Improving math achievement

Bar chart on chalkboardIn 1982, in an effort to improve mathematical computation skills among its students, a Pennsylvania school district redesigned its mathematics program for grades K-8. Assistant Superintendent Joseph Yarworth, together with Timothy Schwambach, a remedial mathematics teacher in the Muhlenburg School District, and Robert Nicely at the Pennsylvania State University College of Education, designed a long range plan to increase student achievement.

A volunteer group of elementary and middle school teachers also worked on this project. Over a period of three years, these teachers redesigned the courses of study, integrating the written curriculum with what was being taught and tested. They also developed a system to document student achievement, aligned the regular and remedial programs and implemented a proven diagnostic-prescriptive instructional model.

Taxonomy of math concepts

This teacher group used a taxonomy of math content and concepts called PRIMES to analyze items on the state’s minimum skills test. They then determined how much coverage the curriculum gave to each item on the test and made changes in the curriculum. Throughout the following year in-service in the PRIMES taxonomy and in analysis of achievement scores was provided during the regular school day for all teachers in grade level groups.

The following year (1984-85), a new mathematics textbook series was selected which closely matched the curriculum they had designed. In-service in that year focused on diagnostic-prescriptive teaching methods, particularly the analysis of student errors.

Diagnostic-prescriptive teaching

The diagnostic-prescriptive model that teachers began using that year starts each instructional unit with a whole-class review of the objectives of that particular unit. Following the review, students are pretested for the skills and concepts required in the unit and the results are used to place students in three groups: ‘mastery’, ‘readiness for mastery’, and ‘no knowledge’.

The ‘mastery’ group is assigned to a learning center or an enrichment activity designed to reinforce and extend the knowledge of each member. The teacher also has the option of giving this group an assignment outside the essential curriculum.

Students who are ‘ready for mastery’ are assigned a structured activity designed to give them practice and help them overcome any weaknesses.

Students in the ‘no-knowledge’ group are given basic instruction by the teacher with direct supervision of their practice. Once this latter group is working on a written task, the teacher moves among the other two groups answering questions and monitoring progress.

Just before the end of the period, the teacher uses a whole-class activity to summarize what was studied that day. Finally, the teacher describes the next day’s work.

If the basic instructional (‘no knowledge’) group has demonstrated mastery on the assignment, the entire class is post-tested on the day’s objective. The lesson may be continued the following day if too many students have not mastered it. If only a few students continue to need help, remediation on that objective is scheduled for another time and the class moves on to the next objective. Regrouping of students is done as needed.

In 1986-87, by which time the diagnostic-prescriptive model was being used throughout the district’s elementary classrooms, the failure rate on the state’s basic skills test fell from 9% at the 3rd grade level to 0%, and from 16% at the 5th grade level to 0%. This indicated that at least for the lowest achieving students, this diagnostic-prescriptive model was highly successul.

In reviewing this 5-year project, the success achieved was attributed to the leadership of a knowledgeable teacher (Timothy Schwambach), as well as the preparation work done by a group of interested teachers. This core group provided inservice during staff meetings which enabled the entire faculty to confidently incorporate this new program into its classes.

Editor’s Note: This article did not report on changes in the middle and upper achievement ranges. In the future, the district plans to use a 5-year data base to monitor the scores on the California Achievement Test. It is hoped that a successful instructional model will increase skills throughout all levels of achievement.


“Organizing for Results in Elementary and Middle School Mathematics” Educational Leadership October 1988 pp. 61.

Published in ERN January/February 1989 Volume 2 Number 1.

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