Visual Exponential Patterns

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:

1. Graph/Equation/Pattern
2. Matching Patterns with Equations

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

Becoming great at teaching

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:

1. 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.
2. 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.
3. 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.

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

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

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

1. Table of Contents

1. Table of Contents
2. Exponent Mistakes + Teacher Explanations
3. Let's. Try. Evidence.
4. Results
5. What's Going On Here?
6. 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

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.