By Jim Ysseldyke, PhD
In my last post, I talked about the number of repetitions it takes for students to learn multiplication facts. We learned that some facts (4s, 5s, 6s, and 7s) require significantly more repetition to learn than do others. We also learned that students with lower math skills require significantly more repetitions to learn multiplication facts and that the number of repetitions required decreases with rising grade level, indicating that younger students require more repetitions.
I ended that earlier post asking the question, “What approaches work best for developing computational fluency, and do they work best for all students—regardless of grade or proficiency level?” That is the topic I address here.
There are many instructional approaches to achieving fluent computation, but as with reading fluency or fluency in playing a musical instrument, repeated practice is a necessary component (Binder, 1996). In two related investigations, my colleagues at the University of Minnesota and I examined the effectiveness of alternative methods of teaching students to be automatic with math facts.
In one study published in the Journal of Educational Research (Kanive, Nelson, Burns, & Ysseldyke, 2014), we compared the effectiveness of computer-based practice to conceptual understanding interventions on both computational fluency and word problem solving using measures of math fact retention and generalization. A total of 90 4th and 5th graders who were identified by their teachers as struggling in math and who scored below the 25th percentile on the district math test participated in the study. Students were randomly assigned to one of three treatment groups: (1) computer-based math facts practice using Renaissance MathFacts in a Flash®, (2) conceptual understanding, or (3) business as usual (regular classroom instruction). Single-digit multiplication facts were taught to those in the conceptual understanding group using activities developed by Van de Walle and Lovin (2006) and included the use of manipulatives and games. Specific activities are described fully in the Kanive et al. (2014) journal article.
The following were among the major findings of this study:
The computer-based practice intervention was more effective for increasing fact fluency among struggling learners than classroom instruction only (business as usual).
Students in the conceptual understanding group did not outperform the computer-based practice group on any of the outcome measures.
Learning how to perform a novel skill or procedure with frequent opportunities to practice emphasizes both accuracy and automaticity in mathematics fluency, which may result in higher mean change scores on a generalization measure.
Computer-Based vs. Conceptual Interventions
Students learn a skill by first initially acquiring accuracy, then becoming more fluent with the skill, and eventually generalizing the skill and using it to solve problems. The differential effects and efficiency of mathematics interventions may stem from how well each intervention mirrors the hierarchical way mathematical skills develop and instruction is delivered. That said, in general, technology-based interventions provide several advantages over purely conceptual approaches, starting with the ease of providing students individualized practice within group settings. Computer-based interventions may also incorporate instructional strategies identified as essential elements to the development of mathematics proficiency without expending the limited time and resources in schools that are required for other evidence-based interventions.
Our study also found that the computer-based practice intervention also led to better retention on a measure of computational fluency than did the conceptual intervention or a control condition. This finding suggests that computer-based practice interventions may be more effective for increasing computational skills.
Deficits in basic computational skills are prominent in classrooms, and mathematics computation skills are a critical component to student’s ability to learn higher-level mathematics (Axtell et al., 2009; Gersten, Jordan, & Flojo, 2005). Computer-based practice interventions have potential utility to teachers as an effective approach for those interested in improving mathematics proficiency and as a tool for re-teaching and reviewing basic mathematics facts.
Computer-Based Interventions vs. Mnemonic Strategies
In a third investigation (Nelson, Burns, Kanive & Ysseldyke, 2013) we compared the effectiveness of a computer-based practice intervention with a mnemonic strategies intervention. Mnemonics strategies typically refer to words, sentences, or rhymes designed to enhance storage and recall of facts or processes (Mastropieri, Scruggs, & Levin, 1987; Test & Ellis, 2005), and they have a long history of demonstrated effectiveness for struggling learners, including students with learning disabilities in math (Greene, 1999; Kavale & Forness, 2000).
Specific mnemonic strategies (Mastropieri & Scruggs, 1989) include:
Peg words (words associated with numbers used to remember lists of items)
Keywords (associating a similar-sounding word with a target word)
Acronyms (using the first letter of each word in a list to construct a word)
Acrostics (creating a sentence where the first letter of each word is the targeted information
In practice, mnemonic strategies mirror a model-lead-test approach in that they are typically modeled for students and followed by guided practice before students practice independently with feedback (Test & Ellis, 2005). Previous research suggests that mnemonic strategy interventions may help improve computational fluency for struggling learners by relying on easily remembered cues rather than repetition (Maccini, Mulcahy, & Wilson, 2007). Both rehearsal strategies and mnemonic strategies have been shown to effectively increase math fact fluency, but the two approaches have not been compared and previous mnemonic strategy research did not use computer technology to implement the rehearsal strategies.
The Nelson et al. (2013) study used a randomized controlled trial to examine the effects of a computer-delivered practice-based intervention (MathFacts in a Flash) and a mnemonic strategy intervention (Times Tables the Fun Way—TTFW), on the retention and application of single-digit multiplication facts for students with math difficulties. The researchers examined how these two interventions compared in terms of their effects on two separate tests: a retention measure of single-digit multiplication math fact fluency and an application measure of single-digit math facts to mathematical word problems. Note that Times Tables the Fun Way (TTFW; Liataud & Rodriguez, 1999) is an intervention that uses a method similar to peg word by incorporating pictures and stories as mnemonic devices for recalling basic multiplication facts. For example, 8 × 8 is recalled by thinking of two snowmen (the number 8 looks like a snowman) who are cold and need “sticks for the fire” (Liataud & Rodriguez, 1999, p. 70), which sounds like 64.
The result? Students in the practice-based intervention group had higher retention scores (expressed as the total number of digits correct per minute) relative to the control group. No statistically significant between-group differences were observed for application scores.
The Value of Direct Instruction and Drill via Technology
Although multiple factors contribute to deficiencies in mathematics, poor computation fluency plays a fundamental role. More specifically, children who lack fluency in basic computations are at risk for math difficulties that may arise in elementary school and persist into young adulthood. Moreover, students who successfully store basic math fact information in memory and retrieve it easily are more likely to develop the skills necessary for solving a wide variety of complex problems and interpreting abstract mathematical principles (Patton, Cronin, Bassett, & Koppel, 1997; Shapiro, 2010).
The results of our recent research bolster theories that emphasize teaching math facts directly and providing students with intensive/extensive drill and practice until they become automatic with the facts. Our studies found that technology-based interventions like MathFacts in a Flash were an effective intervention for enhancing the multiplication fact fluency of elementary students, especially those who are struggling with math. Further, they provide educators an efficient and cost effective way to provide that intensive practice.
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.
Binder, C. (1996). Behavioral fluency: Evolution of a new paradigm. The Behavior Analyst, 19, 163–197.
Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293–30.
Greene, G. (1999). Mnemonic multiplication fact instruction for students with learning disabilities. Learning Disability Research & Practice, 14, 141–148.
Kanive, R., Nelson, P., Burns., & Ysseldyke, J. (2014). Comparison of the effects of computer-based practice and conceptual understanding interventions on math fact retention and generalization. Journal of Educational Research, 107(2), 83–87.
Kavale, K. A., & Forness, S. R. (2000). Policy decisions in special education: The role of meta-analysis. In R. Gersten, E. P. Schiller, & S. Vaughn (Eds.), Contemporary special education research: Synthesis of the knowledge base on critical instructional issues (pp. 281–326). Mahwah, NJ: Erlbaum.
Liataud, J., & Rodriguez, D. (1999). Times table the fun way: Book for kids: A picture method of learning the multiplication facts (3rd ed.)Sandy, UT: Key Publishing.
Maccini, P., Mulcahy, C. A., & Wilson, M. G. (2007). A follow-up of math interventions for secondary students with learning disabilities. Learning Disabilities Research and Practice, 22, 58–74.
Mastropieri, M. A., & Scruggs, T. E. (1989). Constructing more meaningful relationships: Mnemonic instruction for special populations. Educational Psychology Review, 1, 83–111.
Mastropieri, M. A., Scruggs, T. E., & Levin, J. R. (1987). Effective instruction for special education. Austin, TX: Pro-Ed.
Nelson, P. Burns, M., Kanive, R., & Ysseldyke, J., (2013).Comparison of a math fact rehearsal and a mnemonic strategy approach for improving math fact fluency among students with math difficulty. Journal of School Psychology, 6, 659–667.
Patton, J. R., Cronin, J. F., Bassett, D. S., & Koppel, A. E. (1997). A life skills approach to mathematics instruction: Preparing students with learning disabilities for real-life math demands of adulthood. Journal of Learning Disabilities, 30, 178–187.
Shapiro, E. S. (2010). Academic skills problems: Direct assessment and intervention (2nd ed.). New York, NY: Guilford Press.
Test, D. W., & Ellis, M. E. (2005). The effects of LAP fractions on addition and subtraction of fractions with students with mild disabilities. Education and Treatment of Children, 28, 11–24.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student centered mathematics: Grades K–3. Boston, MA: Allyn & Bacon.
Jim Ysseldyke, Ph.D., is professor emeritus of educational psychology at the University of Minnesota and a consultant for Renaissance on Response to Intervention (RTI) and Star assessments. He served as director of the Minnesota Institute for Research on Learning Disabilities, director of the National School Psychology Network, director of the National Center on Educational Outcomes, director of the School Psychology Program, and associate dean for research.