A quick one about problem-solving in class

I think that I just noticed another piece of the problem-solving puzzle. I've just finished a really good unit on factoring with my ninth graders, and a lot of the frustrations that I've had with this group have been absent for the unit. I think I know why.

Before, I would have anywhere from a half to two-thirds of the class working hard on problems, and the remainder goofing off. They would either be stuck, not interested in doing any work on the day or just punching each other in the head. (I teach high school boys. They do this sort of thing.)

Anyway, that didn't happen with this unit. They self-organized into helpful little groups. They started saying mature things like "I need more practice on multiplying binomials" or "What's the next step after this?" They taught each other. They shared tips and techniques.

Why? Here's my quick analysis:

1) The unit was set up well. I taught them the concept of factoring when we were doing the distributive property in October. I taught them about multiplying exponents in November. They were all comfortable with the prerequisites to this unit.

That means that they could self-assess. They knew that if they got no x^2 term, that something was up. There was very little of that x * x = 2x sort of thing.

2) More importantly, I think, is that I was clearer than I had ever been about what they needed to learn. After the first day or so, I told them that there were four levels of sophistication that they needed to hit: Multiplying polynomials, Factoring the Diff of Two Squares, Factoring Trinomials, Factoring Stuff Completely. I told them that their job each day was to move themselves up a level. I started class with a "Reality Check" that helped them (and me) assess what they needed to work on. I had problems and activities available for every level. Students got to choose what to work on, so if a kid was feeling frustrated he knew exactly where to head back to.

The lesson for me is this: always share the map with the students. If they know where they're going, they'll feel more empowered to make sure that they get themselves there.

How I Plan: The Triage

Prompted by this, here's what I've figured out so far about planning. I'm a bit embarrassed to post this, because I've always figured that lesson planning is the sort of thing that people learn how to do well during ed school, and that my inability to plan effectively was because I'd never been schooled in the art of schooling. But I think I'm at the point where I'm comfortable enough with what I'm doing that it's worth sharing.


In short: The most important thing I currently do in my planning is reflect on what makes the lesson difficult, and then to figure out some way to react. I write up a reflection on the hard parts, and then anything else that I do is built on top of this.


What I used to do.


Planning was a mess during my first semester of teaching. I would sit down to "plan" and end up googling stuff for 3 hours. I would find something cool, and then try to figure out how it worked so that I could use it in the classroom. Then I would find a problem, and then start looking for another resource. I printed out stacks and stacks of files. I downloaded lots of Mr. Meyer's stuff.


This was the first stage of my lesson planning, when I didn't understand teaching well enough to prepare in advance. If teaching were basketball, I didn't understand the game well enough to conceptually isolate offense from defense, shooting from dribbling.


Towards my second semester I started understanding that a lot of learning involved finding something that students did know and then hitching a new idea to that old knowledge. This lead to a new stage of my planning, where I produced a lot of outlines and mini-scripts of questions. These were almost always scrawled on the back of scraps of paper half an hour before class.* These plans were still pretty awful, though.


*In my desk drawer at work I have a huge stack of these outlines and mini-scripts. They're completely disorganized. I can't quite bring myself to throw them out, but they're entirely useless to me at the moment.


To give you a taste of the awfulness, here's one of the rare mini-scripts that I actually saved on the computer.

There is lot of terribleness* on display here. To start, this is inefficient planning. I don't need to plan out every step of a lesson in this way.  Besides, it's artificial and false to plan out every step of a class in sequence. Truth be told, these sorts of outlines were really just to get me thinking in the right way before class -- I almost never used them during the session. At the same time, these were incredibly time consuming.


*Terriblitude? Teribadingness? 


So when last summer came around, the second thing on my todo list was figuring out a better way to plan lessons. 


The Triage
  
Here are my guiding principles for lesson planning:

1. The planning needs to cut the crap, and focus on the crucial bits. I have 4 preps, and zero patience for the sort of purposeless googling that my planning used to involve.
2. The planning needs to be something reusable. So no more scraps of paper.
3. The planning needs to anticipate the fact that someday I'll probably look back on it and hate it. I didn't want my planning to prioritize the creation of documents or slides that I'll likely hate as I learn more about teaching. I wanted my planning to be more robust than that.

In short, I want my planning to be efficient, reusable and distinct from the creation of classroom materials.

Here's an example of a lesson I put together this past Monday morning. It took me about 20 minutes for the lesson plan, and probably a half hour more to put together the problem set, and it was used later that afternoon: 


Quadratics Day 5 Lesson Plan
Quadratics Day 5 Problem Set


In case you don't feel like clicking through, my lesson planning basically involves a hierarchy of activities.


At the bottom of the hierarchy is what I need to do before every lesson to be prepared. And, at the moment, my thinking is that I need to reflect on what the hard parts of a lesson are going to be in order to be prepared. Some days this is reflecting on content, others it's reflecting on management issues that I'm having in class, and sometimes it's whatever. In my mind, this is the crucial core of a lesson.


If I've got that down, and I have more time, then the next most important thing for me to do is to reflect on good questions to ask in class. I also like thinking about the warm up questions that I'll ask in class, because this often gets me thinking about the bigger picture of the lesson. In other words, thinking about how I'm actually going to start the lesson helps me think about what I'm actually building this knowledge on top of.


What's great about this is that it's efficient. Reflecting on the hard parts of a lesson and some good questions to ask gets me pretty prepared to teach, even if it's a day when I can't put together an awesome task or a worksheet or a cool visual. In other words, most days. And if I have that time, then I can build on top of the planning, and the things that I produce are more focused and, um, good.


Because I'm recording things that happen before the lesson I'm pretty confident that this stuff will still be valuable to me. It will give me something to bounce off of, a place to pick up my thinking, even when I recognize that the thinking is no longer as on-target as I once thought it was.


This post has gone on long enough, and if you're really curious about how I plan you can click through those links above. But the important point for me is this: break down your teaching into little chunks. Then, find the little chunks that you need to think about the most before walking into the classroom. Make sure you do those every day. Then do everything else.

VFC: Quadrilaterals

Here is a Virtual Filing Cabinet for resources concerning Quadrilaterals.. This is part of an ongoing experiment in how to better share online teaching resources. If you like this post, then make your own post for a particular topic.

What am I missing here? Point me to your favorite quadrilaterals resource in the comments.


[Last Updated: 2/5/2012]

The Hard Parts


A lot of the traditional proofs of the properties of quadrilaterals depend very heavily on congruent triangles. One of the real challenges in teaching this topic is to change your students' perspectives. When they see a rhombus they should see four congruent triangles; when they see a parallelogram they should see two pairs of congruent triangles; when they see a kite they should see two pairs of congruent triangles arranged differently.


There are lots of challenges for novices that go along with this change in perspective. Students need to see triangles in quadrilaterals, even if the diagonals are absent. Students need to see parallel lines with a transversal even when the sides of the quadrilateral are not extended. 


Another major theme of this unit is the hierarchy of shapes. A square is a rectangle, but it's also a rhombus. They are all parallelograms, though, and so what's true of parallelograms is true of them as well. This, plus a whole slew of new vocabulary.


Quadrilateral Resources


For vocabulary, I like the approach of the Discovering Geometry series. Show kids a bunch of examples of things that are "trapezoids", show them a bunch of things that aren't, and then challenge them to formulate a definition that works. This is a pretty common approach, from what I can tell. Here's a post from misscalcul8 on her version of it.


Once you have the vocab down, you might want to make it more concrete and emphasize the relationships between these shapes. I've posted about an activity that I like where students create "family trees" for quadrilaterals. 


One of the big challenges of this unit is (to my mind) getting students to see quadrilaterals as composed of triangles, as this generates all of the non-obvious properties of the quadrilaterals. I like this activity, which uses a series of tangram challenges of increasing difficult. It literally forces students to compose various quadrilaterals out of smaller shapes, including triangles. This can also serve as a concrete model that can be returned to over the course of the unit.


I'm still looking for resources for the actual nitty gritty of this unit which is the properties of the various quadrilaterals. I'll post resources as I find them, and please let me know if you have resources to add to this page.

Quadrilateral Family Trees

There's not a whole lot to this idea, but it was a good opener for the quadrilaterals unit, and it was a good use of our whiteboards.

One of the things that I'm trying to be more sensitive to in Geometry is just how difficult vocabulary is for students. I feel as if a lot of teachers in my life taught vocabulary as if the hard part was keeping the connection between the concepts and their names straight. That's wrong, though. What's difficult about learning a new term is for the concept to make a clear and distinct impression upon the mind.*

*  "Clear and distinct impressions" is for all my ex-philosophy major brothers and sisters out there.

Anyway, learning vocabulary. It requires real conceptual clarity before we can even talk about these things, let alone prove things about them, and so it's worth the extra time to get those concepts clear.

And it's also important to get the interconnectedness of these shapes clear -- that's a major theme of this unit.

Anyway, start by introducing the concept of a family tree. Know your audience. My boys know about video games, so we drew three big circles containing the words "Sony," "Microsoft" and "Nintendo." They told me video game consoles made by each of those three companies. When we got "Playstation" on the board I asked them whether there were different types of Playstations. Yeah, there are. And all of those things are Playstation machines, and all of those are Sony consoles, etc.

Then they made these.

Nothing ground-breaking here, but it was a good activity.

VFC: Quadratic Expressions, Equations and Functions

Here is a Virtual Filing Cabinet for Quadratic Expressions, Equations and Functions. This is part of an ongoing experiment in how to better share online teaching resources. If you like this post, then make your own post for a particular topic.

What am I missing here? Point me to your favorite quadratics resource in the comments.

[Last Updated: 1/29/2012]

The Hard Parts
One of the hard parts about teaching quadratics is the complicated formulas that often appear. At its worst, a quadratics unit can get mired down in a lot of meaningless a, b, and c's. A lot of folks seem to approach this unit by asking themselves how they can avoid the formulas for as long as possible.

Your bigger sequencing decisions also matter here. If you have covered GCF factoring before touching quadratics, then you have a tool that can be used by your students for understanding things such as the x-intercepts of quadratics or the equation of the axis of symmetry. Will you use quadratics as an application of trinomial factoring, or as a motivation for trinomial factoring? Have you covered multiplying exponents yet?

Quadratic Expressions
There's a good way of deepening and making factoring problems more open over here.


Quadratic Equations
For a higher level math class, PCMI has a series of problems that drives at the connection between the area and perimeter of a rectangle and the solutions to a quadratic equation.

This puzzle requires students to solve lots and lots of quadratic equations by factoring.


Quadratic Functions
James Tanton is just phenomenal here. He starts with transformations, and has a great conceptual procedure for finding the vertex and the axis of symmetry from an equation.

The Exeter Academy problem set begin this topic on page 62. They've introduced factoring early, so that makes it easy to talk about where the x-intercept are for equations where c is zero. This is a nice set-up for a Tanton-style approach for finding the axis of symmetry and the vertex.

Here's a Malcolm Swan domino game for matching graphs with equations. It also has a set for finding the roots.

SBG kills motivation.

I've been testing out a new grading system this year in my classroom, and I'm really excited by it.  I used to pay kids $1 per percentage point on a test (e.g. 75% earns you 75 bucks) but I realized that this was encouraging kids to pay attention to the wrong information. I didn't want them to see that they only earned $60 and think, "Well, I bombed that test." I wanted them to dig into the feedback that I was giving them and figure out what their strengths and weaknesses were, what it would take for them to get better.


This year I've changed the way kids get paid in my class. I give them 35 skills, and they get paid $20 for each skill that they can show me that they've mastered. This has radically changed the way that my students think about grades. Instead of focusing on indistinguishable blobs of assessment they are able to get laser-focused feedback about what they need to work on. I can customize assessments for each student. It's great.


Enough coyness. Here's what I'm getting at: let's say you're you've read Dan Meyer's version of Standard Based Grading and you're about to adapt it to your classroom. You might think some version of this:

Mistaken Hypothesis #1: "I'm going to see way more motivated kids in the classroom. After all, their incentive is now to do things that lead to more learning because they're aligned with points."

You're not going to see more motivation, at least not in the ways that you want. Consider the slanty-word classroom at the top of the page. Would those kids be thinking about money or learning? Will some kids try to cheat the system for more money? Will kids check their bank accounts constantly? Don't kids see good grades as rewards?

Are points significantly different from payment?

Mistaken Hypotheses #2: "That's why you shouldn't use grades to motivate kids. Instead you should create an environment where students are intrinsically motivated to do the learning. I can adapt the grading system so that it's meaningful as a feedback system, and that will transform grades from rewards into feedback. Yeah, kids are going to be addicted to points, but I can break them by teaching them this new way of thinking about assessment."

If you think that this you can break your student's points addiction while still rewarding them with points, you should read Drive, by Dan Pink. The second chapter is called "Seven Reasons Carrots and Sticks (Often) Don't Work." Here are seven reasons, culled from psychological studies, why extrinsic motivators backfire:

1. They can extinguish intrinsic motivation.
2. They can diminish performance.
3. They can crush creativity.
4. They can crowd out good behavior.
5. They can encourage cheating, shortcuts, and unethical behavior.
6. They can become addictive.
7. They can foster short-term thinking.

Points aren't just a distraction -- they can actively undermine our students' intrinsic motivation. In "Drive," Pink reports on psychological research that identifies the most problematic types of rewards as "if-then" rewards. These sorts of rewards are exactly what gets proliferated in an SBG classroom. Should we be worried that SBG kills intrinsic motivation?

Hypothesis #3: "But SBG is a huge improvement on the old system." 

Maybe. Kids definitely like it more. I definitely like that it helps me guide student learning. But there are still tons of extrinsic motivators, and they're way more in-your-face than before. The extrinsic motivators are way better aligned than in the old system, but the proliferation of "if-then" rewards can be corrosive to intrinsic motivation. On the other hand, maybe the old system was killing intrinsic motivation as well.

SBG is probably still better, but points are bad.

Let's sum things up:

• Extrinsic rewards kill intrinsic motivation. Fact.
• In an SBG classroom that grades students there is a proliferation of "if-then" extrinsic rewards. This should make us really freaking worried.
• The old points-based system might be way worse.

Postscript:

Let's assume that Dan Pink's recommendations for businesses can be easily adapted to schools. Let's assume that kids : points  ::  grown-ups: money, and that you're at a school that uses grades. What should you do?

"People have to earn a living...The best use of money as a motivator is to pay people enough to take the issue of money off the table." 

So, if you're interested in motivating your kids give them a good grade at the start of the semester, tell them that you won't think about grades unless you're forced to, and then don't talk about grades ever again.

In 95% of classrooms this isn't feasible for a million different reasons, but, hey, here it is.

Struggling to get better at classroom management

I have to run off to prepare a class or two. This blog is usually quite silent on weekdays, and for a good reason -- I have four preps and no curricula to help me along. But I'm struggling with classroom management, like I have since I started teaching, and I want to write something that I think is true.

Almost everything that I've done better on purpose in classroom management is on the individual level. You're confused by my expectations? I'll make my expectations clearer. You use my punishments as a chance to give your "DON'T TASE ME BRO" routine? I'll stop giving you those opportunities.

There's a whole list here. Cold calling. Eye contact. Being sensitive to my movement around the room. Talking to students outside the classroom. Taking points off for bad behaviors, adding points for good ones. Calling students after school to hear what they're frustrated about.

None of these are helping for the situations that I'm in now. Here's what I think is happening:

The big, huge problem with classroom management that I see all the time is that I want the entire group of 15-25 kids to be doing something, but a kid doesn't want to do it. He* gets in the way of the other 14-24 kids doing something. I have to react, because I want to be able to help the rest of the group. The thing that I do (write name down, ask to wait out of the classroom for me, tell him privately he's out of line, make eye contact) doesn't work. I get frustrated.

* It's always a "he." I teach at an all-boys school.

There are two ways that I can get better at this. The first is to find better ways to respond when the student is out of line. I think that everyone agrees that this is small-ball.

The big-picture issue is, what can I do to stop this from happening?

Right now my answer is that I need to solve the individual vs. group problem. I am always going to have individuals who, some days, are not capable of putting in the work that they need to. Sometimes my kids forget to take their meds. Sometimes they just got pissed off by their 2nd period teacher. Sometimes it's just their 9th hour of school* and they didn't have enough sleep and they are having trouble focusing. And that's OK, but I need to build a classroom where those kids would be ashamed to derail the group.

* They have a lot of hours of school. From 8:30 to 6.

I'm having a lot of trouble with one class right now. We're supposed to have a quiz today, but that's canceled as of now. Today I'm going to focus on planning a positive day that 90% of the class buys and finds worthwhile. I need to rebuild my classroom so that the group's momentum is inevitable, and that no one is willing to get in the way of it.