Life Expectancy, and a lesson that didn't work

Here was something that didn't work with either of my Algebra 2 classes. I'm wondering why it didn't work, and if there's a better draft of this problem to be made.

The lesson was vaguely three-acty.

Act One

How long does the calculator say that a person born in the 90s will live? The 80s? The 1890s? Why the difference?

I asked one class: "When will you have to be born to expect to live to 100?" The other class got "How long can you expect to live if you're born in 2010? 2050?"

Act Two

[Source: CDC]

What do you notice about the table? Take a guess for 1980. My guess is 60 years. You guys like that? Why not?

How good is the following rule: "life expectancy = years * 10"? Is the rule "life expectancy = years * 9" better or worse? How can you find a rule that's better than either of them? What are you changing in the equation?

Can you find an equation that fits it pretty well? How far off would the predictions be?

Act Three

Here's our most recent data. What do your equations predict?

What went wrong?

First, the objective stuff:

• Kids didn't seem into it.
• Kids didn't know where to jump in.
• Kids were confused by the idea that it has to be a rule that gives a line.
• Kids thought it unnatural to make a prediction based only on a few prior data points.

Other issues:

• It wasn't clear to me or them what they were trying to predict. Since we can't check their actual predictions (cuz they're in the future) we have to just limit the data that we make available to them. This seems to be a limitation of the "data analysis of social stuff" type of problem, and an argument for doing regression problems with stuff that we can actually test in the classroom.
• We'd done a similar, and superior, problem last week with data from 100m dash times. I wanted kids to end up with actual equations as models, and I don't think this was different enough to necessitate equations. A lot of kids repeated their tricks from last time: averaging the rate of change, coming up with recursive rules instead of closed-form rules. I didn't feel as if anything, other than my insistence, was pushing them towards closed-form equations.

Improvements?

• Post it as a historical puzzle. Let's say you were in 1960: how far off would your best prediction be for life expectancy in 2010?
• Find a better hook. I needed something like that life expectancy calculator just to make sure kids knew what "life expectancy" means. 

Help? Anyone?

The hard problem of online learning

Idea: Kids learn things on their own, at their own pace.
Problem: They'll get stuck, and frustrated.

Idea: We'll give kids feedback automatically that doesn't need the attention of a person, so that they can still learn stuff on their own.
Problem: That's complicated.

Idea: We'll use computers.
Problem: Computers are pretty cheap, but the quality of the feedback isn't very good.

(See also: Turing Test)

This is a really tough problem. Nobody has any ideas that seem entirely promising. Maybe we just have to wait for the technology to improve.

Here's my pitch for a shift in the way we think about this problem.

In the world of "giving feedback to kids on math" there are two contestants. There's (1) people and (2) machines, and people are beating the stuffing out of machines. Truth is, machines haven't ever shown that they're up to snuff, as far as quality goes.

But humans have issues too. A single human can only be in one place at a time and can only focus on giving feedback to one kid at a time. We people have all sorts of stuff to do besides slavishly providing kids with quality feedback. I mean, unless you're a teacher. But, then you have to figure out a way to give quality feedback to a few dozen kids at once. Humans are limited in a way that machines aren't.

But humans are making progress. Technology is the key, here. Advances in machines have allowed humans to group together to overcome some of the limitations of being a person. If you mess around with Wikipedia,   a human being will find out and fix it. Closer to our discussion, if you ask a question on Math StackExchange you'll get a good, quality response before long. If you ask a teaching question on twitter, you can also depend on getting a good answer.

The lesson here is that the humans are making progress.

Nobody has figured out a way to create a site where math students can get quality human feedback on any topic they're studying. Nobody has figured out a way to get quality feedback out of machines. Pretty much everybody who's working on online learning is trying to figure out a way to help the machines. In the meantime, sites like StackExchange or Physics Forums keep on building and improving their communities.

I've got no idea how to build a site that has a quality, supportive community of students and adults who will provide quality feedback to K-12 students. But there are places on the web that are like this. And if you can sustain a vibrant online community, then you can start creating more difficult tasks for students online. The rate of innovation in online curricula could speed up quite a bit.

Creating a quality community is a hard problem. Creating a piece of software that can give excellent feedback is an attractive and lucrative problem.

The pitch is: focus on the hard problem, not the attractive one.

What kids hate about school

I've struggled in the past to figure out a way to start the conversation about classroom culture with kids. Here's what I hit on this year:

Here's what I learned:

1. Kids hate being bored, love being interested.
2. Kids hate doing homework.

Full responses below:

Best Worst Classrooms

Next step: Post these notes in a very public place.

Update: Ken Templeton has provided us with worldes, and there was much rejoicing.

 

Reality Checks

I just posted this on Math Mistakes. It comes from a piece of work that I collected at the end of class today.

Collecting that first batch of student work was my favorite part of the day. It turned an impossible job into a difficult one. How can I help them learn if I don't know how they think? What it cost me was 3 minutes at the end of class and 10 minutes of analysis after class. And now I know who I'm (likely) going to need to target, who needs me to call home over the next few days, who seems to have their act together.

Consider this an official call for submissions. Collect work from your kids often, and take a picture of some of the interesting stuff and email it to mathmistakes-at-gmail-dot-com.

One last time: collect information at the end of as many classes at possible over the first two weeks. Try to hold in the laughter when your colleagues start complaining about the surprises on those first batch of end-of-chapter tests.

First First Day

I have two first days. First the freshmen come, then the big boys. It's nice for me, because I get to adjust some of my shtick for the second round.

Assigning seats is one of these little things that caused me a ton of stress last year, because I always left it for the last minute and I always screwed things up like forgetting to give a kid a seat or giving a kid four seats. Anyway, here's how I fixed it this year:

Plus, the coordinate plane is in the curriculum.

As long as I'm posting, here's some other assorted stuff from today:

The above is the white board in the computer lab before I applied an hour and a bottle of rubbing alcohol to it. The board hadn't been cleaned in (get this) 5+ years, and it was tough work to get that ink out.

Here's my math classroom. Because of various weirdnesses about our school there are lots of times when students are in the classroom without an adult, and we've gotten ourselves a reputation for not being able to have nice things because the kids can rip them apart. 

This year I want to try out the opposite approach. I'm going to try to keep filling the room with nice (but inexpensive...) stuff, like plants, storage boxes where I can keep materials, posters and calendars. Basically, I want to mark my territory.

We solved these problems today, all of which were happily stolen from Park Math:

Handout p2d1

Which already yielded this happy misunderstanding:

Post-mortem on a mistake I made

James Tanton's latest newsletter is phenomenal. But I got stuck on this part:

What was driving me nuts was that I thought that this argument was too loose. So I tweeted my question to the author:

@jamestanton I read and loved that piece. But I'm still confused. Based only on paper-sharing intuition why wouldn't my 2 pals get .5 page +
— Michael Pershan (@mpershan) September 2, 2012

@jamestanton + eventually if I give each of them 1/9 of my page, and then 1/9 of what's left.. so that 1/9 + 1/81 + ... = 1/2 ?
— Michael Pershan (@mpershan) September 2, 2012

Then, I got a series of very helpful tweets from Justin Lanier:

@mpershan @jamestanton You're solving a different problem than you mean to. 1/2 = 1/9 + (1/9)(1-2/9) + (1/9)(1-2/9-2((1/9)(1-2/9))) + ..
— Justin Lanier (@j_lanier) September 2, 2012

But he wasn't answering my question! He didn't understand me. I had to restate the problem I was having. Justin patiently repeated his argument. Why didn't he get it? How could I explain my issue better?

Then, James Tanton offered some help as well, giving a nearly identical argument as Justin's:

@mpershan Does it not? Each of your friends gets (1/9) + (1/9)(7/9) + (1/9)(7/9)^2 + (1/9)(7/9)^3 + ... and this must be half the paper.
— James Tanton (@jamestanton) September 2, 2012

That's when it hit me.

All of a sudden everything that Justin and James Tanton had said made perfect sense. My mistake was clear. I saw where I went wrong. I had messed up an argument, and not an especially tricky one. Besides, this was high school math -- the thing that I'm supposed to be teaching. I felt embarrassed.

Here are some teaching lessons that I want to take away from this experience, assuming that what I experienced is true of others too:

1. When someone understands most of something, they're equipped to turn a misunderstanding into an objection, and it's much harder to convince a person that their objection is wrong than it is to correct a misunderstanding. The only thing that worked for me was (a) having a second person explain it to me (b) after a break from thinking about the problem. Which, by the way, means that
2. kids shouldn't be forced to work through problems in order, unless that sequence is necessary. Moving between problems often helps when you're in a rut.
3. Doing and thinking about math during the year is an important part of teaching. Moments like these remind me a lot of what it felt like to be clueless during high school and college in my math and science classes.
4. This post is still true. My insecurities about learning could easily bleed into my teaching, but I shouldn't let them.

Afterword

There was a more self-indulgent post that I wanted to write. I'll throw it in the afterword, though it probably belongs elsewhere.

Here's what I remember about math and science classes in high school and college:

• Asking dumb questions that everybody else understood in the back of Algebra.
• Finishing last on Calculus tests.
• Having to go to office hours every night for Multi-Variable Calculus in college.
• Trying to understand how my friends got their answers for Mechanics.
• Not understanding any math lecture that I went to in college, ever. 

I'm slow. Some people are sharp, quick-witted, and that's just not the sort of thing that has ever really been true about me. The kind of difficult I had above is the kind of difficulty I've been having my whole life, as far as learning stuff goes.

To turn it back to students, some of them are sharp, some of them are slow. And I think it's important to remember that nearly every aspect of school celebrates the quick and sharp intelligence over the slow one. Let's not pretend that waiting 30 seconds before taking an answer to a question is enough (though it helps) to even the playing field. If you're slow you tend to do worse on timed tests and on homework. Your sharper classmates will solve more problems than you during class. If you're slow then you finish class and your notebook seems foreign.

Though, maybe being sharp and quick is part of what it means to be good at math. Thoughts?