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Assessing the Instructional Level for Mathematics: A Comparison of Methods

Implications for Practice and Policy

Matthew K. Burns
University of Minnesota

Amanda M. VanDerHeyden
University of California at Santa Barbara

Cindy Jiban
University of Minnesota

Betts (1946) originally hypothesized that presenting a learning task that a student was sufficiently familiar with but still provided some degree of challenge led to optimal learning. This appropriate level of challenge was called an instructional level. Too little challenge or too great of a challenge was referred to as independent and frustration levels respectively.  The term instructional level is frequently used in practice and although many have a general understanding of the term, few understand how to operationalize it in an instructionally useful way.

An appropriate level of challenge (or instructional level) is one of the essential components of an effective learning environment, but research has yet to adequately define an instructional level for mathematics. Gickling and Thompson (1985) suggested an accuracy approach in which mathematics assignments should contain 70% to 85% known items to represent an instructional level task. Deno and Mirkin (1977) suggested that the instructional level for mathematics be determined with fluency (i.e., accuracy plus speed) measures instead of accuracy data alone. They further estimated that 10 to 19 digits correct per minute (dc/min) would represent an instructional level for students in first through third grade, whereas 20 to 39 dc/min would equal an instructional level for children in fourth through 12th grades. This study compared mathematics performance of 434 second, third, fourth and fifth grade students to these accuracy and fluency criteria and found that fluency measures were more reliable and had better evidence for validity. All participants were exposed to a standard protocol-based intervention and progress was monitored using single-skill probes. Children demonstrating the strongest growth given a standard intervention were identified. Average mathematics fluency at baseline was computed for these students. Second and third grade children who showed the strongest growth given intervention performed between 14 to 31 digits correct per minute at baseline. Fourth and fifth grade children who showed strongest growth given intervention performed between 24 to 49 digits correct per minute. Functionally, these ranges represent empirically-derived instructional ranges or ranges of performance that are associated with optimal growth given intervention or instruction.

The two instructional levels derived from the current data, although somewhat similar to Deno and Mirkin’s (1977) criteria, were higher than those previously suggested. Moreover, the number of children for whom the task represented an instructional level varied significantly based on which criterion was used. Finding an instructional level for mathematics could be important because interventions could occur with children who experience mathematics difficulties until their skills reach an instructional level, at which point the child could participate in general instruction and be expected to experience improved learning outcomes (Gickling, Shane, & Croskery, 1989; VanDerHeyden & Burns, 2005). In other words, instructional level criteria can be used for instructional placement decisions or as criteria to work toward during interventions. Moreover, practitioners could use these criteria to evaluate if a child’s problem is specific to the child or class of children (VanDerHeyden, Witt, & Naquin, 2003). A fluency probe could be obtained for all children in a classroom by allowing them 2 minutes to complete a mixed-skill worksheet. Next, the average fluency rate could be computed and divided by two to obtain a digits correct/minute metric. If the average fluency rate meets or exceeds the grade-specific instructional level, then any difficulties experienced by a child are probably specific to that child. A class average that falls below the instructional level suggests that the classroom as a whole is experiencing difficulties and may require a classwide intervention. However, most importantly these data suggest that the current mostly commonly used criteria for an instructional level could be problematic and additional research is needed.

References

Betts, E. A. (1946).  Foundations of reading instruction.  New York:  American Book. 

Deno, S. L., & Mirkin, P. K. (1977). Data-based program modification: A manual. Reston, VA:

Council for Exception Children.

Gickling, E. E., Shane, R. L., & Croskery, K. M. (1989). Developing math skills in low-achieving high school students through curriculum-based assessment. School Psychology Review, 18, 344-356.

Gickling, E. E., & Thompson, V. P. (1985).  A personal view of curriculum-based assessment.  Exceptional Children, 52, 205-218. 

VanDerHeyden, A. M., & Burns, M. K. (2005). Using curriculum-based assessment and

curriculum-based measurement to guide elementary mathematics instruction: Effect on individual and group accountability scores. Assessment for Effective Intervention, 30, 15-31.

VanDerHeyden, A. M., Witt, J. C., & Naquin, G. (2003). Development and validation of a

process for screening referrals to special education. School Psychology Review, 32, 204-227.

Additional Resources

Burns, M. K. (2004).  Using curriculum-based assessment in the consultative process: A review of

three levels of research.  Journal of Educational and Psychological Consultation, 15, 63-78.

Gickling, E. E., & Armstrong, D. L. (1978).  Levels of instructional difficulty as related to on-task behavior, task completion, and comprehension.  Journal of Learning Disabilities, 11, 559-566.