Rational Expressions

How I Plan: The Triage

2012-02-10T07:41:00.000-08:00

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:

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.
The planning needs to be something reusable. So no more scraps of paper.
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

2012-02-05T18:32:00.000-08:00

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

2012-02-05T18:13:00.000-08:00

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

2012-01-29T17:34:00.000-08:00

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.

2012-01-22T13:37:00.000-08:00

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:

They can extinguish intrinsic motivation.
They can diminish performance.
They can crush creativity.
They can crowd out good behavior.
They can encourage cheating, shortcuts, and unethical behavior.
They can become addictive.
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

2011-12-21T05:31:00.000-08:00

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.

When you get down to it, who knows how long I'll be in this profession.

2011-12-11T09:15:00.000-08:00

Things that I like:

I like the idea that -- for however many hours a day I'm thinking about my job -- I'm thinking about how I can help out some 80-odd people.
I like thinking about the big-picture things. I like planning semesters and units in advance, and thinking about the major themes of my classes.
I like planning lessons that really nail what's hard for kids about a new idea, draw it out, give us a chance to talk about it, give the kids a chance to kick it in the face several times and then challenge them to move on to an even more powerful idea.
I like designing a classroom.
I like designing an assessment system.
I like that I'm getting better at getting people to think, just by asking a question. That stuff is like mind control.

Things that I don't like:

The work that I'm doing is really short-term. I plan a lesson, hours later I try out the idea, then I file it away for a year. I have to wait a long time to get a chance to improve on my first drafts.*
The kids don't seem to care about the things that I do. The kids, for the most part, don't really know what's good for learning, and their praise for teachers is often weird. His explanations make sense, and he'll take as many questions as you need is praise for a teacher who only lectures. He covers material really quickly means that he moves from topic to topic without going in depth. He makes you work hard means that you have to cram for his tests if you want a chance of passing. I'm not interested in that sort of praise.**
I don't like entering numbers into spreadsheets. It's boring.
I have a hard time finding things to feel a sense of accomplishment in. The school year doesn't culminate -- it fizzles out, and the last time I see these kids is when I'm collecting their finals. I never feel satisfied after a good lesson or a good unit, because holy cow here comes the next bunch of stuff to teach and I've got to think about that instead of my success.
The discriminant stinks. Degree/Minute/Seconds notation stinks. Regents exams stink. Sometimes I feel like I'm playing for the losing team.

* Part of this is because I teach so many different curricula. I teach Algebra 1, Geometry, Algebra 2 (2 sections) and Computer Programming.
** When I run a high school, kids are going to take a class on pedagogy and educational psychology during the first semester of high school. They'll know what research says about how people learn, they'll learn study habits, and they'll be able to judge their teachers according to the standards that we want to be judged by.

Killing my ego in the classroom

2011-11-20T09:36:00.000-08:00

I have some honor students who are really strong. Way stronger than I was in high school. Most of the time that's an awesome challenge for me as a teacher, but sometimes it forces me to confront my insecurities.

I've been doing math on my own for fun.* It's great. Highly recommended. I've been going through old PCMI problem sets and having tons of fun with them. Last Thursday I gave my Algebra 2 students a quiz. I projected a problem on the board that I had printed out for myself and told students that I was stuck on number 3, and asked for some help. As they finished up the quiz, some of the stronger students grouped up and started battling it out among themselves.

*Yes, I read comic books and like Community. Yes, I teach math and computer science. But I don't want to be a big nerd. No -- I want more. MORE. I want to be the biggest nerd. Ever.

Here's the problem, by the way. I hadn't had a chance to spend a ton of time on it when I showed it to the kids.

That night I got an email from one of those students with a solution. He was right-on. He made a fantastic observation that had eluded me when I was working on it.

My first reaction was pleasure in the collaborative relationship I had fostered with my student. Oh, wait, actually my first reaction was the exact opposite of that. Thoughts went through my mind, like, "He got it before I did!" and "Is my student smarter than me?" My old insecurities as a learner rushed back to the fore and I felt ugly and embarrassed. And in that moment I worried, "Should I be teaching these kids?"

I have colleagues who think that appearing smarter than students is an important part of their job. They've spoken about how it's important to appear knowledgeable to students, and they have plans about what to do when a student asks a question that they don't know.

And though I've given lip service to a different conception of the teacher/student relationship in the past, now I'm really facing up to it. Because my students really do know things that I don't. Not just about math, but about everything. My job is not to be smarter than my students. My job is to make my students smarter. And if you think that teaching is all about transferring knowledge, then I guess I should worry just a bit about what I can provide extremely bright students. But I'm finally really getting what it means to facilitate learning, and it's in my gut, running through me like adamantium.*

* HUMONGOUS NERD.

Teaching, as I understand it now, involves checking my ego at the door. Sure, I know some things. But only those things that will help my students grow.

Saturday night I got a message from that student about the problem he had solved. He was having trouble proving that his observation held. Outside of that interaction, yeah, I knew a proof. But as soon as we started talking about the problem I no longer knew the proof. Instead, I just knew a general strategy that is helpful for such situations and, hey, here's a link that you might find interesting, and let me know if you figure anything out. Five minutes later, I've got a proof in my inbox.*

*By the way, I've ripped off this asterisk/footnote thing from Joe Posnanski, who you should read if you like sports. You should also read him if you hate sports, because he's got a shot at changing your mind. Linkety link: http://joeposnanski.blogspot.com/

How I stumbled onto problem-solving

2011-11-10T22:29:00.000-08:00

This is a story of how I ended up with problem-solving at the (sometimes) center of my classroom. It happened, more or less, by accident, and now I'm trying to figure out what to do with it.

Here's the short version: I used to start by teachering* for about 10 minutes, then easing students into practice work with choices. That was going just OK. Now, though, I'm starting with exploratory problem sets and reacting with small-group instruction and explanations and activities that react to how folks are doing on the problems. That's going way better.

*Teachering is like plain old teaching, but with my teacher voice. You know teachering when you see it.

This past September I told kids that every day there would be a "Warm Up" assignment ready for them when they walked in. I encouraged them to work with partners, and I gave feedback on how well they were working during the opening assignment. At that stage, the "Warm Up" was for reinforcing or subtly extending old ideas, or taking a pass at prerequisites for the day's main lesson.*

*I'm teacher-uneducated, so my teaching education has really been scrapped together from experience, books and blogs. My slides originally were just ripping off Dan Meyer's. Now I find that my slides still look like his. It's Tahoma's fault.

Some kids started finishing the Warm Up early, and wanted to know what they were supposed to do. I started putting challenges and extensions into the opening problem set. The problem sets started getting longer and I started telling kids that I didn't expect them to finish everything. I started putting more interesting, weirder problems towards the end of the opening problem set.

*I remain pretty fond of those last few composition problems, especially the second to last one. I think that playfulness is an important quality in hooky problems.

Kids started complaining when I told them that we had to move on from the Warm Up. They wanted more time. So I started giving it to them. The Warm Up problem set started stretching out, and so I started putting more and more of the day's new content into there. It started becoming a nuisance to put it up on the projector -- I was running out of room on the slide -- and I still wasn't writing problems that were interesting to the quickest students.

I had a very, very clear problem in the classroom, and it was the classic one: kids all need different things. I didn't know how to handle it.*

Well, I knew about giving choices to students, but I didn't really know what choices to offer them other than something stupid and lame like "You can either do these 4 practice problems or these 4 practice problems." "Oh, really Mr. P? Really? Oh boy oh joy. I get to choose!" I shouldn't joke. It was fine -- it just wasn't working super well.

It was around this time that I started playing around with old PCMI problem sets. I started staying up really late to do math problems, and I started thinking about the difference between what I was providing students and what was keeping me up at night.

I thought about it over a weekend, and I came back wanting to rip off PCMI. So I tried to.

This has been going on for three weeks now, and here's my new cycle:

1. Students work on problems in small groups. Usually pairs.
2. I circulate, observe, and help.
3. I come back for whole group stuff. This is either a short explanation/demonstration (which is how I try to weasel out of the word "lecture"), an activity, a question, a game, whatever.

To contrast, here was my old cycle:
1. Warm Up
2. Relatively short whole-group instruction, with pair activities interspersed.
3. Activity, usually practice problems
4. Something else whole-group.

Take a guess: Which cycle is more efficient? Which cycle do students enjoy more? Which cycle gives me more face time with more students? Which forces students to learn how to justify themselves mathematically?*

*The fact that I can't be there to help all of them is a feature, not a flaw, in having problem solving sessions. They have to figure out the right answer, and I literally can't be there to help. I don't even have to be annoying about it.

I mean, there were a lot of problems with my old cycle. It's not like I thought that I had it all figured out. But I do feel like I understand a really important component of teaching that I didn't understand before. This whole process is more efficient and effective if my job is to react to the students, once we're in the classroom. It's the difference between me imposing information on uninterested bystanders, versus me lending a hand to people in the grip of a problem.

I still need to figure out how much support to offer, and I still need to figure out how to find and write really grabby, hooky problems. But, for now, I'm sold.

One last thought: I don't think that problem sets are right for every lesson, though I am leaning on them quite a bit at the moment. I do think that, fundamentally, it's better for me to be in reaction mode rather than performance mode. At the moment, problem sets are the way that I'm giving students a chance to encounter important or interesting things. I'd like to have a few more tools under my belt for getting students working hard on math other than problem sets.

Blog posts are supposed to end with a snappy line, but this one just peters out.

What makes a good problem?

2011-10-19T13:00:00.000-07:00

Let's start things off with a short taxonomy of problems:
1. Valuable problems are problems that worth doing. You'll learn something from engaging and eventually solving them.
2. Hooky problems are problems that keep you up at night. They aren't boring -- they're what you do when other things are boring. In short, they're fun.
3. Hard problems are difficult to solve.

There are clearly more types of problems out there, but this is what I need to start with.

As a teacher, I want to assign valuable problems to help my students learn. The problem is that a lot of valuable problems are hard, and people sometimes get frustrated with hard things. When people don't get frustrated by hard problems, it's often because the problems are hooky. So I have an interest in understanding what sort of things teenagers find hooky.

Anyway, I hope to continue thinking about this in a series of posts. Here's the first marker of a hooky problem that I have to share.

Here's a question that is, you know, it's just fine. It's not bad.

Here's one that I worked on for more time than I care to admit:

Here's one thing that makes the second problem hookier: it provides its own feedback. You know if you've found a rule, or if you haven't had a rule. You know how to check your answer. It's self-checking. That gives you the opportunity to try things and fail and retrench while sitting secluded in your room or your desk.

In contrast, the top problem doesn't offer much help. If you know how to find inverses and domains and range, then you'll feel confident. If not, you're sunk.

To be clear, whether a problem is self-checking is neither inherent to the problem nor absolute. If a high schooler has been taught to confirm domain and range with graphs on the graphing calculator, then the top problem might be self-checking in a way that approaches the bottom problem. Whether a problem offers its own feedback depends on the base of knowledge someone brings to the problem.

But the bottom problem is still hookier, and that's because the base of knowledge it requires is exceedingly low for anyone attempting the problem. All it requires is for the solver to be able to feel comfortable working with fractions.

Anyway, I need to continue this so that I can become a better problem writer. I hope to refine my thoughts on playful and deceptive problems so that I can try to understand how these relate to hooky problems.

Brilliant readers: what makes a problem hooky?

What I need to work on for Week 2

2011-09-09T10:45:00.000-07:00

Next week I need to focus on stimulating more of my students.

In almost all of my classes I have students asking, "Is the whole year going to be this easy?" I've collected work from students a couple times this week, and the students on the lower end are just where they need to be. That is, they don't feel lost but the work is challenging enough that they're still making (good) mistakes. Everyone else feels altogether too comfortable.

So the big thing that I want to focus on in my planning this week is making sure that there are proper challenges for all students.

My big strategy for stimulating more people this year is to switch activities at least twice a day. I envision a lesson as a warm up exercise, followed by three activities, and an exit ticket. Lecture/Classroom discussions are hard to differentiate, so it's falling on the other two activities of the day to challenge students.

I've done a think, pair, share exercise with all of my classes, and the first one went very well. But yesterday my questions didn't seem rich enough to be worth asking. That's probably a function of my limited capacity to come up with great questions, and limitations in the structure of the activity. If I'm going to get away with continuing with these things I need those questions to be more challenging while still remaining wide open for every member of the class. I need to think of extensions before hand.

This week I'm also going to be more careful about offering students choices. I did it yesterday in Algebra 1 and I thought it went well for independent work. What was really important was having some really good, deep questions on hand for students really looking for a challenge. I need to do a better job searching those out beforehand so that they're ready for students to practice.

So this week I'm going to firm up the structure of my lesson plans so that I can get better at differentiating within that structure. Once I've got that down, I can consider experimenting with the structure. The structure is going to be:

1. Warm Up
2. Lecture
3. Pair work
4. Practice with worked examples
5. Exit ticket

How I spent my summer vacation

2011-08-31T22:39:00.000-07:00

"Hey, how was your summer? Do anything exciting?"
"Ha, no. Just stayed at home. Relaxed, mostly."

Well, sort of. On the eve of a full day of faculty meetings, here's what I did this summer.

Thought through the sequences of my courses:
Last year I was just hoping to hit everything, and I didn't even manage to do that. This year, I have a plan. I've given some thought about the sequence in which ideas are introduced. I tried to separate related concepts and topics to allow for reinforcement. I tried to put functions and graphing before solving equations. I tried to integrate extrapolation and prediction into the more "algebraic" topics. What I came up with is saved in three SBG-style skills lists.

Algebra 1 Skills List
Geometry Skills List
Algebra 2 Skills List

These lists are only for the first semester, and they're a bit over-stuffed. Remind me to write a post contrasting two very different styles of standard-keeping in an SBG system. Anyhoo...

Put together a Computer Science class
My programming skills? At the level of a beginning undergrad. Is the plan fully formed? No, not quite. But I did my best to piece together a plan and sequence for an intro Computer Science class. As it stands, we'll spend a week playing with Scratch (mostly for sequences, conditionals and loops), then move to Greenfoot (objects, classes, methods, debugging) and then move into straight Java in BlueJ. I'd like to end the year with a month or two of PHP, but first I'd probably have to learn PHP. I thought of some interesting, helpful, non-programming assessments for Computer Science. It's enshrined in this VERY under-construction document.

Computer Science: Topics and Units

It's changing a lot, and daily. I still need a list of more specific skills and projects. Still, the hardest part of the designing stage is done. Now it's time to implement.

Rethought learning
Plainly stated, my job is to get kids to learn math. I wasn't very good at this last year, so it was time to rethink everything. How, exactly, do I expect kids to learn? Clearly classroom, homework, skills quizzes and final exams are involved, but how? What's the plan? Why should I expect kids to retain information?

As always, this is a work in progress, but here's what I came up with:

How learning happens in my classes.

Rethought the classroom
How do I use the classroom to support learning? This is something I didn't really understand last year, and it showed. I didn't understand classroom management or how to use the classroom to teach kids stuff.

It's time to bounce back. I read the excellent Discipline in the Secondary Classroom. I thought through what I care about, and what values are important. I came up with a discipline plan. I have a list of procedures to teach (on paper, sadly) and over the next few days I'll draft a plan for teaching them. I read about psychology of attention and pacing . I read about differentiation. I came up with a list of activities and tasks that actively engage learners. This year I'm going to have a list of activities and tasks, and I will switch between them every 10-15 minutes.

Changed the way I plan
Lesson planning was chaotic last year. I felt as if I couldn't plan in advance, because I didn't know what sort of plans would be helpful. It was time to think about what's crucial in lesson planning. What's the sort of thinking that I can't do on the spot? What's the sort of planning that will allow me to improvise well when I don't have time to put something awesome together?

The answer that I landed on was: Plan the units in advance, and then plan lessons with a focus on two questions - "What are the hard parts?" and "What are some good questions to ask?" Again, here's the documents to prove it:

Alg2 Functions Unit
Alg2 Functions with Tables

Relaxed
Well, yeah. I did relax. I took walks and runs, I visited family and read books. I blended. Bought a new bookshelf.

And I thought about teaching. I'm not dedicated to this profession -- at least not yet. I can't tell you if I'll be in this game in a year. To be honest, I have a hard time seeing myself enjoying this job year after year. But I'm trying to balance keeping my eyes open with a full-blown commitment to improving as a teacher.

Right now I have no students. In a week I'll have 80. Those students deserve a better teacher than last year's version of me. I've been working hard to give them that.

One last plank in the HW plan

2011-08-20T19:08:00.000-07:00

Oh, yeah. A problem that I always had last year was with assigning homework in advance. I was assigning work from the day's lesson, as more practice. The problem was that I had trouble assigning work in advance, since I never knew in advance how much or what I was going to cover. And I had to make print outs in advance. but what I'll do this year is stagger the homework so that it's more practice on things that we've already done in class.

I came across this (in retrospect, fairly straightforward) idea when reading this awesome piece from Henri Piciotto.

I like this because it gets at two of my problems at once: my organization and clarity issue, and my retention issue.

The 703rd Homework Plan Posted by a Math teacher

2011-08-16T12:53:00.000-07:00

Seriously, homework is a pain in the neck for everyone. But here's what I'm going to do this year, I think.

Why don't you just give me the summary at the beginning of the post, instead of the end?

Fine.

1. Accountability, moderately tied to grades, with open-notebook homework quizzes. You've seen these before from Sam and Kate.
2. Instead of handouts and worksheets, students get a link to a site with questions, answers and explanations. I happen to like RegentsPrep.org.
3. We don't spend a ton of class going over homework, but I do start the day with a Warm Up exercise with some problems that are pretty similar to homework.

Ok. Now, the details.

A. Students get a link to an online problem set: http://bit.ly/re2zfC. The problem set has solutions and explanations. Students are assigned some problems and are expected to answer the problems fully in a separate homework notebook.

B. I've got three students: Rachel, Jimmy, and John.

Rachel already knows how to do this stuff, and doesn't think that she needs the practice. She doesn't do the homework.

Jimmy doesn't know how to do this stuff, so he tries a problem. It's wrong -- the site tells him that -- and so he clicks around until he finds the right answer. Then he reads the explanation. He copies this into his notebook. Then he tries another, similar question. The hope is, now Jimmy might as well try it again, and he gets it right.

John doesn't know how to do this stuff, and John doesn't do the homework.

C. A couple times a week I spend 5 minutes at the end of class giving a HW quiz. The HW quiz can be on ANYTHING that has been assigned for homework since the beginning of the year. They're allowed to use their homework notebooks, and I tell them the date-assigned and number of the questions on the quiz.
Rachel does fine, since she knows the material. (Or she doesn't, and she realizes that maybe she needs more practice.)

Jimmy does fine, since he has his homework notebook to help. He sees the problem for another time, which is helpful for Jimmy.

John doesn't do well on the homework quiz. I try to figure out why, and then I try to help John.

D. I could even build in some meta-cognition into the homework quiz. Maybe, for each question, they're asked "Could I do this without my homework notebook?"

We'll see how this goes, but I'm more confident with this plan than with my non-plan from last year.

[WDIDWT] Short Student Conversations

2011-08-04T07:24:00.000-07:00

"Now, with your partner to your left..."

I found it confusing to just talk about "group work." Group work can mean a bunch of different things. For this edition of "When do I do (with) this?" I want to talk about an especially structured, baby-step flavor of collaborative learning. In this sort of instruction the teacher is not helping students learn by explaining something. Rather, students are having conversations with each other. They're usually assigned a single task, and are assigned a partner to work out the task with.

For example, one structure for this type of instruction is "Think-Pair-Share." Here's a description of how this works:

In think-pair-share, the instructor poses a challenging or open-ended question and gives students a half to one minute to think about the question. (This is important because it gives students a chance to start to formulate answers by retrieving information from long-term memory.) Students then pair with a collaborative group member or neighbor sitting nearby and discuss their ideas about the question for several minutes. (The instructor may wish to always have students pair with a non-collaborative group member to expose them to more learning styles.) The think-pair-share structure gives all students the opportunity to discuss their ideas. This is important because students start to construct their knowledge in these discussions and also to find out what they do and do not know. This active process is not normally available to them during traditional lectures. (Source: National Institute for Science Education)

Why would I do this?

A bunch of reasons.

First, because the brain needs processing time. Here's a quote from Teaching with the Brain in Mind:

Alcino Silva discovered that mice improved their learning with short training sessions punctuated by rest intervals. He says that the rest time allows the brain to recycle CREB, an acronym for a protein switch crucial to long-term memory formation...This asociation and consolidation process can only occur during down time, says Allan Hobson of Harvard University.

The same thing is pretty much summarized here.

Another reason for using this instructional strategy is that you can use it to get students to actively engage with a concept that they might have only previously had a surface acquaintance with. A good question is important here, such as "Show me an example" or "What's the difference between..."

It also provides students an opportunity to evaluate their understanding, since they're faced with a partner who has a different understanding than they do.

When do I do this?

Do this when students have just encountered a new idea, or you've lectured for 10-15 minutes, because your students need
(1) processing time
(2) a more active learning style
(3) an opportunity for self-assessment.

Here are some ways to vary this activity. I'll add more as I find them.
* Think-Pair-Share
* Mini-quiz
* Quickwrites

Nu?

This should be part of the every day routine and part of the planning process. It's also a good mini-step into group work. For me, the progression is something like:
Level 1: Short Student Conversations (Pairs)
Level 2: Longer Student Conversations (Pairs)
Level 3: Longer Student Conversations (Larger groups)
Level 4: Group problem-solving

My hope is to start off the year at Level 1, and only ascend when I feel totally in control of the previous levels.

[WDIDWT] When do I do with this? Lecture Edition

2011-08-01T21:23:00.000-07:00

[WDIDWT]

Folks, it's a joke. Just ignore it and move on to the post.

I'm hoping that this will be the first in a series. One of the things that I'm struggling with right now is that I don't have a good sense for when the various instructional strategies are appropriate.

When do I lecture?
When do I have students explain ideas to each other?
When do I assign a worksheet?
When do I have students solving an application problem?
When do I question with the entire class?
When do I have the class look for procedural mistakes?

Feel very, very free to leave your thoughts in the comments.

I'm going to start with lectures, since it's what I spend most of my time doing in the classroom. (Scorn! Scorn!)

First, though, I suppose we should define what it means to lecture. For various reasons (explained here), I think that it's helpful to take the word "lecture" to essentially mean "explaining." I take lecturing to be a specific flavor of explaining -- lecturing to me is explaining something to a larger audience. Clearly, sometimes explaining is a good way to help a person learn new information. After all, we spend much of our communication with others explaining things. The question is, how should it work in the classroom?

When should I lecture?

Here's a neat little summary of some papers that I found on a neat little website:

Johnson, et. al (1991) and Bonwell and Eison (1991) highlight several uses where lectures are appropriate:
To disseminate information to a large number of people in a short period of time.
To present concepts too difficult for students to process on their own.
To gather information from a variety of sources that may take the students a long time to gather.
To arose interest in the subject.
To teach auditory learners.
To present information unavailable to the public such as original research.

This list rings true with me. But even when lecture is appropriate, it needs to be used carefully.

"Research shows that students listening to a 50-60 minute lecture are unable psychologically or physiologically to concentrate on the content and retain it. One study found students could recall 70% of the content from the first 10 minutes of the lecture but only 20% from the last 10 minutes (Hartley and Davies, 1986)"

Lesson: don't lecture for long. Combine that with the unique capacity of verbal communication to inspire and captivate, and there's a strong case for starting each lesson with a short lecture that does some of the following things:
1. Motivates the content.
2. Provides necessary background information.
3. Takes a first pass at explaining a concept.

The trickiest thing is the third, though. A lecture should hit as many people as possible with a concept, but it's crucial that students not feel as if they understand the entire concept from the lecture. This is a subtle thing, but if you lead off with an explanation, that's going to feel like learning to the student, and make it harder to follow up with instructions that aims to deepen that knowledge. That's probably more something to be aware of for introducing those sorts of activities, rather than lecture, though. Lecture can't be the whole story, but there's no reason to force it to do less than it's capable of. That's just being silly.

Nu?

In summary:
1. Lectures are good at arousing interest in a subject. It's good to lead with that.
2. No reason not to try to hit as many people with the hard/essential/fundamental concept of the day with the lecture. Just make sure that you follow that up with an activity that takes a second pass at this concept, and make sure that you don't allow students to convince themselves that lecture gave them all that they need. (Try, asking a question whose answer is a straightforward, but unintuitive, consequence of the new concept).
3. Don't lecture for long without breaking it up with some sort of pause procedure.

Sources: (will hopefully update this as I discover more sources)
1. http://www.wcer.wisc.edu/archive/cl1/cl/doingcl/lecture.htm
2. http://blog.mrmeyer.com/?p=10652#comment-291577

How I want to spend my August

2011-08-01T13:35:00.000-07:00

It's only a month until school starts up again, and I want to be ruthlessly efficient in these next few weeks. So I need to spend some time clarifying what I need to do in order to get ready for school. So, here's my list of things that I need to get done.

(0. Various side projects and relaxing. I haven't forgotten these at all.)

1. Homework -- I need to figure out how it's going to be assigned and assessed. I didn't grade it at all last year, and students didn't do it, so I stopped bothering to assign it, essentially. I need to choose some sort of structure to at least start things off with. I'm leaning towards bi-weekly homework quizzes, and 4-8 problems assigned nightly. I've also been playing around with the idea of reading assignments.

2. Grading -- My big question is will I grade behavior, and (if I do) how will I grade behavior? Also, need to figure out where homework and classwork fit into here. Also also, I'm hoping to do skills quizzes and also summative tests, so I need to figure out how that will work. I also need to set up my gradebook before school starts, as there were aspects of my set up last year that made it hard for me to make changes. (Can Google please develop a free product aimed at schools so that I can just use GoogleDocs, puh-lease? I need something that integrates well with Excel, at the very least.)

3. Ice Cream -- Eat ice cream. Lots and lots of ice cream. This is going great.

4. Skills quizzes -- I need to finalize my skills lists, enter them into the gradebook for the kids, and figure out how I'm going to work on retention this year. Also, I like the idea of grading being dynamic both for better and for worse, but I'm not sure how to work that out. Maybe by adding the two scores? Maybe by making the first time they see the skill worth one point, and the second time worth 4? And when exactly am I going to give these skill quizzes -- weekly? When we finish a topic? (Last year I just did a skill quiz when we finished a topic, but that didn't allow for retention checks since we were already asking pretty time-consuming questions.)

5. Small-group work -- I want to implement some small group work into my classes in a more formal, regular way, but I'm not sure how or where to start. I'd like for it to be a regular, structured feature of learning in my classroom.

6. Blend -- So I bought a blender. Ideas on things to blend? Everything I've made so far has just been good, not great. I'm aiming for greatness, immortality, milkshakes, etc.

7. Good questions and important points -- I want to start compiling a list of good questions to ask students for my different units, as well as compiling a list of the hard parts that my students struggle with. I think that this is the best sort of prep that I can do, in terms of lesson planning. It'll make me more efficient over the course of the semester. This, rather than compiling activities, seems to be a good use of my time. It will also make it easier for me to lay bare the connections between different topics.

8. Binders -- Plan on how I'm going to use binders with students. I think that there should be dividers for things such as calculator instructions, a different divider for notes, a different divider for handouts, and a folder for their quizzes/tests. Then they should also have a list of the skills that they need to know. I'd like to make a model binder for myself.

9. Buy a stapler. I often need to staple things.

10. Don't blog when I'm hungry. This is really just a note for myself, but, man, I'm really hungry. I'm going to go blend something.

How the elephant got its lifespan.

2011-08-01T07:35:00.000-07:00

[Insert adorable elephant photo.]

This post is about a linear regression lesson. Nothing Earth-shattering, but something to add to the list of data to analyze. In short: here is some data that is pretty linear, and the relationship is graspable by students. But since the data is only pretty linear, it forces us to have a discussion about outliers, error-bars and correlation coefficient. So, better to use data that is only sort of linear (r = +/- 0.6 sounds right) for linear regression.

First, some facts:

* An opossum pregnancy lasts about 15 days, and their children live for about a year.
* Dogs spend about 60 days gestating, and they can expect to live for 10 years.
* Elephants spend, on average, 624 days in the womb.

When I consider these three facts, I find myself really bugged by one question – how long does the elephant live?

The first thing we need is more info about the relationship between gestation and birth.

After that, we need to figure out a way of representing this info in a clean and easy to read way. Then we want to see if we can figure out any pattern that would help us predict how long an animal will live, depending on gestation period.

Here we hit a problem: There does seem to be a general tendency for animals to live longer when they spend more time gestating. But there’s no perfect pattern here. Some animals seem to fit the pattern, while others don’t.

And this is what many of my students find confusing about statistics. They’re used to dealing with perfect patterns and absolute relationships, but what do we do with half-patterns and tendencies?

We make an educated guess, and make sure that we are clear about how much we’re guessing. We make a guess, in this case, by drawing a line that more-or-less passes through the data points.

So what about elephants? If this line correctly describes the pattern, then we would expect elephants to live for about 40 years, on average. As it turns out, the average elephant lifespan is 40 years. In this case, at least, the guess is right on.

Frank Noschese's TED Talk*

2011-07-18T21:19:00.000-07:00

* DISCLAIMER: I'm making this up.

[Dramatic TED music.]
[Applause.]

Frank:

Thank you everyone. We've heard some fascinating talks today from some very knowledgeable people. Not only are the ideas great, but the lectures have been funny, exciting and engaging. And they really need to be. Every speaker here wants everyone to walk out of the room having learned more about their research, their work, or their writing.

I wonder, though: is a lecture the best way to do this?

Full disclosure: I'm a high school physics teacher, and there are often debates between those who think that the future of education is taking short lectures and distributing freely and widely, and those who think that there is something better than lecturing.

So what I thought we'd do today is run a little bit of an experiment. Let's do a quick poll to start things off. Text your answer to this number, and we'll get the poll results immediately and put them up.

[The question is a counter-intuitive implication of a point made by one of the previous speakers. Like, maybe the previous speaker.]

OK, so we've got the results and here they are.

They're not bad. Most folks got the answer correct. A big group didn't. That's not entirely surprising. Research supports the limitations of lecturing, and the research is constantly confirmed by any classroom teacher.

Now, if you're an educator, you might say, well, what can you do? Most of the people listening to the talk got the point, and you'll get the rest through remediation and tutoring. Send them to Khan Academy or keep them after school.

The issue is, that's pretty inefficient. There are other issues as well (etc.)

It would be great if we could do better than this. So that's what I propose we spend the next 15 minutes doing. Let's try something, and you won't have to believe me that it's better than lecturing. You're going to see for yourselves that it is.

First, I need to tell you an important fact about something called the modeling method. At its heart is the modeling cycle, which begins with modeling development, and is followed by modeling deployment. Here's what those terms mean... Now, remember this, because this is important.

Now, let's try something all together different.

[Insert: inquiry-based, modeling learning here. It can be about anything that you like. Here's how you pull it off:
* Start with another text poll to figure out how many folks get the answer
* Recruit a bunch of educators to help you.
* Have the educators pass out materials.
* Educators go around to groups and ask Socratic questions to groups of people in the audience. ]

OK, now let's all come back together. Sir in the front, what did your group's experiment show, please come up and show us.

etc.

Now, let's end with two polls. Right before the experiment, I told you an important fact about how modeling instruction works. Let's see how many of you remember that fact, which I told you, and I told you was very important.

[Poll results.]

OK, I'd say just OK. Now, let's ask another question, based on what you learned through inquiry and experiment.

[Show results.]

Let's not do lecturing better -- ladies and gentlemen, let's do better things.

Algebra 2: "How do we predict the future?"

2011-07-13T08:18:00.000-07:00

I took Kate's Algebra 2 standards and I reorganized the first semester as an answer to the question, "How can we predict the future?"

Here's the results. Nothing earth-shattering, but feedback would be appreciated.

Here's the doc: Algebra 2 Rejiggered: Predicting the future.

A waste of time?

2011-07-13T07:23:00.000-07:00

I'm not a curriculum developer. It's insane to think that I can do this. But I'm devoting a lot of time to making sense of my curriculum, writing lessons over the summer, etc. What's the point? There are, out there, better lessons and course-wide structures than I could possibly produce. I'm devoting a lot of time, right now, to thinking through Algebra 2. Why bother?

Here are my reasons:
(1) Thinking through curricular issues will make me more sensitive to student needs. I'll know which questions to ask, which points to emphasize, which ones to let go more easily.
(2) I don't have access to all those resources, and it's not a given that my school can provide me with them. I am, still, on my own.
(3) My school year has about 130 math teaching days, so I'm forced to make curricular decisions. I can't rely on someone else's curriculum.
(4) My students take the Regents exam, so my challenge is right there: I need to make tough curricular decisions while still getting my students to pass NY's Algebra 2 Regents.

Still, it's exhausting and doesn't have much to do with the actual practice of helping kids learn these things. Sigh.

How to put modeling at the center of NY State's Alg2/Trig

2011-07-11T11:56:00.000-07:00

Q: What new skill should Algebra 2 students leave with?
A: The ability to model a system mathematically.

NY's A: Yeah, that. Also, how to solve an absolute value equation, how to employ Degree, Minute, Second notation to represent an angle, how to graph an inverse cosine curve...

And that, basically, is the challenge in reorienting the Alg2/Trig course around a single question or theme.

Still, I've been working on reorienting NY's Alg2 curriculum along these lines. This isn't exactly ground-breaking: Kirk Weiler's e-text, for instance, points at such an orientation. He starts by discussing functions, and then introduces different families of functions that end with a regression and modeling unit.

Here's what I would like to do differently in organizing the curriculum:
1. I want to bring the modeling and regression to the beginning of the function unit, to motivate our study of the function family.
2. I want to discuss a concrete example of a function, such as the familiar linear functions, before talking about functions in the abstract.
3. I want all the other stuff -- and boy oh boy is there a lot of other stuff -- to fit into the larger discussion about modeling.

1 and 2 are doable. 3 is hard. Still, there are some things that can be done to integrate the various skills of the course. For instance, much time is spent in Alg2 solving equations. By the end of the year, students should be able to solve absolute value equations, radical equations, quadratic equations, trig equations, exponential equations, log equations and rational equations.

These sort of skills, however, become necessary when you've mathematically modeled a system, found a representative function, and now wish to extrapolate. You're either going to be evaluating an expression, or solving an equation. If you think about the curriculum in this way, you have functions at the center of the curriculum, the functions are there for modeling, and a clear distinction between evaluating an expression and solving an equation will be constantly reinforced.

Ditto for inequalities.

So functions, modeling, and solving equations are taken care of. They fit into the larger framework. What's left over is all the stuff that has to do with manipulating expressions. For instance: simplifying radicals, simplifying complex exponents, simplifying complex fractions, exponent rules, etc. How do these things fit into the larger framework?

The best that I can do now is to say that these are upgrade packages, so to speak. The ability to manipulate expressions will allow us to have an easier time evaluating function expressions for a value, or expressing answers to function equations. So I think what I'm going to do is be explicit that these areas don't directly fit into our modeling narrative -- they're not used to describe or extrapolate based on data -- but they're excurses, upgrade packages that will allow us to model certain relationships more accurately.

In summary: 1) Bring statistics and regression to the foreground, to motivate the study of functions. 2) Put extrapolation at the center of function units. Extrapolation motivates both the evaluation of expressions and the solving of equations. 3) Explicitly bring out all the leftovers into upgrade packages, that will assist us in our next modeling exercise.

My next post will organize Alg2 standards into this framework. The post after that, hopefully, will reflect critically on this and think about what some of the problems of this will be.

Numbers for 9th graders

2011-07-04T15:09:00.000-07:00

I was tutoring a kid the other day. I'm introducing her to Algebra2, and we spend the hour talking about relationships between numbers. Once we've got a few of these relationships pinned down, I tell her that they're called functions, and then we talk about some other functions. She asks, "So Algebra2 is pretty much about functions?" Yep, that's right.

She pauses, and thinks. "It's weird. When I took Algebra the first time it was just all these random topics that we needed to know, and I knew them, but they were all different. I thought that Algebra 2 would be the same, but I guess I'm wrong, it's all about functions."

That's satisfying. Because she knows what the question is she knows what's important (general foundational stuff about functions, stuff that relates to the nature of these functions) and what, relatively, isn't. She knows what she's studying, she'll know how to integrate the new knowledge. We'll introduce each new function with a similar "big" question ("When will the missile fall?"; "Are we going to overpopulate Earth?"; "Why does Albany want us to study DMS notation?")

So, what's the question for Algebra 1? At first I thought Algebra 2 was the challenge, but now I'm having trouble constructing a meta-narrative for Algebra 1 and teasing out a question that introduces that narrative. Clearly a lot of the course is driving towards the concept of a function/2-variable equation. But at the beginning of the year we're still doing arithmetic, so how do I describe the endgame early on?

The best I'm doing right now is thinking about the question, "What counts as a number?" I'm imagining this as a mini-arc that develops, with care, the concept of what we're going to treat as a number in Algebra while also giving me a chance to brush up their arithmetic skills. I'd like the answer of this question to involve integers, fractions, properties of real numbers, square roots, expressions and variables. I could add a historical subplot to the story, revealing info about when this stuff was thought up ("People invent math? WTF?").

Here's what I have so far:

More on "big questions"

2011-06-28T05:32:00.000-07:00

A "big question" is supposed to do two things in a math class. First, it’s supposed to help students situate knowledge. Second, it’s supposed to make the content more meaningful to students. How does a question have this effect?

My (totally made up) analysis is that we’re trying to bootstrap our math content onto a question that students can quickly recognize as meaningful, and an approach to answering the question that students can quickly recognize as natural. We’re hoping that they care about the question (giving our, say, Alg2 content value) and that they’ll remember the natural approach to answering the question (so that they can associate our, say, Alg2 content with the approach).

A question such as “What simple functions are there?” is no help to students because (a) they’re not interested in the answer and (b) they don’t have any idea how to go about answering it. As a consequence, the question (a) is unable to make Alg2 more meaningful to students and (b) unable to provide students with a framework for their knowledge.

Even excellent metaphors or analogies won't necessarily make great "big questions." Take the idea that functions are analogous to relationships; just as we could catalog human relationships, we could catalog the numerical ones. In question form, that looks like, “What kinds of relationships can numbers have?”

But what makes for a great metaphor or analogy, in this case, doesn't lead to a great question. It fails at both the tasks that a "big question" is supposed to excel at: (a) I don’t think my kids will think that it’s worth answering and (b) I can’t think of a natural way to go about answering the question.

The next step for me is to pick anything--anything at all!--from the three math courses that I teach and try to get some practice finding big questions. Then I'll try to take on NY's Algebra 2.

The value of big questions

2011-06-26T16:11:00.001-07:00

What makes big, course-spanning questions so great is not that they motivate students with a tantalizing question. No question that I ask is going to be able to motivate students 5 months after I ask it. Any moment of curiosity will have passed. For motivation and engagement, we need daily questions, curiosities and (for +10 Meyer points!) perplexities.

I think that the big unit/course spanning questions are wonderful because they provide meaning to the curriculum. It's harder to ask the question "Why are we learning this?" when what we're learning is clearly situated in a larger, obviously meaningful framework. For instance, the question "Is there life on other planets?" naturally leads to the questions "What are the conditions for life?", "How hot are other stars?", "How far away are planets from stars?" and "What's in the atmospheres of alien planets?" Bam, there's your calculus-based intro Astronomy course. And while students in a different class might wonder, "What good are absorption lines?" my bet is that students in this class will (a) be more likely to situate them correctly as helpful in determining temperatures of distant stars or atmospheric content of exoplanets and (b) won't think that astronomy is useless and boring. So that's what we're going for here, I think.

And now, a problem. Let's partition the world of course-spanning questions into the purely mathematical and applied mathematical questions. Let's take an applied mathematical question such as "Can we predict the motion of a basketball?" or "How do electronics work?" or "Can we beat the stock market?" If we really and honestly pursue these questions, we're going to have to go beyond our mathematics, since we're going to need to use the tools of physics, or economics, or engineering. In other words, doggedly pursuing non-mathematical questions quickly leads us out of the mathematical domain.

On the other hand, rich mathematical questions don't typically do the work of being obviously meaningful to students. The best that I can think of is "What's a number?" which I imagine as a narrative arc spanning the first bits of a first year of Algebra.

This is a long-winded way of saying that I think we're either looking for mathematical questions that are big and basic enough to motivate this month-long investigation, or applied mathematical questions that are closed under honest inquiry.

Virtual Filing Cabinet: The Blog

2011-06-05T14:58:00.000-07:00

I have a lot of summer projects, as this is the end of my first year of teaching. I'm teaching (gulp) Algebra 1, Geometry and Algebra 2 again next year, in addition to a computer programming course. This is way more than I can handle with excellence, but I teach at a small school so I just have to suck it up. I didn't have much of a choice about this.

Anyway, I need to think through all my curricular assumptions, now that I've actually gone through this stuff once, and I need to work it through on paper. I'm going to do as much as that as I can on this blog. My goal is to do some thinking on a ton of things, and to collect as much from the internets as I can. In short, I'm hoping to make this blog a VFC for curricular stuff, with thoughts about the advantages and disadvantages of various resources.

This has the added benefit of potentially helping others, and making a helpful contribution to the online math world. Since my last idea on online sharing didn't really catch on (I still like it, though!) maybe this will prove helpful to some.

Principles of Online Sharing

2011-03-02T15:37:00.000-08:00

Thanks so much to Pat B for helping to publicize the Stack Exchange idea on his blog. Here's an elaboration of why I think Stack Exchange might be the right platform for online sharing.

It’s fairly uncontroversial that the math/science education community has created,
collectively, a dazzling array of resources and ideas, but that the online presence is sprawling and unorganized. It’s unfortunate that there isn’t a better way of discovering great resources.

My intention is to convince you that there’s a promising platform for resource sharing that we haven’t really explored yet, and that the community should give it a shot. That platform is a Stack Exchange Q&A site.

I attempted to create a short list of principles that ought to guide online sharing. Here’s what I came up with:

1. Anybody should be able to participate.
2. The good stuff should be easy to find.
3. Resources are only helpful if you know how to use them.

Anybody should be able to share. You shouldn’t need to be able to write an awesome
blog or maintain a twitter cohort to be able to get help with problems that you’re having. Also, if you have an awesome idea people should be able to get access to it, even if you’re not well-known as a great resource. The second principle is that there should be some sort of mechanism for elevating the most helpful material above the rest.

Finally, we shouldn’t just be in the business of dumping our resources (read: documents) onto the world. That’s not helpful. What is helpful is explaining the important pedagogical decisions that go into the resources that we’ve created, and perhaps providing a link to a worksheet or a set of slides. But sharing documents without their context just isn’t helpful sharing. Our online platform should encourage pedagogical thoughtfulness and educational problem solving, and discourage a plug-and-chug approach to teaching.

I think that a Q&A site is a pretty good approximation of my ideal sharing site. For those of you unfamiliar with the Stack Exchange platform, here’s how it fulfills my principles:

--Anybody can sign up to ask or answer questions. A typical question would be
something like a question that Kate asked a few weeks ago, (I have never had
success getting the cherubs to see the connection between the coordinates on
the unit circle and sine and cosine.”), but anybody could ask it.
--You are able to subscribe to various topics that you’re interested in via RSS. For
example, you might only follow questions on high school geometry, or only on
SBG implementation. When you see something that interests you, you chime in.
You can also search the questions for topics you’re interested in. For instance,
if I’m working on an Intro to Trig sequence, I might search the questions
for ‘trigonometry’ and see a discussion about the best way to introduce the
functions, with links to a few blog posts or worksheets that implement interesting
approaches.
--Anybody can vote on the best questions or responses, immediately sorting out
approaches or resources as quality ones.
--As more people vote for your questions or comments, you get awards or points or badges and other fun stuff. So there’s definitely a fun factor, as well as a
meritocracy built in.

I took the first step and created a pilot program for such a Q&A site. In order to launch we need a bunch of folks to sign up and participate. This is worth trying. Please give it a shot.

Here’s the link: http://area51.stackexchange.com/proposals/29616/teaching-and-tutoring

[Other concerns:
Q: Aren’t there other sites that do this?
A: Well, yeah, but they’re either defunct or about much more than math and science pedagogy, making them less fun to participate in.

Q: Is this really worth investing my time, since it might not work out and then we’ll have another half-useful resource just laying around the intertubes?
A: Consider this: if this Q&A platform doesn’t work out in the long run, we’ll still have organized at least some of the math resources out there, and this will be a first step towards some other attempt at organizing the sprawl.

Q: What’s something sorta like what you’re proposing?
A: http://math.stackexchange.com/

Q: What’s the process like for launching this thing?
A: http://area51.stackexchange.com/faq

]

Bringing it all back home -- a Stack Exchange Proposal

2011-02-21T11:11:00.000-08:00

Good job, math teachers! We have a thriving online community where people are regularly sharing content and solving problems together.

Problem: Good stuff is scattered all over the place. How is someone supposed to find all of the good content out there in a productive way?

Proposed solution #1: Curriki or Better Lesson.
Problem: They aren't very good. Lots and lots of fluff. Hard to find the good stuff.

Proposed solution #2: Virtual Filing Cabinets
Problem #1: Everyone has their own, and they're far from exhaustive. A well cultivated VFC can be very helpful. For instance, I turn to Sam Shah's all the time, but he doesn't have much on Algebra 1. So I go somewhere else for that. We're still not efficiently matching my questions with the answers that are already out there.
Problem #2: The cultivator is my middle man. If I'm going to find something good I have to rely on him to find it and tell me. That's inefficient. Better if everyone I know could tell me what's good and what's not, so that I don't have to wait on him to discover something good.

Proposed solution #3: Tons and tons of blog subscriptions.
Problem: On the one hand, that sorta works. But what if I have a problem that nobody I read has thought of, or nobody that I read has posted about? There's all this knowledge out there that I can't tap.

Proposed solution #4: So start a blog and get readers who will help you.
Problem: That's not easy. You earn readers by having a unique and interesting perspective, and not everybody has one. Plus, that's just a very, very inefficient way to get answers to questions from a community. And I'm likely only to have a few readers, when I want as many people as possible to help me with my educational problems so that I can improve.

Proposed solution #5: Teaching and Tutoring Stack Exchange

This is a Q&A site for teaching and tutoring. It's focus is on questions of how to get ideas into little human brains--not on how to get little human butts to sit in little wooden desks. This would effectively, and gradually, impose order on the huge spiraling galaxy of material that we have orbiting in the blogopher.

Problem: I have no influence in the teacher-blog-o-world.

So please consider my proposal and consider linking to this new Stack Exchange on your blog so that we can get some support to try this idea. For the good of mankind!

Peace out.

MBP

I like butts

2011-02-01T17:43:00.000-08:00

Wait, there's a unit of measurement called a butt?

And I want to give my students a review of dimensional analysis before introducing radians?

Oh, ok. Got it. Here's a link to what that looks like.

Slope

2011-01-30T20:17:00.000-08:00

I'm still not sure how to use this blog. Frankly, I would like lots of readers to read about my problems and have them be so gripping that a lot of learning--on both sides of the keyboard--is happening around this place. But my struggles are so routine and rookie-ish that I feel strange assuming that there's any insight to be worked out from them. But, you know, forget it, it's my blog and I'll be boring on it.

OK, here's how I'm introducing slope to my geometry class. I'm trying to get better at easing students into concepts, though this is a concept that my students have seen before.

Core idea: Slope measures how steep a line is.

1. Start with drawing two lines with a stick figure standing on them. Ask them, which line is the guy more likely to fall off of? (Draw a floor with a dotted line to give some orientation.)
2. How can we make this idea more precise? (Some students will remember the formula for lines, and just push them into concepts at this stage.)
3. Draw a unit forward, and ask, "how much higher up is the guy when he walks one foot forward?" for each line. Ask for guesses of numbers.
4. (Re-)Introduce slope as the ratio of your height up when you walk forward.
5. But how do we describe walking forward and going up? Introduce the (familiar?) formula.
6. Time for practice calculating slope from two points. Then, at the end of that set, draw a line given a point and the slope.
7. At this stage, students are able to draw lines given a point and the slope. Now they're off to practice given a bunch of problems containing parallel lines and perpendicular lines. They need about 15-20 minutes to work on these.
8. We come back together to discuss the relationship between parallel, perpendicular lines and slope.

Full contact math

2011-01-25T21:28:00.000-08:00

Did I mention that this is my first year and I'm straight out of college and don't really know anything about how to help people learn things? OK, just getting that out of the way.

There are better and worse ways to help people understand things. Here is some of what I've learned about that over the past few months.

1) Verbs are important. When you see 4 x's in the numerator and 2 x's in the denominator are you inclined to cancel, simplify or unmultiply the fraction? One of these verbs is noxious, the other annoying and one tells a student exactly what they should think about doing. This stuff matters. Man, if I have to tell another Algebra 2 student not to cross out a summand in the numerator even though I thought we cancel stuff when it's on the top and bottom of a fraction I swear to God that larynxes will be torn out of children's...happy place, happy place...OK, I'm in control. I'm just going to be careful when I talk to my freshmen, that's all. They're not going to hear the word "cancel" once. But they will hear me talk about unmultiplying fractions, and they'll know exactly what I want them to do.

2) Some procedures are better than others. So, how do you solve a rational inequality, or an absolute value inequality, or a quadratic inequality? You want to give them a procedure that will yield them the correct answer. But you also want them to understand why a procedure works. So you choose, as your procedure, to teach them to solve the inequality like an equation, plot those points on a number line, and then to test each region between those points to see if it satisfies the inequality. And, since you want them to understand all of this, you don't give them the procedure before you explain to them how it works. Right?

Blech. Students quickly forget your explanation, and just rely on the procedure, which is a brainless algorithm that doesn't put the mind in contact with the relevant concepts? But what else can you do?

Well, I've learned that I can design my own procedures, and that with subtle changes I can design them so that they force a student to come in contact with actual math. So instead of the above procedure for solving inequalities, I now teach students to split up the inequality into two functions, to graph both of them, solve the system of equations to find where the two curves intersect and then graphically intuit which regions satisfy the inequality. That's the same procedure as above, in case you're following, just rewritten from code into math for humans.

I suppose that, in sum, my basic moral from my first semester of teaching is: put students in contact with the right ideas early and often.

Mandelbrot Set for Algebra 2

2011-01-25T20:46:00.000-08:00

For my Algebra 2 final exam I wanted to give students a chance to experience putting ideas together to learn something new. I also didn't want them to freak out. So I gave them a short intro to the Mandelbrot set (that was basically ripped off from this article).

A bunch of students thanked me for it, which was nice. They thought it was cool, and not boring. I sent them the link to the Jonathan Coulton song after the final:

Good news: Lots of them were able to tell if a complex number is in the Mandelbrot set and used previous knowledge (function notation, composition of functions and multiplying complex numbers) to learn something new. It also made for a more interesting final.

I wonder how this would work as a full fleshed-out lesson next year. I'm not sure that the Mandelbrot Set is really helpful for leading students to create the math that they ought to be learning. Meaning, I'm not sure how much would be gained from showing them the graph of the set and seeing what questions interest them. Anyway, here's the doc:

Mandelbrot Set

Coming soon....

2011-01-11T23:20:00.001-08:00

Throngs of faithful readers: hi!

Anyway, coming up soon is the end of the semester for me (yeshiva high schools don't get the regular Christmas break) and I'll be posting a lot of my resources and thoughts on some of the things I learned about Algebra 2 this semester.

My procedure for giving procedures

2011-01-01T17:00:00.000-08:00

Step 1: If possible, don't.
Step 2: If necessary, decompose the problem into all its conceptual parts. Students come in contact with all aspects of the problem before being given the procedure that solves it.
Step 3: Design a procedure that yields the solution, but requires the student to come in contact with the conceptual parts from Step 2.
Step 4: Give them the procedure.

My rational inequalities lesson(s) worked well this way, and I think teaching students to factor trinomials using diamond problems fulfills Step 3, but in teaching it I skipped Step 2. I did an OK-not-great job of forcing them to confront where the terms in the trinomial come from in the multiplication of binomials. Now it's too late, I think, because they're comfortable factoring trinomials and aren't really in the mood to be retaught something they already know how to do. In other words, I think the opportunity for them to just absorb why the procedure works has passed.

Teaching isn't hard?

2010-12-21T18:39:00.001-08:00

Then how come there are so many bad teachers?

Fumble

2010-12-20T08:23:00.001-08:00

Student: Did you grade the tests over the weekend?
Me: Nope.
Student: Isn't it your job to grade them?
What I should've said: Sometimes I need a break over the weekend. Also, I was a soup kitchen yesterday and I came back exhausted. On Saturday night I was working on computer science for my independent study, and when I got back last night I worked on planning lessons.

What I said: What did you do this weekend?
Student: Hung out with friends?
Me: You have friends?
Other Student: Ha! He went there!

I've got a lot of patience to gain.

Factoring Resources

2010-12-19T19:57:00.000-08:00

If I have anything to add as a factoring resources, it's that my students get way more multiplying polynomials questions right if they multiply the polynomials in a repurposed Punnet Square. I don't feel at all guilty about this for two reasons:

1. My strongest students ignore the box since they can do the multiplication much more quickly without it.
2. Me weaker students LOVE it, since it keeps them from making sloppy mistakes.

I like it because it reinforces their geometric intuitions about area and makes a nice connection between algebra and geometry. The downside is that it doesn't really do a great job setting them up for factoring, but we'll see how that goes.

I'm not sure whether to teach them factoring by grouping or whether to just focus on getting them the ac/a+c intuition for trinomials where a=1. The advantage of teaching them grouping is that it'll make Alg2 much easier for them. And some teachers swear by the grouping.

Here's a rolling post on factoring resources:

CONTEXT
http://samjshah.com/2009/08/13/factoring-schmactoring/

DEFENSE OF FACTORING
http://mathmamawrites.blogspot.com/2010/12/my-math-alphabet-f-is-for-factoring.html

SOME TRICKS THAT MIGHT BE USED TO MOTIVATE IT
http://blog.mrmeyer.com/?p=8713#comments

USING GROUPING TO FACTOR TRINOMIALS
http://jd2718.wordpress.com/2007/09/28/trinomial-factoring-nice-site-and-last-detail/

SOME FACTORING GAMES/WORKSHEETS
http://algebra.mrmeyer.com/week26/handouts.zip
http://algebra.mrmeyer.com/week27/handouts.zip

E-TEXT WORKSHEETS
http://www.teacherweb.com/NY/Arlington/AlgebraProject/photo3.aspx

Aaaaaaaauuuuggggghhhh!!!!! (Or, my 3 opposing inclinations on how to teach exponential functions/logs)

2010-12-19T18:13:00.000-08:00

I'm going to begin by stating some standard-issue frustrations, some non-standard-issue frustrations, then I'm going to reflect on three ways I could teach exponential functions and logs.

First, the personal baggage: This is my first year teaching, I just got out of college, I've never taken a class in education, and I'm teaching at a yeshiva high school in NYC with 3 preps.

What this means is that I'm feverishly thinking about the entire three-year math sequence in New York. I'm really committed to teaching this stuff in a way that reveals its true value and combats my students grumpiness about math. But because I'm teaching at a yeshiva that means that I also have way less time to teach the same curriculum as everyone else does. The average public school has 180 days for teaching math--I have 124.

That means that I'm going to do a worse job in the classroom, and there's just nothing that I can do about that. What's lost in those 56 days is so much of the context and the meaning behind math. (Or, to put this another way, Sam Shah spends 13 days on exponential functions?!)

What does that mean for me? It means that I can't spend three days on exponential functions, including continuous growth. NY State gives me one day, more or less.

Which brings me to the decision: how do I teach it? I have three opposing inclinations.

1) Teach them science. Teach them population growth, or--even better--simple differential equations. Teach them dynamics! Get them to understand how these things are actually used in the world everyday by people--that is, by scientists.

PRO: It's true. They'll believe it and appreciate it.
CON: It'll take too long, both for me to prepare and also for them to understand. It isn't a standards-efficient activity.

2a) Teach them finance. True, you don't need to understand exponential functions to operate in the real world. But it'll help you understand credit cards and APR . I could start with exponential growth in general and then move into continuous growth.

PRO: The kids will like that it has to do with the "real world." "Oh, this stuff is actually useful!" Also, the amazing blogo-verse has already provided me with worksheets ready to go. In addition, it's a bit more standards-efficient then what I would cook up.
CON: When kids say "This stuff is actually useful" they're talking about today's lesson, not Algebra II. And they mean "Unlike everything else that we've learned." We have to be careful to distinguish "real-world" and "everyday-life." An example is "real-world" if people use it to understand the world. By this test, almost everything in Alg2/Trig passes. Almost. (I'm looking at you absolute value inequalities...) But very few of the "real-world" applications are "everyday-life" applications.

2b) Teach them this stuff: http://www.mathalicious.com/?cat=98. I forgot about mathalicious.

3) Teach them problems. Rules. Methods. Algorithms. This is what they're used to, but they find it boring and I find it SUPER boring.

PRO: I won't fall farther behind the pace.
CON: *sigh*

And this is my choice every time I sit down to figure out what to do in the classroom. It's a fight between science, the everyday, and, *sigh*.

Functions and Uncountable Sets

2010-12-07T17:25:00.000-08:00

So I've got no idea how to offer a scientific context for functions in general, which is what I'm working on now in my Algebra 2 class. So I gave myself a bit of a challenge. I spent today trying to get my class excited and confused about the idea that besides for the countable infinity there is a larger, uncountable infinity. Then I told them that once we learn functions they have everything that they need to understand the proof.

I might have dug myself into a hole here. My plan was that I could lay the groundwork for the proof as I introduce them to domain, range, one-to-one, onto and bijection. Then I figured at the end I'd devote a bunch of class time to trying to help them grasp Cantor's diagonalization proof. This is problematic, though, because I'm going to need to devote almost a full period to help them grasp the diagonalization argument, and the confusing parts aren't the Algebra 2 parts, and I'm already crunched for time with this curriculum.

But I'm a desperate guy. Almost all my students think that what we're learning is worthless. I need to do something!

UPDATE: This might help.

Thin Lens Equation/Rational Equations

2010-12-01T18:15:00.000-08:00

This lesson went well with my honors students this week, and I'll see how it goes with my lower track students next week. The idea is that the only legitimate context for much of the material in Algebra II is its scientific context, and I used this to give my students this context while practicing solving rational equations.

In my mind this is a step up from the "average rate" questions that dominate most of the applications of rational equations that I've seen. This is also easier for students (and this teacher) to understand and explain than resistor and circuit problems.

Here's my power point presentation, which I used to tease them ("Ever wonder why objects in your sideview mirror are closer than they appear?"), leading into the worksheet.

Thin Lens--Rational Equations

View more presentations from mpershan.

They split up into pairs and worked on the worksheet together.

Thin Lens Worksheet

Rational Equations in Physics, as I find them

2010-11-29T10:13:00.000-08:00

This is a post that I'll update as I find more material. The idea is to take one of these topics, give a scientific introduction and then present rational equations as the way of solving a scientific problem.

Circuits:

http://kalamitykat.com/2010/02/21/solving-rational-equations-project/
http://samjshah.files.wordpress.com/2010/02/rationalcircuits1.jpg
http://samjshah.files.wordpress.com/2010/02/rationalcircuits2.jpg

Lensmaker's Equation:

http://en.wikipedia.org/wiki/Lens_%28optics%29#Lensmaker.27s_equation

Additive velocities and Special Relativity:
http://en.wikipedia.org/wiki/Velocity-addition_formula#Special_theory_of_relativity

Algebra II troubles

2010-11-28T15:38:00.000-08:00

The “take everyday stuff and bring it into the classroom” shtick just doesn’t work for me when I’m preparing Algebra II lessons. And I think it’s because by the time we get to Algebra II we’ve reached a new point in the education of our students. We’ve exhausted the material that we think everybody out on the street ought to know, and we’ve started introducing specialized mathematics that not everyone needs to know. That is, our broader goal in Algebra II isn’t to provide people with the math they need to be average working folks, but rather to make more specialized education in the maths and sciences both attractive and feasible. That is, we teach it so that we attract kids to more math and science, and also so that it’s possible for kids to be prepared for more math and science.

As far as I can tell there are two reasons why we teach kids stuff:

(1) We think that they need to know it, even as a non-specialist. So comfort with percentages, ratios, rates, averages, comfort with numbers, abstract thinking--these are all skills that we want our students to have no matter what they do in life.
(2) We want to recruit and prepare students for a specialty. We, as a society, need mathematicians, physicists, doctors, engineers, accountants and all sorts of other professions that require more mathematical comfort than your average citizen, and therefore need more training. If we don't teach higher math, then our students won't be prepared for the training requisite for these jobs. Further, part of our job is to make working with math enticing enough that we're able to recruit workers into fields that require a good deal of number work. So our job is dual when we're in this mode: to prepare and recruit.

Basically, I think that Algebra I mostly falls into Category 1, and Geometry is half and half, but Algebra 2 is firmly in Category 2. So much of that curriculum is either preparatory for Calculus or of application only to scientists. Which is NOT a knock on it. But it just means that we can't use the same approach to teach Algebra 2 as we do Algebra 1.

So we need to think about the best way to do Algebra 2. I think where we end up is what so many teachers are already doing: integrating scientific material into our Algebra 2 courses. This is difficult for me, since I don’t have a great physics background beyond mechanics. But I think that this is the direction where I’m heading: our job in Algebra 2 is to make math, and its applications to science, seem attractive while simultaneously preparing students for their future.

Not sure exactly how this shows itself day-to-day, but I think we need to present material with scientific motivations. For instance, maybe the proper way to introduce complex numbers isn't as most general solutions to polynomial equations, but rather scientifically. Maybe we integrate complex numbers into our trigonometry so that we can ask "How can we model trigonometric fluctuations algebraically?" or something. (Truth is, I'm just learning about complex integration now, so I don't understand the applications of complex numbers at a depth greater than wikipedia browsing).

XKCD by Math Topic

2010-11-23T08:19:00.001-08:00

Math Topics

Trigonometry: http://xkcd.com/809/
Negative Numbers: http://xkcd.com/571/
Prime factoring: http://xkcd.com/5/ ; http://xkcd.com/247/
Irrational numbers: http://xkcd.com/10/ ; http://xkcd.com/487/
Poisson Distribution: http://xkcd.com/12/
Fractals: http://xkcd.com/17/ , http://xkcd.com/124/
Fourier Transformations: http://xkcd.com/26/
Self-reference: http://xkcd.com/33/ ; http://xkcd.com/688/
Four-color theorem: http://xkcd.com/52/
Binary: http://xkcd.com/74/ , http://xkcd.com/99/
Pythagorean Theorem and Shortest Distance: http://xkcd.com/85/
Sierpinski triangle: http://xkcd.com/95/ http://xkcd.com/543/
Riemann-Zeta Function: http://xkcd.com/113/
Derivative: http://xkcd.com/128/
Imaginary Numbers: http://xkcd.com/179/ ; http://xkcd.com/410/
Matrix transforms (rotation): http://xkcd.com/184/
Turing Machines: http://xkcd.com/205/
Right-hand rule: http://xkcd.com/199/
Recursion: http://xkcd.com/244/
Logic: http://xkcd.com/246/
Exponents: http://xkcd.com/271/
Complexity theory: http://xkcd.com/287/ ; http://xkcd.com/399/
Fibonacci Sequence: http://xkcd.com/289/ ; http://xkcd.com/587/
Linear equations: http://xkcd.com/314/
Coordinate plane: http://xkcd.com/388/
Ring integration: http://xkcd.com/415/
Purity of math: http://xkcd.com/435/
Young mathematicians: http://xkcd.com/447/
Incompleteness: http://xkcd.com/468/
Metric system: http://xkcd.com/526/
Statistics: http://xkcd.com/539/ ; http://xkcd.com/552/ ; http://xkcd.com/563/
Erdos Number: http://xkcd.com/599/
Linear relationships: http://xkcd.com/605/
Infinitude of primes: http://xkcd.com/622/
Derivative: http://xkcd.com/626/
Tautology: http://xkcd.com/703/
Principle of explosion: http://xkcd.com/704/ ; http://xkcd.com/816/
Collatz Conjecture: http://xkcd.com/710/
4D: http://xkcd.com/721/
Venn Diagram: http://xkcd.com/747/ (I missed another one earlier) http://xkcd.com/773/
Simplifying radicals: http://xkcd.com/759/

Girls sucking at math: http://xkcd.com/385/ ; http://xkcd.com/775/
Teaching math: http://xkcd.com/263/
Scantron: http://xkcd.com/499/
Homework: http://xkcd.com/519/

[(Normal approach): http://xkcd.com/55/]

Pendulum: http://xkcd.com/755/
Three-body problem: http://xkcd.com/613/
Period/Frequency: http://xkcd.com/594/
Periodic Table: http://xkcd.com/18/
Kepler: http://xkcd.com/21/
Schrodinger and Indeterminacy: http://xkcd.com/45/
Blackbody radiation, Planck's Law: http://xkcd.com/54/
Gravitational mass: http://xkcd.com/89/
Special Relativity: http://xkcd.com/103/ ; http://xkcd.com/265/ ; http://xkcd.com/675/
Centrifugal force: http://xkcd.com/123/
Motion Problems: http://xkcd.com/135/
Angular momentum: http://xkcd.com/162/
String Theory: http://xkcd.com/171/ ; http://xkcd.com/397/
Expanding Gas: http://xkcd.com/200/
The Hamiltonian: http://xkcd.com/230/
Gravity/motion problems: http://xkcd.com/226/
Resonance: http://xkcd.com/228/
Supercolliders: http://xkcd.com/253/
Electromagnetic Spectrum: http://xkcd.com/273/
Tesla coil: http://xkcd.com/298/
Satellite dish: http://xkcd.com/316/
Gyroscopes: http://xkcd.com/332/
Resistors: http://xkcd.com/356/
LHC: http://xkcd.com/401/ ; http://xkcd.com/474/ ; http://xkcd.com/564/ ; http://xkcd.com/637/ ; http://xkcd.com/702/
Phase shift: http://xkcd.com/368/
(Karnaugh map? http://xkcd.com/62/)
Quantum teleportation: http://xkcd.com/465/
Pluto not a planet: http://xkcd.com/473/
Scale: http://xkcd.com/482/
Induced current: http://xkcd.com/509/
Ohm: http://xkcd.com/643/
Electron Orbitals: http://xkcd.com/658/
Frictionless vacuums: http://xkcd.com/669/
Gravity wells: http://xkcd.com/681/
Dimensional analysis: http://xkcd.com/687/
Mars spirit rover: http://xkcd.com/695/
Circuits: http://xkcd.com/730/
Green flash: http://xkcd.com/766/
Exoplanets: http://xkcd.com/786/
Physicists: http://xkcd.com/793/
Lift: http://xkcd.com/803/

Biology--

Genetics: http://xkcd.com/419/
Biologists: http://xkcd.com/520/
Tagging animals/Migration: http://xkcd.com/585/

Science--

Scientific process: http://xkcd.com/683/

Virtual Filing Cabinet Virtual Filing Cabinet

2010-11-23T08:18:00.000-08:00

Hooray, meta! Anyway, here's a filing cabinet of useful resources on the web:

General lessons resources
http://www.mathalicious.com/

Virtual Filing Cabinets:
http://samjshah.com/worksheets-projects/

Entire Curricula:

Lesson Resources:
Math for America