## Sunday, February 3, 2013

### Homework Problems

Earlier this week, a friend told me about an awesome problem her daughter (third grade) had for math homework.  Loosely paraphrased, it runs like this:
Annabel and her friends share a pan of brownies.  Everyone gets the same number of brownies, except that Annabel gets two brownies less than each of her friends.  If they have 33 brownies, how many friends does she have?
There's so much in this problem:  opportunities to try possible solutions, develop conjectures, figure out possible intermediate stages, practice multiplication and factoring, and think about functions.  There's just one thing: it's not a homework problem.

Why not?  First of all, while it may use skills students have already developed, the context is new.  It's an extension--in several different directions simultaneously--of ideas about products and factoring.  And there's no obvious place to start working the problem, because there are two unknowns--that is, both the number of brownies in each share and the number of friends--and givens from which it's hard to work backwards.

What is homework good for?  In Classroom Instruction that Works, Marzano points out that homework can help achieve three goals:

• Improve learning of specific skills through practice.
• Develop of self-discipline and independence.
• Improve metacognition, in this case, awareness of what the student does or doesn't know.
But these three goals are only served if the task is something on which students can reasonably expected to make progress working independently.  My friend tells me that, in her child's class, every child got help from parents, who were often at least as flummoxed as the child.  In the classroom, I would use this problem to teach problem-solving strategies:  using systematic trial-and-error to develop observations, conjectures, and theorems; solving the given problem; and then developing generalizations of the problem.  But it's not reasonable to expect any of that to happen at home. And so we achieve the opposite of our original goals:
• Because there's little or no repetition on the child's work (which would happen if the child could work on multiple cases), there's little opportunity to practice specific skills.
• Kids learn to rely on their parents to interpret and answer questions on homework.
• Kids don't get a sense of what they should reasonably be able to do on their own; they do learn that complex tasks are impossible for kids to do.
I'm not advocating a return to long worksheets containing dozens of instances of essentially the same exercise--those only kill the joy of mathematics, and teach kids to stop thinking.  In the best-case scenario, such sheets force kids who are pretty good at a skill to practice it so much that it loses all interest.  Of course, if the kids aren't competent at the skill, they just practice it wrong--or learn that there's yet another thing they can't do.  So those old worksheets aren't the answer.  But problem-solving is a collection of skills and habits that have to be developed--before students are asked to produce them on homework.

The key aspect of homework--the one this assignment overlooks--is that it's something kids should be able to do and learn from at home. What can be done at home?  There's no single answer that works for every subject and every class.  But a first cut might be to ask the following questions:
• What skills are required to be successful on the homework?  Have those been taught in class?  What evidence is there that kids can apply them successfully?
• What habits does the homework build on?  What evidence is there that students in the class already have those habits?
• If a student is stuck or discovers he/she is mistaken, what resources would be necessary to unstick the student or work through the misconception?  Are those resources available at home?
• How can students monitor their own success or failure through the assignment?

1. I suppose it depends if you want students practicing an algorithm or practicing their problem solving. To practice problem solving you need something new.

But the other principles you talk about still apply, support, feedback, metacognition all apply. I do think we need to be clear with students about what they're learning or practicing.

2. Seems like there was either a pretty unusual shape to the pan, or a pretty unusual way of cutting the brownies if you ended up with 33. The wording is a litte clunky as well.

Otherwise I think it is a reasonable question to ask a third grader to explore at home. All the better that it is an easily explained problem that the parents have some trouble figuring out - what a nice opportunity to learn something with your child. I certainly wouldn't focus on whether they figured it out or not.

I really don't agree with formal homework in third grade, especially "practice" type of homework.

3. Well, I actually mostly agree with both of you. @John, I think the key phrase is "practicing problem solving." If the kids haven't done *problems* (versus exercises) in class, then there's nothing for them to practice ... which given what my friend told me, sounded like what was the case here.

@Hodge, clunkiness was mine. I think it's a good problem to explore, but I think in class is a better place to explore it until the kids get used to exploration-type of problems. One thing I've seen (at all levels of K-12 math) is teachers giving lots of traditional practice-y type work in class, and then slamming kids with exploratory problems at home (heck, I've done it myself), and then the kids simply don't know what's even a good place to start or what's expected of them.

My sense is that three problems of a given type is probably enough practice for a single night, so I would try to avoid giving more than that .... just clarifying that I'm not a big fan of the "piles of practice" homework assignments, either.