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Teaching: Generalize and abstract

Depending on time available to answer the motivating questions from the Summary phase, you might divide the questions among different groups, or have each group (or perhaps the class as a whole) take on every question. Different groups might consider the questions in different orders. You can get ideas for how such discussions might be structured by revisiting Applications, Algorithms, or Abstractions from the Navigation window.

Questions about other applications of spanning trees can be answered by having students look at problems in Applications. Keep pushing students on this tour to think about and pursue the In general questions in the list that lead to study of the concepts emphasized on this tour.

At least one In general question will motivate touring Algorithms. As needed, remind students of what's meant by algorithm, and encourage them to examine increasing levels of detail in the algorithms that are presented. For help in developing algorithms, refer students to Applications, and have them play with lots of examples they make up, as in Abstractions. Have them test their algorithms on models of the other problems in Applications.

Many In general questions concern the mathematical ideas themselves, independent of applications. Finding properties of the mathematical objects being studied can lead to appreciation of the beauty of mathematics and can prove useful in solving problems. This is the approach that is developed in Abstractions. Try to get students to develop conjectures as they tour here, and to give convincing logical arguments for them, perhaps supported by algorithms.

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