Which Instructional Methods Are Most Effective for Mathematics?

Deciding which instructional methods are most effective for increasing students’ learning continues to be one of the great challenges for educational research. Should teachers use Method A or Method B? Which one will show the best results?

An important truth about the effectiveness of instructional methods is that particular methods are not, in general, effective or ineffective. Instructional methods are effective for something. Educators always need to be clear about what this something is when they talk about the effectiveness of instructional methods.

Focusing on the following two learning goals, we ask which instructional methods are most effective: conceptual understanding, which is the construction of meaningful relationships among mathematical facts, procedures, and ideas, and skill efficiency, which is the rapid, smooth, and accurate execution of mathematical procedures. These two learning goals are central to mathematics learning and have often competed for attention.

Conceptual Understanding

Research conducted over the past seventy-five years has spanned a wide range of mathematics topics, age groups, and class settings. The results point to two important features of teaching that can help the development of students’ mathematical understanding.

Work and Analyze

Teachers and students should intentionally and explicitly talk about, and work on, important mathematical relationships.

At least some time during each lesson should be spent on the following activities:

• Examine relationships among facts, procedures, and ideas within a lesson and across lessons. Is one problem a special case of the preceding problem? How are the problems solved today similar to and different from the problems considered yesterday? How do linear graphs, tables of ordered pairs, and linear equations all represent the same linear function?
• Explore the reasons why procedures work as they do. In addition to practicing procedures, students should examine and discuss why the procedures work, especially when new procedures are being introduced. Why do we usually add from right to left? When we solve an equation, why must we do the same thing to both sides?
• Solve problems using different procedures and then examine the similarities and differences between them. How is Jack's procedure different from Martha's procedure? Note: It is not necessary for students to practice multiple procedures for solving similar kinds of problems, but comparing different procedures is beneficial.

Work and Wrestle

Teachers should provide opportunities for students to wrestle with key mathematical ideas and ensure that students do some of the important mathematical work in the lesson.

Allowing students to work hard to make sense of mathematics does not mean standing by while they become unnecessarily frustrated and confused, nor does it mean presenting problems that are well beyond their reach. But it does mean providing time during the lesson when students are allowed to work on problems they don't immediately know how to solve, and it does mean resisting the temptation to jump in and tell students how to do something at the first sign of uncertainty.

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