The Role of Review in Mathematics Instruction. ERIC/SMEAC Mathematics Education Digest No. 2

What are the reasons for reviewing? When should reviewing be done? What instructional procedures for reviewing are effective? Answers to these questions, drawn from research and other literature, are outlined in the sections that follow.

REASONS FOR REVIEWING

In addition to promoting retention, review should be incorporated into mathematics instruction for the following, somewhat overlapping reasons:

--Review promotes continuity and helps students to attain a more comprehensive view of the mathematical topics covered. It helps them summarize main ideas, develop generalizations, and get an overall view of what they have been learning piecemeal.

--Review helps students to assimilate or consolidate what they have learned, enabling them to fit ideas into new patterns.

--Review serves as a diagnostic tool, revealing weaknesses and strengths to students and teachers. It helps teachers identify what is already known and what is not yet known; then reteaching can be planned.

--Review assures that the prerequisites needed for learning new content have been mastered.

--Review adds to students' confidence in their ability to move successfully to new mathematical topics.

TIMING OF REVIEWS

Most textbooks incorporate review sections at both the beginning and the end of most chapters. Saxon (1982) takes exception to this pattern, terming it "spastic." He argues that review should be continuous. Instead of presenting 25 problems on a new topic, he suggests that only three or four be given, along with 25 review problems. He claims that his data show that this approach helps students attain mastery.

Research clearly indicates that review should be systematically planned and incorporated into the instructional program. Before a new topic or unit is begun, an inventory can help the teacher ascertain whether any prerequisite knowledge is missing. Such a review also helps students to pull together the mathematical ideas they will need for the new topic. The inventory should include the range from simpler skills and concepts to the most difficult in order to pinpoint those which need to be retaught.

Daily review of homework alone is not sufficient; it often concerns only a small portion of the needed prerequisites for a new topic.

After a topic or unit is taught, key points or objectives should be reviewed. Students thus become aware of the major highlights of the lesson, so they can focus on the mathematical skills or concepts that will be needed in future lessons. It should be made clear to students that this is not simply a collection of exercises and problems: the review includes those topics which are the most important to remember.

Long-term retention is best served if assignments on a particular skill are spread out in time, rather than concentrated within a short interval. Review immediately after instruction consolidates the ideas from that instruction, while delayed review aids in the relearning of forgotten material. One researcher (Gay 1973) worked with eighth graders on four principles from algebra and geometry. She found that, if there were to be two periods of intensive review, it was better to space them at one and seven days after instruction, rather than on either the first two days after the topic was taught, or after an interval d 7 days.

Additionally, research has indicated that short periods of intensive review are better than long periods. Interspersing review throughout the textbook or curriculum is better than concentrating review at one time.

TYPES OF REVIEW

Outlining

The process of outlining forces students to organize ideas and provides a structure that will help students put ideas together. When students use the outline they have made as an aid in reviewing, restructuring or recall of the mathematical ideas is promoted.

Questioning

Several researchers have focused on review questions, finding them effective for revealing content that has not been meaningful to students. Burns (1960) studied whether thought-provoking review study questions, spaced throughout instruction in grade 6, would result in greater achievement with fractions and decimals.

Students who used the review questions scored significantly higher than those using textbook drill pages. Pupils' reactions were good; they reported that they found the questions useful and interesting.

In another study on review procedures, Lee (1980) explored the effects of different types of review questions in a mathematics textbook. Seventh-grade students were given work on module-seven arithmetic. A review passage consisting of a summary of rules and examples was placed immediately after each original passage. A review question, consisting of either words or calculations, was then given.

Review questions were found to facilitate retention by promoting comprehension. Feedback helped students to consolidate what they had learned in the course of answering the review questions, and they developed their ability to transfer problem-solving skills. The word-type review questions required a thorough understanding of the concepts and rules and the ability to apply them to new situations. On the other hand, the calculation-type review questions required only comprehension of a narrow range of concepts and rules, often focused on rote learning.

When review questions were added to review passages, inattentive reading often resulted. Students perceived this as too much reading; they felt they had relearned from the passage and therefore became bored with answering the questions. Thus, Lee concluded, only one type of review, passage or questions, should be used at a time. If only one is used, it should probably be review questions because they aided learning more.

Testing

A study with students in grades 6, 8, 10, 11, and 13 sought to determine if the learning and retention of mathematical knowledge was affected by any of three types of review procedures: testing, testing-with-explanation, and unit review (Clayton 1974). For the unit review, a list of behavioral objectives for the topics taught was given to students, with a worked example and an optional exercise identical to the corresponding test item.

The test group was only given a test and then the answers. Both the unit review and testing-with-explanation groups were given worked-out solutions for test items and optional exercises. The reviews enhanced the learning and retention of mathematics, with testing-with-explanation the most promising and testing alone the least promising. As in other studies, the usefulness of feedback in promoting achievement was apparent.

Homework

The role of differing types of homework designed for review in first-year algebra classes was explored by Friesen (1976). One group had homework consisting of exercises relating to the topic taught that day. Another group was assigned fewer exercises relating to the topic taught that day, plus exploratory exercises on content taught each of the prior two days and review exercises on the first and third days after the teaching of a topic. The group having exploratory and review exercises achieved and retained better than the group having exercises related only to the daily topic.

Difficulty Level

Summerfield (1975) gave fifth and sixth graders a brief review and an extensive review inserted into instructional materials on factors and primes. No significant differences were found. However, high scores on the achievement test made it appear that the review material was too easy. It may be that review is only helpful for those who need review. This fits with Gay's finding that it was better for students to choose the amount of review they were to have, rather than giving everyone the same amount.

Games

Bright and colleagues (1980) found that games provide effective review. In a motivating situation, students can focus on the mathematics and enjoy the process of recalling and restructuring.

CONCLUSION

Review is an important component of the mathematics instructional program. It can't be neglected -- and it can be made interesting as well as profitable.

FOR MORE INFORMATION

Burns, Paul C. "Intensive Review as a Procedure in Teaching Arithmetic." ELEMENTARY SCHOOL JOURNAL 60 (January 1960):205-211.

Clayton, McLouis. "The Differential Effects of Three Types of Structured Reviews on the Learning and Retention of Mathematics." DISSERTATION ABSTRACTS INTERNATIONAL 35A (August 1974):904-905.

Friesen, Charles D. "The Effect of Exploratory and Review Homework Exercises upon Achievement, Retention, and Attitude in a First-year Algebra Course." DISSERATION ABSTRACTS INTERNATIONAL 36A (April 1976):6527.

Gay, Lorraine R. "Temporal Position of Reviews and Its Effect on the Retention of Mathematical Rules." JOURNAL OF EDUCATIONAL PSYCHOLOGY 64 (April 1973):171-182.

Good, Thomas L., and Douglas A. Grouws. "The Missouri Mathematics Effectiveness Project: An Experimental Study in Fourth-Grade Classrooms." JOURNAL OF EDUCATIONAL PSYCHOLOGY 71 (June 1979):355-362.

Lee, Hyoja. "The Effects of Review Questions and Review Passages on Transfer Skills." JOURNAL OF EDUCATIONAL RESEARCH 73 (July/August 1980):330-335.

Pence, Barbara. "Small Group Review of Mathematics: A Function of the Review Organization, Structure, and Task Format." DISSERTATION ABSTRACTS INTERNATIONAL 35B (December 1974):2894.

Saxon, John. "Incremental Development: A Breakthrough in Mathematics." PHI DELTA KAPPAN 63 (March 1982):482-484.

Summerfield, Jeanettta O. "The Effect of Review of Prerequisite Skills and Concepts on Learning a New Topic in Mathematics." DISSERTATION ABSTRACTS INTERNATIONAL 35A (June 1975):7627.

This Digest was prepared for the ERIC Clearinghouse on Science, Mathematics, and Environmental Education, 1984.