Consider the data below.
How could you use this data to forestall failure and make the problem solving process transparent to students?
Consider the typical errors students make in problems solving according to Linda Acitelli, Beverly Black and Elizabeth Axelson from the Centre for Research, Learning and Teaching at the University of Michigan.
Sources and Types of Errors in Problem Solving
Inaccuracy in reading
reading the material without concentrating strongly on its meaning
skipping one or two unfamiliar words
losing one or more facts or ideas
failing to reread a difficult section
starting to work the problem before reading all of the material
Inaccuracy in thinking
starting to work the problem before reading all of the material
not placing a high premium on accuracy (above speed or ease)
not taking enough care in performing some operation
interpreting words or performing operations inconsistently
not checking a formula or procedure when feeling some uncertainty
working too rapidly
drawing conclusions in the middle without sufficient thought
analyzing problems inactively
failing to break a complex problem into parts; not using the parts that are understood to figure out more difficult parts
not drawing upon prior knowledge and experience in trying to make sense of ideas that are unclear
not defining key terms correctly
not actively constructing a representation of ideas on paper (when a representation would have helped in understanding the material)
lacking perseverance
lacking confidence and giving up easily
choosing an answer based on a superficial consideration of the problem (having a feeling of what might be correct--guessing)
solving the problem in a mechanical manner without much thought
reasoning the problem part way through, giving up, and jumping to a conclusion
using the "one shot" approach in solving the problem and when that doesn't work giving up
Students need training and practice in problem solving ...Your goal is not to show them how to do problem number two, but to teach them how to go about solving problems and how to think while solving problems. In other words, you must get students to do the thinking and help them modify their thinking by having them slow down and use good problem solving techniques. Some ways to get students to slow down and reflect on their thinking processes when solving a problem follow:
Have students read the problem aloud and tell you what is needed to solve it before they start to work on the problem.
Get students to work problems while "thinking out loud." Encourage students to constantly talk about what they are doing and why. This will slow down the thinking process and make it more explicit and more accurate. You can often help students check their own reasoning and find their own mistakes by having them express exactly what they know about a problem. Comments or questions that can help students clarify their thinking might include some of the following:
What are some possible ways you might go about solving this problem?
Tell me what you know about the problem.
How might you break the problem into small steps?
Please tell me how you got from step one to step two?
What are you thinking right now?
I don't understand your reasoning behind that step. Will you please explain?
Sometimes you will find it necessary to model good problem solving techniques. You may have to demonstrate how you would go about reading and understanding a question before starting to work the problem. You may show how you would solve the problem, making the process clear to the student (e.g., working step-by-step, backing up if necessary when things don't work out, breaking a complex problem into parts and using the parts you understand to figure out the more difficult parts, actively constructing a representation of ideas on paper, etc.). After modeling the process, require students to work through a similar problem to make sure they understand the process.