The difference between given and found problems

There are all sorts of subtle ideas that lurk behind the veil of explicit communication. Every once in a while, it's worth taking stock of what those subtle ideas are to make sure that we're communicating wisely.

Every day, I give my students problems to solve. When I do that, I'm also telling them:

• "This is a problem that you can make progress on."
• "This is a problem that it is worth your time and effort."
• "This problem is new and not the same as what you've studied before."

Because I teach my lessons in units, there's an additional implication present:

• "This problem is connected to what we studied yesterday."

Are these things that I want to be communicating to my students? I don't know. (Those wiser than me will hopefully chime in on this.) But I do have a few observations to make:

1. All of these implications make the problem easier to solve. My assurance that the problem is solvable and that it's worthy of their time helps them take the leap into the problem with confidence. The implication that this question has something to do with what we have been studying significantly narrows the available tools and techniques to choose.
2. This is fundamentally different from the way problems are outside of the classroom. When I find a question that I want to make progress on, I have no assurance that it has a good or interesting solution or that I'll be able to do it. I have no idea what tools I'm going to need, and making those decisions is part of the difficulty of the problem.

What do we do about this? What can we do about this? We can try to present problems to students that are more akin to how they're found in the world, but the mere fact that we're offering them invests the problem with all of the implications detailed above. If we want to eliminate the suggestion that a problem is connected to the previous day's ideas we could eliminate units and integrate our topics more densely. 

Other than that, to really help students solve problems as they're found out there, we need to create more opportunities for them to find problems on their own. 

Not like I do that, but hey, aspirations, right?

Harvest Collegiate is a Pretty Awesome Place

Stephen Lazar is on my must-read list. Last semester he taught a course called "Looking for an Argument" at the new Harvest Collegiate High School.

By his description, it sounds like it was an amazing course:

At Harvest, Looking for an Argument gives a government credit, and all 9th graders take it during the year. The structure is relatively simple. Each week focuses on a different controversial issue. Ours’ ranged from the NYC Soda Tax to the presidential election to Stop and Frisk. The week starts with two teachers debating the issue. The students choose who has which side, establishing that the class is not about being right, but rather about constructing the best possible argument.

And here's why it worked:

 Part of the genius of the course is its simplicity. While provocative topics keep the students engaged week to week, students are practicing only four core academic skills — note-taking, reading with annotation, self-reflection, and timed argumentative writing — over and over again. While students do gain a tremendous amount of knowledge about the world through a variety of topics, that knowledge is never assessed; it’s all about the skills.

Harvest Collegiate does the "one-semester themed courses" in all subjects, including math and science. They have their course list posted online, and the three courses listed for math in the spring semester are:

• Algebra
• Global Youth Trends
• Designing Harvest City

In the course descriptions you get the sense that they're struggling to make these things work with the state standards, which must be an immense challenge. I'd imagine it's the largest impediment to the school experimenting with anything as radically skills-based as the "Looking for an Argument" course.

Which is a shame. I bet that there are all sorts of amazing courses we could design for kids if we had a bit more freedom. I know that Henri Picciotto has designed a whole smattering of math electives. In Brooklyn, Saint Ann's school offers all sorts of stuff, including Mathematical Art. 

Let your imagination go wild: if you were allowed to spend a course focusing exclusively on mathematical skills, what would it look like? Would you spend each day solving different problems? Teach a subject that's normally left for college?

My quick list of dream-courses looks something like this:

• Paradox, Self-Reference and Philosophy of Math
• Prime Numbers
• Biology and Math

All of which are pretty blatant attempts on my part to imagine if I could try out my most positive mathematical experiences on students.

I'd love to hear what you've got in mind.

Something for me, something for you

There have been a series of really good posts recently about the teaching profession. None of the posts really got at the pressures that I'm feeling, though, so I wanted to take a stab at telling my story.

Freshman year, I was enrolled in Philosophy 8, and I got a B- on my first paper. I was a bit taken aback, since that was not the feedback I was expecting on the paper.*

* The words "See me in my office" are scrawled across the top of the page. Cautiously, I knock on the professor's door in Emerson Hall. She tells me to come in. "Sit down, Michael, please." She explains how my paper has changed her views on Descrates' skeptical argument in his First Meditation. "Have you considered a career in philosophy?" she says. "Really, you must. The field needs you, Michael."

I really wanted to do better on the next paper. I was taking a walk between buildings, and trying to run through Descartes' argument for the existence of the external world. My mind was fuzzy and distracted, but I pushed the noise away and I was able to see the argument in my mind.

By pushing away the distraction I got myself into a sort of locked-in focus. The moment was one of clarity. I held the argument in my mind for the rest of the day, and wrote the paper, which ended up getting great feedback.

I come back to that moment often because it was so satisfying. There was a challenge, a breakthrough, followed by a well-earned clarity. As a student I tried to push myself toward those sort of moments. There are a handful more times that I had those sort of breakthroughs*, usually as the result of long walks where I tried to train my thinking on one particular problem for an extended period of time.

* By the way, by "breakthrough" I just mean that I understood something that I was supposed to, for a class or a paper. I don't mean that I made lasting contributions to anything except my own understanding of some subject.

That rush is all about feeling a challenge and just plowing right into it, and pushing past it. It's one of the most satisfying feelings that I know.

Dan and Patrick went back and forth last week about whether teaching remains a challenge for good teachers. My metric for how the "challenge" afforded by an activity is the frequency with which it can create those breakthrough moments for me. And by that measure, teaching is just doing OK, not great. Yeah, it's challenging, but not in the ways that create those moments.

I don't know why this is. Part of it is that I usually don't have time to dwell on an idea for long enough to really tackle it. Because of the quick turnover time of units and lessons, I often find that my best ideas come quickly and accidentally, instead of through deliberation and purpose. I struggle to construct my schedule in a way that gives me more time to work on long-term problems or projects, but (so far) that itself has been frustrating. And those breakthrough moments, when they do happen, are irrelevant to many of the partners in my students education.*

* "So why not get yourself in grad school or something? The teaching profession isn't about making sure you get your highs in life." True. Nobody needs to bend the profession to meet my needs.

But there is a sort of deep satisfaction that I get out of my job that has nothing to do with intellectual breakthroughs. I really love teaching because of students. I love that good teaching requires me to make myself invisible. I love how teaching requires patience and a deep concern with the ideas of others. I love how it's all about helping people get excited about things. I love how it's about community. And I find all of these things completely satisfying.

So, in short: the pressure I feel is that I like teaching because I think that it's a fundamentally good activity, but that there's something that I get out of it also. Every once in a while this work gives me those breakthrough moments, when I'm able to plan a unit that I'm really proud of, or design a lesson that nobody else has ever thought of. I'm able to design a classroom that works. At these moments, teaching is the perfect job. It allows me to help others, while indulging my addiction to flow.

But, overall? Teaching requires a great deal of professional selflessness, and offers me a lower rate of selfish intellectual indulgences than I experienced as a student. That's the pressure that I feel.

I apologize -- at the end of this post, I'm starting to think that it needs a few more minutes in the cooker. (Which I can't afford, because of grading, lessons, units, see above.) But I think the takeaway is that since teaching requires a certain degree of selflessness from me, I need an extra bit of selfish satisfaction from it. If I had a job in finance, I'd be well-paid and have plenty of time to pursue my own intellectual interests and challenges. If I had a job as a doctor, I'd have good job security, a great deal of respect from peers, and time and energy for the myriad things that could give me some sort of selfish satisfaction.

But I work nights and weekends, am earning 5 figures for you know FOREVER and my friends, parents and students constantly ask me what I'm doing teaching when I could be doing X. So I need a little bit more from teaching than it's giving me right now.*

* But, like I said, it's no one's job to give me a dream job. Hence all the hand-wringing about how to get better, and the interest in Cal Newport, etc.

Strong Kids v. Weak Kids

Me first:

It pisses me off that the reward for experience is teaching smarter, easier kids, and that I'm going to take that reward, before long.
— Michael Pershan (@mpershan) February 13, 2013

Some people agree:

@mpershan Not only will be "easier", but you'll get paid more for it too.
— Jim Adamson (@GlenviewMath) February 13, 2013

@jybuell @mpershan I agree! Most tchrs of honors kids are fully convinced they are fabulous tchrs. It's a long day tchg all "low" classes.
— s reilly (@reilly1041) February 13, 2013

@mpershan @jybuell I teach both. Smarter kids are easier and more fun; needier kids are more rewarding.
— Chris Shore (@MathProjects) February 13, 2013

@mpershan I agree with you about this 100%. Behavior is NOT achievement.

— CheesemonkeySF (@cheesemonkeysf) February 13, 2013

@mpershan Didn't we have a big discussion about this a while back? It's incredibly difficult teaching weak kids. How is this not a given?
— EdReal (@Ed_Realist) February 13, 2013

But, tons of folks disagreed:

@mpershan @ddmeyer "Smart kids are easy to teach" is more teacher prejudice than classroom reality, and a disservice to "smart" kids.
— Patrick Honner (@MrHonner) February 13, 2013

@mpershan the reward is "choice". Don't think it's easier, but probably easier to fool yourself into thinking you're doing a good job.
— Jason(@jybuell) February 13, 2013

@mpershan I also hate that people view "smarter" as "easier".
— Greg (@sarcasymptote) February 13, 2013

@sarcasymptote @mpershan Compliant kids are less "trouble" which makes for an "easier" classroom. That has nothing to do with teaching.
— Andrew Carle (@tieandjeans) February 13, 2013

@mpershan Teaching honors is awesome! But teaching weaker students can be, too. Take a step back. All they need is a different approach.
— Adrienne Shlagbaum (@shlagteach) February 13, 2013

@sarcasymptote @mpershan At my sch, diff. behavior issues with diff levels. More cheating at higher levels, more class disruption at lower.
— Fran Poodry (@MsPoodry) February 13, 2013

Can someone explain what's going on here?

Can anyone explain how there's a disagreement this wide across the profession? Why does it seem straightforward to me that teaching students of low ability is harder, more challenging than teaching students of high ability? Why does it seem straightforward to others that this is a pernicious belief that ought to be challenged?

The Hard Parts

I have about 3 hours a day that I spend planning lessons. I'm currently trying to figure out ways of incorporating hard practice into my professional life. Last week, I experimented by allocating half of that time towards careful thinking about what students would struggle with during my lessons. I write these things down in a document that I call the "hard parts" document.

Overall, I was happy with the first week of this experiment. My lesson quality was better than what I had been averaging in the first semester, even though it was a tough week for me (I was sick and didn't sleep much from Monday to Wednesday). Focusing on the hard parts of each lesson felt difficult and required constant mental effort. Forcing myself to write down observations made the whole process visible, and kept me from getting in a rut. 

Here's a selection of my "Hard Parts" document I produced on Tuesday, a day that went very well for my first period students.

 
Concrete results came out of my time spent thinking about the most difficult aspects of my lessons. It was only on reflection that I realized that my students didn't understand the utility of closed-form equations for patterns, and I successfully created a plan that attacked that issue. It was during my reflection on what's hard about the unit circle definitions of sine and cosine (for my third period students) that I started worrying about the differences between heuristics and procedures. The solution: instead of directly having my students use the special right triangles to find values around the unit circle, I would start by having them estimate the values of a new function, s(theta), around the unit circle, and only make the connection with exact values later.

Unlike Tuesday, Thursday's class was a disaster for first period, even though I also devoted substantial time to my exercise.

The core of the lesson was the "Skype monthly plan" problem:

 Given the above, how much do you pay per month, and how much do you pay per minute? I thought that this problem would be productive, though difficult, for my students. 

Here's the "Hard Parts" document that produced this lesson plan:

So, what went wrong on Thursday? If I did the exercise faithfully, how did my lesson get so muddled?

A careful read of my "Hard Parts" document reveals some issues with my planning:

• My goal subtly changed over the course of my planning. I had just a fuzzy idea of what I wanted class to be about at the beginning. I originally was going to focus class on a problem involving plumbers. The difficulties that I began thinking of for the "plumber problem" were no longer relevant by the time that I had chosen the "cell phone problem" instead.
• Because I didn't focus on the "cell phone problem" I completely missed a student issue that I should have anticipated -- they didn't understand how monthly + per-minute fees worked. That could have been easily taken care of by giving them a chance to evaluate how much you would have to pay under various given plans. 

• The students also didn't get that November and December referred to the same plan. I could have anticipated that -- it's a misconception that I've seen before, and this is a relatively weak group -- but I didn't devote time specifically to the "cell phone problem."

What I'm realizing is that I have to be careful about plans or problems that coalesce toward the end of my allotted focusing time. Because I'm at the end of my time, and because I've already devoted a lot of thought toward the lesson, I'm tempted to skimp on mental effort for the actual problem that I want students to work on. That mistake can be a fatal one for my lessons.

I can help myself more by trying to come up with a rough plan on the weekends for what content I'd like to cover over the week. That's a bit of planning that I had been doing, but let slide. I think that will allow me to start each day with a bit more focus, so that I can spend the planning time anticipating issues with the day's core exercises.