mostly doing it through line-by-line copying from the teacher’s example), and actually trying it on their own. More than half the students spent their time mimicking, instead of really trying.
This is an example for only one task, but it still brings up some really interesting questions to me, as a Montessorian. Is the range of studenting behaviors the same in a Montessori classroom? We certainly see all of these behaviors in a our environments! I’ve observed my students “slacking” (rolling around on the floor, disturbing others, chatting about nothing in particular), stalling (asking to go to the bathroom in the middle of a lesson, spending half an hour at snack, etc), faking (using the material as a prop), and frequently mimicking (following the exact steps I’ve demonstrated for a material, without ever thinking about what’s going on), as well as thinking, of course. Are there other non-thinking behaviors? Are some of them productive?
The remainder of the book is dedicated to explaining 14 key practices that Liljedahl and his team determined are crucial to promoting thinking in math classes. They identified them by taking practices that are so thoroughly institutionalized in schools that they are more or less universal (at least in North America), and frequently taken for granted. Then, they turned those practices on their heads. For example, since it is typical for students to sit in the classroom, they had students stand. Then, they observed what happened.
The irony of this is that, coming from the perspective of a Montessorian, some of these universal norms are irrelevant, and others seem to be missing. For example, the question of how we give grades is largely irrelevant to Montessorians (not entirely, as some Montessori schools are required to give grades, for one reason or another). On the other hand, Liljedahl takes for granted that everyone in a class with be doing math at the same time, and working on the same topic, if not precisely the same problem. From the perspective of a traditional school, this makes sense, as most teachers presumably don’t have the freedom to change this, and yet it creates a challenge for applying the 14 practices to a Montessori classroom, because this is not a reasonable assumption for Montessori, at least until you get to the middle/high school level.
So this brings me to my fascination with Liljedahl’s book: what can we take from it, as Montessorians, and what lines of questioning does it open up? I take for granted that the typical ways we teach math in Montessori classrooms are good, but they can be improved. I’ve seen all of these non-thinking behaviors in Montessori classrooms (including my own), and I’ve seen students come into middle school still struggling with basic number concepts. At the same time, I am adamant about not making changes to my own teaching practice if they conflict with the foundational principles of Montessori. We can change lessons, materials, writing surfaces, and so on, but only if they do a better job promoting the child’s self-construction.
So here are the 14 practices, and my Montessorian questions about them.
Practice 1: Give Good Problems
You can’t expect students to think if they don’t have something worthwhile to think about. In particular, they need problems that allow for problem-solving: that is, figuring out what to do when you don’t know what to do. Problems that are so simple they are rote don’t allow for problem-solving, but neither do problems that are so complex the child has to mimic or memorize the procedure for solving them.
I think we spend far more time thinking about the methods/materials than about the problems we present. This is how I was trained, and it has certainly been central to my practice. The lesson is about how to present the material, and the actual problems we present are an afterthought.
I’ve written multiple posts about this topic in other places, as have many other authors, so I’m going to leave this one here. However, if you have problems that have led to wonderful discussions in your classroom and want to share them, please send them to me!
Practice 2: Form work groups randomly, visibly, and frequently
Rather than having students to pick their work partners or assigning work partners based on our own social or academic agenda, Liljedahl’s research showed that one of the most powerful changes teachers could make was to assign groups randomly, and to make sure the students know the groupings are random. This randomization, especially when it is changed frequently, prevents the children from falling into particular roles (i.e. the “smart one”, who does all the thinking, or the “dead weight”, who waits for other children to do most of the work.)
This creates a problem for Montessori teachers, because we don’t really have the option of randomizing our groups. When all the children in a class are working on, say, finding equivalent fractions, it’s easy enough to group them randomly, but when one group is skip counting by two and another group is dividing fractions by fractions, you can’t very well randomize between these groups.
I have, however, been experimenting with randomizing within a lesson group for practice. Of course, this is a bit different during remote learning, where everyone has their own materials, or no one has any materials at all, but it has been huge success. Once I present the beginning of a lesson, I pull up random.org and share my screen with the children, then I randomize them into groups and put each group in its own breakout room. I jump between rooms to give them hints, additional problems, and so on. They love it. More importantly, they usually work hard, no matter which children are together, and when they don’t, it’s because emotional drama has arisen, and this is a valuable opportunity, too.
I will say that in my class, where children haven’t yet developed the conviction that I manipulate groups for my own purposes, the visible part of the random grouping is mostly irrelevant. After the first few times I shared my screen to show how I was making the random groups, they asked me not to share, and just put them in their random groups so they could be surprised when they got to their breakout rooms.
Some Questions for investigation:
Does randomization matter in a Montessori context? My instinct is that it is helpful. Children are still children, and I absolutely observe children in Montessori classrooms (and not just my own), falling into roles within a working group, some of which are not conducive to learning. I don’t think the children’s independent work collaborations should be randomized this way: independent follow up serves the same role as homework (or “check-your-understanding problems”) in a traditional classroom, and Liljedahl argues that how and whether students do this work is not really the teacher’s business.
While we can’t completely randomize math lessons, because these are strongly based on prerequisite skills, what would happen if we visibly randomized stories, science demonstrations, and so on in our classroom? I can already imagine what sort of chaos this could cause, as students end up in a lesson group where they haven’t had the prerequisite, or have already heard the story, or the like. On the other hand, perhaps this would force us guides to give up some unnecessary control. My trainer told me that it didn’t really matter who gets the presentations on science experiments or who hears the history stories. So long as these are alive in your classroom, the children will teach each other.
We could also try randomizing within a lesson group, for brief problem solving work. While this might not work well for lessons that are heavily dependent on materials, it could potentially work when we are asking children to solve problems that don’t rely so much on materials (something I think we should be doing more frequently). We could also randomize who gets a lesson when, if we have a large group of children who need the same lesson. For example, say I have six children who need a lesson on adding fractions. I can plan to give this lesson twice, maybe once in the morning and once in the afternoon. In a gathering, I can randomly draw names to assign children to one of the two lessons.
These questions really would be areas for some formal action research, preferably coordinated between multiple classrooms. I don’t know the answers, and I don’t know what might happen if we tried them. Fortunately, they are safe things to try. If we randomize some of our lesson groupings for two weeks and it’s a disaster, we can easily return to our typical system with no harm done, except maybe a little mild chaos.
Practice 3: Work at a vertical whiteboard
Liljedahl writes that in all his research on thinking classrooms, the single most effective intervention was to have groups of children work while standing at a whiteboard (and sharing a single marker) rather than sitting at tables and working in notebooks or even on a whiteboard sitting at the table. Actually, the type of surface doesn’t matter, as long as it’s erasable, so he calls them “vertical non-permanent surfaces” (VNPSs).
Reading about this gave me a lightbulb moment. Of course children work better standing at a whiteboard or a chalkboard, rather than sitting around a table. So do mathematicians. In my experience, the vast majority of meaningful, professional mathematics gets done while standing around a chalkboard. (Fun fact: there is a Japanese brand of chalk, called Hagoromo, that has something of a cult following among mathematicians. For a time, mathematicians were hoarding their chalk, as there were rumors that the company went out of business, although that has since turned out not to be true). My suspicion is that when most people picture mathematicians, they imagine some boring guy with glasses sitting at a desk computing sums. The glasses part is mostly true, but much of what mathematicians do is stand around whiteboards and debate.
Liljedahl believes VNPSs work so well because a) they don’t allow students to “hide”, b) they encourage tentative, non-linear, messy thinking (after all, you can erase ideas that don’t work), and c) they encourage groups of students to learn from other groups (what is apparently called “knowledge mobility” in the ed research biz).
So how might this impact a Montessori classroom?
The first, and most obvious question that comes up for me is: how does this work in combination with materials? I am curious to know what non-Montessori teachers who use a lot of manipulative have come up with. My suspicion is that they just use the VNPSs when that makes sense and the materials when that makes sense. While it may not be as good as standing, perhaps students could use portable whiteboards on easels that can be brought to a table and used alongside materials. I’m laughing right now as I picture a classroom with child-standing-height tables along walls, with whiteboards (or a window) right next to them. The children could work with their materials at the table and write on the whiteboard. Certainly, the next time I have a physical classroom (I teach online right now), I’m going to expand the vertical writing options I provide to the children and observe closely.
But getting back to Liljedahl’s three key features, how might these operate in a Montessori classroom? Certainly, Montessori classrooms have the same challenge that traditional classrooms have with work in groups: some children do most of the work, while others tune out, roll on the floor, wander off, or disrupt focus. Having children stand could solve the problem of children rolling on the floor and the like, but I do worry that in a Montessori classroom, where children have a great deal of freedom of movement, this might make it easier for those same children to just wander off.
What about the non-linear thinking? I will have an entire blog post about this soon, but, in short, I think we ought to be encouraging a lot more messy math in our classrooms. In my Montessori training, as well as my recollections from my childhood, there was always a great emphasis on tidiness in math lessons. While there is certainly a place for this (you really do want to line up your columns correctly when you are adding numbers!), so much of math requires making sense of complicated ideas, and that is a messy process. Trying to make sense of math, while simultaneously only writing the finished ideas is a recipe for constricted thinking.
And what about knowledge transfer? This is an interesting one. From what I can tell, the idea of a group of students looking around the room, seeing what another group is doing, and borrowing an idea from them is rather scandalous in traditional schools, and so this is a big shift for many teachers and students. (I have spent very little time in non-Montessori classrooms, so someone please correct me if I’m wrong. That Liljedahl felt the need to address this concern several times in his book suggests that I am correct.) Learning from other children is a deeply held value in most Montessori classrooms, and so this doesn’t feel strange at all.
So I really have two questions about knowledge mobility and VNPSs in Montessori:
Will vertical surfaces actually encourage knowledge mobility? And are they necessary to facilitate it? Liljedahl’s work is based on a context where all the children are starting out on the same problem, working on the same topic, and solving more or less the same questions throughout a time-bounded lesson period. That’s not at all true in a Montessori classroom, where most likely, only one group is working on a particular problem at any given time (in fact, the limited materials in the classroom practically guarantee this, and that’s done on purpose).
On the other hand, this fact has never seemed to be a barrier to knowledge mobility in Montessori classrooms with a healthy culture of freedom. Children frequently observe each other, ask each other about their work, and try to imitate each other’s activities. Moreover, older students frequently stop to help out, if they notice a younger child struggling with their work. So my suspicion is that, while children aren’t likely to borrow ideas from each other “in the moment” in the way they might in a classroom where everyone is working synchronously, they’re still likely to learn plenty from each other, and having work visible on a vertical surface is bound to make the work more obvious and interesting to other children.
From my reading of Liljedahl’s book, these are the three highest impact changes a teacher can make to encourage thinking in a (math) classroom, but there are still 11 others. I’m going to save those for my next post, so that this one doesn’t take hours and hours to read, but come back soon for more!