Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Defining Problem Solving
|Introduction | Connecting to Other Problem-Solving Experiences | Teacher's Role | Monitor and Reflect on Problem Solving | Providing Rich Problems | Your Journal|
"Good problem solvers become aware of what they are doing and frequently monitor, or self-assess, their progress or adjust their strategies as they encounter and solve problems (Bransford et al., 1999). Such reflective skills (called metacognition) are much more likely to develop in a classroom environment that supports them. Teachers play an important role in helping to enable the development of these reflective habits of mind by asking questions, such as 'Before we go on, are we sure we understand this?' 'What are our options?' 'Do we have a plan?' 'Are we making progress or should we reconsider what we are doing?' 'Why do we think this is true?' Such questions help students get in the habit of checking their understanding as they go along. This habit should begin in the lowest grades. As teachers maintain an environment in which the development of understanding is consistently monitored through reflection, students are more likely to learn to take responsibility for reflecting on their work and make the adjustments necessary when solving problems" (NCTM, 2000, pp. 54-55).
Often, teachers can talk aloud to share their thinking as they solve a problem. This can help students find words to express similar thoughts; it's also a way to review important concepts, spark ideas, and demonstrate that even teachers take some time to solve problems. While "thinking aloud," it is important for teachers to verbally express their ideas and in particular to show how they've modified their ideas or changed their approach. Reflection on the validity of a conjecture or a particular approach in solving the problem is also beneficial. Generally, a persevering outlook toward a significant problem will always be helpful.
The following example shows how one teacher shared her thoughts and choices as she worked on the following problem: If tortillas come in packages of 35, and people eat an average of 3 tortillas each at a school supper for 200 people, how many packages of tortillas should be purchased?
Teacher: "I see that each package will take care of almost 12 people, because 3 times 12 is 36. But it's hard to figure out how many groups of 12 people are in 200 people. Instead, I'm going to figure out the total number of tortillas first, because that's how many tortillas I'll need to have in the packages. So, three tortillas each for 200 people is 600 tortillas, because 3 times 2 hundreds is 6 hundreds. I'll make a diagram to help me think about the packages.
"I can make a guess and then check it. Let's see -- if every package has 35 tortillas, I'll try 10 packages. But 10 times 35 is 350; that's too low.
"I'll put in four more packages. Thirty-five and 35 is 70; 70 and 70 is 140 more tortillas. That makes 490 tortillas. I know that two more packages is 70 more, and that gives me a total of 560. Just one more package makes 595. But to really have 600, I need one last package.
"Let's see -- I have 10 plus 4 plus 2 plus 1 plus 1 packages. That's 18 packages. This seems like a division problem; I could have divided 600 by 35 on paper, or I could have used a calculator to divide 600 by 35. (Uses a calculator and writes 17.142857.) That's right; it's a little more than 17 packages, so 18 packages is correct."
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