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Students learn in different ways:

- Some like to have a clear roadmap and to gradually fill in the details.
- Some like to build up small chunks of understanding, gradually building up bigger and bigger chunks.

What characterized professionals is the expectation that there is a whole understanding to have and the constant attempt to fit a new understanding into a bigger picture. Professionals also have a big picture to put their new understandings into and/or to compare their new understandings to. (Is there an MPEX category for this.)

Even for experts, the big picture understandings differ from person to person. This diversity is what gives society the ability to move forward more quickly than the individual can move forward.

We help students build to a professional big picture by building in the constant opportunity for reflection at many levels.

Ways in which students don't know how to be reflective:

- have the habit of constantly asking what the significance of a result is
- ask themselves what an equation is telling them about the relationship of physical quantities to other physical quantities
- ask whether the dimensions of an equation or answer make sense
- automatically recognize and name functional relationships (is 1/r a linear relationship since the r in the denominator is to the first power?)
- ask themselves what physical relationship is being represented when they look at a graph.
- multiple representations (Kerry's results)
- same mathematics in different physical settings (g vs. E, eigenstates, concepts related by differentiation or integration, linearity and superposition)

Ways for students to reflect:

- Small white board activities
- Concept maps
- Breaking small group activities into small pieces (Katherine's project)
- Wrap-up to small group activities
- Reflection on homework: “What does the solution of this problem tell you about the physical world that you would like to remember?”
- Writing (Drew's project, student quote)
- Students working together outside of class
- Study for exams
- Go over exams

An analysis of six groups working on a particular upper-division E&M problem found that most students at some point in the problem-solving process generated an incorrect equation. However there were some students who recognized their errors and others who did not. In every case that students accepted their incorrect equation and went on to use that incorrect result elsewhere in the problem, the students had not done a dimensional analysis nor stopped to think about the physical meaning of their equation. In every case where students checked the dimensions and/or checked the physical meaning of their result, they did not “settle” on incorrect answers and continued to work toward a correct solution. this illustrates the importance of helping use these tools to become successful problem solvers.