The Two-Minute Probability Lesson

How do we share our lessons with other teachers?

You can share every document you make -- lessons, worksheets, activities, warm ups, exit tickets, etc. -- but this way lies ridiculousness. What percent of your work is new or interesting? Is anyone really just going to use all your worksheets? Sharing everything is sharing nothing.

Writing about lessons is a better way to share. You only put out the interesting stuff, and you get to make the case for the quality of your lesson. You can include snapshots and pictures of kiddies creating the things that you helped them create. This is good. 

I wanted to try a new way of sharing a lesson, though. I took 43 minutes of classroom time, and I squished it into 2 minutes. I tried to faithfully convey what I thought the crucial bits of class were. Here's what (I think) I ended up with:

• A representation of a lesson that would've made more sense to me as a first-year teacher than a written post about the class. Actually seeing parts of this thing happen make it seem a bit more doable.
• A sense of what it looks and sounds like when kids talk in my classes. That's the thing that interests me most when I visit other classes, and something that's especially difficult for me to pick up on from a written explanation.
• An entirely contextualized lesson. This is what this version of this lesson looks like in this classroom. In writing up a lesson report we tend to abstract away lots of the concrete details of teaching, but these things matter in situating what someone else has created in relation to your own work.
• It's quick. That matters too.

In the end, I created the sort of thing that I'd like to see others make. I hope that you do.

Afterword:

• I'm submitting this video to mathagogy.com, because this is exactly the sort of thing they want to do.
• It hardly seems worth sharing my files, but pipe up in the comments if you want 'em. 
• For the record, my impression is that my approach to teaching probability is pretty typical. I have a few nice curricular touches -- the Plinko board is a great scene-setter for the entire unit -- but this whole thing is more about the video than the content, imho.

We're wrapping up the year here, and I don't know if I've got another two-minute experiment up my sleeves. The next stage of this, for me, might be in the fall. 

Taking a careful look at Treisman's talk

Here is one of Uri Tresiman's slides from his talk at NCTM:

Quotes from Treisman's talk, on this slide:

• "You can see from this that if you control for child poverty, we're pretty much on the top of the world."
• "The PISA scores mask the fact that child poverty rates were the principal factor in performance, not the particular structures of the country's education systems." 

I tried to track down his data. PISA scores are easy enough to find, and the PSID, CNEF, UNICEF stuff comes from a UNICEF report, "Measuring Child Poverty." Cool. I dug in.

I noticed something on the Treisman slide. I saw Finland and Switzerland among the top scorers, but I didn't see any of the East Asian countries that regularly appear at the top of the PISA scores. It turns out that the report doesn't have child poverty rates for all of them, but that UNICEF report does have Japan's child poverty rates. I also noticed that Treisman's slide doesn't include all of the PISA or UNICEF countries, so I thought that I would put all of the data from PISA and UNICEF into an excel spreadsheet and graph them all.

This tells a bit of a more complicated story, no?

Also:

•  As far as I can tell, these are the countries missing from Tresiman's slide:

• From the UNICEF report: "Underlying weak monitoring is the lack of any robust public or political consensus on how child poverty should be defined and measured." Some people think that relative child poverty should be measured, i.e. percentage of children below 50% of the median income of the country. Others want to use absolute poverty measures. There's one called the Deprivation Index that is determined by things like "Percentage of Children with Three Meals a Day" and "Percentage of Children that have some new clothes." Which measure of poverty matters for education: relative or absolute poverty?
• There's another child poverty report out there, with data from South Korea, another PISA high-flyer. The report is from the OECD. I didn't include their data in this graph, but South Korea is above the mean, near Ireland and Slovakia, by their measures.
• We don't have good data on child poverty for some of the other PISA rockstars. That includes a lot of the East Asian data points: Hong Kong, Shanghai, Singapore, Taiwan. If you think that they're better on child poverty than similar scoring countries, then this helps Treisman's point. If you think that they're worse on child poverty than similar scoring countries, than this hurts his point.
• Hey, take my spreadsheet. Mess around with it. (If, someday in the future, this link is broken, email me and I'll send it your way.)

Comments comments comments comments below.

How kids think about multiplying polynomials

The kids are confident

I recently gave my students another survey. At the time of the survey, the students had just studied exponents and were about to begin multiplying/factoring polynomials. To emphasize: they've never studied multiplying polynomials in a formal setting. The second half of this survey is what you folks call a pre-assessment or something.

This response from a student -- let's call him Mike -- is especially interesting.

Mike nails the first three questions. A conscientious kid, he even expresses modest, self-aware doubt on the Question 3, though he uses the Distributive Property like a pro in his response. In short, Mike is a student who knows how exponents and the Distributive Property work. 

So far, we haven't taken Mike out of his comfort zone. Take a look, though, at what happens when we push him out of the boundaries of what he's studied formally (this year):

The first thing to note is Mike's confidence. He expresses more confidence in the stuff he's never studied in school than the stuff that he studied a week ago. 

Two numbers, not one

Turn next to his particular responses to Questions 4 and 5. Isn't it interesting that he provides different answers to Questions 4 and 5, even though he analyzes (a+3)(a+3) as (a+3)^2? Why could that be?

Here's one take on his contrasting responses: in responding both 9a^2 and x^2 + 49 to two questions that Mike himself takes to be similar, he betrays that he doesn't see those two responses as especially different. In other words, Mike is operating on the "x" and the "7" as two independent numbers, he's doing the same thing with "a" and "3", and as a consequence there isn't that much of a difference between 9a^2 and x^2 + 49.

Debunking(?) the Distributive Property of Exponents

Let's talk, for a moment, about that x^2 + 49. It's by far the most common response to this question on the survey. It's also a very common mistake, one that we've cataloged on mathmistakes.org before -- twice, actually. Here is a sample of some of the comments that teachers left on those mistakes:

"I think the prior comments have essentially nailed it. The trouble is the “distributive property”." 

"I have to say that I don’t get why we talk about “the distributive property” of multiplication over addition, but not that of exponentiation over multiplication. If naming the distributive property is helpful in one circumstance, why wouldn’t it be helpful in the other?" 

"We don’t teach why multiplication distributes across addition, and we talk about a power to a power means you multiply, so it seems very hard to distinguish between this mistake and when you can distribute."

(Heck, I even named one of those posts "The Distributive Property of Exponents.")

But let's look at some of the other responses from this survey:

 

Suppose that the Distributive Property of Exponents theory is right. That theory has nothing to say about (a+3)(a+3). There's no reason why (a+3)(a+3) should provoke a misapplication of the Distributive Property of Multiplication. I mean, there's no visual resemblance, and it's not at all clear to me how students would be misusing the Distributive Property to arrive at a^2 + 9. (Unless they're going through (a+3)^2 along the way, which strikes me as unlikely.)

Maybe the Distributive Property explanation is helpful in some cases, but there's something else going on here. 

So, what is going on here?

Let's look at one more response, before wrapping this thing up:

Why would this student multiply the 7's together, while adding the 3's together?

I used to think that consistent mistakes were the most interesting, but now I'm pretty sure that inconsistencies are far more revealing about what students are actually thinking. They're like tensions in a story or conflicting experiments. 

I don't know how to explain these (fairly typical) responses, but here are the principles that will guide me thinking:

• I won't try to attribute to students well-thought out theories about how multiplication polynomials work. That's inconsistent with the idea that these are intuitions, pre-reflective ideas. (See previously, here.)
• I won't try to blame everything on the Distributive Property. This is bigger than that.
• I'll keep a close eye on the tendency of students to see polynomial multiplication as a series of operations on the individual numbers, rather than taking a more holistic view.

That's it for now. Go forth and comment -- here about the overall analysis, and over at mathmistakes.org for the last survey result from this post.

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.