A presentation of the following article was given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

Many of the recent changes in mathematics curriculum and pedagogy focus on increasing student proficiency in discussing mathematics, writing about mathematics, and using technology appropriately. One area that is often overlooked is increasing student proficiency in reading mathematics. Typically, most students never read a mathematics textbook. At most, the book is used as a means of finding examples similar to assigned homework problems. Yet if our goal is to have our students become self-learners, it is essential that they learn to carefully read technical material. As with any skill, this is something which takes practice and guidance.

I have developed a method of teaching which has resulted in a high percentage of my students reading the text on a regular basis. For each chapter, I produce Reading Outlines for my students. A typical Reading Outline consists of three types of questions:

• Basic questions on symbolism or terms
• Concept questions
• Application questions

Most students find the first type of question fairly easy and can usually answer these directly from reading the text. The concept questions typically involve writing and the application question are similar to homework questions. Most students have difficulty completely answering these type of questions on their own.

I tell my students that it is important that they read the book and fill out the Reading Outline before coming to class. I tell them that the purpose of reading the text is not to completely understand the material on their own, but rather is so they become acquainted with the terms and symbols we will be using that day and so they have a fairly concrete idea of what they do not know. I start class by asking "What questions do you have from the reading? What is it that you do not know or understand?" Students rarely ask about the basic questions on symbolism and terms which means I rarely spend class time going over such material. Instead, all or almost all of my class time is spent discussing conceptual questions and looking at applications.

I make it clear to my students that I will respond to their questions, but if they do not ask, I do not discuss. This puts the responsibility on them for knowing what it is they do not understand from the section they read. I instruct my students that if they want to know what I consider important information from a section, they should be guided by what I ask in the Reading Outline. I also give them a summary list of computations and concepts at the end of each chapter. I let my students determine how much time we need to spend on a particular topic by continuing to answer questions until no more are asked. Rather than causing my class to be behind those taught by other instructors, I find that I am usually ahead of or keeping pace with the others. Part of the reason for this is that my students already have some understanding of the material before coming to class.

My teaching style is discussion rather than lecture. Most of my time is spent questioning the students. I typically ask 40 questions in a 50-minute period. I do not ask for volunteers to answer but rather randomly select students (this past year I used cards of students names to select who would answer the question). I tell my students that they are allowed to respond with "pass" if they cannot answer a question. Again, the questions assume that they have done the reading and are familiar with the basic ideas. If one student asks a question from the Reading Outline, I typically ask another student to begin the discussion by defining the terms used in the question. From there, I'll turn to another student to elicit information that will get the class started in answering the question. Using this technique, it becomes clear to the class that everyone is individually responsible for reading the material and will be held individually accountable. It is also easy for me to determine who has not looked at the book by the way they stumble over answering the basic questions. Also, to once again emphasize the importance of reading the text, I will usually have an unannounced quiz once a week. This quiz is at the beginning of class, covers the material they were to have read but which has not been discussed, and they are allowed to use their Reading Outline but not their book.

I have found that all of my students read the book at least 50% of the time and that approximately 85% read it everyday. Although most students feel they are working harder than students in other sections (because of having to read the text), most of them also feel they are learning more and appreciate the flexibility they are given in determining what should be discussed and how much time should be spent in the discussion. The use of the Reading Outlines and the method of discussion involving randomly calling on students receives overwhelmingly favorable review from all but one or two students each semester. I have found that the number of questions asked by students has increased greatly. Not only do students ask questions such as "How do you do question three?" but they also ask questions about why things work that way and what is really happening in this problem. One of my favorite student questions, and one which I get frequently, is a "What if...?" question. Often, this type of student-generated question leads to a much more productive and interesting discussion than I would have anticipated when writing the Reading Outline questions for that section.

I have found that preparing Reading Outlines is no more time-consuming than preparing lecture notes. In fact, the process is similar in that the goal is to determine the important information and how to present it. Rather than going to class with lecture notes, I bring my Reading Outline and make sure I am comfortable discussing the questions I have asked. There are times that a student will ask a question which depends on material which we have not yet discussed, but since my method of teaching is discussion with significant student input, I may take most of the class period answering the one question, making sure that we cover all of the preliminaries adequately.

If you would like more information about the use of Reading Outlines or would like to see samples of Reading Outlines which I have created, please contact me.

Janet Andersen
Dept. of Mathematics
Hope College
Holland, MI 49422-9000
andersen@math.hope.edu