Mathematics Interventions and Strategies

Since the National Council of Teachers of Mathematics released its Standards in 1989, reform of mathematics curriculum to increase students’ understanding of mathematical concepts has been a major issue of discussion. Most agree on the goal; however, fierce debates have arisen about the best instructional methods to reach those goals. Some have stressed discussion of problem-solving strategies and the emergence of mathematical discussion between students as the most critical elements in mathematics reform. [Page 309]Others have feared that by minimizing the teaching of arithmetic computation and simple word problems, many students—especially those with learning disabilities, individuals with psychological processing or learning difficulties, and other learning problems—would never develop conceptual understanding.

In 2001, the National Research Council issued a report on lessons learned from experimental, development, and case-study research on teaching and the learning of mathematics. This document, Adding It Up, concluded that mathematical proficiency includes:

• Understanding mathematical concepts
• Fluent and accurate computational ability
• Strategic competence
• Adaptive reasoning

The latter two reflect insights gained from cognitive science and developmental psychology. Strategic competence is the ability to use either words or pictures to represent a problem and potential solutions, and the ability to develop multiple strategies for solving mathematical problems. Adaptive reasoning is the ability to justify strategies, and to analyze strengths and weaknesses of solutions proposed by others. These are ambitious goals, but they should be the objectives of a mathematics intervention.

Between 1996 and 2002, several researchers synthesized the experimental research on effective mathematics interventions for students who struggled with mathematics. Although the body of research is small, and many of the measures only tap one or two of the elements of mathematics proficiency noted previously, the syntheses suggest several features of instructional interventions that are likely to enhance achievement for these students. Some of the more promising interventions feature mediated verbal rehearsal, the use of visual representations, peer-assisted learning, and strategies for the efficient retrieval of basic arithmetic facts. These would seem to be indicators of teaching situations or interventions that are likely to be successful for this group of struggling students.

One technique, sometimes called mediated verbal rehearsal, that seems to be effective is taught in the following sequence:

• Providing examples
• Demonstrating proficient math solutions to students
• Requiring students to solve similar types of problems
• Requiring students to verbalize their justifications for their solutions

This technique appears to work when the teacher gives students feedback about the solution and justification. This approach attempts to build what the National Research Council calls strategic competence and adaptive reasoning in students.

The intervention research also supports the use of visual representations (e.g., figures, drawings, diagrams) of mathematical problems. This would seem to support the idea that students with learning problems are helped when they are required to represent mathematical relationships and problems in multiple ways.

Students with learning problems benefit from peer-assisted instruction. Typically a student who is struggling is paired with a more proficient student. However, both play the role of tutor and tutee. This enables the less proficient student to carefully watch, monitor, and question what and why the peer is solving a problem in a particular fashion. It also allows the more proficient peer to ask the student with a learning problem to articulate why the decision was made, to suggest that the student draw out a picture of the problem or put his or her reasons for a strategy choice into words. Tutors and tutees can be trained in a variety of techniques that promote this type of dialogue about mathematics. There is some evidence that more expert tutors or interventionists can use a similar methodology, especially when students are working on more complex mathematical material.

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