Math Stories

Like most math classrooms, I've got a number line. My kids use the hell out of it after we learn about negative numbers, some up to the end of the school year.

We've got a week and a half left, and the kids are restless. I didn't like the number line I bought this year, So I decided to have the kids help me out.

The templates were just a half sheet with the line and a tick mark across the top, along with a dashed rounded rectangle to provided them a guideline for where to draw.

I also printed up little slips with numbers for them to do, so I wouldn't end up with all 3s, 7s, and 13s. The range is from -20 to 20 not including 0¹.

It turns out that across all of my classes, I've got enough for just over 3 full sets. Count on a couple of absences, and I get a choice of 3 to chose from for each number.

Some of them were quite artistic, but because legibility is a prime requirement, I'm not sure all of these will make the final cut.

¹ I probably should have included it, but I think I want to keep the center printed, to emphasize the importance. (back)

SamJShah posted about factoring up, and his concerns that it relies on some magic that hides deeper mathematical understanding.

I share his concerns about that method. Fortunately, I've found a way to do it that builds on prior knowledge, rather than relying on legerdemain.

It goes back to teaching binomial multiplication - I rely mostly on the area model, though I allow students to use other methods , but make sure they have at least an understanding of it.

For those not familiar with the area model, it works like this:

This is the algebra tile version of (x+2)(2x+3). After a while, the kids get tired of drawing rectangles, and you introduce the following shorthand:

You let them learn this, and test them on it, and do some projects where they have to make use of it.

Then, when you're ready to introduce factoring, you give them a trinomial. Something like . If you've carefully left an example of one of the above binomial multiplication problems on the board by accident, and draw the blank area model boxes, they should pretty quickly figure out how to fill out half of the box:

The other half is easily filled in using the usual diamond puzzles. Except I don't use diamonds, I just use Xs. This worksheet is pretty good at developing the skills for the crosses (I've got a slide deck that steps through the first two Xs to explain what's up, after that the kids tend to take off on their own.)

Once the inside boxes are filled up, it's simple matter to work backwards to find what the two binomials are.

Aside from the trick of the area model, I find that the kids make pretty good connections with the lesson arc.

61¹ techniques. Not strategies. Techniques. Things that can be continually practiced and improved. Things that don't define what you teach, but by being good at them allow you to do it easier, and get the end result closer to what you want.

These 61 techniques are distributed across 9 chapters

• Setting high academic expectations
• Planning that ensures academic achievement
• Structuring and delivering your lessons
• Engaging students in your lessons
• Creating a strong classroom culture
• Setting and maintaining high behavioral expectations
• Building character and trust
• Improving your practice: additional techniques for creating a positive rhythm in the classroom
• Challenging students to think critically: Additional techniques for questioning and responding to students.

Each of these chapters has 5-10 techniques, and each of those techniques has 3 or 4 variations that give it some depth. For example, the much cited "No Opt Out" technique (I keep wanting to call these patterns) may range from simply having a stuck student repeat the answer given by the teacher, to a lengthier process of having other students provide cues.

As with the earlier pattern languages - these techniques exist as a whole. There is no obvious starting point, no linear progression. So reading the first chapter leads to a bit of a disconnect - it feels as if there is a bit of missing structure that hasn't been presented yet. I would strongly guard against digesting each piece individually - My current approach is to try to blast through everything to get a large overview, and then revisit the sections as I see fit. It is a credit to Lemov's writing that I keep getting sucked into the nuances of each technique.

It seems that, given the lack of a starting point, that Lemov simply started with the most important things first - everyone talks about high academic expectations, few people actually can describe what it means. His techniques provide a concrete outline for how that might look in the classroom.

The lesson planning section could be seen as just a reiteration backwards planning, but it's how he suggests the planning be done that makes a difference. The focus on what constitutes a reasonable objective, as opposed to standards derived afterthought used to justify the lesson is a fresh breath from every other backwards planning description I've read.

The structuring your lessons section focuses on how students learn, how their brains work, and how they respond to what you do as a teacher. Once again, the techniques are not presented in the abstract, but as concrete actions, complete with motivations and desired outcomes (which allow you as the teacher to judge whether you are implementing them adequately).

(I can't comment on the later sections, because I'm still working my way through the book)

There are additional resources at the end: a DVD of videos demonstrating the techniques in action (often times in a variety of fashions), and interviews with the teachers in those videos which allow you to see that those techniques are deliberate, and not just some happenstance.

I have two closing comments.

Firstly: It is very easy for teachers to see something that is somewhat familiar, and say "I do that already". I would caution against this - even if you do something well, you can revisit it, the motivations behind it, and how well you can apply it in exceptional cases. Throughout this book so far, even for those things I already did comfortably, I found new insights and food for thought. This is a goldmine, and the small nuggets in the cracks can do as much to benefit you as the big previously undiscovered veins.

Secondly: The power I see in this book is less the ideas, and more the common language used to discuss those ideas. Much as the software pattern community continued on development of the ideas started in the gang of four book, I see this as a potential starting point for what constitutes good teaching, to discuss amongst ourselves. In order for this to work, the language needs to make it into the mainstream in our profession. Please get a copy of this book (I get nothing for this) and start using the language with your colleagues. See where it goes.

¹ The title say 49. The last two chapters have a dozen bonus techniques for refining those 49. (back)

There's this trope, mostly appearing in martial arts fiction, of a learning process where tasks that appear menial and pointless are actual lessons that impose the skills or knowledge needed without the student being aware that they're learning anything.

My favorite of these is in The Matador series by Steve Perry¹. In it, neophytes try to follow an Arthur Murray style sequence of footsteps. Each successive step is not only more difficult than the previous, but it build and expands on the skills needed in all of the previous steps.

In my dreams, this is how I could teach math: A sequence of problems, carefully constructed, such that each problem provides a small measure of growth from the previous problem - so close that no instruction or direction is needed, but enough of an advancement that the student needs to make a cognitive leap to get there.

I realize this is a pipe dream, for several reasons. Students learn at different rates and with different styles, some need a lot more reinforcement of concepts than others do - there is no one ideal path.

Furthermore, a lot of what we teach in math is not the enlightenment of the concepts, but conveniences of notation. There is no real intuitive way to grasp those - we as teachers need to impart those conveniences onto their existing understanding.

Nevertheless, I believe there is great power in providing just the right problem to a student. Rather than having to convince and cajole a student into knowledge, a well designed question can shake up their assumptions, and provide the spark of inquisitiveness that so many teachers wish their students had.

Key to developing these questions is an understanding of how students develop theories and understandings, and how they can develop incomplete or flawed theories that nevertheless function adequately for a significant subset of the problems they face².

I'm hoping to keep this as an ongoing subthread in this blog, and I hope that it may actually someday develop into a fleshed out philosophy that is concrete enough to share and defend.

¹ I think I just outed how horrible my taste in recreational literature is. I have no apologies - crappy SciFI is a staple of the geek world. (back)

² cf Polya. (back)

Had a discussion with another math teacher here the other day. We were both in favor of increased observations in our classrooms, both by administrators, and by other teachers. We both think that having more feedback would allow us to focus our efforts at growth.

Where we differed was in how we'd like to have observers behave. I was in favor of straight observation - primarily empirical evidence, along with possible suggestions of roots or paths to follow.

The other teacher was in favor of spontaneous coteaching. I've certainly been in situations (especially early on) where having another teacher jump in provided strong learning opportunities, and It's not that I don't think I can't learn anything from anyone else.

It's just that my teaching style has drifted towards being less helpful. As long as a kid is working something through, I don't want to give them the answer. I want them to figure it out.

I've spent the past 5 years learning to be really patient - I can sit for minutes with an attentive observing look on my face, without giving away anything about whether what they're doing is right or wrong. I've also gotten pretty good at jumping in with a question whenever progress seems to slow down, anything from "What do you need to do next?" to "Why did you do these last three steps?" All of which serve to focus, without actually giving anything away.

I've had an administrator come into a room and try to help out the kids in this situation. It seemed to me that it was almost more a display of the administrator being able to solve a problem than it was of them being able to teach it. Sure, the answer got on the paper faster, but they'd done just about everything except tear the pencil out of the students hands and write the answer down themselves.

I had to let them be proud of their assistance, but then I had to work up a new problem for the kids to do, and let them go through exactly the same struggles they would have before.