School Psychology Quarterly 2015, Vol. 30, No. 3, 398 – 405

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Number of Repetitions Required to Retain Single-Digit Multiplication Math Facts for Elementary Students Matthew K. Burns

Jim Ysseldyke

University of Missouri

University of Minnesota

Peter M. Nelson

Rebecca Kanive

Pennsylvania State University

University of Minnesota

Computational fluency is an important aspect of math proficiency. Despite widely held beliefs about the differential difficulty of single-digit multiplication math facts, little empirical work has examined this issue. The current study analyzed the number of repetitions needed to master multiplication math facts. Data from 15,402 3rd, 4th, and 5th graders were analyzed using a national database. Results suggested that (a) students with lower math skills required significantly (p ⬍ .001) more repetitions than more skilled students; (b) across all students, single-digit multiplication facts with 4s, 5s, 6s, and 7s required significantly (p ⬍ .001) more repetition than did 2s and 3s; and (c) the number of practice sessions needed to attain mastery significantly (p ⬍ .001) decreased with increase in grade level. Implications for instructional planning and implementation are discussed. Keywords: Math fluency, multiplication facts, repetition

Recent data from the National Center on Educational Statistics (NCES) indicate a minimal increase in mathematics proficiency for 4th-and 8th-grade students, and that 19% and 29% of students are below basic levels in the 4th and 8th grades, respectively (National Assessment of Educational Progress, 2010). Improving the early mathematics proficiency of students is critically important in light of its association with later educational outcomes such as college graduation and the ability to solve increasingly complex mathematics problems (Kilpatrick,

This article was published Online First November 24, 2014. Matthew K. Burns, College of Education, University of Missouri; Jim Ysseldyke, Educational Psychology, University of Minnesota; Peter M. Nelson, Educational Psychology, Counseling, and Special Education, Pennsylvania State University; Rebecca Kanive, Educational Psychology, University of Minnesota. The current study was supported by a research grant from Renaissance Learning in Wisconsin Rapids, WI. Correspondence concerning this article should be addressed to Matthew K. Burns, 109 Hill Hall, University of Missouri, Columbia, MO 65211. E-mail: burnsmk@ missouri.edu

Swafford, & Finell, 2001; National Mathematics Advisory Panel [NMAP], 2008). Moreover, students who store basic math fact information in memory and quickly retrieve it easily are more likely to develop skills necessary for solving a variety of complex problems, interpreting abstract mathematical principles, and successfully living independently (Patton et al., 1997). Computational fluency is the efficient and accurate completion of math calculations (National Council of Teachers of Mathematics, 2000), which is enhanced by automatic recall of basic facts (Gersten & Chard, 1999). Math facts are considered automatic when students solve math problems more quickly by recalling the answer than by performing the necessary mental algorithm (Logan, Taylor, & Etherton, 1996). For example, automatic recall occurs when a student can look at 3 ⫻ 3 ⫽ and quickly recall that the answer is 9 rather than use some other strategy such as counting by threes. Many students who are not proficient in advanced math problems lack fluency of the basic skills within them (Houchins, Shippen, & Flores, 2004; Wong & Evans, 2007). Much like the automatic processing theory of reading (Samuels, 1987), computing basic math opera-

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REPETITION OF MULTIPLICATION MATH FACTS

tions without devoting much cognitive energy to the task allows students to dedicate their resources to more advanced applications within the problem. Teaching component skills consistently led to increased performance of more advanced skills such as multidigit multiplication or word problems (Dehaene & Akhavein, 1995; Singer-Dudek & Greer, 2005), and students with advanced computational fluency exhibited increased engagement in math activities (Axtell, McCallum, Bell, & Poncy, 2009). Thus, the speed with which students complete basic math operations (e.g., single-digit multiplication and division) could serve as an intervention goal for some students, especially given that students with math difficulties frequently struggle to quickly recall basic math facts (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Gersten, Jordan, & Flojo, 2005). Increasing computational fluency is also a core component of quality core instruction (NMAP, 2008). Many elementary schools place less emphasis on instruction and practice of basic math facts as students progress through the curriculum because they assume that students have acquired a sufficient level of computational fluency from previous instruction to proceed with related or more complex tasks (Isaacs & Carroll, 1999). There are many instructional approaches to achieve fluent computation, but repeated practice with the skill seems to be a necessary component (Binder, 1996). Previous mathematics intervention research found that when students identified with a learning disability in math extensively practiced multiplication facts, they retained them, generalized them, and increased fluency to a level that was typical for their grade (Burns, 2005). However, “few curricula in the United States provide sufficient practice to ensure fast and efficient solving of basic fact combinations and execution of the standard algorithms” (NMAP, 2008, p. 26). Empirical and descriptive literature supports the distinction that some basic multiplication math facts are considered relatively easy, whereas others are considered more difficult and require considerably more practice to learn (Woodward, 2006). Specifically, doubles, Times 5 patterns, Times 9 patterns, and square numbers are believed to be easier facts for students to learn (Garnett, 1992; Thornton, 1990). Thus, multiplication fluency for simple math

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problems may increase at differing levels depending on the numbers used for multiplication and the instructional strategy used (Wood, Frank, & Wacker, 1998). Moreover, recommendations for completing certain skills by particular grade levels (e.g., NMAP, 2008) are based on curricular progressions toward a particular goal (i.e., completing Algebra), but little is known about the effect of development on those reported benchmarks. Research has consistently shown that younger students retain less information than do older students (Burns, 2004; Fry & Hale, 2000), but the relative difficulty of various math facts across developmental levels is unknown. Increasing awareness in regard to developmental differences may result in added attention to the length of time spent on certain math facts at various grades as well as the type of instruction provided. Despite the potential advantages that knowledge about the differential difficulty of multiplication facts holds, few projects have sought to empirically determine which math facts require more time and practice. Furthermore, recent math interventions research highlights the important role that initial fluency level may play in intervention design (Burns, Codding, Boice, & Lukito, 2010; Codding et al., 2007), but more work is needed to better establish the instructional needs of individual and groups of students. The purpose of the current study was to examine the effect that grade level, type of math fact, and previous math performance has on the number of practice sessions needed to master a set of multiplication facts. The following research questions guided the study: (a) What effect does type of math fact being learned have on number of repetitions needed to fluently compute single-digit multiplication facts; (b) What effect does math skill have on number of repetitions needed to fluently compute single-digit multiplication facts; (c) and What effect does grade level of the student have on number of repetitions needed to fluently compute single-digit multiplication facts? Methods Participants and Setting Data were obtained from an extant database compiled by Renaissance Learning (RL). Participants included a national sample of 15,402

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elementary aged students (Grades 3–5) of which 49% were female and 51% were male. The sample consisted of 3,666 students in 3rd grade, 5,973 students in 4th grade, and 5,763 students in 5th grade. Ethnicity data were available for 81% of the participants included in the study, 41% of participants were Caucasian, 21% were Hispanic, 12% were African American, 6% were Asian, and 1% were American Indian. Measures Number of attempts. Data from the national database were collected with the software program Math Facts in a Flash (MFF; RL, 2003a). MFF is a software program designed to enhance computational fluency in the four basic mathematics operations. Math facts are hierarchically arranged by MFF into 62 levels of increasing difficulty, and the software program records the number of attempts needed to master a given level. The number of attempts needed to master a level served as the primary dependent variable in the study. Students begin each level by taking a timed 40-item baseline test that includes feedback on accuracy and provides the correct answer for missed facts. If students answer all items correctly on the baseline test, then the level is skipped. If not, then students spend approximately 5 to 15 min practicing independently. The practice set is not timed and students answer at their own pace. Each practice set consists of a 40-item randomly determined set of problems from the level on which the student is working (e.g., addition of 4 and 5, subtraction of 8 to 10, multiplication by 4 and 5, etc.). All items within the level were practiced, including ones that were answered correctly during the baseline assessment, and those that were mastered in early levels through the commutative property (e.g., 8 ⫻ 3 would be practiced if 3 ⫻ 8 was already mastered). During the practice session, students receive feedback on each answer choice. If the wrong answer is selected, the correct answer is highlighted and the student is allowed to repeat the item. After successfully completing the practice session (i.e., correctly completing all of the items), students complete a 40-item timed test. If all items are answered correctly within 2 min, then the level is considered mastered and the

student begins the subsequent level during the next session. The number of attempts included the combined number of practices and tests for a particular level, which provided an indication of how much total practice a student needed to master an MFF level. For example, a student may have completed the practice set six times for a given level and required two tests to pass the level, which would have resulted in an attempts score of 8. To master a level, students had to correctly answer 40 problems in 2 min, which is an average response rate of 3 s per problem. Math proficiency. Math skills were assessed with Star Math (RL, 2003b), which is a computer-adapted assessment system. Students complete 24 items generated on a computer, from a bank of almost 2,000 items. Student responses are immediately scored by comparing the data to a norm group of over 29,000 students, and converted to an estimate of overall proficiency in math. Reliability estimates ranged from .78 to .88, and STAR Math was described as a technically sound instrument (Garner, 2005). Students at or below the 25th percentile on Star Math were identified with below average math skills, those between the 26th and 75th percentile were identified as average range, and those above the 75th percentile were considered to be in the above average range for math skills. The 25th percentile is a commonly used criterion for determining students who are at risk for developing academic difficulties (Burns, Klingbeil, & Ysseldyke, 2010; Torgesen et al., 2001; Vellutino et al., 1996). Procedure Although the MFF database includes data on all 62 levels of difficulty, only data on single digit (0 –9) multiplication skills were used for this study. Thus, four levels were used in the study, which included single-digit multiplication by 2s and 3s, by 4s and 5s, by 6s and 7s, and by 8s and 9s. All multiplication items are included in each level without regard to the commutative effect. In other words, the level that addressed 4s and 5s included 0 ⫻ 4 to 9 ⫻ 4, and 0 ⫻ 5 to 9 ⫻ 5. MFF includes two levels between the 4s and 5s level and the 6s and 7s level that are reviews of previously mastered

REPETITION OF MULTIPLICATION MATH FACTS

levels, but those levels were not included in the current analyses.

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Results The distribution of the data was evaluated by examining a visual display of the data and the numerical estimates of kurtosis and skew. There was a substantial positive skew for each multiplication fact level (range of 3.43 to 9.06) and leptokurtic distribution (range of 19.52 to 166.54). Therefore, the research questions were addressed with nonparametric analyses. We used an alpha level of .001 to determine significance because of the multiple analyses for each question and because of the large sample size and accompanying power. The first research question inquired about the effect of math fact type on the number of repetitions needed to fluently compute single-digit multiplication facts. The mean number of repetitions needed to successfully complete each level, student math skill, and student grade level are listed in Table 1. Students practiced multiplication most often for digits 4 through 7. Fewer attempts were recorded for levels focusing on digits 2 and 3, and 8 and 9. The data were analyzed with a Friedman Test for related data, but the level that included 8s and 9s was not included in the analysis because the commuta-

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tive property suggested that all but 8 ⫻ 8, 8 ⫻ 9, 9 ⫻ 8, and 9 ⫻ 9, were practiced in earlier levels. The Friedman Test resulted in a significant effect ␹2(2) ⫽ 402.79, p ⬍ .001. The second research question inquired about the effect of math skill on the number of repetitions needed to fluently compute single-digit multiplication facts. As shown in Table 1, the mean number of attempts decreased as math skill increased. The effect of math skill number on repetitions was evaluated with KruskalWallis test and resulted in a significant effect for all four groups of items: 2s and 3s, ␹2(2) ⫽ 570.54, p ⬍ .001; 4s and 5s, ␹2(2) ⫽ 529.58, p ⬍ .001; 6s and 7s, ␹2(2) ⫽ 381.87, p ⬍ .001; and 8s and 9s, ␹2(2) ⫽ 426.67, p ⬍ .001. The above average group had the lowest mean rank for all four analyses and the below average group consistently had the highest mean rank. The final research question addressed the effect of student grade level on the number of repetitions needed to fluently compute singledigit multiplication facts. Similar to math skill, there was an inverse relationship between grade level and mean number of repetitions. The effect of grade level on number of repetitions was evaluated with Kruskal-Wallis test and resulted in a significant effect for all four groups of items: 2s and 3s, ␹2(2) ⫽ 1246.38, p ⬍ .001; 4s

Table 1 The Mean Number of Repetitions Needed to Successfully Complete Each Level, Student Math Skill, and Student Grade Level 2s and 3s

Grade 3 Below average Average Above average Total Grade 4 Below average Average Above average Total Grade 5 Below average Average Above average Total

4s and 5s

6s and 7s

8s and 9s

n

M

SD

M

SD

M

SD

M

SD

460 1,939 1,267 3,666

11.24 6.46 4.11 6.25

17.41 7.84 5.07 9.12

11.24 8.45 6.37 8.03

17.41 9.60 7.55 9.50

7.87 7.36 5.84 6.90

10.01 8.93 6.62 8.40

6.17 5.07 3.93 4.82

6.91 6.21 4.27 5.76

910 3,068 1,995 5,973

5.89 3.21 2.14 3.26

9.87 4.34 2.27 5.26

6.21 4.12 2.82 4.00

8.32 4.99 3.21 5.23

6.17 4.03 2.94 3.99

8.27 5.33 3.65 5.53

4.53 3.38 2.23 3.17

5.36 4.36 2.28 4.06

968 2,870 1,925 5,763

3.89 2.31 1.76 2.39

5.67 2.79 1.52 3.25

4.86 2.88 1.90 2.89

7.13 3.48 1.86 4.09

4.98 2.85 1.96 2.91

8.06 3.71 2.13 4.50

4.40 2.40 1.80 2.53

7.05 2.67 1.59 3.68

Note. Number of attempts as an indicator of difficulty by level, grade, and math skill status.

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and 5s, ␹2(2) ⫽ 1800.12, p ⬍ .001; 6s and 7s, ␹2(2) ⫽ 1445.21, p ⬍ .001; and 8s and 9s, ␹2(2) ⫽ 948.51, p ⬍ .001. Data from the 5thgrade students had the lowest mean rank for all four analyses and the 3rd-grade students consistently had the highest mean rank.

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Discussion Results from the current study provide preliminary support for the impact of several relevant variables on the number of repetitions necessary to fluently compute single-digit multiplication facts. Students’ grade and level of math skill tended to affect repetitions in a predictable manner. That is, as student grade and math skill increased, the number of repetitions tended to decrease. Moreover, students tended to require more practice to master singledigit multiplication facts with digits 4 –7 than those with digits 2–3. Given the size of the extant database used for analysis, the current study provides an unprecedented view into single-digit math fact difficulty for elementary-aged students across multiple skill levels. The observed differences in attempts by (a) grade, (b) skill level, and (c) math fact difficulty could provide a future framework to inform instructional planning and presentation. The significant differences across initial math skill levels (below average, average, above average) could have implications for practice. For example, many schools are now screening math skills of all students with standardized measures (National Council of Teachers of Mathematics, 2006; NMAP, 2008), and these data support measuring math skills to predict the amount of instructional and independent work time students may require to master computational fluency. However, the inverse relationship between math skill and repetitions should be considered preliminary and additional research is needed to determine the instructional implications. Instructional assessment generally informs planning decisions related to what and how to teach (Hosp & Ardoin, 2008). Fluency data are often used to determine whether instructional material is too difficult (frustration level), appropriate (instructional level), or not difficult enough (mastery or independent level) for students. Original work in this area by Gickling and Armstrong (1978) demonstrated increased

on-task behavior, task completion, and task comprehension when material was matched to the student’s instructional level. The importance of assessing instructional level for reading instruction has been well-documented and despite less research in math, there is promising work on the positive effects of matching instruction to student need for math (Burns, VanDerHeyden, & Jiban, 2006; Daly et al., 1996). Evaluating the specific skills that students possess and lack can inform the development of interventions that are individually appropriate (Poncy & Skinner, 2006). The current study supplements this research by affirming the predictive value of readily available data on math performance. Mastering basic computational fluency is an important goal for instruction, especially among students with learning difficulties in math (Geary et al., 2007), because fluency in math facts within an information processing model is fundamental for progression to more complex and abstract math skills (Woodward, 2006). Moreover, early competence in basic skills predicts later success with more advanced skills (Geary, 2011), perhaps because as children develop math skills, they tend to replace computation with memory retrieval by activating different areas of the brain (Price, Mazzocco, & Ansari, 2013). Although the current study did not examine ways to enhance math fact fluency or why fewer repetitions were needed to master facts for older grades (i.e., 5th grade), the developmental trend found in the current crosssectional data was consistent with previous longitudinal research. Limitations and Directions for Future Research Results of the present study should be interpreted with consideration of several limitations. The dataset used for the current analyses was from a national sample and may not generalize to small subgroups of students. Demographic data from the entire database were not analyzed, but lack of ethnic diversity in the sample set from Grades 3, 4, and 5 should also be recognized. The large standard deviations observed for the number of attempts make practical interpretations of the results somewhat difficult. Nevertheless, a large differential in skills of students within each grade level might account for the

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REPETITION OF MULTIPLICATION MATH FACTS

broad range of attempts, something that further emphasizes the significance of instructional match for successful student achievement. Moreover, the study did not take the order with which the facts were rehearsed or the commutative property into account, which makes interpretations about math facts with 8s and 9s difficult to interpret given that mastery of some of the facts within 8s and 9s were likely due to practice with earlier sets (e.g., 9 ⫻ 3 was rehearsed as 3 ⫻ 9). In addition to not considering the commutative effect, the study did not systematically vary the order with which the levels were presented. The level containing 2s and 3s preceded the level with 4s and 5s, which preceded the level with 6s and 7s, which was followed by the level with 8s and 9s. Some students completed sets with one attempt, but the order that the levels were presented did not change. Finally, despite reaching mastery level performance for single-digit multiplication skills, retention and maintenance cannot be assumed. Students may be able to accurately and fluently respond to stimuli without the ability to recall it later. This is especially relevant given the fact that retention and maintenance of computation skills is important in the sequence of an instructional hierarchy (Haring & Eaton, 1978). The current study provides data on differential math fact difficulty that could be useful for instructional planning, but more research is needed concerning the reasons for these differences. In addition to addressing the limitations discussed above, future researchers could examine why some math facts required fewer repetitions than others to be mastered, and to what extent variables within the practice session influenced the number of repetitions needed to reach mastery such as practicing fewer facts in each set or requiring less than 3s for each problem to demonstrate mastery. Moreover, the number of repetitions needed by skill level could provide a potential framework for differentiation within the classroom after additional research. In addition, the number of repetitions required was substantially higher for Grade 3, which was when students were first learning the material. Thus, the effect of other instruction that occurred in addition to the math fact practice could be examined.

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Conclusion The current study provides preliminary support that some math facts require more repetition to learn than do others, and that students at younger grades or with math difficulties require more repetition that students in older grades or with higher math skills. These data could have potential implications for practice and could contribute to the literature on individual and group differences in student learning. However, future research should seek to both clarify and improve upon current methods of instructional planning through replication and practical application across unique environmental settings. References Axtell, P. K., McCallum, R. S., Bell, S. M., & Poncy, B. (2009). Developing math automaticity using a classwide fluency building procedure for middle school students: A preliminary study. Psychology in the Schools, 46, 526 –538. http://dx.doi.org/ 10.1002/pits.20395 Binder, C. (1996). Behavioral fluency: Evolution of a new paradigm. The Behavior Analyst, 19, 163– 197. Burns, M. K. (2004). Age as a predictor of acquisition rates as measured by curriculum-based assessment: Evidence of consistency with cognitive research. Assessment for Effective Intervention, 29, 31–38. http://dx.doi.org/10.1177/073724770402 900203 Burns, M. K. (2005). Using incremental rehearsal to practice multiplication facts with children identified as learning disabled in mathematics computation. Education & Treatment of Children, 28, 237– 249. Burns, M. K., Codding, R. S., Boice, C. H., & Lukito, G. (2010). Meta-analysis of acquisition and fluency math interventions with instructional and frustration level skills: Evidence for a skill by treatment interaction. School Psychology Review, 39, 69 – 83. Burns, M. K., Klingbeil, D. A., & Ysseldyke, J. (2010). The effects of technology enhanced formative evaluation on student performance on state accountability math test. Psychology in the Schools, 47, 582–591. http://dx.doi.org/10.1002/ pits.20492 Burns, M. K., VanDerHeyden, A. M., & Jiban, C. (2006). Assessing the instructional level for mathematics: A comparison of methods. School Psychology Review, 35, 401– 418. Codding, R. S., Shiyko, M., Russo, M., Birch, S., Fanning, E., & Jaspen, D. (2007). Comparing

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Number of repetitions required to retain single-digit multiplication math facts for elementary students.

Computational fluency is an important aspect of math proficiency. Despite widely held beliefs about the differential difficulty of single-digit multip...
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