Spring Math
Study: VanDerHeyden et al. (2015)

Describe how students were selected to participate in the study:

All fourth and fifth grade students attending any of the 7 schools in a southeastern U.S. city were eligible for participation. Inclusion criteria were (1) enrolled in the system at the time of spring testing in the spring of the intervention year (spring score available), (2) not categorized as limited English proficient according to state criteria, and (3) participating in general education mathematics instruction. From the original sample, 254 fifth graders and 283 fourth graders met inclusion criteria. Among these included students, 186 fifth graders and 188 fourth graders had a preceding year’s spring test score available (pre-test score) on the year-end state test. Thus the final sample for the distal measure outcomes was 186 5th graders and 188 4th graders. The final sample for the proximal outcome measures was 254 fifth graders and 283 fourth graders.

Describe how students were identified as being at risk for academic failure (AI) or as having emotional or behavioral difficulties (BI):

Classes for which the median score was in the at-risk range on the fall occasion proximal measures were considered as eligible for intervention. 100% of classes screened met the risk criterion in both the control and intervention groups.

ACADEMIC INTERVENTION: What percentage of participants were at risk, as measured by one or more of the following criteria:

• below the 30th percentile on local or national norm, or
• identified disability related to the focus of the intervention?

0.0%

BEHAVIORAL INTERVENTION: What percentage of participants were at risk, as measured by one or more of the following criteria:

• emotional disability label,
• placed in an alternative school/classroom,
• non-responsive to Tiers 1 and 2, or
• designation of severe problem behaviors on a validated scale or through observation?

%

Specify which condition is the submitted intervention:

Classwide mathematics intervention. A sequence of skills was identified for each grade level. A standard protocol was followed to conduct a 15-min classwide intervention each day. Within treatment classes, students were grouped into dyads based on the beginning of year screening data such that higher-performing students were matched with lower-performing students and middle-performing students were matched with each other. The standard protocol included a period of guided practice with peer coaching and feedback for each member of the dyad, a timed interval of independent practice, corrective feedback (including teacher-led re-teaching, error correction, and “think aloud” problem solving where the student had to explain to his/her math buddy how he or she corrected an error), goal setting, and a group contingency for improved class performance. On Friday of each week, a probe of the skill being targeted during intervention was administered following standard CBM procedures. If the class median score surpassed 80 digits correct per 2 min (Deno & Mirkin, 1977), the entire class advanced to the next level of intervention difficulty (i.e., the next skill in the sequence) and intervention continued at that level the following week. When the class was working on a basic fact (e.g., multiplication 0-12), flashcards were used during the guided practice period of the intervention as indicated by the intervention protocol. When the class was working on a skill that was not a basic fact (e.g., multi-digit addition and subtraction), practice worksheets were used during the guided practice interval as indicated by the intervention protocol. At each grade level, the intervention skill sequence included 14 skills. In theory, classes could have finished the intervention in 14 weeks but this would have represented very rapid progress (VanDerHeyden & Burns, 2008). Out of 26 intervention classes, only 3 completed all 14 skills during the 29 weeks allotted to intervention. The intervention was similar in format to classwide peer tutoring (Greenwood, 1991) and peer assisted learning strategies (Fuchs, Fuchs, Mathes, & Simmons, 1997), which have been experimentally evaluated and shown moderate effects on mathematics achievement (e.g., +.24 and +.33, Slavin & Lake, 2008). The intervention strategies emphasized explicit, direct instruction on computation and procedural mathematics skills emphasizing whole number operations, numbers and operations in base ten, and numbers and operations with fractions and decimals. The use of guided practice, immediate corrective feedback, multiple opportunities to respond, incentives for improved performance, and gradually increased task difficulty based on mastery of easier, related skills are strategies that have been widely studied with moderate to strong effect sizes on achievement (Hattie, 2009; Slavin & Lake, 2008) for all children and for children who are at risk (Kavale & Forness, 1999). When these strategies are implemented as part of an intervention package to advance mathematics performance, users might reasonably expect to observe performance improvements (Bryant et al., 2011). Classwide math intervention is a component of Spring Math (www.springmath.com). Spring Math is a web-based system that directs screening for students K-8 in fall, winter, and spring; summarizes and interprets the data by grade and class; recommends a classwide intervention if needed; and recommends individual students for intervention. Spring Math provides all follow-up assessment and follows decision trees to identify the intervention skill difficulty level and the type of intervention needed (acquisition versus fluency-building). Spring Math provides all materials needed to conduct the intervention for one week including the intervention protocol with a teacher script, scripted activities to develop and expand conceptual understanding, and a follow-up assessment for a targeted and generalization skill. Spring Math graphs weekly performance gains, interprets data, and adjusts the intervention packet for the following week. An administrator and coach dashboard tracks implementation metrics and recommends actions to support high-quality intervention use in all classrooms. The classwide intervention emphasizes fluency building for essential skills that are foundational for progress at a given grade level. Individual intervention is the next layer for students who require more assistance to attain key understandings. Individual interventions include both acquisition protocols and fluency-building protocols. The protocol used depends upon the student’s assessment(s).

Specify which condition is the control condition:

Control classrooms

If you have a third, competing condition, in addition to your control and intervention condition, identify what the competing condition is (data from this competing condition will not be used):

Using the tables that follow, provide data demonstrating comparability of the program group and control group in terms of demographics.

Grade Level

Race–Ethnicity

Socioeconomic Status

Disability Status

ELL Status

Gender

What method was used to determine students' placement in treatment/control groups?

Random

Please describe the assignment method or the process for defining treatment/comparison groups.

Experimental Design and Group Assignment: This study used a nested, between-groups experimental design to evaluate the effects of an intervention on (a) year-end statewide accountability test scores, and (b) differences in growth on three curriculum-based measures (CBM) administered on three occasions. Assignment occurred at the classroom level. Classrooms were randomly assigned to the intervention or control group within each grade within each school. In cases where the number of classes at a given grade level represented an odd number, a greater number of classes were assigned to intervention than control. Class assignments and pre-intervention average scores on experimental measures are shown in Table 3. Assignment procedures resulted in 10 control classrooms and 13 intervention classrooms for both the fourth and fifth grade samples. In the fourth grade sample, there were 16 teachers for the 23 classrooms. Five teachers taught only the control curriculum, six teachers taught only the intervention curriculum, and five teachers taught both curricula. In the fifth grade sample there were 15 teachers for the 23 classrooms. Five teachers taught only the control curriculum, six teachers taught only the intervention curriculum, and four teachers taught both curricula. Thus, for some of the teachers treatment was crossed with teachers and for other teachers, teachers were nested in treatment. See Table 3 on p. 1259 of the AERJ manuscript.

What was the unit of assignment?

Other

If other, please specify:

Classes. For some teachers, treatment was crossed with teachers, whereas for other teachers, teachers were nested in treatment. Please see details at bottom of p. 1258 of AERJ manuscript.

Please describe the unit of assignment:

Assignment procedures resulted in 10 control classrooms and 13 intervention classrooms for both the fourth and fifth grade samples. In the fourth grade sample, there were 16 teachers for the 23 classrooms. Five teachers taught only the control curriculum, six teachers taught only the intervention curriculum, and five teachers taught both curricula. In the fifth grade sample there were 15 teachers for the 23 classrooms. Five teachers taught only the control curriculum, six teachers taught only the intervention curriculum, and four teachers taught both curricula. Thus, for some of the teachers treatment was crossed with teachers and for other teachers, teachers were nested in treatment.

What unit(s) were used for primary data analysis?

Schools
Teachers
Students
Classes
Other
If other, please specify:

Please describe the unit(s) used for primary data analysis:

How was the program delivered?

Individually
Small Group
Classroom

What were the background, experience, training, and ongoing support of the instructors or interventionists?

Teachers, RtI coordinators, and administrators were trained to implement the intervention using a combination of antecedent and live coaching strategies. Following a series of trainings specific to principals, the first author traveled to each school to conduct a 1-hour training with teachers whose classes were assigned to the intervention condition. Additionally, each school had an RtI coordinator and that person was charged with receiving and organizing weekly data to provide to the first author via an electronic spreadsheet program designed to organize the data and present graphs of class progress each week during the intervention. In the didactic training session, an overview of the rationale for the intervention program was shared with teachers using the district’s data reflecting low mathematics achievement. Details of the intervention were provided, including sharing the intervention protocol, describing how the intervention would progress based on student mastery of skills within a pre-established hierarchy of skills, showing effects on mathematics achievement obtained in other districts using the same intervention, and showing short video clips of the intervention being implemented in classrooms in other districts. An opportunity to discuss and troubleshoot intervention implementation was provided to teachers at this time. Teachers were provided all materials needed to implement the intervention each week by the on-site RtI coordinators. School principals agreed to conduct implementation integrity checks via direct observation as part of the intervention plan (described in greater detail in the next section). The consultant organized feedback on district progress for district administrators and school principals bimonthly during the year. Graphed feedback on each class’s progress with the intervention was provided to principals and district administrators. The consultant met in person with the district leaders and principals, reporting the number of skills mastered by teacher and identifying implementation errors. Additionally, the consultant communicated directly with principals and RtI coordinators providing a list of teachers whose classes were growing at a slower pace relative to other classes in the same school and encouraged an intervention integrity check in those classes. Finally, on a bimonthly basis (four total occasions), the consultant conducted integrity observations in classrooms with each principal and modeled for school principals how to troubleshoot intervention implementation with the classroom teacher. School principals agreed to conduct implementation integrity checks via direct observation as part of the intervention plan. Principals or on-site RtI coordinators agreed to conduct four integrity observations each week with approximately half of those occurring during regular mathematics instruction within control classrooms in an attempt to capture contamination between control and intervention conditions. The intervention integrity checklist listed each step of the intervention in observable terms and administrators were trained to observe and note the occurrence of each step of the intervention. The trained observer used the scripted intervention protocol to note correctly and independently completed steps of the intervention. Where deviations from the protocol were observed in intervention classrooms, principals and/or RtI coordinators provided corrective feedback on implementation and assisted the teacher to troubleshoot barriers to effective implementation.

Describe when and how fidelity of treatment information was obtained.

We examined fidelity in three ways. (1) We conducted a survey at the midpoint of the intervention of all treatment and control teachers. The survey documented by teacher report characteristics of core instruction in mathematics (number of minutes, type of supplemental intervention if any, number of minutes supplemental intervention was provided, etc.). These findings are reported on p. 1257 of the manuscript under the heading, “Instructional Context” and in Table 2 on p. 1258. Contamination was documented with two teachers in the control group reporting that they had implemented the classwide intervention. (2) We also examined use of the intervention in treatment classrooms using permanent products. We examined the number of skills mastered during intervention by class since classes were equivalent at the start of intervention and the intervention used a standard protocol with classes advancing through a fixed sequence of skills as a class to a new skill based upon the median score reaching a mastery criterion associated with the skill. Because a weekly score was recorded for all students in all treatment classes and the class was expected to advance based upon a specific decision rule, we also computed number of deviations from the treatment plan during the 29 weeks of the study (intervention score recorded, class advanced or did not advance as prescribed). (3) Trained observers (coach or principal) conducted a total of 406 direct observations of the intervention across all classrooms (treatment and control) during math lessons and documented occurrence of the steps in the standard protocol for classwide intervention. Each step was coded as having been implemented or not and the total number of steps implemented was divided by the total number of intervention steps in the protocol and the quotient was multiplied by 100%. 281 observations were conducted in treatment classrooms representing about 8% of opportunities for intervention among intervention teachers (26 weeks of intervention completed on average times 5 sessions per week times 26 intervention teachers). 125 observations were conducted in control classrooms. Average integrity in treatment classrooms was 96.69% (83.5%-100%) and estimated average implementation in control classrooms was 2.69% (0-20%). No feedback was provided to teachers following completion of the observation checklist. What we learned from direct observation data was that the most common integrity error was failing to use the intervention at all, which we suspected direct observation was less sensitive to given that the presence of the observer likely cued or reminded the teacher to implement the intervention. When the intervention was used, teachers tended to implement all of the steps of the intervention correctly. Thus, in our two published articles, we emphasized permanent product estimates of integrity including weeks of intervention and skill progression (which we thought of as trials to criterion data).

What were the results on the fidelity-of-treatment implementation measure?

Teachers completed on average 26.1 weeks of intervention (range, 20-29). Intervention teachers deviated from the intervention plan on average 1.4 times during the intervention (range, 0-5). On average, correct decisions (advance a skill level or not) were made for 94% of decision-making occasions (range, 80-100%). These permanent product indicators indicated that the intervention was sufficiently used to measure its effect on average although between teacher differences were apparent. Results on p. 1272 of the article under the heading “Intervention Integrity Effects” indicate that fourth grade classes successfully mastered more skills compared to fifth grade classes. There was a positive, but not statistically significant relationship between integrity estimates and post-intervention year-end state accountability scores. There was a positive and statistically significant relationship between integrity estimates and CBM growth in the intervention group.

Was the fidelity measure also used in control classrooms?

The survey fidelity measure was used in the control classroom. Direct observations were conducted in control classrooms.

For any substantively (e.g., effect size ≥ 0.25 for pretest or demographic differences) or statistically significant (e.g., p < 0.05) pretest differences, please describe the extent to which these differences are related to the impact of the treatment. For example, if analyses were conducted to determine that outcomes from this study are due to the intervention and not pretest characteristics, please describe the results of those analyses here.

In data analyses conducted for the paper, CBM pre-test program and control comparisons were conducted using a multilevel model, which took nesting into account. None of these comparisons were significant. This is reported on page 1269 of the article: “Of note, treatment and control groups were not statistically different on the CBMs at the first time point before intervention,” and these data are shown in Table 7. There were no significant pre-test (time 1) differences on any of the CBMs at grade 4 or 5. Please note that “simplify fractions” is referred to as “reduce fractions” in the article. Year-end test scores were analyzed with multilevel ANCOVA with the covariate (pre-test means) centered. This provides the best analysis for the structure of the nesting in which “class means are used as the basic observations and treatment effects are tested against variations in these means” (Campbell & Stanley, 1971, p. 23). Given a randomized cluster design, determining pre-test differences on the basis of mean differences analyses that ignore the nested structure of the dataset may be misleading.

Please explain any missing data or instances of measures with incomplete pre- or post-test data.

If you have excluded a variable or data that are reported in the study being submitted, explain the rationale for exclusion:

Describe the analyses used to determine whether the intervention produced changes in student outcomes:

Please see pp. 1262-1266 of the AERJ article for the complete description including models and model terms: Multilevel linear modeling (MLM) was used to account for the nesting of students within classes and classes within teachers. Fourth- and fifth-grade data were analyzed separately. The statewide test analyses were designed to investigate a treatment difference on students’ 2009 statewide test math scores while controlling for 2008 statewide test score. Follow-up analyses examined the difference between groups on students’ numbers and operations subscale of the statewide test while controlling for 2008 statewide test score. Students with a 2008 and 2009 statewide test math score were included in these analyses (referred to as sample b in this application). We used two different dependent variables for each of the fourth-grade and fifth-grade samples. The following descriptions apply to both samples and both dependent variables. Our final specification of residuals included a residual for the teacher level for the intercept and a within-class residual for the Level 1 model. For each of the four combinations of dependent variable and grade, the final model resulted in the smallest Akaike Information Criterion (AIC) fit index (Akaike, 1974) among the models for which estimation converged. Group mean centering (GPMC) based on classroom mean was used at Level 1 to protect against spurious cross-level interactions (Enders & Tofighi, 2007). A multilevel repeated measures analysis, including treatment and occasion as factors, was used to test for a Treatment x Time interaction and account for scores nested within students across occasions, students nested within classrooms, and classrooms nested within teachers for each CBM in each grade. An unstructured covariance matrix (UCM) was specified for each component of the model. That is, for each component the variances were permitted to be unequal for different occasions and the covariances were permitted to be unequal for different pairs of occasions. For models that did not converge with unstructured covariance matrices for student, class, and teacher components, we estimated new models that allowed either UCMs for student and teacher components of the residual and a compound symmetric covariance matrix (CSCM) for the class component or UCMs for the student and class components and a CSCM for the teacher component. We used the AIC fit index (Akaike, 1974) to select the model with the most appropriate variance-covariance structure. For models with a significant Treatment x Occasion interaction, we estimated treatment effects at each occasion. When the Time x Treatment interaction was significant, we estimated an MLM with a treatment effect and a Treatment x Pretest interaction at each of the last two occasions to evaluate if the treatment effect varied with the pretest score. The covariance structure was selected by following the same steps that we used for the repeated measures analysis. To evaluate the relationship of student scores to intervention integrity, the number of skills mastered and the percentage of correctly followed decision rules were used as indicators of correct intervention implementation. To investigate the relationship of 2009 statewide test scores to intervention integrity, we estimated two-level means-as outcomes (MAO) models (Raudenbush & Bryk, 2002) for students in the treatment group only. To investigate the difference in CBM scores by intervention integrity, we estimated two three-level intercepts-and-slopes-as-outcomes (ISAO) models for each CBM probe for students in the treatment group only. Occasion was the predictor at Level 1, and an intervention integrity indicator was used as the predictor at Level 2 for each model. In the first model, the number of skills mastered was the intervention integrity indicator; in the second, percentage of correctly followed decision rules was the indicator.