The "Lecture Method" in the math classroom is a means by which "the expert" presents the material of the course in an organized way to "the learners", going from theory to examples and back again. The learners typically take notes and try to pick up as many ideas and insights as they can from the expert during those class hours when the method is in use. In math classes extensive use of a blackboard during a lecture is common.
The learners usually use the senses of sight and hearing to absorb information. They are active in the sense that they write notes which are, in math classes, usually annotated versions of the material the student sees on the board. Also, as time allows, students ask questions of the expert to clarify and extend points or examples and clear up confusion or possible errors.
To be effective the method requires that the expert and the learners possess both background and skill sets. Among other things, the expert should possess mastery of his or her subject way beyond the details of the course, as well as skills of exposition and delivery-with-style. Sensitivity to how the lecture is being perceived is vital for a lecturer, and the ability to adapt a presentation "on-the-fly" to address unexpected problems is important too. The learners must come to the class with a solid foundation in course prerequisites and have the ability to concentrate for many minutes in a row. They must have the skill of effective note taking and be well organized.
The lecture method has been around for thousands of years. I have been inspired and informed and had the socks blasted right off my feet by gifted lecturers most of my life. And yet I can say that I myself have NEVER walked into a classroom cold as a learner and absorbed anything of real significance about a subject I previously knew nothing about during the course of a lecture.
Is this a contradiction? Far from it.
It has to do with the way I (and others) seem to learn things. It takes me three or more passes through a body of difficult technical material before I really "get it." The first time through I see or hear the vocabulary and get a sense of the issues involved in the subject. I am at "stage one." The second time through I begin to understand the basic examples and a bit about how the topic is connected to other things I know. I am at "stage two." The third time through, if I have worked hard, things begin to crystalize. I can adapt the methods under study to related problems. I make lots of connections to related subjects. I might have one or more of those wonderful "AHA" moments. I am at the sock-blasting "stage three." Mastery of the subject may follow.
The lecture method can be of at least marginal benefit to learners in all stages.
People who come in cold will hear the vocabulary used by the instructor and see the pictures drawn on the board. Using the lecture for this purpose is practically no different from having the book read aloud. Though there is a tiny benefit from this, generally it is a poor use of the lecture time.
People at stage two might get a bit more out of it. In addition to the book-read-aloud aspect, they can begin to benefit from another aspect of the lecture method that is often overlooked. People are "natural" mimics. Seeing a practitioner "model" expert behavior - as apprentices do in other areas - really helps the learner to become an expert too. Human beings absorb behavior patterns from other humans easily, automatically and even unconsciously. To promote this "apprentice effect," a lecturer must consciously expose his or her thought processes as an issue or problem is considered in the class. "Stage one" folks would think of this as rambling. "Stage two" folks might understand enough about the examples to absorb the expert's thinking or attitudes toward the subject and at least can get the feeling that it is something a human can do, even if it is still confusing and connections to other subjects are murky.
However a lecture is only really beneficial (better than time spent studying a book alone, for example) to learners who are near stage three or beyond. Getting to these later stages by no means requires special gifts at mathematics or anything else. The folks at these later stages are quite simply those who have done the hard spade work in advance that will allow them to take advantage of what a skilled lecturer can offer - those things that make the lecturer more than a book reading robot. Insight. Emphasis. Connections. Attitudes. Math as a creative and very human endeavor. A model of "expert behavior." These things are lost on or even confusing to folks who choose to walk into a lecture unprepared. If they do not have a self-reflective bent, they may well think a careful and deliberate lecturer is trying to snow them, or impress them with arcana, or that the lecturer is delivering a string of non-sequiturs.
So in addition to the technical skills of subject-mastery and listening and note-taking and exposition required of the team and mentioned above, the lecture method requires that learners spend two to three hours (a typical number for college math classes) preparing themselves to extract the most from each lecture. This is not "extra time" needed to accommodate the lecture method since the great bulk of the learning hours MUST take place outside the college classroom anyway. That is because there is a colossal stack of topics that someone who desires to understand a technical subject must master. The issue is how best to manage the class (contact with the "guide") time for those who are willing to take a legitimate shot at mastery.
There really is not enough class time for learners in many subjects such as the sciences to "discover" or "invent" or "construct" their way to mastery in the college classroom. Even if we could afford it, the students simply don't have enough years to do it this way. Education theorists who insist that the lecture method is old-fashioned take a pauper's view of the task in front of those who wish to learn any technical subject, neglect many of the benefits of the lecture method and misunderstand the purpose of a commodity that is scarce in college: class time. In college, unlike grade school, only a small fraction of the "learning time" can be spent in the class room itself. It should be spent doing things that cannot be organized as homework problems.
To paraphrase a famous quote: "If we wish to see farther we must stand on the shoulder's of giants." This is absolutely true in the Sciences and Mathematics.
Those of us who have chosen to try to "stand in" for the giants should take heart. No single method of instruction answers all the problems involved in organizing a classroom. But the lecture method works well when applied with skill and care in a college environment, and is a proven winner over many hundreds of years.
Larry Susanka
The Mathematics Department Pages
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