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?