Stacy R. Karl

A Randomized Trial of the Effects of Schema-Based Instruction on Proportional Problem-Solving for Students With Mathematics Problem-Solving Difficulties

This article reports results from a study investigating the efficacy of a proportional problem-solving intervention, schema-based instruction (SBI), in seventh grade. Participants included 806 students with mathematical difficulties in problem solving (MD-PS) from an initial pool of 1,999 seventh grade students in a larger study. Teachers and their students in the larger study were randomly assigned to an SBI or control condition and teachers in both conditions then provided instruction on the topics of ratio, proportion, and percent. We found that students with MD-PS in SBI classrooms scored on average higher than their counterparts in control classrooms on a posttest and delayed posttest administered 9 weeks later. Given students’ difficulties with proportional problem-solving and the consequences of these difficulties, an important contribution of this research is the finding that when provided with appropriate instruction, students with MD-PS are capable of enhanced proportional problem-solving performance.

Investigating a Tier 1 Intervention Focused on Proportional Reasoning: A Follow-Up Study.

This randomized controlled study investigated the efficacy of a Tier 1 intervention—schema-based instruction—designed to help students with and without mathematics difficulties (MD) develop proportional reasoning. Twenty seventh-grade teachers/classrooms were randomly assigned to a treatment condition (schema-based instruction) or control condition (business as usual). Participants included 373 students, of whom 253 demonstrated MD. A measure of proportional problem solving (PPS) was administered at pre- and posttesting and at 11 weeks following treatment, along with a general mathematical problem-solving measure at pre- and posttesting. For the full sample, posttest differences favoring the treatment group were statistically significant for all measures. For students with MD, posttest differences favored the treatment group for the PPS posttest and PPS delayed posttest but not for general problem-solving posttest.

Using Regression Discontinuity to Estimate the Effects of a Tier-1 Research-Based Mathematics Program in Seventh Grade

The present study used a regression discontinuity approach to test whether schema-based instruction (SBI) was effective for students identified with varying levels of mathematics difficulties (MD) who received Tier-1 instruction. The performance of these students was compared to similar students who received business-as-usual Tier-1 instruction. Results indicated SBI on average raised scores regardless of whether a student was categorized as having MD on outcome measures of mathematics problem solving, suggesting that SBI is effective for a wide range of proportional reasoning skills and general mathematical proficiency, which broadens the population of students who could benefit from SBI. Standardized slopes for the treatment effect also provide evidence of the effectiveness of SBI. Implications for practice and research are discussed.

Automated assembly of optimally spaced and balanced paired comparisons: controlling order effects.

To control order effects in questionnaires containing paired comparisons, Ross (1934) described an optimal ordering of the pairings. The pairs can also be balanced so that every stimulus appears equal numbers of times as the first and the second member of a pair. First, we describe and illustrate the optimally spaced, balanced ordering of pairings. Then we show how the optimally spaced, balanced order can be used to implement a matrix-sampling design or a fully incomplete design when the number of stimuli n is so large that respondents cannot reasonably be expected to judge all n(n – 1)/2 pairs. The algorithm for balancing and optimally spacing the list of pairs is described.