Answering an easier question

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:

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

Encryption and Inverse Functions: First Draft

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

"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

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.

Rational Expressions Announces $1 Million Prize for Solution of Teaching Problems

Prizes and contests spur innovation. Think of the Millenium Prize Problems or the X-Prize. The other thing that spurs innovation? Lists of problems. Like Hilbert's. Or Jay-Z's.

In the spirit of all these lists and contests, I'm happy to announce the Really Important Teaching Problems (RITP). These are some of the most difficult, knotty problems that teachers are grappling with in this new century. Work on these problems continues because of their importance and seriousness.

Successful solution of any of the RITP problems will be awarded with the following:

1. A blog post about your solution
2. One-million dollars

(At this point it seems necessary to mention the generosity of the Gates Foundation.)

Attempts were made at succinct and direct statements of the problem. Problem were selected with input from our board of advisors. Without any further delay, I present to you the RITP problems:

1. The (Much) Better Lesson Problem: Is it possible to use the internet to create a free curriculum of the highest quality?
2. Khan's Conjecture: Can classroom learning be personalized to the stage that it performs as well as a high quality tutor?
3. Meyer Theory : How close can a classroom teacher get to completely engaging every student with every topic?
4. The Theoretical Problem: Can good math teaching be well-described, understood, and taught to new teachers?

[The board has approved changes to this list of problems, contingent on convincing arguments being dropped into the comments of this post.]

So, get on it, everybody. I don't want to keep all this money for myself.