Amanda has been doing the Art of Problem Solving Pre-Algebra course. It's about the opposite of, "do 100 problems almost the same using this technique. Now do 100 more problems on the next technique." They stress problem solving, applying techniques in creative ways, and learning through struggling with the topic. To begin each chapter they give 5-6 problems that illustrate the topics in the chapter. First the student struggles with the problems and maybe (or maybe not) discovers the methods used in the chapter to solve the problems and why they work. Then the student reads the chapter to connect what she's learned through struggling with the mathematical theorems.
Some of the problems are really hard. Sometimes Amanda sits staring at a problem not doing anything. I've encouraged her to write something and she seems to be picking that up. But it's not always obvious what to do so sometimes I help her to start.
The problem is that sometimes she's just on the cusp of solving a problem and my help actually detracts from her own solution, confusing the situation. Sometimes my method isn't her method and gets in the way. Sometimes she's just not ready. So I need to do something different.
What I've worked out is that I can literally bite my tongue as she does the problem. I can nod and say, "mmm-hmmm" when she's right, or signal that she's made a mistake. But I don't give her any directions. And she (often) solves the problem.
It's hard for her and hard for me. Someone once said that the butterfly needs to struggle while getting out of the cocoon: if you help it, its wings don't fill and it will die. But (in the case of children) struggle sometimes seems like a bad thing. I have to remember that the struggle is part of the process, just as much as the answer. Maybe even my struggle to remain silent is part of the process.
Some of the problems are really hard. Sometimes Amanda sits staring at a problem not doing anything. I've encouraged her to write something and she seems to be picking that up. But it's not always obvious what to do so sometimes I help her to start.
The problem is that sometimes she's just on the cusp of solving a problem and my help actually detracts from her own solution, confusing the situation. Sometimes my method isn't her method and gets in the way. Sometimes she's just not ready. So I need to do something different.
What I've worked out is that I can literally bite my tongue as she does the problem. I can nod and say, "mmm-hmmm" when she's right, or signal that she's made a mistake. But I don't give her any directions. And she (often) solves the problem.
It's hard for her and hard for me. Someone once said that the butterfly needs to struggle while getting out of the cocoon: if you help it, its wings don't fill and it will die. But (in the case of children) struggle sometimes seems like a bad thing. I have to remember that the struggle is part of the process, just as much as the answer. Maybe even my struggle to remain silent is part of the process.
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