This study evaluates the effectiveness of a new intervention program with primary school children who show difficulties in solving simple multiplication problems. 64 fourth-grade children completed an eight-week long training with a pre-, post-, and follow-up test. Performance on a standardized test for arithmetic operations and a computer test were used to measure children’s knowledge. Results show the overall and long-term effectiveness of a holistic approach to multiplication fact training. All participants of the experimental groups show a significant increase in correct answers and faster reaction times, with the effects being stable over time. The program helped children build a structured multiplication fact network, with untrained facts being solved more quickly and correctly after the intervention.
KeywordsMultiplication ProblemsInterventionArithmetic Fact TrainingMath DifficultiesLearning DisabilitiesElementary School Children
Allen-Lyall, B. (2018). Helping Students to Automatize Multiplication Facts: A Pilot Study. International Electronic Journal of Elementary Education, 10, 391-396. https://doi.org/10.26822/iejee.2018438128
Ashcraft, M. H. (1987). Children’s Knowledge of Simple Arithmetic: A Developmental Model and Simulation. In J. Bisanz, C. J. Brainerd, & R. Kail (Eds.), Formal Methods in Developmental Psychology. Springer Series in Cognitive Development (pp. 302-338). Springer. https://doi.org/10.1007/978-1-4612-4694-7_9
Baroody, A. J. (2003). The Development of Adaptive Expertise and Flexibility: The Integration of Conceptual and Procedural Knowledge. In A. J. Baroody, & A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills (pp. 1-34). Taylor & Francis.
Burns, M. K. (2005). Using Incremental Rehearsal to Increase Fluency of Single-Digit Multiplication Facts with Children Identified as Learning Disabled in Mathematics Computation. Education & Treatment of Children, 28, 237-249.
Burns, M. K., Aguilar, L. N., Young, H., Preast, J. L., Taylor, C. N., & Walsh, A. D. (2019). Comparing the Effects of Incremental Rehearsal and Traditional Drill on Retention of Mathematics Facts and Predicting the Effects with Memory. School Psychology, 34, 521-530. https://doi.org/10.1037/spq0000312
Burns, M. K., Ysseldyke, J., Nelson, P. M., & Kanive, R. (2015). Number of Repetitions Required to Retain Single-Digit Multiplication Math Facts for Elementary Students. School Psychology Quarterly, 30, 398-405. https://doi.org/10.1037/spq0000097
Butterworth, B., Marchesini, N., & Girelli, L. (2003). Multiplication Facts: Passive Storage or Dynamic Reorganization? In A. J. Baroody, & A. Dowker, (Eds.), The Development of Arithmetical Concepts and Skills (pp. 198-202). Lawrence Erlbaum.
Campbell, J. I. D., & Robert, N. D. (2008). Bidirectional Associations in Multiplication Memory: Conditions of Negative and Positive Transfer. Journal of Experimental Psychology: Learning, Memory, and Cognition, 34, 546-555. https://doi.org/10.1037/0278-7393.34.3.546
Chapin, S. H., & Johnson, A. (2006). Math Matters. Math Solutions Publications.
Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta-Analysis of Mathematic Basic-Fact Fluency Interventions: A Component Analysis. Learning Disabilities Research & Practice, 26, 36-47. https://doi.org/10.1111/j.1540-5826.2010.00323.x
De Visscher, A., & Noel, M.-P. (2014). The Detrimental of Interference in Multiplication Facts Storing: Typical Development and Individual Differences. Journal of Experimental Psychology, 143, 2380-2400. https://doi.org/10.1037/xge0000029
DeBrauwer, J., & Fias, W. (2011). The Representation of Multiplications and Divisions Fac, ts in Memory. Experimental Psychology, 58, 312-323. https://doi.org/10.1027/1618-3169/a000098
Dehaene, S., & Cohen, L. (1997). Cerebral Pathways for Calculation: Double Dissociation between Rote Verbal and Quantitative Knowledge of Arithmetic. Cortex, 33, 219-250. https://doi.org/10.1016/S0010-9452(08)70002-9
Delazer, M., Ischebeck, A., Domahs, F., Zamarian, L., Koppelstaetter, F., Siedentopf, C., Kaufmann, L., Benke, T., & Felber, S. (2004). Learning by Strategies and Learning by Drill—Evidence from an fMRI Study. NeuroImage, 25, 838-849. https://doi.org/10.1016/j.neuroimage.2004.12.009
Devine, A., Soltész, F., Nobes, A., Goswami, U., & Szücs, D. (2013). Gender Differences in Developmental Dyscalculia Depend on Diagnostic Criteria. Learning and Instruction, 27, 31-39. https://doi.org/10.1016/j.learninstruc.2013.02.004
Domahs, F., Delazer, M., & Nuerk, H.-C. (2006). What Makes Multiplication Facts Difficult. Experimental Psychology, 53, 275-282. https://doi.org/10.1027/1618-3169.53.4.275
Domahs, F., Lochy, A., Eibl, G., & Delazer, M. (2004). Adding Colour to Multiplication: Rehabilitation of Arithmetic Fact Retrieval in a Case of Traumatic Brain Injury. Neuropsychological Rehabilitation, 14, 303-328. https://doi.org/10.1080/09602010343000246
Geary, D. C., Hamson, C. O., & Hoard. M. K. (2000). Numerical and Arithmetical Cognition: A Longitudinal Study of Process and Concept Deficits in Children with Learning Disability. Journal of Experimental Child Psychology, 77, 236-263. https://doi.org/10.1006/jecp.2000.2561
Gomides, M., Martins, G., Alves, I., Júlio-Costa, A., Jaeger, A., & Haase, V. (2018). Heterogeneity of Math Difficulties and Its Implications for Interventions in Multiplication Skills. Dementia & Neuropsychologia, 12, 256-263. https://doi.org/10.1590/1980-57642018dn12-030006
Grolimund, F. (2012). Mit Kindern lernen. Konkrete Strategien für Eltern. Hans Huber.
Haberstroh, S., & Schulte-Körne, G. (2019). The Diagnosis and Treatment of Dyscalculia. Deutsches Ärzteblatt International, 116, 107-14. https://doi.org/10.3238/arztebl.2019.0107
Haffner, J., Baro, K., Parzer, P., & Resch, F. (2005). Heidelberger Rechentest, HRT 1-4. Hogrefe.
Huang, J., & Chao, L. (1999). Specific Intervention for Multiplication Skill with Learning Disabled Students. Psychological Reports, 84, 662. https://doi.org/10.2466/pr0.1999.84.2.662
Huijsmans, M. D. E., Kleemans, T., & Kroesbergen, E. H. (2021). How Cognitive Strengths Compensate Weaknesses Related to Specific Learning Difficulties in Fourth-Grade Children. Frontiers in Psychology, 12, Article 552458. https://doi.org/10.3389/fpsyg.2021.552458
Ise, E., Dolle, K., Pixner, S., & Schulte-Körne, G. (2012). Effektive Förderung rechenschwacher Kinder. Eine Metaanalyse. Kindheit und Entwicklung, 21, 181-192. https://doi.org/10.1026/0942-5403/a000083
Kadosh, R. C., Henik, A., & Walsh, V. (2009). Synaesthesia: Learned or Lost? Developmental Science, 12, 484-491. https://doi.org/10.1111/j.1467-7687.2008.00798.x
Kaufmann, L., Handel, P., Delazer, M., & Pixner, S. (2013). Wie Kinder rechnen lernen und was ihnen dabei hilft: Eine kognitive-neuropsychologische Perspektive. In M. von Aster, & J. H. Lorenz (Eds.), Rechenstörungen bei Kindern (pp. 15-38). Vandenhoeck & Ruprecht. https://doi.org/10.13109/9783666462580.231
Kaye, D. B. (1986). The Development of Mathematical Cognition. Cognitive Development, 1, 157-170. https://doi.org/10.1016/S0885-2014(86)80017-X
Krinzinger, H., Kaufmann, L., & Willmes, K. (2009). Math Anxiety and Math Ability in Early Primary School Years. Journal of Psychoeducational Assessment, 27, 206-225. https://doi.org/10.1177/0734282908330583
Kroesbergen, E. H., & Van Luit, J. E. (2003). Mathematics Interventions for Children with Special Educational Needs. Remedial and Special Education, 24, 97-114. https://doi.org/10.1177/07419325030240020501
Lee, J.-E. (2007). Making Sense of the Traditional Long Division Algorithm. The Journal of Mathematical Behavior, 26, 48-59. https://doi.org/10.1016/j.jmathb.2007.03.001
LeFevre, J.-A., Bisanz, J., Daley, K. E., Buffone, L., Greenham, S. L., & Sadesky, G. S. (1996). Multiple Routes to Solution of Single-Digit Multiplication Problems. Journal of Experimental Psychology, 125, 284-306. https://doi.org/10.1037/0096-3445.125.3.284
Mabott, D. J., & Bisanz, J. (2003). Developmental Change and Individual Differences in Children’s Multiplication. Child Development, 74, 1091-1107. https://doi.org/10.1111/1467-8624.00594
McCallum, E., Skinner, C. H., Turner, H., & Saecker, L. (2006). The Taped-Problems Intervention: Increasing Multiplication Fact Fluency Using a Low-Tech, Classwide, Time Delay Intervention. School Psychology Review, 35, 419-434. https://doi.org/10.1080/02796015.2006.12087976
McCrink, K., & Spelke, E. S. (2010). Core Multiplication in Childhood. Cognition, 116, 204-216. https://doi.org/10.1016/j.cognition.2010.05.003
Morsanyi, K., van Bers, B. M. C. W., McCormack, T., & McGourty, J. (2018). The Prevalence of Specific Learning Disorder in Mathematics and Comorbidity with Other Developmental Disorders in Primary School-Age Children. British Journal of Psychology, 109, 917-940. https://doi.org/10.1111/bjop.12322
Nelson, P. M., Burns, M. K., Kanive, R., & Ysseldyke, J. E. (2013). Comparison of a Math Fact Rehearsal and a Mnemonic Strategy Approach for Improving Math Fact Fluency. Journal of School Psychology, 51, 659-667. https://doi.org/10.1016/j.jsp.2013.08.003
Pixner, S., & Dresen, V. (2021). TIGRO 2—Training in Grundrechenoperationen. TIGRO Fördermaterial.
Pixner, S., & Schlessing, I. (2021). Die Rolle der positiven Lernatmosphäre. LEDY, 3, 18-23.
Rickard, T. C., & Bourne, L. E. (1996). Some Test of an Identical Elements Model of Basic Arithmetic Skills. Journal of Experimental Psychology: Learning, Memory and Cognition, 22, 1281-1295. https://doi.org/10.1037/0278-7393.22.5.1281
Roussel, J-L., Fayol, M., & Barrouillet, P. (2002). Procedural vs. Direct Retrieval Strategies in Arithmetic: A Comparison between Additive and Multiplicative Problem Solving. European Journal of Cognitive Psychology, 14, 61-104. https://doi.org/10.1080/09541440042000115
Schulte-Körne, G., & Mathwitg, F. (2009). Marburger Rechtschreibtraining. Hogrefe.
Shrager, J., & Siegler, R. S. (1998). SCADS: A Model of Children’s Strategy Choices and Strategy Discoveries. Psychological Science, 9, 405-410. https://doi.org/10.1111/1467-9280.00076
Siegler, R. S. (1988). Strategy Choice Procedures and the Development of Multiplication Skill. Journal of Experimental Psychology: General, 117, 158-175. https://doi.org/10.1037/0096-3445.117.3.258
Siegler, R. S. (2007). Cognitive Variability. Developmental Science, 10, 104-109. https://doi.org/10.1111/j.1467-7687.2007.00571.x
Sleeman, M., Friesen, M., Tyler-Merrik, G., & Walker, L. (2021). The Effects of Precision Teaching and Self-Regulated Learning on Early Multiplication Fluency. Journal of Behavioral Education, 30, 149-177. https://doi.org/10.1007/s10864-019-09360-7
Sokol, S. M., McCloskey, M., Cohen, N. J., & Aliminosa, D. (1991). Cognitive Representations and Processes in Arithmetic: Inferences from the Performance of Brain-Damaged Subjects. Journal of Experimental Psychology, 17, 355-376. https://doi.org/10.1037/0278-7393.17.3.355
Spector, F., & Maurer, D. (2013). Synesthesia: A New Approach to Understanding the Devolpment of Perception. Psychology of Consciousness: Theory, Research, and Practice, 1, 108-129. https://doi.org/10.1037/2326-5523.1.S.108
Symonds, P. M, & Chase, D. H. (1992). Practice vs. Motivation. Journal of Educational Psychology, 84, 282-289. https://doi.org/10.1037/0022-0663.84.3.282
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 Galen, M.S., & Reitsma, P. (2011). Transfer Effects in Children’s Recall of Arithmetic Facts. Cognition and Instruction, 29, 330-348. https://doi.org/10.1080/07370008.2011.583368
Van Galena, M. S., & Reitsma, P. (2011). Transfer Effects in Children’s Recall of Arithmetic Facts. Cognition and Construction, 29, 330-348. https://doi.org/10.1080/07370008.2011.583368
Wong, M., & Evans, D. (2007). Improving Basic Multiplication Fact Recall for Primary School Students. Mathematics Education Research Journal, 19, 89-106. https://doi.org/10.1007/BF03217451
Wood, D. K., Frank, A. R., & Wacker, D. P. (1998). Teaching Multiplication Facts to Students with Learning Disabilities. Journal of Applied Behavior Analysis, 31, 323-338. https://doi.org/10.1901/jaba.1998.31-323
Zhang, D., Xin, Y. P., Harris, K., & Ding, Y. (2014). Improving Multiplication Strategic Development in Children with Math Difficulties. Learning Disability Quarterly, 37, 15-30. https://doi.org/10.1177/0731948713500146