Rational Expressions

Visual Exponential Patterns

2013-05-10T19:30:00.002-04:00

The kids. They have no intuitions about the exponents. Especially when we get going backwards down the number line. And they just cannot walk out of my classroom only knowing how to graph and write equations for linear stuff.

Solution: ripping off Fawn Nguyen and Dan Meyer (Week 10).

You know the drill: find the missing things.

I actually liked my second version of this activity a bit more. I gave them a bunch of patterns and equations and had them match them up on the whiteboards.

This significantly lowered the barrier to entry, and I had some good conversations with some struggling Algebraists.

(A close eye will notice that in this second activity not all the patterns match up with equations, and not all the equations match up with the graphs. This was a good idea, because it was a nice twist, required some creativity in the pattern making, asked kids to review linear modeling, and took away some of the pigeonholing that often annoys me about matching activities.)

Here lie the files, in case you want 'em:

Extensions and spin-offs in the comments, if you please.

Becoming great at teaching

2013-05-08T22:11:00.002-04:00

I picked the wrong exercises

In January I wrote a post lamenting the plateau that many teachers encounter after their first few years. I ended that post with a commitment to avoid that plateau with intellectually taxing exercises, and I suggested three such exercises:

Daily journaling about the hard parts of the material that my students were learning that day. This drill would lead me to think more carefully about my lessons.
Blogging more often about failures, on the theory that there's more to learn from my failures than from my successes.
Great novelists read widely, and (on analogy) I committed to observing more teaching.

I tried each of these. I started planning my lessons by anticipating the hard parts of my lessons. I joined on to (the excellent) Productive Struggle blog and posted more regularly about my failures. I got myself in a bunch of classrooms.

None of these has worked particularly well for me. The exercises didn't feel like they were helping me much, and the more I thought about them the less sense that they made.

I've thought a lot about it, and I think I misfired because I failed to understand what makes great teachers different.

"What do you mean by great teacher?"

Yeah, very fair question. Let me put a few of my assumptions on the table:

Great teachers aren't necessarily influential, but influential teachers are usually great. Analogy: There might be some great, undiscovered novelist out there, but Faulkner is pretty freaking influential so it's worth taking his books seriously.

Therefore, it's legit to look at the careers of influential teachers when attempting to figure out what makes a teacher great.

Read that previous line again. I'm not saying that great teachers are famous and give big talks at things and write books and do PD and whatever. I'm just saying that the community of teachers find these people valuable, and a sensible way to try to figure out how to get good is to look at the careers of valuable teachers.

"All teachers are valuable." Yeah, I know. Yesterday was Teacher Appreciation Day, it was great. That's not what I'm talking about.

Anyway,

I started looking at people who do really valuable work in math education. I started by thinking about the teachers whose work has influenced me the most. I thought about the names with the biggest "buzz" in math and science teaching. I thought about the people with the most popular blogs and books. I tried to think about the things that I had done that had gotten me the most positive feedback, both from students and from other teachers.

This two-pronged hypothesis is where I landed:

If you want teachers and students to love your work, you've got to create amazing curricular materials and share them.
If you want the general public to love your work, you've got to express your ideas through the lens of technology.

Being the most thoughtful guy in the world about classroom management is great, but it's not what's most valuable to teachers and students. Assessment (and assessment reform) is really cool, but it's not what gets teachers and students really pumped up. Standards reform is kind of its own beast, but it's not the key to the heart of your teachery friends.

There's one big thing that matters to teachers, and it's having someone help them make their lessons better. Every other aspect of teaching matters less than that one. That's your core source of value as a teacher. If you want to be great, produce the sort of lessons that people will get excited about. (See how carefully I phrased that? Excitement about your work is the heuristic -- it's not the goal.)

But if you want people outside the profession to admire your work? For better or for worse, tech is the way to go. People eat that stuff up.

A better set of exercises

Being great means doing great things in the classroom, but my three exercises didn't really help me get better at creating interesting curriculum, which is what my students and peers really value. The exercises didn't work because they weren't sufficiently focused on what actually matters to my career.

I put too much value on the idea of blogging about failures. Now that I realize how important creating quality curriculum is, sharing my successes seems less about bragging and more about getting crucial feedback on the stuff that matters.
I got the analogy wrong. Great novelists read lots of books, and I thought great teachers need to consume lots of teaching. I was wrong. Since the primary value of teachers is their curricular work, great teachers need to consume lots of curricular materials. (More in a second on how to do that.)
Meditating on and anticipating the hard parts of a math topic is good, but it's focused on the content and not the lesson. This isn't necessarily a problem, but the drill just doesn't produce great lesson ideas. That's been my experience, at least.

That's the bad news. But, good news, everybody! Here's my updated list of exercises, and I feel a lot better about committing to these guys:

Creating great stuff is hard. These things take time and noodling around, and it's tough to create the good stuff when I'm planning for Tuesday on Monday evening. Instead, I need to ruthlessly devote much more of my planning time to the medium-term future while (temporarily) ignoring the short-term. This extends the time that I'm thinking about a unit, and makes it more likely that I'll come up with something good for the kids.
I had it all backwards -- I should be sharing the lessons that I'm excited about, not the duds. (Unless the duds are interesting.) By sharing my successes I'll have a better shot of getting positive feedback about my work, and other teachers will help show me when I'm on to something.
This one's my favorite. Every once in a while I come across a teaching idea that seems awesome, but also undoable, for all sorts of reasons. It's too crafty. I don't really do games. It requires too much cutting. I've never really used group work like that. My kids wouldn't appreciate it. I'm like a painter that's limited by my brush technique, and I need to push through and try other teacher's lessons in my classes, particularly when the lesson is unlike one that I would teach. (I'm looking at you, Fawn Nguyen!)

God, I hope that made sense. I feel a lot better about this than I did after my earlier post. Let's end with some quotes that seem sorta relevant but mostly I just like them.

"So much for endings. Beginnings are always more fun. True connoisseurs, however, are known to favor the stretch in between, since it's the hardest to do anything with". - Margaret Atwood, Happy Endings

"Every creator painfully experiences the chasm between his inner vision and its ultimate expression." - Isaac Bashevis Singer

As always, start some trouble in the comments.

Now that I think about it, I can't remember why I chose ducks.

2013-05-01T23:01:00.001-04:00

Here's a lesson that went better than it was supposed to.

Grab a whiteboard, and grab your partner. Draw Step 4 of this pattern. Then draw Steps 0, -1 and -2. If you finish that, find a rule for Step n. You finish that, graph the rule.

Here was my favorite:

Ah, who am I kidding, they were all my favorites:

Credit:

Frank Noschese, whose posts convinced me to get a bunch of white boards even though I had no idea how to use them. 2 years later: they're for producing things that can easily be shared, and sharing actual work is crucial for the sort of things I'm trying to pull off in class. Starting conversations is just way easier with shareable work.
Paul Salomon, whose image I blatantly ripped off and made 1000% worse by using ducks instead of circles.
Fawn Nguyen, for rocking my world with visualpatterns.org.

Thanks for making my "just OK" days a bit better, guys.

Speaking of which: any ideas for improvements, people? Drop a note in the comments with ideas.

You can't really reassess an individual skill

2013-04-24T22:09:00.002-04:00

Depending on how you define Standards Based Grading, it gets a lot of things right. You're more likely to get an accurate picture of what someone knows by assessing a skill more than once. What you know now matters more than what you didn't in October. Students need an accurate picture of what they're studying, and "Test #4" doesn't provide that.

Great. But here's something about SBG that's been bugging me for a while.

There's something wrong here, but what is it?

The kid showed that she knows all the triangley stuff, but dropped the ball on the square root side of things. She gets a 5/5 on Finding Sides of Right Triangles, but gets a 2/5 on Understanding Square Roots.
The kid got a question about right triangles wrong, so she gets a 3/5 on Finding Sides of Right Triangles.

Neither of these ideas is quite right. Knowing how to find the square root of 1 is not an all-or-nothing affair. Understanding isn't binary. Rather, understanding comes in degrees, and if a piece of knowledge is weakly understood then it's especially likely to falter when under pressure.

If you aren't super-comfortable working with right triangles, trying to solve a right triangle problem will be mentally taxing, and when you engage in mentally taxing behavior, you mess things up. But you don't mess up the things that are rock-solid. I doubt that I'll mess up single-digit addition when working on a Calculus problem. Rather, when you're using up mental resources it's the infirm and tentative knowledge that falls apart.

It's the sort of thing that we see all the time on mathmistakes.org.

This student said something silly, but it's artificial to attribute this to either his understanding of solving quadratic equations or his understanding of what the equation symbol means. It's both.

Would you ask this student to reassess on Doing Arithmetic with Negative Numbers or Finding Equations Given 2 Points? Neither? Both?

There's a larger point here. The idea that you can create a quality assessment that targets an individual skill is a myth. Take the slope question above. You could make the numbers easier so that the arithmetic probably wouldn't be a problem. For instance, you could use (0, 4) and (2, 10). But this is far too easy -- understanding means being able to apply a skill to a difficult context. So you toss in more difficult numbers, but then you're no longer purely assessing a kid's ability to find a line that passes through two points.

I don't know what this means for SBG or reassessing, and I hope that (in addition to challenging the premise of my post) we hash this out in the comments. Maybe this is an argument for fewer, but more substantive standards, like "Doing Stuff With Lines." I'm not sure, though.

Good Writing on Exponents

2013-04-22T22:14:00.000-04:00

There's been a lot of good writing about exponents recently, some of it in response to my most recent post. Here's a sampling:

Christopher Danielson wonders whether the problem with exponents is that we introduce it as repeated multiplication. He's thinking that "number of doublings" might be better, and he tests this theory on his kid. (The comments on the post are great also. Check them out.)
Chris Robinson modified the exponents survey for his own students, collected a ton of responses and offered his own analysis of what kids are doing with exponents. I still have to dig into his students' survey responses carefully, but I think that my favorite snippet so far is this:

Exponentiation sometimes defaults to multiplication, and multiplication sometimes defaults to addition.

... and some more survey results! Thanks Mrs Reilly! This effect is for realz, guys.
Andrew Stadel wraps it up with some extremely solid lesson plans that put these mistakes right in front of the kids noses. His theory? Part of the way to change kids intuitions about this stuff is to draw out and directly challenge their previous ideas. I stole his stuff for class, and it went quite well.

Exponents are cool.

Why Kids Mess Up Exponents

2013-04-09T22:34:00.002-04:00

1. Table of Contents

Table of Contents
Exponent Mistakes + Teacher Explanations
Let's. Try. Evidence.
Results
What's Going On Here?
Sequels

2. Exponent Mistakes + Teacher Explanations

Anyone who teaches exponents is familiar with a series of closely related mistakes:

Students like to treat exponentiation like multiplication. Why?

Thanks to mathmistakes.org, we have a nice collection of attempts to explain this phenomenon.

"Most kids just try to cram in the fact that negative exponents do ummm, something to the base. Without some conceptual hangar to place this fat they are left wildly guessing. "

"But really, as much as I try not to write these off to “autopilot”, I can’t think of any other explanation."

"Often, if you can get a student to slow down and be more present in a problem, they can avoid mistakes they would be prone to make otherwise. We need ways to differentiate true misunderstandings from these sorts of automatic pilot errors."

"I think that this idea is attractive because many students think that raising a number to a power is the same as multiplying the base by the power as in “4^2 = 8.″

We've got votes for autopilot, wildly guessing, and the idea that students are operating with an incorrect conceptual model for exponentiation.

How can we sort this out?

3. Let's. Try. Evidence!

This survey was given to students in a first-year Algebra class. They're studying exponents, but have never seen negative powers before, and they've certainly never seen non-integer powers.

What would you expect these kids to answer?

If you think that kids are wildly guessing, then they ought to report a relatively low level of confidence in their answers.
If you think that kids are mistaken about what exponentiation means, then they shouldn't get the first question right. After all, if they know what exponents mean with one problem, shouldn't they know what they mean just seconds later?

4. Results

This stuff is pretty fascinating. Here's everything, and here are some quick observations:

Overall, kids answered 50 to the third question, and had a good deal of confidence behind their answer.
Answers to the second question were more varied, but nobody just multiplied the base and the power together, like they did for 3a (or even 4a!).
Overall, kids had more confidence with rational exponents than with negative exponents.

(You might be wondering whether these observations are a fluke, which they might be, but they're at least a fluke twice. Here's a repeat of the experiment.)

5. What's Going On Here?

These kids are not guessing. Or they are, and they're lying, because they're telling you that they have confidence in their answers. So you can knock that theory out, it's not what's going on.

These kids are coming into your classroom with ideas about negative and rational exponents. So it's not about rules or memorizing or whatever, these kids have ideas about powers and are pretty confident about them.

These kids do sometimes treat exponentiation as multiplication, even when they're just dealing with plain old positive powers. To me, this supports the idea that when faced with a difficult exponentiation problem, sometimes the mind skips right to multiplication.

These kids are especially confident about rational exponents, where they seemed comfortable answering "50" to the third question. This could be because they've got a fuzzy story they're telling themselves about fractional powers, or it could be because of something more intuitive. Those are your options, and I'm not exactly sure what the difference is between them or how to test for this.

These kids don't just multiply the base and the exponent together when dealing with negative exponents. Instead, they tend to do treat the negative exponent as a positive one and then just tack on the negative to that result. (I suspect that this has something to do with the way we teach kids to do multiplication of negative numbers: do the multiplication, and just tag on the sign at the end.)

Your problem, as an Algebra teacher, is far more serious than autopilot. It's not just thoughtlessness that's responsible for these mistakes. It's a substantive intuition about what the answer to these things should be. It's such a strong intuition that it exists among students who have never even seen these concepts before, in a classroom or (presumably) otherwise.

6. Sequels

The survey file is here, but it's easy enough to make one of your own. I'd love to see what your students respond to math questions that they've never seen before. This all seems like a fruitful way to plumb the images that our kids bring into our classes.

I think the two live explanations for this sort of student work are that (a) kids have explicit, mistaken models about how exponents work and (b) there's a sort of intuition about what exponents should be, and this intuition operates below the level of consciousness.

I don't know how to tease those two possibilities apart, though all my experience in doing and teaching math leads me to think that it's all about intuition, and not really about explicit, mistaken models.

I was surprised that large numbers were sufficient to trigger multiplication of the base and the exponent in a lot of cases. I was also surprised that this didn't happen with negative powers. I'm not exactly sure what to make of that. I wonder what else triggers multiplication.

And does this sort of thing happen with other operations? Is this a general phenomenon or is something special about exponents?

NSF grant proposals in the comments, please.

Technology lets you say mean things

2013-04-03T22:35:00.002-04:00

This whole piece is really good. Here's the goodest bit:

This is why I get so apoplectic when people talk about MOOCs as disruptive technology. There is not a single thing this “New University of California” does that could not have been done technologically in 1898. Has online education suddenly improved to the point where people can gain never-before-seen levels of competency without attending classes? Hardly. Most MOOCs I’ve looked at are poorly designed even by late 90s standards, and besides, education’s killer technology — the book — has made independent learning possible for at least 500 years.

The real question to ask is why policy proposals like this — formerly the domain of fringe elements — are increasingly seen as innovation. What has changed? The answers to that are complex, and have little to do with technology. But understanding the reasons behind *that* is what is crucial to understanding where we are headed and why we are headed there. I think that “authorized to contract with qualified entities” clause is a piece of it. But the story goes much, much deeper than that….

Here's a story about a teacher. His name is Jerry. Some people don't like the way that Jerry teaches, but they don't want to say so. There's a lot of reasons why they don't want to criticize Jerry. The kids like him, and so do the parents. He's a devoted teacher. He's very not-awful, and there are lots of people that teach like him. And Jerry isn't so into change. He's seen the trends come and go. He's not so into learning the new edu-jargon that is research-based with pie charts and things.

And then some new technology comes out. Jerry's principal gets excited. The people from Teen Einstein (c) have all these awesome ideas about how you can get kids more involved, and they're talking about students taking control and being engaged and personalized whatever, and Jerry's principal is saying Yeah, that's how I'd like Jerry to teach.

And what's Jerry going to say? "No, I don't want to learn how to use that tech." Nah, Jerry's just got to admit that there's something to learn here, that the technology is new to him and so there's something worth looking at. So Jerry's principal is a big fan of technology. He's predisposed to calling it "revolutionary" or "a real game-changer."

Technology solves another problem for Jerry's principal -- how do you tell people that you're improving without admitting that you've got room to improve? You can't just walk behind a podium and tell everyone that you're ending things like hour-long lectures. That's not just change, that's an indictment of your teachers, your district, and everyone else's school experiences.

But technology is (by definition!) new and unanticipated. It's a chance to change without any of the responsibility of inviting change.

---

There are Jerry's in other areas of education, people or institutions that folks are too polite to tear to shreds, and I think that's what's going on with the MOOCs.

Of course (some) people can learn (some) things on their own. Of course, by the time you get to college in a lot of places there isn't a lot of difference between the classroom and learning on your own. But this is not an attractive argument to make because taking this up means indicting the college experiences of everyone, along with the quality of America's college teachers, along with the institutions themselves.

That's not very nice.

But there's this radically new technology that could really change the game. With the internet there has been a revolution in information distribution, and it's changing the way people learn. Something something youtube. Something something social. Something something personalization.

So, don't worry -- nobody's getting hurt. This change isn't about pedagogy, it's not about "You don't teach well" or "What does it mean to learn something anyway?" It's about technology.

Until we figure out a way to convince pretty much everyone that there's good teaching and there's bad teaching and that you can tell the difference between them just by watching carefully, we're going to need -- as a society -- technology to give us a chance to say what we're really thinking.

Is this a function?

2013-03-30T22:24:00.002-04:00

This is clearly a function. Functions are patterns.

But there's more. This is a function:

This isn't:

Functions are like the top graph. Non-functions are like the bottom graph.

What's the difference between the top graph and the bottom graph? One difference, for sure, is that the bottom graph has loops. But why do the loops matter? Can you tell a story that matches the top graph? Can you tell a story that matches the bottom graph?

It's harder to tell a realistic story about the bottom graph. Try making other graphs with the string. What if you make a "C"? A "U"? Are these more like the top graph or the bottom graph?

---

Here's another function:

Google Translate is a function.

But people's ears and minds are not functions. To us, the words "I love you, man" can mean an immense variety of things. A single utterance can mean "I love you" or "I don't love you" or "I think you're funny" or "Let's just stay friends," and it all depends on a million difference things like the tone of his voice, the tone of her voice, whether you're at a carnival with friends or on sitting alone on a couch.

Functions are automatic translators. Non-functions aren't always sure how to understand a sentence.

And, by the way, a confused boy, unsure how to interpret "I love you, man" is sort of like an impossible graph, no? He knows what time it is, but he just isn't sure how happy to be.

---

But now we're getting a bit melodramatic. Let's tone it down a bit.

With credit to Christopher Danielson, this is a function:

But this isn't:

Functions are reliable machines. Non-functions are unpredictable machines. You always put in the same number of tokens, but tons of different things could happen as a result.

---

Now, let t stand for time, and let h stand for happiness. Is t^2 = h a function? Is t = h^2?

(In the comments, Gregory Taylor rightly points out that the question is whether time is a function of happiness, or happiness is a function of time.)

---

And, after all that, is this a function?

If we tell our kids that functions are machines, then the only question a kid can ask himself is "Can you think of this as a machine?" But with a richer set of images to draw on, "Is this a function?" becomes connected to a series of more reliable and helpful questions:

"Is this more like a slot machine or a change machine?"
"Is this more like Google translate or a confused boy?"
"Would its graph look more like a loopless graph or a graph with loops?"

The point here isn't to be precise. When we want precision, we'll use the formal definition. The point is to provide students with a set of images around that formal definition that guides their thinking in helpful ways.

Let's try to find a richer set of images for both functions and non-functions. Let's also be more intentional about bridging the gap between linear, quadratic and exponential things and the sort of semi-arbitrary pairings that we want students to recognize as functions.

Postscript:

This post constitutes my final project for Christopher Danielson's really wonderful functions course. He's going to offer stuff like this in the future, and it will definitely be worth your while.

There's more to say, but I'll save it for another post or the comments.

Update (3/31/13): After some helpful criticism on twitter, I edited the post for quality.

Some conversations are more important than others

2013-03-18T23:57:00.002-04:00

Parent-Teacher conferences are dumb, right? I've got 5 minutes for each parent, 4.5 hours of conferences, a list of things to do tomorrow that's getting longer and longer as the night goes on. The kids start to bleed into each other. Everyone is doing well. Except the kids who aren't, but how do I say that to the parents in a way that will end up being productive for my relationship with the kid?

Conferences are dumb, but that's not really what this post is about. This post is about the kid that I just got off the phone with.

I met with his parents tonight, and I told them that he seemed unhappy with the way class was going. I said that I wanted to talk with him, and they seemed OK with a phone conversation so I asked for their home number. After conferences I gave him a call.

"Look, you don't seem completely satisfied with how class is going. So I was wondering if you had any feedback or anything."

He did.

He thought that class was kinda boring and awfully repetitive. He wasn't a fan of the worksheets that I give out on most days, and he thought that the Warm Up was becoming a distraction. He wanted more notes, because they keep kids from just being spoon-fed information. He wants the notes to be more step-by-step. He's willing to stop by during lunch to show me what he means. He's worried that we're not going to be prepared for the Regents.

"Can you think of a day that went really well in class, a day where you felt I did a lot of good things?"

He thought Trashketball was good and fun. He thought that a lot of days were good, but that it depended a lot on his mood and how into it he was. He likes when we're reviewing for a test.

More on worksheets: it's really easy to copy the work of others, and when I'm walking around it's hard for me to tell the difference, so it doesn't feel like you really have to do the work. He had an idea for the Warm Up -- what if everyone had a different question, so that you couldn't copy others and it felt like everybody owned their own work?

"OK, but I've got a question. Like, I'd get it if you said that you didn't think that the problems in class were helpful or worthwhile, and that's why you don't do them. But it sort of sounds like you're saying that they might be helpful, but you just aren't motivated to do them."

Well, when he's confused by something he tends to shut down. And can he move his seat to the front? That works better in his other classes.

There's a lot that's amazing about this kid. His willingness to talk with a teacher for a while on the phone during his free time. His candor and seriousness is something that most kids can't pull off, and it's clear that he's not satisfied with a class where he's just farting around.

(It's also remarkable that you can get this far into the conversation and still put all the responsibility for motivation at the teacher's feet. And I told him something like that.)

Almost every single one of his critiques resonates with me in some way, and I think this really points out that I've got a lot to learn about creating experiences that work for the 24 different people that occupy Room 312 between 2:05 and 2:48.

But there's a bigger point here, and it's that the way I do customer feedback is all wrong.

I do surveys, anonymous and otherwise. I ask for thoughts. I talk to kids during and after class. But all of these systems have the same flaws: (1) they treat every voice equally and (2) they're quick. What was amazing about my conversation tonight is that it was with someone who is just not clicking with my class, and I got a ton of time with him.

So back to conferences: they suck. But what would've been way better is if I could've had a half hour interview with 9 of the students who are having the worst time in my class. I'm pretty sure that I can identify them. I'm pretty sure that we could make some progress in a longer conversation, or at least hit a mutual understanding and figure out some way to move forward.

Here's my commitment: I'm going to draft a list of my 8 unhappiest students, and I'm going to call them and keep them on the phone for as long as I can. I'm going to figure out what my unhappiest students think about what's going on, and I'll try to just shut up and listen.

Slope and Sunset

2013-03-04T21:21:00.003-05:00

I wanted the kids to understand slope as a rate of change, and I also wanted them to use slope to understand something interesting. I also didn't want class to be boring. It went OK, but I still feel as if the payoff is a bit weak, and I'm hoping you all can help out a bit.

Last week we developed a metric for slope using a version of Fawn Nguyen's (version of Malcolm Swan's) slope activity. (Note: talking about sports statistics with these 11 boys helped the idea of defining a metric go down smooth.) I also showed them mountains and asked them to rank those, and we considered the various advantages of measuring steepness as "width divided by height."

Today we were going to study sunset at different times and places, and I just wanted something cool that would get kids ready to think about astronomy and stuff. I ended up with this barely related video:

It's great and beautiful and I showed it to the rest of my classes today too.*

* Most wanted to know what the green glow is, and I don't really have any idea how that stuff works. One kid had a pretty good explanation along the lines of "something something force shield." He also knew about solar storms. I'm getting off track here, but kids are tons of fun 95% of the time.

From there I asked them what they knew about sunset. They knew that it happens, that it gets earlier and later depending on the time of year.*

* Pro tip: When living in NYC, don't assume that the kids know anything about nature.

I asked them how much it changes per week. Their answers ranged from 1 minute to 7. I asked whether that rate was the same all year long. It took a few tries, but I finally got the question across to everyone, and there was a bit of disagreement.

Using the USNO site I made them a bunch of graphs, and asked them to find the slopes between the points. Here's what they got:

February NYC by mbpershan

We needed to remind these kids how to calculate slope, and they moved pretty slowly, so most of them only got through 3 of the graphs today. Some kids had trouble finding the height at first, reasoning that the highest height on the graph was the height we needed for slope. ("Look, I'm holding this paper 7 feet in the air. Does that mean the paper's got a length of 7 feet?")

I was fairly happy with the way class went, though I was worried by the fastest kid who made it through a bunch of the slopes and told me that he didn't see any patterns or interesting stuff emerging. And while the kids were doing better on slope and making progress on interpreting the numbers as a rate, there were warning signs as we tried to wrap things up. (Warning sides include, boredom, confusion about the questions I was asking, difficulty interpreting the units involved in the rates.)

So, the follow up is tomorrow in class. At the heart of this lesson is a really cool idea: that where you're living on Earth radically impacts the patterns of your life. How can I make this pop, while giving my kids good practice with their skills?

Caveats:

Yeah, I know that it's a bit false to ask for the "slope" when we've got non-linear patterns. But we talked about it, and we agreed to just search for representative points.
It seems to me that there aren't a lot of good problems hanging out there for kids who need a bit more practice with the connection between slope and rates. We've talked about speed, and they've looked at graphs. We'll do more of that. But I was searching for something a bit more, I dunno, worldly and interesting.
This lesson is a spin-off of an Exeter problem.

Oh, and one more thing:

My kids are having a really hard time solving systems of equation through substitution. I tried approaching this really slowly, but we're stumbling on the landing. Any ideas?

The difference between given and found problems

2013-02-24T19:52:00.000-05:00

There are all sorts of subtle ideas that lurk behind the veil of explicit communication. Every once in a while, it's worth taking stock of what those subtle ideas are to make sure that we're communicating wisely.

Every day, I give my students problems to solve. When I do that, I'm also telling them:

"This is a problem that you can make progress on."
"This is a problem that it is worth your time and effort."
"This problem is new and not the same as what you've studied before."

Because I teach my lessons in units, there's an additional implication present:

"This problem is connected to what we studied yesterday."

Are these things that I want to be communicating to my students? I don't know. (Those wiser than me will hopefully chime in on this.) But I do have a few observations to make:

All of these implications make the problem easier to solve. My assurance that the problem is solvable and that it's worthy of their time helps them take the leap into the problem with confidence. The implication that this question has something to do with what we have been studying significantly narrows the available tools and techniques to choose.
This is fundamentally different from the way problems are outside of the classroom. When I find a question that I want to make progress on, I have no assurance that it has a good or interesting solution or that I'll be able to do it. I have no idea what tools I'm going to need, and making those decisions is part of the difficulty of the problem.

What do we do about this? What can we do about this? We can try to present problems to students that are more akin to how they're found in the world, but the mere fact that we're offering them invests the problem with all of the implications detailed above. If we want to eliminate the suggestion that a problem is connected to the previous day's ideas we could eliminate units and integrate our topics more densely.

Other than that, to really help students solve problems as they're found out there, we need to create more opportunities for them to find problems on their own.

Not like I do that, but hey, aspirations, right?

Harvest Collegiate is a Pretty Awesome Place

2013-02-23T21:02:00.001-05:00

Stephen Lazar is on my must-read list. Last semester he taught a course called "Looking for an Argument" at the new Harvest Collegiate High School.

By his description, it sounds like it was an amazing course:

At Harvest, Looking for an Argument gives a government credit, and all 9th graders take it during the year. The structure is relatively simple. Each week focuses on a different controversial issue. Ours’ ranged from the NYC Soda Tax to the presidential election to Stop and Frisk. The week starts with two teachers debating the issue. The students choose who has which side, establishing that the class is not about being right, but rather about constructing the best possible argument.

And here's why it worked:

Part of the genius of the course is its simplicity. While provocative topics keep the students engaged week to week, students are practicing only four core academic skills — note-taking, reading with annotation, self-reflection, and timed argumentative writing — over and over again. While students do gain a tremendous amount of knowledge about the world through a variety of topics, that knowledge is never assessed; it’s all about the skills.

Harvest Collegiate does the "one-semester themed courses" in all subjects, including math and science. They have their course list posted online, and the three courses listed for math in the spring semester are:

Algebra
Global Youth Trends
Designing Harvest City

In the course descriptions you get the sense that they're struggling to make these things work with the state standards, which must be an immense challenge. I'd imagine it's the largest impediment to the school experimenting with anything as radically skills-based as the "Looking for an Argument" course.

Which is a shame. I bet that there are all sorts of amazing courses we could design for kids if we had a bit more freedom. I know that Henri Picciotto has designed a whole smattering of math electives. In Brooklyn, Saint Ann's school offers all sorts of stuff, including Mathematical Art.

Let your imagination go wild: if you were allowed to spend a course focusing exclusively on mathematical skills, what would it look like? Would you spend each day solving different problems? Teach a subject that's normally left for college?

My quick list of dream-courses looks something like this:

Paradox, Self-Reference and Philosophy of Math
Prime Numbers
Biology and Math

All of which are pretty blatant attempts on my part to imagine if I could try out my most positive mathematical experiences on students.

I'd love to hear what you've got in mind.

Something for me, something for you

2013-02-22T15:11:00.000-05:00

There have been a series of really good posts recently about the teaching profession. None of the posts really got at the pressures that I'm feeling, though, so I wanted to take a stab at telling my story.

Freshman year, I was enrolled in Philosophy 8, and I got a B- on my first paper. I was a bit taken aback, since that was not the feedback I was expecting on the paper.*

* The words "See me in my office" are scrawled across the top of the page. Cautiously, I knock on the professor's door in Emerson Hall. She tells me to come in. "Sit down, Michael, please." She explains how my paper has changed her views on Descrates' skeptical argument in his First Meditation. "Have you considered a career in philosophy?" she says. "Really, you must. The field needs you, Michael."

I really wanted to do better on the next paper. I was taking a walk between buildings, and trying to run through Descartes' argument for the existence of the external world. My mind was fuzzy and distracted, but I pushed the noise away and I was able to see the argument in my mind.

By pushing away the distraction I got myself into a sort of locked-in focus. The moment was one of clarity. I held the argument in my mind for the rest of the day, and wrote the paper, which ended up getting great feedback.

I come back to that moment often because it was so satisfying. There was a challenge, a breakthrough, followed by a well-earned clarity. As a student I tried to push myself toward those sort of moments. There are a handful more times that I had those sort of breakthroughs*, usually as the result of long walks where I tried to train my thinking on one particular problem for an extended period of time.

* By the way, by "breakthrough" I just mean that I understood something that I was supposed to, for a class or a paper. I don't mean that I made lasting contributions to anything except my own understanding of some subject.

That rush is all about feeling a challenge and just plowing right into it, and pushing past it. It's one of the most satisfying feelings that I know.

Dan and Patrick went back and forth last week about whether teaching remains a challenge for good teachers. My metric for how the "challenge" afforded by an activity is the frequency with which it can create those breakthrough moments for me. And by that measure, teaching is just doing OK, not great. Yeah, it's challenging, but not in the ways that create those moments.

I don't know why this is. Part of it is that I usually don't have time to dwell on an idea for long enough to really tackle it. Because of the quick turnover time of units and lessons, I often find that my best ideas come quickly and accidentally, instead of through deliberation and purpose. I struggle to construct my schedule in a way that gives me more time to work on long-term problems or projects, but (so far) that itself has been frustrating. And those breakthrough moments, when they do happen, are irrelevant to many of the partners in my students education.*

* "So why not get yourself in grad school or something? The teaching profession isn't about making sure you get your highs in life." True. Nobody needs to bend the profession to meet my needs.

But there is a sort of deep satisfaction that I get out of my job that has nothing to do with intellectual breakthroughs. I really love teaching because of students. I love that good teaching requires me to make myself invisible. I love how teaching requires patience and a deep concern with the ideas of others. I love how it's all about helping people get excited about things. I love how it's about community. And I find all of these things completely satisfying.

So, in short: the pressure I feel is that I like teaching because I think that it's a fundamentally good activity, but that there's something that I get out of it also. Every once in a while this work gives me those breakthrough moments, when I'm able to plan a unit that I'm really proud of, or design a lesson that nobody else has ever thought of. I'm able to design a classroom that works. At these moments, teaching is the perfect job. It allows me to help others, while indulging my addiction to flow.

But, overall? Teaching requires a great deal of professional selflessness, and offers me a lower rate of selfish intellectual indulgences than I experienced as a student. That's the pressure that I feel.

I apologize -- at the end of this post, I'm starting to think that it needs a few more minutes in the cooker. (Which I can't afford, because of grading, lessons, units, see above.) But I think the takeaway is that since teaching requires a certain degree of selflessness from me, I need an extra bit of selfish satisfaction from it. If I had a job in finance, I'd be well-paid and have plenty of time to pursue my own intellectual interests and challenges. If I had a job as a doctor, I'd have good job security, a great deal of respect from peers, and time and energy for the myriad things that could give me some sort of selfish satisfaction.

But I work nights and weekends, am earning 5 figures for you know FOREVER and my friends, parents and students constantly ask me what I'm doing teaching when I could be doing X. So I need a little bit more from teaching than it's giving me right now.*

* But, like I said, it's no one's job to give me a dream job. Hence all the hand-wringing about how to get better, and the interest in Cal Newport, etc.

Strong Kids v. Weak Kids

2013-02-13T09:30:00.003-05:00

Me first:

It pisses me off that the reward for experience is teaching smarter, easier kids, and that I'm going to take that reward, before long.
— Michael Pershan (@mpershan) February 13, 2013

Some people agree:

@mpershan Not only will be "easier", but you'll get paid more for it too.
— Jim Adamson (@GlenviewMath) February 13, 2013

@jybuell @mpershan I agree! Most tchrs of honors kids are fully convinced they are fabulous tchrs. It's a long day tchg all "low" classes.
— s reilly (@reilly1041) February 13, 2013

@mpershan @jybuell I teach both. Smarter kids are easier and more fun; needier kids are more rewarding.
— Chris Shore (@MathProjects) February 13, 2013

@mpershan I agree with you about this 100%. Behavior is NOT achievement.
— CheesemonkeySF (@cheesemonkeysf) February 13, 2013

@mpershan Didn't we have a big discussion about this a while back? It's incredibly difficult teaching weak kids. How is this not a given?
— EdReal (@Ed_Realist) February 13, 2013

But, tons of folks disagreed:

@mpershan @ddmeyer "Smart kids are easy to teach" is more teacher prejudice than classroom reality, and a disservice to "smart" kids.
— Patrick Honner (@MrHonner) February 13, 2013

@mpershan the reward is "choice". Don't think it's easier, but probably easier to fool yourself into thinking you're doing a good job.
— Jason(@jybuell) February 13, 2013

@mpershan I also hate that people view "smarter" as "easier".
— Greg (@sarcasymptote) February 13, 2013

@sarcasymptote @mpershan Compliant kids are less "trouble" which makes for an "easier" classroom. That has nothing to do with teaching.
— Andrew Carle (@tieandjeans) February 13, 2013

@mpershan Teaching honors is awesome! But teaching weaker students can be, too. Take a step back. All they need is a different approach.
— Adrienne Shlagbaum (@shlagteach) February 13, 2013

@sarcasymptote @mpershan At my sch, diff. behavior issues with diff levels. More cheating at higher levels, more class disruption at lower.
— Fran Poodry (@MsPoodry) February 13, 2013

Can someone explain what's going on here?

Can anyone explain how there's a disagreement this wide across the profession? Why does it seem straightforward to me that teaching students of low ability is harder, more challenging than teaching students of high ability? Why does it seem straightforward to others that this is a pernicious belief that ought to be challenged?

The Hard Parts

2013-02-03T18:39:00.000-05:00

I have about 3 hours a day that I spend planning lessons. I'm currently trying to figure out ways of incorporating hard practice into my professional life. Last week, I experimented by allocating half of that time towards careful thinking about what students would struggle with during my lessons. I write these things down in a document that I call the "hard parts" document.

Overall, I was happy with the first week of this experiment. My lesson quality was better than what I had been averaging in the first semester, even though it was a tough week for me (I was sick and didn't sleep much from Monday to Wednesday). Focusing on the hard parts of each lesson felt difficult and required constant mental effort. Forcing myself to write down observations made the whole process visible, and kept me from getting in a rut.

Here's a selection of my "Hard Parts" document I produced on Tuesday, a day that went very well for my first period students.

Concrete results came out of my time spent thinking about the most difficult aspects of my lessons. It was only on reflection that I realized that my students didn't understand the utility of closed-form equations for patterns, and I successfully created a plan that attacked that issue. It was during my reflection on what's hard about the unit circle definitions of sine and cosine (for my third period students) that I started worrying about the differences between heuristics and procedures. The solution: instead of directly having my students use the special right triangles to find values around the unit circle, I would start by having them estimate the values of a new function, s(theta), around the unit circle, and only make the connection with exact values later.

Unlike Tuesday, Thursday's class was a disaster for first period, even though I also devoted substantial time to my exercise.

The core of the lesson was the "Skype monthly plan" problem:

Given the above, how much do you pay per month, and how much do you pay per minute? I thought that this problem would be productive, though difficult, for my students.

Here's the "Hard Parts" document that produced this lesson plan:

So, what went wrong on Thursday? If I did the exercise faithfully, how did my lesson get so muddled?

A careful read of my "Hard Parts" document reveals some issues with my planning:

My goal subtly changed over the course of my planning. I had just a fuzzy idea of what I wanted class to be about at the beginning. I originally was going to focus class on a problem involving plumbers. The difficulties that I began thinking of for the "plumber problem" were no longer relevant by the time that I had chosen the "cell phone problem" instead.
Because I didn't focus on the "cell phone problem" I completely missed a student issue that I should have anticipated -- they didn't understand how monthly + per-minute fees worked. That could have been easily taken care of by giving them a chance to evaluate how much you would have to pay under various given plans.

The students also didn't get that November and December referred to the same plan. I could have anticipated that -- it's a misconception that I've seen before, and this is a relatively weak group -- but I didn't devote time specifically to the "cell phone problem."

What I'm realizing is that I have to be careful about plans or problems that coalesce toward the end of my allotted focusing time. Because I'm at the end of my time, and because I've already devoted a lot of thought toward the lesson, I'm tempted to skimp on mental effort for the actual problem that I want students to work on. That mistake can be a fatal one for my lessons.

I can help myself more by trying to come up with a rough plan on the weekends for what content I'd like to cover over the week. That's a bit of planning that I had been doing, but let slide. I think that will allow me to start each day with a bit more focus, so that I can spend the planning time anticipating issues with the day's core exercises.

Trying to avoid the plateau

2013-01-25T16:51:00.002-05:00

Here's a quick summary, because you're all busy people and this is the internet where distractions are bountiful

Teachers don't know how to get better past their first few years.
Even teachers who think that they're getting better aren't really.
I'm committed to searching for ways to introduce hard practice into my teaching life.

My Piano Skills

I took piano lessons for 5 years. I kept on getting better. I played harder and harder pieces. My teacher helped me train my ear. I wasn't winning awards or anything, but I was clearly on the upswing.

High school started, and I dropped out of lessons. I stopped getting better. Sad face.

I made some friends in high school. Some of these friends played music together, and they invited me to join them. They were into rock, and I played an electronic keyboard.

I sucked. All I knew how to play were sonatinas and Czerny's "Finger Exercises for the Weak." They were handing out chord sheets and politely requesting that I try something that sounds less awful. ("Let's take it from the top. Michael, could you turn down your volume?")

The biggest problem was my ear. They would make changes in the song in the middle and I couldn't pick up on them. One person would start playing a song and everyone else just played along. I couldn't do that. We'd have to stop in the middle of songs so that someone could explain to me what we were doing.

I started sitting down at the piano, outside of practice, and trying to play songs that I'd heard from memory. I'd try to improvise in ways that sounded a tiny bit like that thing that Elton John does on this one song. (Lots and lots of Sus2 and Sus4 is half of most rock piano.)

After I got over that bump, it was awesome. I could play with these guys, and I could keep up with them. I played all the time now, at home and with friends, just for fun. I would play by ear, I would play my favorite songs, and I learned more and more songs.

But a funny thing happened: I still couldn't play the harder songs. I sort of assumed that as I kept on practicing -- and at this point, I was playing songs pretty much every day because I enjoyed it so much -- that I would be improving. And I was improving, sort of. After all, I knew way more songs that I ever did. But I couldn't improvise any better than I used to -- I still didn't sound anywhere like the people did on the songs I listened to.

Why did I stop getting better?

Learning to Teach

The first few years of teaching are truly difficult. With little support you get thrown into a classroom and you just have to figure it out. Teachers get so much better during those first few years of teaching.

And then there's this plateau in teacher growth that creeps up on teachers, some time around years 3-5.

What's the cause of this plateau?

I think teachers plateau for the same reason that I'm no better at piano than I was five years ago. I got good enough to get by, and improving past that point requires a focused, systematic commitment to improvement that I didn't have. Or, as Cal Newport puts it, "if you just show up and word hard, you'll soon hit a performance plateau beyond which you fail to get any better."

Practice and Hard Practice

How can I avoid the plateau in teaching quality? It's not about practicing more. I was practicing piano all the time but not getting better. Well, that's not entirely fair. I was getting better in some misleading ways that hid from me the fact that I was plateauing. After all,

I was learning new songs
I was slowly picking up new tricks

Maybe this is just the way progress has to be after the first few years. Maybe progress is about expanding your repertoire and slowly picking up some new tricks.

And maybe mid-career improvement as at teacher is the same. You learn how to teach and then you expand your repertoire. You teach courses that you've never taught before, and that's pretty difficult. After all, it's awfully time-consuming. You make tweaks to your lessons. You take on leadership roles.

But that can't be right -- after all, there are musicians who become great at a much faster pace than I was moving at. They have systematic ways of practicing that are much more difficult than what I was doing. They drill themselves. They sit at the radio and transcribe solos. They seek feedback on their playing from people more accomplished than their high school bandmates.

There's a difference between practice and hard practice. Hard practice makes you better quickly. Practice lets you, essentially, plateau.

Most teachers plateau because they don't do hard practice.

Ah, so blogging and sharing on twitter is the key. That's hard practice, if there is any, right?

Reading and writing blogs and tweeting is* not hard practice. Having deep conversations about teaching and math is* not hard practice. Learning a new technology is* not hard practice.

* usually

Blogging and tweeting is like expanding the list of songs that you can play. You read blogs and become aware of things that other people are doing. There isn't enough time to read everything deeply, so you are drawn to people whose work is consistently useful to you. Inevitably, these are the people whose approaches most closely resemble your own. You write blog posts to share work that other people can benefit from. Inevitably, these posts reflect the things that you're already good at, because those are the lessons that tend to go well.

Blogging and tweeting are great for many, many things. Sharing is good. Expanding your repertoire is good. But they don't constitute hard practice.

Put it like this: do you feel like you're a 1st year teacher when you blog? Does your brain hurt? Do you feel as if you're lost, unsure how to proceed, confused?

If not, you're not engaged in hard practice.

Hard Practice for Teachers

A lot of my work over the past year comes from the idea that hard practice matters, a lot. I worry about whether my students are practicing hard enough. I worry that online learning lets kids off the hook. And I've been searching for ways to avoid the teaching plateau, though I think that my initial attempts have mixed results.

It's time to redouble my efforts. I'm half way through my third year, and this would be a great time for me to ease into a comfortable routine of expanding my repertoire without improving my skills.

I'm going to commit to finding things that are intellectually taxing that are central to my teaching. It's going to require experimentation to find the right combination, but I think this search itself constitutes a sort of hard practice.

Here's my starting set of exercises:

Last year I would often start planning my lessons by spending several minutes thinking about where my students would get tripped up by the topic of the day. I eventually dropped this practice, because I didn't need it to get by. Now I'm thinking that dropping this practice was a mistake, and the fact that I lazily dropped it is evidence that it's the right sort of thing for me to be doing. So I'll be journaling daily about what my students are going to have trouble with in the day's topics and lessons.
Writers read great novels. Musicians hear great music. Craftsmen apprentice under master craftsmen. But I stopped consuming teaching several years ago. On the analogy of other crafts, I need to consume more great teaching to get better. I'm going to commit to watching experienced teachers twice a month. Watching a teacher will sometimes involve watching a video, though I hope to get myself into some actual classrooms, both in my school and out of school.
Blogging isn't usually a difficult exercise, but I think that it could be. I'm going to attempt to flip the usual blogging process by writing about at least one failed lesson a month. I'm confident enough now to post my worst work for the world and attempt to come to grips with why it failed and why I thought it would work.

One last tentative thought: Teacher training programs tend (tend!) to be poor. Teacher improvement past their 3rd year tends (tend!) to be poor. Is this a coincidence? If we knew how teachers can become great, would that transform our ability to systematically improve the quality of beginning teachers?

Share exercises that you've used on yourself or your students in the comments, please. And, as always, please let me know if you've think I've gone wrong here.

The Best Blogging of 2012

2012-12-24T22:33:00.004-05:00

Here is the best stuff that I saw in 2012:

Fawn Nguyen - Her Google Form for reassessments has changed my life for the better. I really like her Foxy Fives problem set. Her love of the Shell Centre materials has sent me digging through their lessons and tasks. This year, her blog became my favorite resource-sharing blog. And she also has a 180 blog? So prolific, and so good. Her work has pushed me to be more organized, and to plan more carefully.

Christopher Danielson - His hexagon investigation might be my favorite post of the year. He's consistently thoughtful and sharp on twitter. Logarithms are another strength of his work. His online course is an exciting experiment. He wrote -- somewhere -- that he constantly asks his education students to make explicit the pedagogical assumptions of a text, lesson or activity, and that line rings in my head nearly daily. It was an excellent year for his blog, and I can't wait for what comes next.

Justin Lanier - I've always enjoyed Justin's work over at I Choose Math, but it's his 180 blog that really opened up his classroom to me. His Geometry class digs into beautiful investigations, and I loved his simplification of the proof that the square root of 2 is irrational. His work shows how exciting class can be when argument, proof and just plain-old thinking are at the center of things. My attempt to emulate his and Paul Salomon's work has been the biggest change in my tone and style this year.

Dan Meyer - Always Dan Meyer. What can be said about his work that hasn't already been said by so many others? A lot of my thinking starts with, "How does Dan pull that off?" His influence is just remarkable, and I get the sense that a lot of people are watching his career very carefully and taking notes.

Paul Salomon - Over the summer Paul shared his introduction to exponent properties with me. Here's a video of him explaining it. (Ignore the goofy redhead.) I walked away from that conversation realizing that I'd been neglecting proof and argument in class, and that it was time to bring them back to the center of our discussions.

Kate Nowak - She's the one that mentioned that she was bringing her CME texts with her to her new gig and got me interested. She's the most creative activity planner out there. Log Wars. Laser Kids. Line of Best Fit. Her work is just so impossibly and consistently good.

Finally, not really a blogger, but whatever:

CME Project - I bought the teacher's editions over the summer, and their work is just phenomenal. Owning these texts has raised my baseline performance by giving me high-quality material to lean on. The texts have also shown me how to teach so that proof and structure become visible and integral.

Teaching complex numbers

2012-12-14T13:57:00.000-05:00

I've been thinking a lot about complex numbers over the past few weeks. I wasn't happy with any of the available introductions to complex numbers. I wanted to put transformations of the plane at the center of it all.

I've been experimenting in the classroom. I started by defining (0, 1) as a transformation as the unique rotation that takes (1, 0) to (0, 1). In other words, a 90 degree rotation. By similar reasoning, (-1, 0) is an 180 degree rotation. And then I asked kids to figure out what (0, 1) applied to (4, 1) would be. (They studied rotations in Geometry, and didn't need much review.)

Then we generalized further: Let (a, b) be the unique transformation that maps (1, 0) to (a, b). It's pretty intuitive that you can accomplish this in the plane with just a rotation and a dilation.

That's where things got interesting for us. This approach is still very much a work in progress, and my Thursday lesson flubbed. It's annoying, because I've got a bunch of kids going across the Atlantic on an exchange program and I didn't get to wrap things up for them. So I wrote this letter to tie together the loose ends.

I want to write more about this later, but at the moment this stands as my clearest statement of how I want to approach introducing complex numbers to kids.

Feedback please?

A Letter to My Algebra 2 Students on Complex Numbers

Making Mathematical Decisions

2012-12-09T22:27:00.001-05:00

[The first draft of this post was over at the Global Math Department, where I presented about mathmistakes.org. Listen to the full recording for some analysis about the way users interact with the site.]

Evidence

Here are some math mistakes (source: MathMistakes.org). What do you notice?

To draw things out, kids are doing the following:

3 ^ 9 = 27
5 ^ 2 = 10
x ^ 0 = 0
100 ^ 1/2 = 50

3 Theories

Why do kids do this? Here are 3 options:

They don't understand exponents, and are just guessing.
They are reasoning consistently within a model, but a mistaken model. They think that 5^2 means "You have 2 5's. That gives you 10." They think that 2^0 means "You have no 2's. That gives you 0." They think that 100^1/2 means "You have half of an 100, which is 50."
Kids are not reasoning explicitly at all, but rather have a strong intuition that exponentiation should be solved using multiplication.

In short, we could be dealing with guessing, reasoning and intuitions. Which of these is right?

I dismiss the first option. If kids were guessing, then why don't they ever add the exponent and the base? Why don't they ever subtract? At best this explanation is incomplete.

The second option is more attractive. A kid answers exponentiation problems by saying "3^2 is 9, because two 3's make a 9" and extends that incorrectly: "3^0 is 0, because no 3's make a 0." The idea is that kids are reasoning about exponentiation in an explicit way, but that this explicit way is mistaken.

The third option is that kids are not reasoning explicitly. A search for a model would be beside the point -- kids have a strong intuition, in certain contexts, that exponentiation should be treated as multiplication.

How can we distinguish the second and third options?

If the second option is right, then kids should need to pause to reason before incorrectly evaluating an exponent such as 100^1/2 as 50. If the third option is right, then kids should just be able to shout that answer out very quickly.
If the second option is right, then kids should be able to explain their reasoning. Relatedly, we'd expect students without any strong model for exponents to be unable to provide any answer at all to unfamiliar exponentiation problems.
If the second option is right, then kids should operate consistently. They shouldn't sometimes reason according to the model and sometimes not. (Or, alternatively, the fact that they inconsistently apply this model would require explanation, one potentially provided by the third option.)

My Claim

I don't have all the evidence that I need to knock out the second theory, the idea that kids are explicitly reasoning about exponents. But, from what I've seen, kids have answers to the exponents questions WAY too quickly for it to be explicit reasoning. I think that there's something to the theory that this is an intuition.

So what's going on? There is a strong connection between exponentiation and multiplication. Everyone learns this strong connection. And in unfamiliar contexts the brain falls back on the intuitive connections between exponentiation and multiplication, and answers the question "What's the base times the exponent?"

Why? For reasons that I've tried to articulate before, I think that kids sometimes see harder problems as easier ones.

Predictions

How could we prove or disprove this specific idea? Here are some predictions of my lil' theory:

There should be other strongly connected operations, and we should similar mistakes when we ask kids to do tough things with those operations. A likely suspect would be subtraction of negative numbers, which asks kids to take subtraction into unfamiliar territory. There's even a bit of evidence that they treat that stuff like addition in these contexts.

We might even find evidence of more of this stuff in the early years of schooling, as kids are just learning their operations. MathMistakes.org could use some Elementary School submissions.
The best way to support my idea would be to artificially induce the sorts of mistakes that I'm talking about in students. The idea would work like this: I would define a new operation. Kids would show proficiency with it. Then, I'd define another operation in terms of the first. Kids would show proficiency with that one too. The next part is fun. Then I'd ask kids to extend the second operation in an unusual way, and see if they spit out the value of the first operation.

I'd like to see more stuff like this

I took a bunch of examples of student errors, I tried to unify them under some sort of theory that would make sense out of them. I tried to think through the theory to consider its competitors. I'm considering what would count as evidence for and against my theory. I'm trying to find testable predictions of my theory.

In other words, I'm trying to participate in the science of how kids learn stuff. And I think that more teachers should do that. Especially since understanding student errors would be widely valuable outside the classroom, but is easiest to theorize about when in the classroom and in interaction with warm bodies. Especially since digital cameras make it easy to collect lots and lots of evidence of how kids mess stuff up while you're grading.

Especially because it's fascinating, and I want to know more about it. Come up with a theory and write about it. How do people reach mathematical decisions?

7 Best, 5 Worst

2012-12-03T22:38:00.002-05:00

It's time for a quarterly review. Here's the good and the awful from this year's teaching.

7 Best

1. Paul Salomon's Introduction to Proving Stuff about Exponents - The idea is to use function notation to prove things about exponents without being distracted by the repeated-multiplication model that we (rightly) inculcate our kids with in the early years. Maybe the best thing about this problem is that it really does force everybody in the room to use proof. There's no mushiness, no intuition to fall back on, just cold reason. The second best thing might be getting student work that looks like this:

2. Exponents for Functions - And all it took was a little bit of nudging to get kids to understand why the hell f^-1 should refer to the inverse of f. It was beautiful. Then we drew the analogy farther, figuring out what a rational "power" of composition would have to mean.

3. Encryption and Inverse Functions - Not huge, but it gave me a language for talking about invertibility. Plus, it was a ton of fun. ("Can you give us, like, enough time to actually figure the code out?")

4. Swap and Solve with Equations - My kids were struggling with equations. They could handle anything that you could undo the steps on, but that thing don't work if you've got variables on both side of an equation. I wanted to share with them the "you've got equal weights on a balanced scale" thing, but I couldn't make it snappy.

This was a blast. I gave everyone an index card with a number on it, and they had to write an equation that had that number as its solution. Then, they gave their equation to a pal and asked them to solve it.

Why did this work? Because if you want to stump your friend you need to write a hard equation. And once some jerk reveals what makes x + 30 - 20 + 4 - 7 + 1 = 10 a pretty easy problem ("Oh, come on Mr. P, you gave it away!") you have to up your game. To use fancy man language, there was a load of intellectual need in that room.

5. 100m Dash/Stratos Space Jump - We used the 100m dash to talk about linear regression, and the Space Jump to break it. Both of these problems fundamentally worked as contexts for using the line of best fit to make predictions.

6. Height v. Shoe Size - I love making graphs on the white board. This was a particularly fun way to introduce two-variable data to my Algebra students. They put the post-its at their height and shoe size. Hey, look, there's a trend there. And we can talk about outliers too. The next day I took this picture and abstracted everything but the datapoints, leaving a scatterplot. (Explicitly imitating this guy.)

7. Constructing Number Tricks - This was pretty similar to my swap and solve activity with equations, and it worked in a similar way. Kids like coming up with their own things.

5 Worst

1. Guess-Check-Generalize - This was a boatload of frustration for me. Guess-Check was an easy sell for me; I'm still looking for buyers on Generalize. I tried lots of problems, drawn from CME and Park Math, and they did hook kids in, but every time that I brought in any abstractions I lost the crowd. My one minor success was with this pretty on-the-nose worksheet. Next time I teach this I'm going to try that sort of on-the-nose stuff earlier, and I might also wait until all my kids are extremely comfortable solving equations to attempt teaching this strategy.

2. Life Expectancy - I blogged about this guy already, but it bears repeating: this was a huge disaster lesson for me.

3. Graphs of Inverse Functions - No idea how to teach this. I'm, like, 1 for 6 in attempts to teach this thing, and I'm pretty sure that the one win was a fluke. Maybe the issue is that I just find it really cool that the graphs of a function and its inverse reflect across y = x, and I expect kids to find it as cool as I do. That very well might be the problem, since I tend to teach this by asking kids to graph and bunch of functions and their inverses and keep an eye out for something cool.

Or maybe the issue is that they're not comfortable with technology and graphing interesting functions is cumbersome? Whatever it is, I don't know how to make what really should be a cool idea pop for students.

4. Defining New Symbols - So promising! I love the problems, some of my kids love the problems, and it seems like a great way to practice evaluating expressions while also ramping-up the sophistication for the stronger kids.

It was way too hard for the kids just getting used to variables and expressions, and my attempts at explaining this stuff were just met with blank stares. (We lost a day to me trying, like, three different ways of explaining this to a eerily quiet room.) I love this idea, but I'm not yet sure how to make it work.

5. Percentage/Fractions - Don't know how to teach 'em, especially quickly, especially to Algebra students who have never quite gotten them and need to know them for more advanced topics. I tried a bunch of stuff, and it all kind of failed. The one thing that I'm feeling better about is division by a fraction, which I'm pretty sure that I know how to teach now.* The issue is everything else.

* Next year you can be sure that I'm going to draw out the distinction between two different division models very early. Is 10/2 = 5 because 10 split up into 2 even groups would have 5 members each, or because there are 5 groups of 2 in 10? Only one of these models really works well for 10/0.5.

Bonus: Solving Equations, in General - I don't know how long it takes most teachers to get kids up to speed on solving linear equations, but holy cow it took me a while. We've got to speed things up, I think.

Soapbox

I wouldn't mind seeing your "X Best and Y Worst" post. I think that would be fun.

How not to teach it: division by zero

2012-11-09T01:12:00.003-05:00

What doesn't work

We all agree that this is unsatisfying:

You aren't allowed to divide by zero, because it's a rule.

And many of us (read: me, yesterday) think that this is better:

What's 10 divided by 2? It's 5, because 10 split up evenly into two groups has 5 in each. 10 divided by 1? 10 in 1 group has 10. But 10 in 0 groups? What would that even mean?

And, then, having clearly and elegantly explained why dividing by zero would be a very, very silly thing to do, we go back to the day's main topic:

So 2 to the negative 3rd power is 1/8...

Wait, hold on.

I get it: if you think of division as evenly grouping items, then dividing by zero makes no sense. But that's just the normal wear and tear of a mathematical model. We ask kids to believe that exponentiation is like repeated multiplication, and then we ask them to forget that when we introduce negative powers. Multiplication is repeated addition until you throw in "2.3 times 5.1" and then everything goes to hell. We freaking give quadratics imaginary solutions, and our failure to imagine what "0 groups" looks like is stopping us from dividing by zero? Yeah, right.

(Also, wouldn't zero groups have no items in them?)

And another thing: we tell kids that division by zero is undefined. Skeptically, they take out their calculators and punch some keys and get an error message. "Wait! He's right. It gives you an error."

When it comes to dividing by zero, there is a lot wrong with the standard teachery maneuvers:

Any sort of "wtf would 0 groups mean" argument does not show that division by 0 is non-sensible. All that it shows is that this particular model of division -- the grouping model -- breaks down for non-integers. That's normal in math. Kids should be regularly creating and discarding conceptual models.
"Undefined" is the best we can do? Language matters, and saying that 5/0 is undefined makes it sound like, shoot, well, we were going to get around to it but we just chose to let it slide.
The kids are checking their calculators to see if division by zero makes sense. For crying out loud, that's not math. They're wondering whether to believe you or not, because what you're saying doesn't make sense. Hell, everyone knows that 5 divided by 0 is 0. It just makes so much sense...

This works better

March in front of the classroom. "What's 3 divided by 0? Someone tell me NOW," you say.

If your students are a bunch of sissies and nerds they'll shout "You can't divide by zero!"

"Oh don't give me that math teacher stuff. Who says that I can't divide by zero? Give me a real answer."

That's all it takes. Really. They've been waiting in every math class since they were 8 to get this off their chests.

"3 divided by 0 is 0."

OK, cool. Now we've got something to work with. Ask the class, agree or disagree?

But if 3 divided by 0 is 0, and 5 divided by 0 is 0, then wouldn't this follow?

So 5 = 3, right?

They squirm. They try something else. Maybe 5 divided by 0 is 5? You can handle that too. The point is to lead them to contradiction, and let them grapple with that tension. There are other ways to tug out the contradictions.

Here's why this is better:

We shouldn't be telling kids that the reason that we don't divide by zero is because an intuitively pleasing model fails. That should never stop a good mathematician.
Telling kids that division by zero is "undefined" sounds lazy. It's more accurate and informative to say that division by zero leads to contradiction.
How do you help kids see that it leads to contradiction? Take suggestions from the kids of what division by zero should mean, and then let them see the implications. Let them try to make things consistent. Make the choices clear. Sure, we can think of division that way. We'd just have to refine our rule for multiplying fractions. So, what's your new rule for multiplying fractions?

All of this is more authentic than what I used to do. ("What I used to do"? I'm talking about yesterday. Things move fast around here.) The biggest change in what I'm doing this year is slowing down and having kids make arguments in class. Proof and argument is how I'm helping my kids make sense of this stuff. There's no shortcut or substitute.

Answering an easier question

2012-10-30T17:31:00.002-04:00

There are a wide variety of mathematical errors, and it's worthwhile to try to find patterns and themes that stand behind the particular mistakes.

The following quote is from Thinking Fast and Thinking Slow:

"I propose a simple account of how we generate intuitive opinions on complex matters. If a satisfactory answer to a hard question is not found quickly, [the intuitive capacity] will find a related question that is easier and will answer it. I call the operation of answering one question in place of another substitution. I also adopt the following terms: the target question is the assessment you intend to produce; the heuristic question is the simpler question that you answer instead."

Not all errors involve some sort of substitution. Sometimes we avoid jumping to a conclusion, we grapple with the difficult problem, but we still make a substantive error. This is just one type of mathematical error.

Questions for homework:

Do you agree with the author of this post?
Are there other categories of mathematical error that you can identify?
What can teachers do to effect the usage of heuristic questions by their students?
Is this the sort of question the domain of psychologists? Of teachers? Of both?

Encryption and Inverse Functions: First Draft

2012-10-23T18:37:00.000-04:00

Here's what kids usually see when they walk in to my room:

Here's what they saw today:

I told them to break the code. It didn't take long, especially because there was a huge hint up there. But the point was that I wanted to talk about codes, encryption and reversible functions today.

After they broke the code, I asked them to explain the encryption process in terms of functions. We ended up with G(a), which takes letters and spits out numbers, and f(n), which takes a number and gives you three more than that.

Then I gave them another encryption.

This time I told them the key. It was g(n) = absolute value(n - 10).

"Wait, it could be two letters."

"It's 'HELLO' but it could've been 'FILLO'."

It's a lousy code, because it's ambiguous. The information about the starting letter is ambiguous.

The rest of the lesson was sort of lousy, with some good moments. I teachernated that if a function is reversible, then it makes a good code. And another way to say that it's reversible is that it has an inverse function. Most of the rest of class was spent trying to figure out if various functions had inverses. But there were some highlights:

"It's only a bad code if you use all the alphabet." We talked about restricting the domain artificially.
"So any code that has two different letters with the same number is lousy." Nailed it, kid.

Basically, I'm sold on the idea of using encryption as a context for motivating the distinction between one-to-one and non-one-to-one functions, and I'm also sold that this can motivate functions versus non-functions. (Just try imagining what the inverse of one of those non-invertible functions would look like.)

But I feel like I didn't nail this lesson. The concept seems solid, but I don't think I made it really interesting or especially challenging. Any ideas on how to improve it? I'm giving it another shot next week with 11th graders.

"Substituting for x" is a subtle killer

2012-10-10T12:11:00.000-04:00

"So I just swap the number and then treat it like arithmetic? Oh, that's easy!"

Here are some common mistakes kids make when evaluating expressions or functions:

Able to evaluate forward, but unable to undo the evaluation, even when given something like f(230).
They'll swear to you that a^2 is -1 when a is -1, because -1^2 is 1, even though they know that (-1)^2 = 1, and that -1 times -1 is 1.
They'll make weird calculation errors when evaluating expressions that they wouldn't make if they were just doing the arithmetic.

I think that if you want to help your kids avoid these mistakes, you're not doing them any favors by talking about swapping, replacing, substituting or blanks. All of this language support a "mystery value" picture of expressions and functions, where variables stand for particular numbers, and every variable is just waiting to be revealed as standing for a particular mystery number.

Instead, it's helpful for kids to think of expressions and functions as operations to be done on any number. Number tricks are a nice way of setting this up, but I think that you can undercut things by talking about swapping/replacing/blanks when dealing with expressions or functions. The reason is (and this is subtle, and possibly wrong) because swapping says "this expression is just about particular numbers."

Better language would be applying the expression/function to a number. This emphasizes that the expressions says something about numbers in general, which can be applied to any particular number. (Evaluating is fairly neutral language, but not if you define evaluating as "substituting.")

Postscript:

There's a further difficulty when teaching function notation that I want to get off my chest. If you introduce function notation with evaluation, and define evaluation as swapping, kids miss out on the subtleties of the notation. Why do they miss out? Because evaluation with swapping is too easy -- you just ignore the random letter before the parentheses, take the number inside the parentheses and swap any variables with that number.

But does that f stand for something? And what are the parentheses doing? What is f equal to? What if you have an f inside the f? Is that like f times f? And what does this have to do with outputs and inputs? Does f stand for the output?

Evaluating functions with swapping doesn't give kids enough friction to force them to notice the weirdness of this notation. And that means that they're missing out on the move from seeing functions as processes to seeing them as mathematical objects, the sorts of things that we can use adjectives and predicates to describe.

Productivity Experiments

2012-10-06T21:46:00.004-04:00

Everybody with an internet connection is participating in a massive experiment. The experiment goes like this: what is the effect of a distraction machine on the human race? (The Amish are the control group.)

I'm incredibly nervous, all the time, about how effectively I'm doing stuff and getting better at doing stuff. And -- right now -- curbing my internet habits is the major front of that effort.

Here's what I've done so far:

Killed Facebook.
Got my inbox size down. Way down. I print out messages that I'll need to respond to later and post them on a bulletin board.
I've bought a bulletin board, by the way. It's great. I post my monthly budget and emails that I need to respond to. I'm less nervous about losing track of stuff. My mind is more settled.
I've set up a filter to eliminate the different between read and unread messages. So far? The results aren't great. I'm still checking my email very often, though. We'll wait and see on this one.
I'm pretty excited about this one: I've eliminated Google Reader and replaced it with FeedDemon. There are two reasons why I think this is going to make me a more effective blog consumer. First, my RSS reader is no longer in the browser. That means that I can't access it from any computer other than the one that I leave at home. It also allows me to set up filters so that I can try the read/unread experiment with my blogs also. (You can't do that in Google Reader, I think. And it costs $20 for the license to set that up in FeedDemon.)
I also unsubscribed from blogs that post often enough that I could hope to gain something by checking my reader more than once or twice a day.

Overall, the goal is to start batching my consumption of online stuff.

I think that this stuff matters. A lot of folks recommend subscribing to hundreds of blogs and scanning them quickly to find the important stuff. Same with twitter. (Which I struggle with too.) That might work for some folks, but being distracted doesn't support my goal of being a thoughtful teacher that (eventually) comes up with some really good stuff. So they have to go.