1.1 Importance of whole number knowledge

Mathematics fluency is the automatic recall of basic mathematics facts which is measured by determining whether student responding is rapid and accurate (National Council of Teachers of Mathematics [NCTM], 2014). From a theoretical perspective, building fluent responding is a prerequisite for the retention, maintenance, and generalization of the skill to other academic tasks (Maki et al., 2021). The instructional hierarchy, which is a learning hierarchy to describe how a learner develops a new skill (Haring et al., 1978), is supported by research which shows that proficiency with basic facts is a significant predictor of later mathematics achievement such as rational number knowledge, algebraic principles, and word problem solving (Fuchs et al., 2016; Marcelino et al., 2017). Specifically, research indicates that proficiency with basic facts plays a fundamental role in the acquisition of higher-level mathematics by reducing the cognitive load of advanced mathematics problem-solving (Price et al., 2013). Moreover, the development of basic fact fluency is associated with positive educational outcomes such as attending college, solving complex mathematics problems, and interpreting abstract mathematics principles (Fuchs et al., 2016; Marcelino et al., 2017). The critical importance of the development of basic fact fluency coupled with the pervasiveness of student underachievement in mathematics calls for intervention to take place at a class wide level.

Within a multitiered system of support framework (MTSS), tier 1 (i.e., universal instruction) includes the delivery of an evidence-based core curriculum and effective instructional practices to all students to ensure that most students (i.e., about 80%) achieve proficiency (McMaster & Fuchs, 2016). Unfortunately, commonly used mathematics curricula do not provide sufficient opportunities to practice for students to become proficient (Doabler et al., 2012), which may contribute to the persistent underachievement of United States (NCES, 2022). Solutions to this problem begin by addressing foundational skill gaps during core instruction. When school level performance falls below 80% proficiency or displays low overall levels of mathematics performance, one of the first steps is to provide class wide interventions that target foundational skill gaps (VanDerHeyden et al., 2012). Thus, educators need to supplement core curricula with evidence-based interventions facilitating adequate and explicit practice opportunities that educators can apply in the classroom (Long et al., 2016; VanDerHeyden & Codding, 2020).

1.2 Evidence-based interventions

The use of carefully constructed practice opportunities and immediate feedback for improving student mathematics performance is supported by decades of research (e.g., Codding et al., 2011; Twyman & Sota, 2016). Technology and peer-mediated interventions allow students multiple opportunities to practice mathematics facts with immediate, corrective feedback (Klingbeil et al., 2020; Tamim et al., 2011). Moreover, these evidence-based interventions can be embedded into regular classroom routines during core instruction (Klingbeil et al., 2020; Tamim et al., 2011). Both approaches offer opportunities to differentiate instruction, require few materials, and minimal training is needed for successful implementation. The alleviation of these implementation barriers coupled with the positive impacts on student performance across skills make these two interventions not only positive for students but also feasible for classroom teachers to implement.

1.3 Comparing evidence based practices

Although the effects of RPT and technology-mediated drill and practice are well documented in research, few studies have compared the effects of the two modalities of intervention on student computation outcomes. Researchers have stressed the importance of comparing research-based interventions (Petersen-Brown et al., 2019). Comparing novel practices such as technology-mediated interventions to existing evidence-based practices is more informative than a no treatment or business as usual control condition because this comparison can provide information regarding how new practices perform in an applied setting (Petersen-Brown et al., 2019). There is also value to determining how a new practice compares to practices known to be beneficial (Kilgus et al., 2016; Petersen-Brown et al., 2019). Researchers have suggested that when comparing interventions, it is critical to ensure that they are as similar as possible to create rigorous research conditions which reduce the likelihood of drawing inappropriate conclusions (Petersen-Brown et al., 2019). As stated above, both technology-delivered drill and RPT have similar core components such as the presentation of a stimulus (e.g., mathematics fact), a response (e.g., verbal or typed), and immediate, corrective feedback.

1.4 Assessment modality

When assessing the effectiveness of technology-based interventions, it is critical to evaluate the modality of the assessment used to measure the intervention outcomes (Duhon et al., 2012). Assessment across modalities refers to measuring students’ skills in a different modality than the one the intervention is provided in (Duhon et al., 2012). Prior research has indicated that when an intervention is delivered by computer, but the progress monitoring tool is paper-pencil, skill transfer is not always observed (Duhon et al., 2012; Rich et al., 2017). Additionally, Rich and colleagues (2017) conducted a study which targeted the subtraction fact fluency of second graders who participated in one of three interventions (e.g., explicit timing [paper practice], computer-based drill and practice, or 4 days of computer drill and 1 day of explicit timing [combined practice]). Rich and colleagues (2017) found that for students in the combined condition, 1 day a week of paper-based practice (e.g., explicit timing) was sufficient to facilitate generalization across assessment types. The findings of these studies have implications for evaluating student outcomes because it suggests that alignment between the intervention modality and assessment tool matters. Specifically, the implications for the results of a combined technology and paper-based intervention facilitating generalization across assessment measures has great practical value because students in schools often encounter both paper-based and technology-mediated assessments at the classroom, state, national, and international levels. The generalization of combined interventions to both assessment modalities could provide students with the necessary practice to generalize their learning across modalities leading to an increase in their performance across assessments and improve the accuracy with which school professionals make decisions regarding student performance.

1.5 Purpose

2.1 Participants and setting

Participants were third grade students recruited from the five third grade classrooms across two elementary schools within a suburban school district located in the Midwest United States. The school district served predominantly white (76.3%) students who were not eligible to receive free and reduce-priced lunch (82.8%). Upon receiving support from the University of Minnesota's IRB and the participating schools’ administrators, the first author invited all five third-grade classrooms across the two schools to participate in the present study (1 from school A and 4 from School B). Of the five classrooms, three teachers agreed to participate. Data collection was completed in the first 3 months of 2020. Passive parental consent forms were sent home with all 87 students in the three classrooms and were collected by the classroom teachers upon return. Of the 87 possible participants, parental consent and student assent were obtained for 76 students. Two participants moved out of the district after the start of the study and two participants consistently missed intervention sessions to receive special education services. Therefore, 72 participants made up this study (the classrooms had 31, 24, and 17 participants). Based on the results of a statistical sample size calculator (Erdfelder et al., 1996) for an analysis of covariance (ANCOVA) model with fixed and interaction effects, the desired sample size is 88 participants to achieve power of 80% with an α level of .0125 and effect size of f = 0.40. Accordingly, the recruited sample is slightly underpowered.

2.2 Measures

The pre- and posttest measures used for all students consisted of two proximal measures of multiplication fact fluency and two distal measures of mathematics achievement. The proximal measures were three forms of a 2 min paper-pencil curriculum-based measure (CBM) and three forms of a 2 min iPad-delivered CBM. The median score for each student on each assessment was used. The two distal measures were the mathematics fluency and mathematics calculation subtests of the Woodcock-Johnson Fourth Edition, Tests of Achievement (WJ-IV; Schrank et al., 2014). The primary dependent variable in this study was student's scores in digits correct per 2 min (DC2M) on the paper-pencil CBM and the iPad delivered CBM at pretest and posttest. Additionally, students were observed during the intervention phase using momentary time sampling to measure their academic engagement and off-task behavior. Lastly, a social validity measure was administered to students during posttest to gather information related to the students’ acceptability of the intervention procedures.

2.3 Procedures

Pretesting occurred on two different school days in each participants’ classroom. Paper-pencil CBMs and the iPad CBMs were delivered to the participants by the first author. Each student was randomly assigned to either complete the three iPad CBMs before the paper-pencil CBMs or vice versa to control for order effects. Both the iPad and paper-pencil CBM probes were administered in a single session lasting 20 min (i.e., 2 min for each probe plus time for instructions and a 3-min stretch break after completing three probes). The classroom teachers assisted the first author and monitored students during the assessment. The second day of pre-testing was reserved for the WJ-IV fluency and calculation subtests. The fluency subtest was administered first (3 min), and the remaining time (approximately 20 min) was reserved for students to work on the calculation subtest.

2.4 Inter-scorer agreement (ISA) and procedural fidelity

ISA was assessed for 20% of sessions and was calculated by summing the total number of digits attempted and dividing by the total digits agreed upon and multiplying by 100. A fourth-year school psychology doctoral student completed ISA. The mean ISA on paper-pencil CBM was 99.65%, (range, 86%–100%) and iPad CBM was 99.82% (range, 93%–100%). The mean ISA for each of the WJ-IV subtests was 99.15% (range, 86%–100%).

Procedural fidelity was assessed for both interventionist procedures and student intervention procedures. Fidelity of interventionist procedures was assessed by the classroom teachers for 20% of sessions. Teachers and interventionists discussed intervention procedures before implementation and a six-item checklist detailing the procedural steps was used to monitor fidelity. Procedural fidelity was calculated by summing the number of correctly implemented steps, divided by the total number of steps on the implementation checklist, and multiplying by 100. Treatment adherence for the interventionist was 100% (range, 83%–100%). Fidelity of student procedures was assessed by the first author using a checklist detailing the procedural steps of each intervention. One pair of students was randomly assigned for a fidelity check before the start of the intervention sessions, and one pair of students were assessed in each classroom on each day of intervention. Treatment adherence was as follows for students in the iPad condition 99% (range, 67%–100%), RPT condition 99% (range, 92%–100%) and combined condition 99% (range, 92%–100%).

2.5 Research design and analysis

A pretest–posttest experimental design with random assignment across three conditions was used. Descriptive statistics of the four mathematics measures including means, standard deviations, and correlations were calculated. Additionally, data was examined using a one-way analysis of variance (ANOVA) and independent samples t-tests to evaluate the presence of pre-test differences between groups on each of the measures and order effects between participants who were tested on the iPad first versus those who used paper-pencil first at both pre- and posttest on the CBM measures.

To determine whether there were differences between treatment conditions (RQ 1) and whether the effects of the intervention modality generalized to cross modality assessments (RQ 4), four one-way between-groups ANCOVA models (i.e., one for each measure) were conducted. The independent variable in each model was the treatment condition (i.e., iPad, peer-tutoring, or combined) and the dependent variable was the posttest scores on each measure. Participants’ scores at pretest on each respective measure were used as the covariate in this analysis. The assumptions of the chosen inferential statistic, ANCOVA, were checked. Due to the multiple comparisons being conducted (e.g., four ANCOVA models for each measure) the Benjamini–Hochberg (B-H) procedure was used. This method was chosen due to its utility in controlling for false discovery rates and power in limiting type 1 errors (Palejev & Savov, 2019). Additionally, the B-H procedure allows the user to sequentially compare the observed p values for each dependent variable, based on their importance or expected level of effect (Palejev & Savov, 2019). Therefore, the B-H corrected level of significance for each dependent variable is as follows; iPad CBM (p = .03), paper-pencil CBM (p = .02), WJ-IV fluency subtest (p = .01), and WJ-IV calculation subtest (p = .001).

To determine whether there were differences in the growth of students across the four mathematics measures (RQ 2) and whether there were differences in the fluency of student responding across assessment modalities on the proximal measures (RQ 3), the rate of improvement (ROI) of each group on the two proximal CBM measures was calculated by subtracting the pre-test scores of each group from the posttest scores and then dividing by the number of weeks of intervention to get an average change in DC2M per week for each group on the measure.

Paired samples t-tests were conducted to evaluate differences in engagement and off-task behavior between groups (RQ 5). Each observation session was categorized as either taking place during the iPad or peer-tutoring practice. Therefore, the combined group was categorized into the type of practice the student was engaged in at the time of the intervention. Mean scores and standard deviations across treatments were used to evaluation social validity of the intervention procedures (RQ 6).

3.1 Between group differences on mathematics measures

There were no violation of the assumptions of an ANCOVA. Normality was evaluated through visual analysis of the Q-Q plots of each measure were not indicative of a violation to the assumption of normality. The relationship the dependent variable and the covariate were best fit through a linear model for each measure. The homogeneity of variances assumption was tested through a Levene's test and for each measure the test was not significant and therefore was not indicative of violations to the homogeneity of variance assumption. The homogeneity of regression slopes was tested by examining the significance of the interaction between the independent variable and covariate, none of these interactions were significant indicating that the homogeneity of regression slopes assumption was not violated.

Table 2 includes the results of the ANCOVA models across the four assessments. The first four ANCOVAs include treatment group as the main effect with pretest as a covariant. With regard to whether there are differences between the treatment groups on the mathematics measures, the results of an ANCOVA indicate that the main effect for treatment was significant for the iPad CBM at the B-H corrected α level of p = .03. Posthoc comparisons using the B-H corrected α indicate that there was a significant difference between the iPad condition and the combined condition on the iPad CBM (M difference = 11.93, SE = 4.65, p = .013), favoring the iPad condition. There was not a significant difference between the iPad condition and RPT condition (M difference = 10.49, SE = 4.78, p = .032) when using the B-H corrected α or the RPT condition and the combined condition (M difference = 1.44, SE = 4.83, p = .767). The remaining three ANCOVAs indicated that the main effect for treatment group was not significant for the paper-pencil CBM, WJ-IV fluency, or WJ-IV calculation assessments.

3.2 Within group differences on mathematics measures

Table 3 includes the means, standard deviations, and ROI for each treatment group across the four mathematics measures at pre- and posttest. ROI data indicate that differences were observed between treatment groups across the four measures from pre- to posttest. On the paper-pencil CBM, the combined group (ROI = 5.16) had a higher ROI than the RPT (ROI = 4.56) and the iPad (ROI = 4.01) groups. On the iPad CBM, the iPad group (ROI = 5.53) has a higher ROI than the peer tutoring (ROI = 3.40) and combined (ROI = 3.21) groups. On the WJ-IV fluency, the iPad group (ROI = 1.02) had a higher ROI than the combined (ROI = 0.73) and the RPT (ROI = 0.53) groups. Lastly on the WJ-IV calculation, the combined group (ROI = 0.28) had a higher ROI than the RPT group (ROI = 0.16). Of practical significance, on the paper-pencil CBM, the ROIs for the iPad condition (4.56 DC2M), RPT condition (4.01 DC2M), and combined condition (5.16 DC2M) surpassed the expected ROI on this measure for students receiving no intervention. According to Pearson (2011) the expected ROI for third grade students on the math measure is a mean of 0.72 DC2M per week.

3.3 Fluency rates across assessment modality

Figure 1 includes a depiction of the rate of fluency across CBM modalities by group. When examining whether there is a difference in the rate of student responding across paper-pencil and iPad CBMs, all three groups demonstrated consistently higher rates of fluency on the paper-pencil measures at both pre- and posttest. The RPT and combined group demonstrated greater improvement in their multiplication fluency on the paper-pencil CBM at posttest than at pretest whereas the iPad group had similar rates of responding across both the paper-pencil and iPad CBMs with a slightly higher ROI (+0.97 DC2M per week) on the iPad CBM.

A paired samples t-test was used to examine differences in scores on the two CBM measures at both pre- and posttest. Results indicate that participants’ scores at pretest on both the iPad and paper-pencil CBM were significantly different (t (67) = 6.53, p = .000). Additionally, the participants’ scores at posttest were significantly different between the iPad and paper-pencil CBM (t (67) = 9.88, p = .000). These results indicate that at both pre- and posttest the participants had higher rates of DC2M on the paper-pencil CBM than the iPad CBM.

3.4 Generalization of interventions across assessment modality

When examining the generalization of single- and multi-modality intervention across assessment modalities, both the RPT and iPad group demonstrated growth from pre- to posttest on the proximal measures indicating that the results of a single modality intervention may generalize to a cross modality assessment. However, both groups demonstrated higher rates of improvement on the matched modality assessment which indicates that a matched intervention and assessment modality may be preferential to demonstrate maximum benefits of the intervention. In terms of skill generalization, all three treatment groups demonstrated improvements in their scores on the WJ-IV fluency which indicate that the effects of these multiplication fluency interventions may generalize to fact fluency across operations (addition, subtraction, multiplication, and division). However, on the most distal measure of mathematics achievement, the combined and RPT conditions demonstrated small improvements in scores from pre- to posttest which indicates that there may be a benefit to combined practice or RPT practice over iPad practice on the generalization of skills to more distal measures of mathematics performance.

3.5 Student engagement and off-task behavior

Students in the RPT condition had significantly higher rates of engaged time (M = 95.67%, SD = 9.18) than in the iPad condition (M = 87.38%, SD = 18.43; t (72.59) = −2.70, p = .009). Moreover, the students in the RPT condition had significantly lower rates of off-task behavior (M = 10.64%, SD = 14.35) than those in the iPad condition (M = 27.91%, SD = 27.98; (t (73.59) = 3.68; p < .001).

3.6 Intervention acceptability

Students in all three treatment conditions reported high acceptability ratings. While no significant differences were observed, the iPad condition had the highest mean acceptability (31.61; SD = 6.24), followed by the RPT condition (M = 29.76; SD = 7.04) and then the combined condition with a (M = 29.22; SD = 7.15). All three teachers indicated that they were highly likely to use both RPT and technology-delivered flashcards in the future.

4.1 CBMs

On the paper-pencil CBM, all groups had significant improvement from pre- to posttest, though there were not differences between groups at posttest. The lack of differences between groups is inconsistent with previous studies which found that the paper-pencil condition outperformed the technology and combined conditions on proximal paper-pencil measure (Duhon et al., 2012; Rich et al., 2017). It is possible that because the time to practice was held constant (10 min) that the potency of opportunities to practice in the iPad condition were higher than that of the RPT condition where students spent 5 min as the tutee and 5 min as the tutor. Previous research suggests that acting as both the tutor and tutee is beneficial for students (Haas et al., 2022); however, it is unclear if it is equally beneficial to be in either role. This, coupled with the idea that the combined group had the opportunity to generalize their learning from the iPad to paper-pencil during the day of RPT practice, may have contributed to the slightly greater ROI observed with the combined group as compared to that of both the RPT and iPad groups on the paper-pencil CBM. Additionally, the ROI on the measure for all groups surpassed the expected growth of 0.72 on the math computation measures with no intervention (Pearson, 2011). Solomon et al. (2020) found higher average math fluency growth rates per session (1.62 digits correct per minute). In the present study, students received four sessions per week and the dependent variable was digits correct per 2 min (DC2M) which means that a rate of improvement of approximately 12.96 DC2M would be consistent with the results of other interventions targeting math fact fluency (Solomon et al., 2020). On the two CBM measures no groups reached a ROI of 12.96 DC2M or higher. Therefore, while the results of the present study resulted in higher ROIs than would be expected for students who are not receiving intervention, the ROIs were not as high as results found in Solomon et al. (2020). The lower ROIs observed in the present study could be due to many factors relating to student characteristics or levels of initial fluency or the large set size used in the included interventions.

On the iPad CBM, all intervention conditions produced significant growth from pre- to posttest. However, a significant difference was observed between the iPad group and combined group favoring the iPad condition. In addition, on the iPad CBM students in the iPad condition had a ROI greater than 5 DC2M per week and students in the combined condition and RPT conditions had a greater than 3 DC2M ROI. Given that the students in the iPad condition yielded the highest ROI compared to all conditions and performed significantly higher than those in the combined condition on the iPad CBM it is possible that typing speed improved for these students because of the repeated practice using the iPad. Another explanation may be that the iPad CBM was the most proximal measure to the iPad intervention, given the matched modality. This is consistent with results from Duhon et al. (2012) and Rich and colleagues (2017) who found that the technology-mediated intervention yielded the highest scores on the technology-delivered CBM. Although there were not significant differences on the iPad CBM between the RPT and iPad conditions, this could be due to insufficient power in the present study due to a small sample size. Lastly, for both the paper-pencil and iPad CBMs pre-test was a significant covariate for posttest scores and explained much of the variance in posttest scores (62.3% and 57.4%, respectively). These results are consistent with research which suggesting that student's initial skill level is a significant predictor of posttest outcomes (Clarke et al., 2019).

Students across all conditions had higher rates of DC2M when completing the paper-pencil CBM than the iPad CBM at both pre- and posttest. The results of the present study support previous research which has found that student response times are slower when typing than writing (Hensley et al., 2017; VanDerHeyden et al., 2022). Another potential reason for this difference could be because the students have more experience with writing than typing; while the participating classrooms utilized technology, the time spent using it was much lower than time spent in traditional activities (e.g., worksheets, paper-pencil tests) involving writing.

4.2 Generalization

Despite the growth noted for students in each condition across the paper-pencil proximal measure, this performance did not generalize to the WJ-IV fluency subtest, which reflects a broad measure of mathematics facts from all four operations. Given participants above average range of performance at pretest, it is possible that students had little growth to make on the assessment suggesting a ceiling effect was present. Similarly, for both the RPT and iPad group there were no generalized effects observed on the WJ-IV calculation subtest. However, the results of a paired samples t-test indicated that there was significant growth in performance for students in the combined condition from pre- to posttest though these effects were not found to be significantly different than the other treatment groups in the ANCOVA model. It is possible that the sample size was too small to detect a difference or that there truly was no significant difference between scores. The lack of gains for may also be because the WJ-IV calculation subtest which is a more distal mathematics measure comprised of problems ranging from simple addition to advanced calculus. Regardless, these scores are of practical significance and provide support for the use of multiple modalities which is supported by previous research that has found that computer assisted instruction is more effective as a supplement to traditional instruction rather than a replacement (Burns et al., 2012; Cheung & Slavin, 2013; Musti-Rao & Plati, 2015).

4.3 Systematic direct observation

Students participating in the RPT intervention were significantly more engaged in the intervention than those using the iPad and, correspondingly, students using the iPad were significantly more off-task than those participating in RPT. No other studies have compared iPad conditions to RPT, albeit Bryant et al. (2015) found that a technology delivered intervention was more engaging for students than teacher delivered small group and worksheet practice. The lack of identified differences in posttest scores between groups despite the differences in engagement could be due to a lack of statistical power or it could be that the similarities between interventions in terms of active ingredients lessened the impact of the differences in engagement.

When examining the list of active ingredients across the iPad and RPT intervention conditions, the only difference is that multiplication facts were typed into the app while the student worked independently on the iPad whereas with RPT students verbalized and wrote their answers. Taiwo (2017) demonstrated a functional relationship between student verbalization or mathematics problem solving and mathematics proficiency. These data suggest that the RPT group may have had an advantage in verbalizing their responses, however this did not translate to differences in growth on the included measures. Perhaps other variables discussed above such as the potency of opportunities to practice impacted the growth of the RPT condition. In other words, the combined group both may have yielded a higher number of opportunities to respond working on the iPad and were more engaged in the peer-tutoring intervention perhaps producing the small difference between ROI on the paper-pencil CBM between the peer-tutoring and combined conditions.

4.4 Intervention acceptability

Each group of students rated their respective intervention as being highly acceptable. These results are consistent with other studies that have examined student acceptability of technology-delivered interventions (Kromminga & Codding, 2020; Musti-Rao & Plati, 2015) and RPT interventions (Greene et al., 2018). There were no significant differences between the conditions in terms of student acceptability, indicating that any of the three intervention conditions might be acceptable intervention options for students. The classroom teachers also rated that they were very likely to use both the technology-delivered and RPT interventions in their classroom in the future. Researchers have found that interventions are generally more acceptable if they are not excessively time consuming, have limited or no negative side effects, and intrusiveness of the intervention (Silva et al., 2020). Therefore, it is possible that the limited time burden on the teachers for both intervention procedures, the short period of class time required to complete the intervention (e.g., 20 min), and the intervention's fit within typical classroom practices led to high acceptability among classroom teachers.

4.5 Limitations and future research

As with all research the results must be taken with the limitations. First, the sample size for the present study was smaller than the power analysis indicated would be necessary to detect an effect. Future replications of this study are needed in future research. Second, the present study did not include a control condition. While this limits the claims, we could make regarding the impacts of the intervention compared to students receiving no treatment, researchers have stressed the importance of including active treatment comparisons to provide more information regarding how two treatments compare in terms of instructional design principles, impacts on student performance, and resources for implementation (Petersen-Brown et al., 2019). Future research may consider including an active control and business as usual control condition to provide comprehensive support for the effectiveness of the treatments. Third, the current sample of third grade students was predominantly white students without disabilities from a middle-class suburb in the Midwest United States. This limits the generalizability of the results to other grade levels, races or ethnicities, disability status, urbanicity, geographical region, or schools with different instructional practices. Future research should replicate this study with different populations of students to determine if these practices could be beneficial to different groups of students. Fourth, due to limitations with how the facts could be presented in the iPad condition (i.e., all 91 facts randomized) the peer condition also practiced daily with the full set of facts randomized. This was done to keep the intervention conditions similar for comparison between conditions. However, it is often recommended to practice facts in small sets of unknown facts rather than all at once (Burns et al., 2016). Therefore, it is possible that the intervention impacts on students’ outcomes may be higher with smaller sets of mathematics facts practiced daily. Future research may also consider examining student growth throughout the intervention rather than simply looking at change from pre- to posttest because growth may not have been linear. Additionally, future research may also examine whether student baseline level of performance results in differential effects of interventions across modalities. Another limitation is that the first author served as an interventionist for each of these classrooms, future studies may wish to examine the effectiveness of teacher implementation of these class wide interventions. An additional limitation is that there was no interobserver data collected on the measure of student engagement and the first author both administered and scored the CBMs, meaning they were not blinded.

4.6 Implications for practice

The results of this study suggest that all students improved their multiplication fluency from pretest to posttest on proximal measures and there were few differences between groups. That said, on the iPad CBM, students in the iPad condition significantly outperformed students in the combined condition, potentially illustrating the impact of matching assessment and intervention modality on student outcomes. A significant difference in engagement was also found, favoring the RPT intervention. Taken together, these results have practical meaning given that reciprocal RPT, iPad delivered flashcards, or the combination of the two were beneficial for improving the multiplication fact fluency of these third-grade students and were highly acceptable to both the students and teachers.