Thoughts on Collaboration

It's the first week of school, and I put in some time rewriting my course policies to articulate why, when, and how to collaborate on problem sets.  This issue is particularly crucial in my advanced Geometry class, where students solve college-level problems, not just exercises. Feel free to use some version of this in your own classes, and -- even more important -- to let me know what you think!
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Collaboration, Research, and Hard Problems

Contrary to stereotype, mathematics is best done as part of a community—not alone in a study.  I hope and expect that students will frequently discuss problems, ideas, and solutions in and out of class.  Some guidelines:

·         Make sure that when you are “working with” someone else, you are both really working and contributing.  Contributions can take many forms—clarifying, questioning, justifying, and restating, to name a few—not just coming up with “the idea”, but each of you should leave the collaboration feeling good about what you, and the other person, contributed to the session.

·         One test for how much you did in solving a problem is whether you can reconstruct the entire solution on your own afterwards.  If you need to look at your notes extensively, or get lost in the middle, you probably should have collaborated more actively.

·         One time when it is useful to be alone in your study is to check for your own understanding.  I recommend that students start problem sets by themselves, to see what they can accomplish and what ideas they can generate individually before working with other students.  I also recommend that students finish and write up problem sets by themselves, to make sure that they really understand the work they and their fellow students did together.

·         Give credit generously: it doesn’t subtract from the points you get (and in the real world of “karma points”, giving credit almost always adds to your own).  Write “Wolfgang suggested this auxiliary line” or “Credit to Seraphina for spotting the similar triangles.”  Taking ideas from other people without attribution is plagiarism; taking ideas with attribution is research.

·         Finally, unless specified in a particular project or assignment instruction, please DO NOT do research on the web (or in books, if you still remember those).  The essence of this course is learning to reason mathematically by solving problems and thinking through solutions, not regurgitating theorems and ideas you learned somewhere else.    

Summer Homework Remix Challenge

So I'm just getting back from three weeks teaching at HCSSiM (which in case you don't know about it, is an absolutely awesome summer math program for high school students), and one thing that really leapt out at me was the number of students trying to cram in several hours of summer homework on top of (literally) 45+ hours of mathematics each week.  (You read that right: four hours of class each morning, quasi-optional lecture at 5pm, three hours of problem session from 7:30-10:30pm.)

This situation is appalling.

First, the kinds of high-performing kids who go to HCSSiM are working harder than ever during the year; they need a break from regular work.  Second, the work that gets sent home is--in general--the worst sort of homework you can imagine.  A sampling:

• Outline n chapters of a bio/history/government textbook, where n ≥ 3. My experience is that few kids, if any, are taught how to outline, so these wind up being lists of section headings.  Teachers typically don't even read or give feedback on this work.  I wonder, too: if a kid can get "enough" out of this type of learning experience, what do teachers think the are adding in the actual classroom?
• Write out, by hand, 100 vocabulary words and definitions.  (My colleague and friend Erica, soon to be a teacher in a Massachusetts middle school, asked "If the work is so low-level that it's impossible to tell whether one kid has copied another's assignment, why would you even assign it?"
• Read a 400 page, mostly stream-of-consciousness novel chosen by the teacher, with little or no guidance.
• Fill out dozens of worksheets practicing math facts, or vocabulary from a foreign langauge, or ...

Assignments like these replicate the worst parts of the school experience--repetitive, low-level work with little attention to context or purpose besides "get it done"--without any of the other experiences that can make going to school worthwhile.  What's more, they put this school-sucked-dry experience into the summer, which kids typically enjoy.

Why do kids who enjoy learning prefer summer to school?  I can think of three reasons:

1. During the summer, kids can choose what they will do and when they can do it.
   
   Of course, choice is more satisfying than constraint.  But study after study has shown that giving kids choices about what learning activities they do increases student engagement and the activities' effectiveness.  So taking the choice out of summer learning activities is a double-whammy.
2. During the summer, kids do things that they find relevant.
   
   In actual summer academic programs--like the ones my kids and my friends' kids do--the learning activities are focused on things kids actually want to know about.  And kids don't choose to do things that they find irrelevant.
3. During the summer, kids do things that are challenging.
   
   Have you seen a kid spend hours practicing a skateboard move, or throwing a football, or playing a videogame?  Kids don't do things they find easy--they find things that are at a "can't-quite-do-it-yet" level, and when they've mastered something, they move on.  

By contrast, the summer homework assignments listed above involve no choice, make little attempt to be relevant to kids' own interests, current issues, or anything else of interest, and are very much one-size-fits-all (in fact, because no teacher is available, they're usually way too easy--which reinforces the idea that they're mostly "get-it-done" work, not an opportunity to actually learn anything).

So here's my remix challenge, in three parts.

I.  Find an actual summer homework assignment passed out to kids in grades 6-12.
II.  Rework it to fit the three criteria--choice, relevance, adaptable challenge.
III. Post it here, in the comments, or via email to me at pjkarafiol@gmail.com.

Three guidelines:

• Obviously, the made-over assignment must address many of the same objectives and issues as the original.
• Obviously, the assignment must be one that students can complete, with some reasonable degree of success, on their own.  The assignment doesn't have to be easy, but (e.g.) feedback might be built-in or easy to obtain.
• Third, the assignment shouldn't take more than 4-5 hours of work, 8 tops.  Really, guys, this is the summer.  If you want the kids to attend summer school, teach an actual class.

As extra credit, ask a student to suggest a summer homework "makeover" of their own.

Here are two to get you going:

1.  For the AP Biology Chapter 1 assignment on basic physics and chemistry: read two articles from one of the following periodicals (Scientific American, New York Times Science Tuesday, Discover Magazine) written within the last year about a discovery or problem in biology.  For each, identify what chemical or physical processes are described in the article, and be ready to give a short (3-5) minute presentation on the underlying chemistry or physics described.

2.  For the vocabulary list -- given the same list of words, find instances of 20 of these words in recent writing (last five years) on the web, in periodicals, or published books.  For each, give the surrounding paragraph, explain what the word means in context, and write a sentence or two evaluating whether the author should have chosen a less-esoteric word (with a suggestion).

Happy end of summer!

PS: Full Disclosure -- I havea a summer assignment of my own, but it involves choice, is not onerous, and shouldn't take more than 3-4 hours to complete.  Here it is:  http://www.wpcp.org/StudentLife/SummerAssignments.aspx