At the beginning of each lecture a professor makes a choice: S/he can choose to make the material presented clear and understandable for the students, or s/he can allow the material to be nearly incomprehensible. Making material understandable often requires a great deal of work and preparation time, but this effort pays off by improving student comprehension of the concepts taught. After a good lecture, students are better equipped to understand and learn from the text book and related homework problems.
Many techniques can be used to make material more understandable. Five of the most important include: justification, review, repetition & examples, introductions, and sub-concepts.
In order for students to understand material, it must be justified. Why is the material important? How will it be used? Why is this new technique better than the technique we learned last week? Justification is important because it gives students a reason to listen to you.
If a new technique will make it easier to solve a certain class of problems, show how difficult it is to solve one of those problems without the new technique. If the material relates to a totally new class of problems, show why that class of problems is important. If the material is related to something currently active on the news, show the relationship. Do everything possible to show that the material is important, relevant, and useful.
Any new concept will rest on a foundation of previously learned ideas and terminology. Review is important because it brings to mind the important concepts that will be needed to understand the new material. Review also helps to link new material to old.
If the topic you are about to discuss requires the use of several vocabulary words, review them and give an example of each. If you are going to use previously learned but unique symbology, review the symbols and their meanings. At the beginning of each class, briefly review the content of the previous class, simply to reload it in your students' minds. If you are going to present a new idea or technique, find the three or four important concepts that are required to understand the new idea, and review those concepts. These three or four background concepts set up a context for the new material. They also bring all relevant information forward so that it is fresh in each student's mind (making it much easier for the student to see and form connections between new and old material). Review shows that the new material is a logical extension of previous knowledge.
Many professors see review as a waste of time: "They should know this material; why should I review it?" Review is important because human beings cannot remember everything they have learned instantaneously, especially if the knowledge is a year or two old. If they don't recall or understand a key concept or technique, then everything you say in class is meaningless, and will not be understood.
People learn through repetition. No idea or fact is completely understood the first time it is seen. For complete understanding, the idea should be shown in a variety of contexts and applications, and should be connected to as many previously learned concepts as possible.
One way to provide repetition is through examples. The first examples should be easy, to make sure students grasp the idea completely and to provide confidence. More advanced examples can show the use of the idea in different contexts. At least one example should show the boundary of the concept (the point where the concept breaks down or no longer applies), because this boundary is either a position where a different concept takes over, or a position where research is currently active. When possible, examples should be spread out over multiple class periods so students have time to digest the material and generate questions.
Many new ideas are, initially, simple and straightforward. They become complicated only because they have been expanded (through research) over a long period of time. When this is the case, introduce the idea so that students can see it is simple. Then add complexity incrementally, occasionally reminding students that the original idea was simple.
A good example from computer science is a concept called "hashing." Hashing is fundamentally simple, but it has expanded to the point that it is now reserved until the junior or senior level. A teacher could easily make hashing seem impossible because of its complexity and multiple variations. But the teacher could also introduce hashing by showing that it is extremely simple, and then add complexity incrementally over several class periods. In the latter technique, students will understand and feel comfortable with the concept; with the former they will tend to understand far less, and will feel intimidated by the material.
Some ideas are fundamentally simple, but others are inherently complicated or messy. Complex concepts require that a number of different techniques and ideas all be understood together before any part of the new concept can be grasped.
When trying to present a complicated concept, it is important that students know and understand each sub-concept individually. This can be done by deciding what the important sub-concepts are, and then presenting each one as a separate topic. The complicated concept can be introduced by referring back to (reviewing) these previously understood sub-concepts and bonding them together into a unified whole.
It is important to tell students you are doing this. For example, you might start by saying, "We are going to begin talking about idea X. To understand it, you need to understand the sub-concepts A, B and C. I will present A, B and C first, and then bond them together to form X." As ideas A, B and C are presented, you can show how they relate to X and to each other--on a complicated subject a lot of repetition is required, so you might as well start early.