Friday, August 24, 2012

White Paper on Problem Solving: Struggle without frustration

I’m pretty confident that if I just give my kids a bunch of really hard problems to solve, the following will happen:
  • First, the class erupts with demands for help.
  • Then, the class collectively gives up. Two kids keep on working, because they’re those kids.
  • Students complain to their parents, who complain to the school, who pass it on to me. (“They say you’re not teaching them anything.”)
  • I fax (fax!) everybody in school a long memo detailing where our profession has gone wrong and how to steer it right.
  • I get fired and lose all my clients save for one hot-headed wide receiver.
  • Together we teach each other the importance of trust, love and commitment in both personal and professional relationships.

And I certainly don’t want that to happen.

In the previous post I explained how, last year, I responded to the pressure of keeping my kids un-frustrated by making my problem sets easier. In the first post, I explained why I shouldn't have done that. In this installment I want to come up with some strategies for helping my kids feel comfortable with struggle. I’ll keep my eye on the comments; if you’ve got something good, I’ll toss it into the post.

But first, yet another picture of a cat with a Rubik's cube. Seriously, how many of these are there on the internet?

Never mind. Stupid question.

How to keep kids from getting frustrated* by difficult problems:

* Frustration is sometimes OK, but is just as often unproductive for a student. For the rest of this post, you can assume I'm talking about the unproductive stuff.

Let's start with a distinction. When a kid gets frustrated in an unproductive way while working on math, her frustration comes in two flavors:
  • social
  • intrinsic
Social frustration comes from feeling as if she's inadequate relative to her peers or relative to the expectations of others. It's real, but it comes from her understanding of other people's views of her. Intrinsic frustration is everything else. It's the stuff that would cause a person to walk away from a problem even in a closed room, with nobody watching. (I don't really know where "privately feeling stupid" fits. But whatever.)

Here are ideas for minimizing the social pressures:
  • Be explicit: One day last year, during a quiz, a kid pointed at me and said, "Mr. P, you put a problem on this that we've never seen before!" And I was, like, yeah that's what I was trying to do. But that was actually a really good moment, when the class came to understand what I was about. I should've talked about that in the first week: "Yo, kids, I know this class is different. But it works and you'll still have support, and it'll be OK." That sort of thing.
  • Find unfinished answers interesting: Last year we never had conversations about the kids' work with the whole group. This year I'll bring to the fore not just correct answers, not just finished answers, but approaches and ideas from unfinished problems. I'll intentionally spread the wealth, so that we're talking about everybody's work, eventually. And we won't do the embarrassing "So where did he go wrong?" questions, at least not at first. Instead we'll celebrate the process by asking, "How could we finish it off using his approach?" We won't force everyone to go through the ringer, at least not at first.
  • Try to build team mentality: This is a bigger classroom management puzzle for me, but I'm going to start by throwing in questions to the problem set that say "Look around. Does anybody need help? Take five and see if you can be of service." 
And here are some ideas for minimizing the intrinsic pressures:
  • Mix up easy (but cool) and hard problems: When things are going well, when you're in the zone, you're in a state of flow. Flow is what keeps most of us coming back for more, even when the going gets rough. By mixing up the satisfying questions with the knotty ones, I'm betting that I can get more kids working for longer. (Also, a really good problem has easy, cool and hard aspects all wrapped up in one neat package. Keep an eye out for those.)
  • Be more interesting: Whenever a kid gives up on a problem, part of the problem is motivation. If the problem was SUPER interesting, he would probably keep chugging away. So I need to do a better job finding more interesting problems and more interesting hooks into those problems. 
  • Interrupt more often: There were times last year when kids would be working on problem sets (i.e. worksheets) for 20-30 minutes without serious interruption. That's great, but one way to release the pressure of frustrated students is to pause and take a deep breath. I'll interrupt them more often when I sense that people are struggling. We'll talk about the problems, get some approaches and strategies up on the board, discuss next steps and then send them off for another 5-10 minutes.
  • Refer to common problem solving strategies: I'm betting that if we have a repertoire of habits of mind/strategies that we can all talk about, things will go down easier. I think we'll probably have a poster up in the classroom (like Daniel Schneider's) that has a list of things to try when you're stuck. We'll start with three: Guess/Check/Generalize, Tinker and Find an Easier Problem.
I fully expect the comments to be awesome on this post. 

Coming up next I'll agonize over whether this sort of problem solving should be an everyday part of my class or whether it belongs alongside other ways of doing things. I'll also post a sample problem set for the first week of class.

Thursday, August 23, 2012

White Paper on Problem Solving: What I did last year

In the first part of this series I defined problem solving as struggling over difficult problems, and I tried to work out what the pedagogical benefits of struggle are. In this post I want to turn a critical eye to the way that I pulled this off last year. Hopefully, I'll end up with a better strategy for this coming year.

But first, here's another cat with a Rubik's cube.

Cats: the silent killers.

What I did last year:
Last fall, two things happened that lead to a change in my classroom.
  1. I noticed that students enjoyed working on my “Warm Up” problems.
  2. I read the book “Drive” by Dan Pink.

What happened as a result is that I put solving problems at the center of my classroom. I felt empowered by how my students enjoyed spending more class time solving problems, and because I read “Drive” I had a framework for understanding why they liked it. I was giving them more autonomy, and people like having choices about how they work.

Here is the sort of thing that I was putting in front of my students with regularity last year:

How proud should I feel of this work?  There are certainly some things that are going right here.
  • This problem set starts with a “Warm Up” section that brings together different problems that connect to the new problem being solved in the “Important Stuff.”
  • The problem set is designed so that students can work on it on their own, and with groups. Kids prefer that to a lecture or an activity that occurs with the whole group. I’ll get more face time with students who are struggling with these ideas, which is also a good thing.
  • The problem set also starts with a question that is pretty concrete and easy, so that the tricky, abstract stuff is built on top of a firm starting point.
But there are also some serious problems lurking under the surface. In particular, I’m doing much of the intellectual work for the students just by sequencing these questions the way that I do. Students knew to expect that there would be a relatively straight line running through the problem set leading to an insight. The result was that a major component of a student’s struggle over a problem was an attempt to figure out the connection between sequences of questions.

Take question 4 in the above document. Just from the sequence of questions students are likely to infer that the way to find the angles whose sine is 0.43 by using the arcsine function to find one angle, and then to find the other using the unit circle.  This kills any chance of multiple approaches to the problem, reduces the difficulty of the problem by many factors, and doesn’t give students a chance to search their memories for a helpful approach.

(I also think that, while it’s admirable that the “Warm Up” contains problems from different topics that connect to the new one, it’s probably a mistake to corner them off into their own section.)

If you see more issues with my approach, please give a shout in the comments.

There’s a reason why I made this problem set the way that I did, though. I was nervous that students wouldn’t be able to figure out the actual new problem of the day on their own. And that’s true. Kids would have gotten frustrated if I just gave them the difficult problem without any context. But the solution that I came up with was to make the problem much, much easier. I need to figure out different ways to keep kids from getting frustrated by difficult problems.

That’s what I’ll write about in the next post. 

Wednesday, August 22, 2012

There is no evidence for the usefulness of math in non-mathematical contexts

"Learning mathematics forces one to learn how to think very logically and to solve problems using that skill. It also teaches one to be precise in thoughts and words. Practice doing that is obviously very useful in many different areas of life." - The Math Forum
Let's make a bold claim. I'm going to claim that there is no evidence that learning math makes you better at other things unless those other things are
  1. math
  2. things that use math, (i.e. they're just math)
I'd love for someone to convince me that I'm wrong on this. Do mathematical habits or skills transfer to other contexts? Drop in the comments if you've got a good argument or some evidence.


Here are some other folks who make claims about the usefulness of math.
"What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis." -Andrew Hacker, "Is Algebra Necessary?"
The argument for algebra rests on the transfer from math to other areas of life, something that has never been proven despite the claims of people such as University of Virginia cognitive scientist Daniel Willingham. -- Roger Schank, "No Algebra Isn't Necessary"

White Paper on Problem Solving: The Why

I'm putting together a short series of posts on problem-solving to get myself ready for the new year. In particular, there are a bunch of changes that I want to make in my classroom and I want to make sure those are properly justified and motivated.

But first, look at this cat:

Such beautiful creatures. But they'll burn you if you're not careful. Anyway,

What do I mean by “problem solving”?
For me it means that students are regularly asked to make progress on questions that they have never been told how to answer. This isn’t an air-tight definition, but it will do for now.

There are a lot of supposed benefits of a problem-solving approach to learning math. Here are a few that come to mind:

  • It’s truer to the work that mathematicians do
  • It’s more fun for students
  • It develops habits of mind that are transferable

I think I agree with the first two ideas, and I’m skeptical of the third, but that's all sort of beside the point. My core responsibility in the classroom is to teach these kids a bunch of skills and concepts in a way that compares favorably to the way they’re learning them next door. If the most effective way to teach is lecturing and drilling, I will teach that way, even if it’s boring and unlike the way that mathematicians work.

The good news is that fun, truth and effective learning coincide in this case.  I think.

I want my students to solve difficult problems in class because I believe it’s the most effective way for them to learn and remember the content. Here are my pedagogical assumptions:

  1. Difficult tasks help organize knowledge: When a person is faced with a difficult task, they search their memory for a way to accomplish the task. They think about the tools that they have and how well they fit the task at hand. This search reinforces a person’s understanding of their tools and how they are used. In math, the tools are the sorts of things we want kids to know: procedures, skills, concepts and habits of mind.
  2. Organization takes the form of connections between topics: It's a pretty solid result that novices organize knowledge by topic, and experts organize them by their underlying structure. A difficult problem doesn’t cue students into the tools that they’ll need to use, and so anything might be relevant. As students attempt difficult problems they need to start organizing what they know into more useful clusters than “first semester” or “lines stuff.” Instead, when presented with an equation they’ll start thinking about what tools they have for solving equations. When presented with a challenging proof they’ll need to think about other problems that they’ve proven in the past and decide which ones are relevant for the current puzzle.
  3. Students will fail often:  Some studies have shown that knowledge sticks better after a person has taken a difficult test and failed. This makes a certain amount of sense – the brain is most attentive when we know we’re missing something. The right answers come in a problem-solving class, but they will always follow failure.
  4. Different approaches invite justification:  It’s helpful for learning to have different approaches to discuss. Multiple approaches create the need for explanation, and explanation and justification also help students organize and remember their mathematical knowledge. When solving a good problem, students will almost never have just one approach. The teacher can skillfully select multiple approaches to bring to the fore.
  5. The mind remembers stories very well:  "If you want to make something memorable, you first have to make it meaningful." But how do you make it meaningful? Stories that connect with the rest of the things that you know can do this. As Dan Meyer has put it, good math stories come in three acts. First comes the hook, where the problem is posed. Then comes the development, where students struggle with the math and run into trouble. Then comes the resolution, which we might talk about as a whole group, or students might discover on their own. The daily story telling comes easily: “Today we tried to do this, and we ran into trouble. Then we discovered X, and then we were able to solve the problem. But what about Y? See you guys tomorrow.” 
These are the things that I think are true. Where possible, I’ve pointed to evidence supporting my assumptions. But they’re empirical assumptions, and I’d feel better if I had more evidence supporting them. Where did I go wrong? Lemme know in the comments.

Coming up in this series I'll point to things that I was doing wrong last year and how I think that I can fix them.

Wednesday, August 1, 2012

Pop Quizzes and Probability

Here's a fun probability problem that worked pretty well in class, and is pretty easy to pull off:

1.  Tell everyone to clear their desks. It's a pop quiz. Use your serious face.
2.  Hand them this:

3.  Read the questions out loud. It gets kind of fun. 
4.  What's the question that everyone wants to know? In my classes, it was "Wait, did I pass?" It was a smooth transition from that question to the question I asked: "Well, what are the chances that you passed?"

Then we're in familiar territory. Take guesses, write them on the board. Take ideas from the whole group. Circulate, offer suggestions, ask questions, give hints as you see fit.

The kids won't let you forget to read out the answers before the end of class.


There was a pretty interesting design question that came up as I was planning this activity: what should the quiz questions be? In order to get random guessing you just need a set of problems that kids couldn't possibly know the answers to. Here were some alternatives that I came up with when planning this problem:

In the end,  I wanted a set of questions that kids could think -- just for a second -- that I actually expected them to know. So I went with some Linear Algebra pulled from a test I took in college. But something else might work for your students.

A quick thought on sequencing within units

Sequencing matters. Sequencing matters within courses. It matters within units, and it matters within problem sets and lectures.

But I think that I've been doing this sequencing thing all wrong. When I plan units I sit down with a blank sheet of paper and try to figure out the best order for kids to learn a topic. I spend a lot of effort trying to turn my unit into a good story. 

I think I'm retiring that way of thinking about things. Here's why:

  • A unit is not a good time frame in which to tell a story. Any tension you "build" in the first lesson is lost by the fifth.
  • Kids don't learn ideas in the order in which it seems natural for me to sequence them.
  • Who says that a story has to be matched with a unit? Maybe these big ideas matter to some lesson months down the road?
So instead of trying to craft my units into well-told stories, what I think I need to do is craft every lesson into a good story, and use my units to develop themes and concepts.

All of these thoughts came up as I was planning my functions unit for Algebra 2. I was stressing out about whether my kids need to know compositions before they can be exposed to inverses, and how I would help my students understand that functions are a whole lot like numbers. But I was all worried, because, while my kids have had no trouble with things like f(g(2)), they've had a terrible time with f(g(x)). How could I preserve the story while also focusing on the skills?

I think that this way of thinking is wrong-headed. Instead of trying to figure out a way to tackle composition and a way to tackle inverses, I'm going to do a series of good problems that have kids interact with both of these ideas at the same time. An earlier day will throw a bunch of functions with numerical inputs at students so that they notice that functions can undo each other, and then they'll try to figure out how to find an inverse of a given function. Next day, we'll use what we know about inverses to bump up the level of sophistication and handle variable inputs, and talk about undoing as an algorithm for finding inverses. We'll tease out a rule for recognizing inverses, stated abstractly. Then we'll use that as our "in" for talking about composing f and g into a new function, and using the ideas of composition of functions with variable inputs to get a second algorithm for finding the inverse.

I guess what's different about this for me is that there's no particular day when I can say, oh, we handled composition today. Learning of any single topic is being distributed over an entire unit. But those units are also connected to each other, so the big themes, ideas and skills need to be distributed in a similar way over the entire course.