Is teaching mathematics an art or a science? If it is an art, then only highly talented people will be successful teachers of mathematics. For an art, which is largely dependent on creativity, can be learned only to a limited extent; the rest must come intuitively. If mathematics teaching is a science, then anyone capable of learning to teach mathematics, regardless of talent, should be able to do it.
We consider the teaching of mathematics both an art and a science. One needs a certain amount of innate ability to teach successfully. With the rarest exceptions, this ability needs to be buttressed in varying amounts with sociological, psychological, philosophical, and common-sense principles. In this book, we provide the prospective and inservice teacher of mathematics a plethora of ideas covering all aspects of the profession, ideas that we believe will support the artist in everyone. In many cases we provide solid, experience-tested suggestions. One could possibly entitle the book Everything You Wanted to Know about Teaching Mathematics but Didn't Know Whom to Ask.
The first part of the book discusses methods of teaching mathematics, considering all aspects and responsibilities of the job. We begin with an overview of the history of mathematics education so that the teacher of mathematics today understands how the teaching of mathematics has evolved. After describing the rudiments of lesson planning, we discuss the aspects of teaching that make for effective instruction. Since one of the most important aspects of mathematics instruction is the students' ability to solve problems both inside and outside the classroom, we have devoted an entire chapter to this topic. We approach problem solving in a variety of ways: from its instructional underpinnings to its recreational and motivational aspects. One difficult issue is the rapidly changing world of technology. There are two problems. First, no sooner will this book be printed than there will likely be changes in the technology available to the mathematics instructional program. At the same time, there are many schools in the United States that are less fortunate than others and do not yet have the technology described here available for teacher use. We have attempted to address the issues and situations to cover the majority of today's schools.
Mathematics teachers have additional responsibilities beyond the regular instructional program. They must concern themselves with enriching instruction for gifted and not so gifted students, and ought to provide their more motivated students with extracurricular activities to further stretch their interest in mathematics. We conclude the first part of the book with a discussion of the professional responsibilities of mathematics teachers.
We make a conscious effort not to tell the teacher how to deal with every situation. Rather we attempt, as much as possible, to provide alternatives to allow the teachers to make professional judgments about their teaching performance. This is as it should be; no one method of teaching is suitable for all teachers. Personality differences among teachers dictate varying methods of instruction, and what works for one teacher may not work for another.
The second part of the book provides mathematics teachers with a collection of enrichment units appropriate for the entire secondary school curriculum spectrum. Each unit states its objective(s), provides a means for pre-assessment, and then provides an in-depth description of the subject so that a reader unfamiliar with the topic can easily learn it. This development is accompanied by suggestions for teaching the topic to a secondary school class. A post-assessment is provided to help determine the level of learning (mastery) of the topic taught. There is a subject matter cross-index provided at the beginning of the section to enable teachers to select units by topic and grade level.
The National Council of Teachers of Mathematics 2000 Principles and Standards for School Mathematics (Standards) are referred to throughout the book. No one knows where the future emphases in the teaching of mathematics will be. Currently, the Standards are determining the agenda. Clearly, the significant technological advances we are seeing today, coupled with a decline in the cost of computers and calculators, will have a major impact on how mathematics is taught in the future. However, in many parts of the United States there is still uncertainty about the direction that mathematics education should take. We have therefore taken a forward-looking yet cautious approach to the teaching of mathematics, relying, when in doubt, on time-tested methods and ideas.
In effect the reader has within the covers of this book a complete resource on how to teach mathematics, including "the tricks of the trade:" Also provided here is an invaluable collection of enrichment material that can be easily adapted to classes of all levels, to enhance instruction and to motivate students to study mathematics with interest and enthusiasm.
The book is intended primarily for two audiences: for pre-service training of mathematics teachers for the secondary schools, and for inservice teachers of mathematics who are seeking to improve their teaching skills and increase their resources through formal coursework. The book should also be available to mathematics teachers who simply want to have a
