Good Instruction in Mathematics

This chapter applies the lessons of Chapter 1 (which discusses how to identify good instruction) to the case of mathematics. There has been much controversy about what makes for good instruction in mathematics. Nevertheless, scientific and humanistic sources do allow us to paint a picture. Some instructional methods are less guided (such as pure discovery learning) and others more guided (like teacher-led instruction); scientific and humanistic evidence are in agreement that general guidance is needed, but should not come at the expense of student cognitive engagement. The evidence also consistently shows that instruction should emphasize genuine understanding of the underlying reasons for mathematical principles. Skills (such as fluency in computations) are not in opposition to concepts, but rather in mutual support. Solving varied and unexpected problems is essential in good mathematics instruction. Mathematical “rigor” (meaning precision in expression) plays an important role in mathematical thought, but should be carefully balanced with accessibility for children. While such principles give general guidance, knowing them is not enough to create excellent instructional programs: they need to be applied consistently in each moment of each lesson. Getting these details right is challenging, and can only be done through years of trial and error. This helps explain why good instructional traditions in mathematics are so rare.

Integrating Manipulatives and Computers in Problem-Solving Experiences

In 1980, the National Council of Teachers of Mathematics stated that “problem solving must be the focus of school mathematics.” In 1989 the Council reaffirmed that belief with the Curriculum and Evaluation Standards for School Mathematics (Standards). Standard 1 for grades K–12 is “Mathematics as Problem Solving.” The Standards also asserts that “a computer should be available in every classroom for demonstration purposes, and every student should have access to a computer for individual and group work.” Also according to the Standards, “manipulative materials are necessary for good mathematics instruction.” In a typical classroom, problem solving may be taught, manipulative materials may be used, or students may be working at a computer. These functions, however, are usually completed as disjoint activities. Integrating these activities is possible, and this article illustrates how it can be done.

Teaching Mathematics with Technology: Common Multiples: Activities on and off the Computer

One feature of good mathematics instruction is to present students with a number of environments that embody the same concept but from different perspectives. Such multiple embodiments permit students to reflect on a particular mathematical concept and form generalizations about aspects of the concept that remain unchanged across representations. Also, particular aspects of an alternate embodiment may allow students to form an understanding that they may not have acquired through their first exposure to the content in its most traditional representation. This month's department offers two alternate representations for the concept of common multiples that can be modeled either on or off the computer.

Review: What Does the Textbook Say?

Of all the aids to teaching and learning in the history of schooling, the textbook is the most venerable and the most disputed. In the school textbook, we meet fundamental problems of didactics. The textbook can aid the reacher in making decisions about instructional content and about pedagogical intention and methodology. The teacher may adopt ideas straight from the book or may modify, or possibly reject, them on the basis of his or her experience, knowledge, or personal concept of “good” mathematics instruction. In any case, the textbook presents essential guidel ines and has a considerable impact on the teacher's activities. At the same time, the textbook is expected to serve as a working manual for the pupil. It ought to motivate pupils, give them a chance to use and experiment with mathematics, and allow them to work out mathematical concepts or problems on their own. Most modern mathematics textbooks indeed claim to be pupils' books, but a closer view shows that they nre really addressed to the reacher. These and other problems concerning how to conceive of, and work with, school mathematics textbooks have seldom been discussed in teacher training programs.