Fostering Multiplication Fluency Skills Through Skip Counting

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Fostering Multiplication Fluency Skills Through Skip CountingMatthias Grünke11University of Cologne, Department of Special Education and Rehabilitation, Klosterstr. 79b, 50931Cologne, Germanymatthias.gruenke@uni-koeln.deAbstract – The purpose of this single-case study was to implement and evaluate atechnique to teach multiplication fluency skills (skip counting) to a 7-year old girlwho had severe difficulties with multiplication tables. A multiple-baseline designacross two fact sets was applied to determine the effectiveness of the approach.Assessment prior to the intervention showed very low fluency skills. During thecourse of the treatment, the student’s performance improved, reaching mastery.Overall, the skip-counting approach turned out to be very successful. The paperends with a discussion of the limitations of the study and suggestions for futureresearch directions.Key Words – Math fact fluency; Math problems; Single-case research; Skipcounting; Special education.1IntroductionFact fluency in addition, subtraction, multiplication, and division is essential for access to and successwith higher-level mathematical concepts. It is important to reduce working memory overload toincrease the amount of energy available for problem solving. In order to master complex tasks, it isnecessary to execute basic arithmetic operations fast and accurately (Codding, Burns, & Lukito, 2011;Nelson, Burns, Kanive, & Ysseldyke, 2013). Thus, if students do not acquire sufficient math fluencyduring their elementary education, they will very likely continue to demonstrate difficulties in thisrespect throughout their lives (Gersten, Jordan, & Flojo, 2005). Automaticity of multiplication factsseems to be especially crucial in this context (National Mathematics Advisory Panel, 2008), as theinability to master multiplication tables has been found to have a very negative effect on long-termperformance in mathematics instruction (Jordan, Hanich, & Kaplan, 2003). For example, students withproblems in this domain will, in all likelihood, struggle with ratio and proportion, fraction concepts,and measurement conversions (National Mathematics Advisory Panel, 2008).Unfortunately, a great proportion of children are not adequately prepared in mathematics by the end oftheir elementary education (Duncan et al., 2007), primarily because many teachers do not incorporatesufficient opportunities in their daily instruction for students to practice basic multiplication skills(Daly, Martens, Barnett, Witt, & Olson, 2007). However, several promising approaches have beenfound to remedy the problem of insufficient fluency. For example, in their meta-analysis, Codding etal. (2011) present a number of methods that have proven to be helpful in building automaticity ofmultiplication facts, including cover copy compare, interspersal techniques, self-management, tapedproblems, peer-delivered feedback, positive practice overcorrection, and various flashcard procedures.One instructional approach that is not featured in this meta-analysis is skip counting (also known ascount-bys strategy). Skip counting involves counting by a number that is not 1. For example, if youskip counts by 4 up to 20, you would count in the following order: 4, 8, 12, 16, 20. Combining the1

International Journal of Basic and Applied Science,Vol. 04, No. 04, April 2016, pp. 1-6Grünkebase (4, in this example) with the number of groups (5, in this example) produces the standardmultiplication equation: 4 multiplied by 5 equals 20 (Wagganer, 2015). Most children findmultiplication by 2, 5, and 10 to be easier than other multiplication facts (Baroody & Dowker, 2003).Skip counting is a means to help students understand the concept of repeated addition and to memorizemultiplication tables other than the x2s, x5s, and x10s. It has been used as a way to teachmultiplication for a long time. For example, skip counting is part of Investigations in Numbers, Data,and Space, a widely used K-5 mathematics curriculum, developed between 1990 and 1998 (Budak,2015).Surprisingly, as of February 2016, the database PsycINFO lists only six publications that contain theterm "skip counting" and only two publications that contain the term "count bys" in their titles.Besides, only three of those studies are empirical (i.e., Duvall, McLaughlin, & Cooke-Sederstrom,2003; Grünke & Calder Stegemann, 2014; McIntyre, Test, Cooke, & Beattie, 1991). Using a singlecase design, the results of each experiment provide impressive evidence for the effectiveness of theapproach. In every instance, a skip-counting intervention of as little as 10 sessions producedremarkable improvements in the multiplication fluency skills of the participants.As mentioned, to date, skip counting has generally gone unnoticed by the research community. Thepurpose of the present study was to extend the body of empirical literature about this technique toteaching multiplication fact fluency to struggling learners. As in the previous experiments, a singlecase design was applied to answer the underlying research question concerning how much anelementary school student’s multiplication skills would improve with not yet mastered fact sets using asimple skip-counting intervention.2Method2.1ParticipantThe participant of this study, Belma, was a 7-year-old girl enrolled in the second grade in a largeelementary school in a major city in Western Germany. Her parents had moved from Turkey beforeBelma, the third of four children, was born. She spoke both Turkish and German fluently and was ableto master addition and subtraction facts without major difficulty. However, when her teacherintroduced the concept of multiplication in the middle of second grade, she did not seem to grasp it.She was mostly unable to solve problems dealing with multiplication tables that her teacher presentedto the students in her class through worksheets. According to her teacher, Belma especially failed with8s and 9s fact sets. Otherwise, she did not demonstrate any serious academic problems.2.2Experimental DesignAn AB multiple-baseline design (Horner & Odom, 2014) across two fact sets (8s and 9s) was applied.During the whole study, 20 weekdaily probes were collected. The beginning of the instruction topractice each fact set was randomly determined within certain specifications: In accordance with thesingle-case intervention research design standards proposed by Alberto and Troutman (2012), theminimum number of data points during the baseline was set at five. Therefore, the instruction had tolast at least five daily sessions. Thus, the intervention could have started any time between the 6 th andthe 16th probe. A drawing of all 10 possible options for each fact set resulted in an arrangementwhereby the intervention to practice the 8s fact set started before the sixth probe (i.e., after the fifthbaseline measurement point), the intervention to practice the 9s fact set started before the 11th probe(i.e., after the 10th baseline measurement point).2.32InterventionInsan Akademika Publications

GrünkeInternational Journal of Basic and Applied Science,Vol. 04, No. 04, April 2016, pp. 1-6Weekdaily training of each fact set lasted for 15 minutes. A 25-year-old female graduate student inspecial education from a large German university served as the teacher. She instructed Belma in aquiet corner of the classroom, while the rest of the children were engaged in independent seatwork.The student teacher presented Belma with large index cards that portrayed the counting sequence forthe 8s (8, 16, 24, 32, 40, 48, 56, 64, 72, 80) and for the 9s (9, 18, 27, 36, 45, 54, 63, 72, 81, 90),respectively. At the start of the intervention, the student teacher repeatedly read the respectivesequence out loud. Then, she encouraged Belma to join her. In order to facilitate memorization, theyreiterated the sequences in form of a rap song.After Belma was able to recite the order of the numbers by heart, the student teacher presented herwith rather simple multiplication problems (e.g., 1 x 8, 2 x 8, and 10x 8) and later moved on to morechallenging ones (e.g., 3 x 9, 4 x 9, 5 x 9, 6 x 9, 7 x 9, 8 x 9, and 9 x 9). The problems were solved bytaking a certain number of steps of the sequence for the 8s or 9s. For example, if Belma needed tocalculate 4 x 9, she had to count the first four numbers of the counting order for the 9s (9, 18, 27, 36).The teacher scaffolded the process and fell back on the index cards as necessary.2.4Dependent VariableThe number of correctly solved multiplication problems in response to worksheets containing all 10tasks for each fact set served as the dependent variable. Thus, each worksheet consisted of 20problems, which were presented in random order. Belma was presented with a different worksheeteach day. During the intervention phase, she worked on the problems after the daily training. The timelimit for finishing the daily assignment was 5 minutes.3ResultsAs shown in Figure 1, the number of correctly written digits in response to the presented worksheetsincreased drastically from initial levels. Specifically, the means of Belma’s baseline and interventionphase measures grew from 0.40 to 7.87 for the 8s, and from 0.50 to 7.10 for the 9s. By the time theintervention ended, she had reached mastery on both fact sets. Visual inspection of the data (see Gast& Spriggs, 2010) clearly showed that Belma’s performance continually improved over the course ofthe treatment.www.insikapub.com3

International Journal of Basic and Applied Science,Vol. 04, No. 04, April 2016, pp. 1-6Grünke8s9sFig.1: Number of correctly answered multiplication problems for the 8s and the 9s.The Percentage of Data Exceeding the Median (PEM) was used to represent the effect size of theoutcomes. It identifies the percentage of measurement points exceeding the median of the baselinephase and varies between 0 and 1 (Ma, 2006). For both fact sets, the PEM was 1, reflecting a highlyeffective intervention (Alresheed, Hott, & Bano, 2013). To determine the statistical significance of thedifferences between baseline and intervention phase measures, a randomization test was applied(Dugard, 2014). Because the two treatment points were selected randomly within a certain range, thismethod could be used to assess whether the improvements were likely due to chance. With the help ofa specific Microsoft Excel macro for AB multiple-baseline designs (developed by Dugard, File, &Todman, 2012, and downloadable at https://www.routledge.com/products/9780415886932), it wasdetermined that the differences between the phases were statistically significant within a standard 95%confidence interval.4DiscussionIn this study, skip counting turned out to be a highly effective intervention by all measures commonlyused in single-case research (visual inspection, effect size, and inferential statistics). In fact, the resultscould hardly have been more positive. The participant improved her multiplication fluency skills in thetwo target fact sets to the maximum level in the course of 10-15 sessions. Thus, the present case reportconfirms findings from previous research, thus suggesting that skip counting is another potent meansto help break the negative spiral of insufficient multiplication fluency skills, difficulties withapprehending higher-level mathematical concepts, and eventually problems with independent living(Patton, Cronin, Bassett, & Koppel, 1997).Nevertheless, the findings are subject to certain limitations. First, as with all single-subject designs, thegeneralizability of the results is relatively problematic. Claims about whole populations are alwaysbased on limited samples. Thus, even the findings from large-group experiments cannot be viewed asuniversally applicable. However, generalizability is a bigger issue if a study involves only one4Insan Akademika Publications

GrünkeInternational Journal of Basic and Applied Science,Vol. 04, No. 04, April 2016, pp. 1-6participant (Kächele, Schachter, & Thomä, 2009). Second, no follow-up data were collected. Thus, itis not possible to make claims about the long-term sustainability of the treatment effects. Theshortness of the data collection period was due to the beginning of school vacation. Even though itseems unlikely that the participant would quickly forget what she had learned during the intervention,it would have been desirable to have collected real follow-up data. The number of correctly solvedmultiplication problems on worksheets can be counted very objectively and reliably. Thus, the way thedependent variable was determined does not constitute a major threat to the explanatory power of thestudy.In summary, the present experiment confirms the usefulness of skip counting as a means to fostermultiplication fluency facts in children. Future research should focus on confirming these results.According to Chambless and Ollendick (2001), four single-case studies with positive outcomes are notenough for a certain treatment approach to qualilfy as being evidence-based. At least nine are needed.The Council for Exceptional Children (CEC, 2014) proposes in its standards for evidence-basedpractices in special education the need for 3 methodologically sound single-subject studies withpositive effects (meaningful change in the dependent variable for at least 75% of the cases) and aminimum of 10 total participants. In addition, 2 methodologically sound group comparison studieswith positive effects and at least 60 participants are mandatory. Hence, group studies are needed toverify the findings of the present and the three previous experiments in terms of the benefits of skipcounting. In addition, it would be helpful to investigate whether this technique is appropriate for peertutoring. Thus, being able to offer invididualized support to struggling learners by experiencedstudents under the supervision of a teacher would help to make assistance more readily available forall children in need of support.ReferencesAlberto, P. A., & Troutman, A. C. (2012). Applied behavior analysis for teachers (9th ed.). UpperSaddle River, NJ: Pearson/Prentice Hall.Alresheed, F., Hott, B. L., & Bano, C. (2013). Single subject research: A synthesis of analyticmethods. Journal of Special Education Apprenticeship, 2(1), 1-18.Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills. Mahwah,NJ: Erlbaum.Budak, A. (2015). The impact of a standards-based mathematics curriculum on students’ mathematicsachievement: The case of investigations in number, data, and space. Eurasia Journal ofMathematics, Science & Technology Education, 11(6), 1249-1264.Chambless, D. L., & Ollendick, T. H. (2001). Empirically supported psychological interventions.Annual Review of Psychology, 52(1), 685-716.Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta-analysis of mathematics basic-fact fluencyinterventions. Learning Disabilities Research & Practice, 26(1), 36-47.Daly, E. J., Martens, B. K., Barnett, D., Witt, J. C., & Olson, S. C. (2007). Varying interventiondelivery in response to intervention: Confronting and resolving challenges with measurement,instruction, and intensity. School Psychology Review, 36(4), 562-581.Dugard, P. (2014). Randomization tests: A new gold standard? Journal of Contextual BehavioralScience, 3(1), 65-68.Dugard, P., File, P. & Todman, J. (2012). Single-case and small-n experimental designs: A practicalguide to randomization tests. New York, NY: Routledge.www.insikapub.com5

International Journal of Basic and Applied Science,Vol. 04, No. 04, April 2016, pp. 1-6GrünkeDuncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., Pagani, L.S., Feinstein, L., Engel, M., Brooks-Gunn, J., Sexton, H., Duckworth, K., & Japel, C. (2007).School readiness and later achievement. Developmental Psychology, 43(6), 1428-1446.Duvall, T., McLaughlin, T. F., & Cooke-Sederstrom, G. (2003). The differential effects of skipcounting and previewing on the accuracy and fluency of math facts with middle schoolstudents with learning disabilities. International Journal of Special Education, 18(1), 1-7.Gast, D. L., & Spriggs, A. D. (2010). Visual analysis of graphic data. In D. L. Gast (Ed.), Singlesubject research methodology in behavioral sciences (pp. 199-233). New York, NY:Routledge.Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for studentswith mathematics difficulties. Journal of Learning Disabilities, 38(4), 293-304.Grünke, M., & Calder Stegemann, K. (2014). Using county-bys to promote multiplication factacquisition for a student with mild cognitive delays: A case report. Insights into LearningDisabilities, 11(2), 117-128.Horner, R. H., & Odom, S. L. (2014). Constructing single-case research designs: Logic and options. InT. R. Kratochwill & J. R. Levin (Eds.), Single-case intervention research (pp. 27-51).Washington, DC: American Psychological Association.Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). Arithmetic fact mastery in young children: Alongitudinal investigation. Journal of Experimental Child Psychology, 85(2), 103-119.Kächele, H., Schachter, J., & Thomä, H. (2009). From psychoanalytic narrative to empirical singlecase research. New York, NY: Routledge.Ma, H. H. (2006). An alternative method for quantitative synthesis of single-subject researchers.Behavior Modification, 30(5), 598-617.McIntyre, S. B., Test, D. W., Cooke, N. L., & Beattie, J. (1991). Using count-bys to increasemultiplication facts fluency. Learning Disability Quarterly, 14(2), 82-88.National Mathematics Advisory Panel. (2008). Foundations for success: The final report of theNational Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.Nelson, P. M., Burns, M. K., Kanive, R., & Ysseldyke, J. E. (2013). Comparison of a math factrehearsal and a mnemonic strategy approach for improving math fact fluency. Journal ofSchool Psychology, 51(6), 659-667.Patton, J. R., Cronin, J. F., Bassett, D. S., & Koppel, A. E. (1997). A life skills approach tomathematics instruction: Preparing students with learning disabilities for real-life mathdemands of adulthood. Journal of Learning Disabilities, 30(2), 178-187.Wagganer, E. L. (2015). Creating math talk communities. Teaching Children Mathematics, 22(4),248-254.6Insan Akademika Publications

Skip counting involves counting by a number that is not 1. For example, if you skip counts by 4 up to 20, you would count in the following order: 4, 8, 12, 16, 20. Combining the .

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