Assessment in math should be an integral part of teaching and its purpose should be to increase learning, not just to assign grades, states the National Council of Teachers of Mathematics in its report: Curriculum and Evaluation Standards for School Mathematics (1989). Research findings have demonstrated that the time-consuming grading of daily classwork and homework assignments by teachers is not an efficient or an effective form of assessment. To determine what is effective, Brigham Young University Professors H. Clifford Clark and Marvin N. Nelson (both former public school teachers and administrators), worked with master teachers to identify diagnostic teaching strategies that increase learning. These strategies, which combine immediate feedback and cooperative learning methods, enable students to learn from their mistakes and enable teachers to spend more time on reteaching and less on grading papers. The strategies used by these teachers included:

1. Red Dot Technique – students work on problems individually while the teacher moves around the room checking work, putting a red dot next to problems that require further attention. Students are encouraged to rework problems and to obtain help from other students, as well as the teacher. The teacher continues checking work, marking those problems which have been successfully reworked with an “O.K.”.

2. Paired Learning – students working in pairs are assigned a problem. They each complete the problem independently and then compare answers. If their answer are different, they work together to determine the error and correct it. The teacher is available for assistance.

3. Student Answer Key – students are provided an answer key for immediate self-checking. Incorrect problems are reworked. Students must show detailed work for each problem to discourage misuse of the answer key. Students are encouraged to help one another.

4. Chalkboard Analysis – four students at a time work on assigned problems at the board, while the rest of the class completes the same problems at their desks. Errors are identified, discussed and corrected by the entire class.

5. Calculator Computation Check – all practice problems are computed with paper and pencil, and then a calculator is used to check accuracy. Incorrect problems are reworked until the correct answer is obtained. Computational detail is checked to ensure students understand how to obtain the answer. Students are encouraged to seek help as needed.

### Immediate feedback

A study was conducted in 1987 to determine if techniques emphasizing immediate feedback increased student achievement. Conducted by Clark and Stahle, this study compared classes which used immediate feedback and correction with classes that followed a traditional instructional routine.

A total of 321 third, fourth and fifth grade students participated in the study. There were three traditional “control” classes and three experimental classes at each grade level. Teachers in the traditional classes generally introduced a skill in lecture format following which students were assigned independent written practice. Problems were checked and graded by the teacher, who returned the papers the next day. There was little evidence of same-day feedback in the traditional classes.

The experimental teachers were randomly selected from a pool of teachers who had received training in informal assessment strategies. The control teachers were randomly selected from the same school districts. Every student was pretested and posttested with the ENRIGHT Diagnostic Inventory to determine their achievement gain on whole number multiplication and division problems.

Results showed that for both multiplication and division operations, the experimental classes showed significantly greater achievement gains. The researchers concluded that students who received immediate feedback and who were encouraged to rework their problems achieved significantly more at each grade level.

Interestingly, observers reported that the atmosphere in the experimental classrooms differed markedly from the traditional classes. In the experimental classes, students tended to interact more often, sharing learning strategies. These classes were characterized by a ‘busy hum” but little off-task behavior was reported. The researchers suggested that students in the experimental classes felt a sense of security because they knew that, through trial and error, they could achieve 100% correct on every assignment. It was also noted that teachers in the experimental group spent a maximum amount of time on reteaching and little time on record keeping.

### Middle school math classes

Middle School math classrooms studied by Clark and Nelson were found to be very similar to the elementary classrooms studied. They report that the informal diagnostic strategies studied in the intermediate grades work equally well at the middle school level. Clark and Nelson believe that immediate feedback and cooperative learning in math classes helps develop a better understanding of conceptual processes and increases achievement. At the same time, these techniques encourage self-confidence and positive attitudes toward learning.

*“Evaluation: Be More Than A Scorekeeper” Arithmetic Teacher May 1991, Volume 38, Number 9 pp. 15-17. “Functional Diagnosis Improves Computational Skill in Elementary Mathematics” School Science Mathematics, February 1991, Volume 91 Number 2 pp. 60-63. “Improving Math Evaluation Through Cooperative Learning Strategies” H. Clifford Clark and Marvin N. Nelson. unpublished manuscript.*

**Published in ERN September/October 1991 Volume 4 Number 4**