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Worthwhile mathematics tasks are those that get students to think about mathematics that is interesting, useful, or beautiful. The mathematical world is interesting enough, sometimes.
It is a world of thought, or visualization, and representation, and then bringing those representations into being, through speech, and on paper.
For over 30 years, with NCTM reforms, we have been talking about “teaching through problem-solving”, rather than teaching *about* problem-solving. I am not sure “teaching through problem-solving” is all that well-defined.
We who are, broadly speaking, constructivist teachers need to answer for how kids will do enough math, to be good at math, in problem-solving classrooms, but that’s a topic for another post. Somewhere, on a dusty shelf, you might have an old binder full of problems.
While you can get students thinking about division as fair sharing easily enough by posing a cookie problem, you could probably also activate their thinking about that same idea in a more interesting way.
Sometimes we mathematics teachers treat our subject like parents treat cough medicine: something that goes down bitterly, but is necessary for our health. Likewise, we are not talking about pointlessly sweetening or watering down our subject matter.
Think about a young child trying to accomplish a goal. Even if it’s a long crawl across the room to pet the cat, they keep going.
Even if there’s a latch on the kitchen cabinet so they can’t open it, they keep trying to get to the pots and pans inside.
From there, students learn how to work together and problem solve issues as the arise.
The video shows segments of several problem solving class meetings in which students problem solve their classroom issues collaboratively.