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A comparison of strategic development for multiplication problem solving in low-, average-, and high-achieving students

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Abstract

The present study investigated the differences of strategy use between low-, average-, and high-achieving students when solving different multiplication problems. Nineteen high-, 48 average-, and 17 low-achieving students participated in this study. All participants were asked to complete three different multiplication tests and to explain how they solved these problems. Results suggested that low achievers used incorrect operation strategies more frequently, indicating a lack of conceptual understanding of multiplication. High-achieving students demonstrated greater flexibility in problem-solving and were more accurate in performing direct retrieval or math algorithm strategies. Results were discussed about improving low achievers’ use of advanced strategies, enhancing their flexibility in choosing strategies and improving students’ accuracy in using direct retrieval or math algorithms.

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References

  • Ashlock, R. B. (2002). Error patterns in computation: Using error patterns to improve instruction (8th ed.). Upper Saddle River, NJ: Merrill Prentice Hall.

  • Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, fall issue, 15–46.

  • Brown, S. (1992). Second grade children’s understanding of the division process. School Science and Mathematics, 92(2), 92–95.

    Article  Google Scholar 

  • Burny, E., Valcke, M., & Desoete, A. (2012). Clock reading an underestimated topic in children with mathematics difficulties. Journal of Learning Disabilities, 45(4), 351–360.

    Article  Google Scholar 

  • Clarke, B., Smolkowski, K., Baker, S. K., Fien, H., Doabler, C. T., & Chard, D. J. (2011). The impact of a comprehensive Tier I core kindergarten program on the achievement of students at risk in mathematics. The Elementary School Journal, 111(4), 561–584.

    Article  Google Scholar 

  • Downton, A. (2008). Links between children’s understanding of multiplication and solution strategies for division. In M. Goos, R. Brown, & K. Makar (Eds.), Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australia (pp. 171–178). Sydney: MERGA.

    Google Scholar 

  • Geary, D. C., Bow-Thomas, C. C., & Yao, Y. (1992). Counting knowledge and skill in cognitive addition: A comparison of normal and mathematically disabled children. Journal of Experimental Child Psychology, 54, 372–391.

    Google Scholar 

  • Geary, D. C., & Brown, S. C. (1991). Cognitive addition: strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children. Developmental Psychology, 27, 398–406.

    Article  Google Scholar 

  • Geary, D. C., & Hoard, M. K. (2005). Learning disabilities in arithmetic and mathematics: Theoretical and empirical perspectives. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 253–267). New York: Psychology Press.

    Google Scholar 

  • Geary, D. C., Hoard, M. K., & Byrd-Craven, J. (2004). Strategy choices in simple and complex addition: contributions of working memory and counting knowledge for children with mathematical disability. Journal of Experimental Child Psychology, 88, 121–151.

    Article  Google Scholar 

  • Geary, D. C., Hoard, M. K., & Bailey, D. H. (2011). Fact retrieval deficits in low achieving children and children with mathematical learning disability. Journal of Learning Disabilities, 45(3).

  • Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morthy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A Meta-analysis of instructional Component. Review of Educational Research, 79(3), 1202–1242.

    Google Scholar 

  • Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20(2), 147–158.

    Google Scholar 

  • Kroesbergen, E., & Van Luit, J. (2002). Teaching multiplication to low math performers: Guided versus structured instruction. Instructional Science, 30, 361–378.

    Google Scholar 

  • Kroesbergen, E. H., Van Luit, J., & Maas, C. J. M. (2004). Effectiveness of explicit and constructivist mathematics instruction for low-achieving students in the Netherlands. The Elementary School Journal, 104, 233–251.

    Google Scholar 

  • Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: contributions to children’s learning of multiplication. Journal of Experimental Psychology. General, 124, 83–97.

    Article  Google Scholar 

  • Lewis, K. E. (2010). Understanding mathematical learning disabilities: a case study of errors and explanations. Learning Disabilities: A Contemporary Journal, 8(2), 9–18.

    Google Scholar 

  • Mulligan, J. (1992). Children’s solutions to partition problems. In B. Southwell, R. Perry, & K. Owens (Eds.), Proceedings of the 15th annual conference of the mathematics education research group of Australasia (pp. 410–420). Sydney, Australia: MERGA.

    Google Scholar 

  • Mulligan, J. T., & Michelmore, M. C. (1997). Identification of multiplicative thinking in children in Grades 1-5. Journal for Research in Mathematics Education, 28(3), 309–331.

    Article  Google Scholar 

  • Mullis, I. V. S., Martin, M. O., Foy, P., & Arora, A. (2012). TIMSS 2011 international results in mathematics. Chestnut Hill: TIMSS & PIRLS International Study Center, Boston College.

    Google Scholar 

  • National Assessment of Educational Progress. (2009). Mathematics 2009: National assessment of educational progress at grades 4 and 8. The Nation’s Report Card. NCES 2010–451. National Center for Education Statistics.

  • National Mathematics Advisory Panel (2008). Foundation for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

  • Oliver, A., Murray, H., & Human, P. (1991). Children’s solution strategies for division problems. In J. Ferrini-Mundy (Ed.), Proceedings of the sixteenth international conference for the psychology of mathematics education (Vol. 2, pp. 152–159). Durham, NH: PME.

    Google Scholar 

  • Opfer, J. E., & Siegler, R. S. (2004). Revisiting preschoolers’ living things concept: a microgenetic analysis of conceptual change in basic biology. Cognitive Psychology, 49, 301–332.

    Article  Google Scholar 

  • Ostad, S. A. (1997). Developmental differences in addition strategies: a comparison of mathematically disabled and mathematically normal children. British Journal of Educational Psychology, 67, 345–357.

    Article  Google Scholar 

  • Park, J., & Nunes, T. (2000). The development of concept of multiplication. Cognitive Development, 16, 763–773.

    Article  Google Scholar 

  • Radatz, H. (1979). Error analysis in mathematics education. Journal for Research in Mathematics Education, 10(3), 163–172.

    Article  Google Scholar 

  • Siegler, R. S. (1988a). Individual differences in strategy choices: good students, not-so-good students, and perfectionists. Child Development, 59, 833–851.

    Article  Google Scholar 

  • Siegler, R. S. (1988b). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology. General, 117(3), 258–275.

    Article  Google Scholar 

  • Siegler, R. S. (1995). How does change occur: a microgenetic study of number conservation. Cognitive Psychology, 28, 225–273.

    Article  Google Scholar 

  • Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 119–233). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Siegler, R. S. (2005). Children’s learning. American Psychologist, 60, 769–778.

    Article  Google Scholar 

  • Siegler, R. S. (2006). Microgenetic analyses of learning. In W. Damon & R. M. Lerner (Series Eds.) & D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology: Volume 2: Cognition, perception, and language (6th ed., pp. 464–510). Hoboken, NJ: Wiley.

  • Siegler, R. S. (2007). Cognitive variability. Developmental Science, 10, 104–109.

    Article  Google Scholar 

  • Siegler, R., & Chen, Z. (1998). Developmental differences in rule learning: a microgenetic analysis. Cognitive Psychology, 36, 273–310.

    Article  Google Scholar 

  • Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), The origins of cognitive skills (pp. 229–293). Hillsdale: Erlbaum.

    Google Scholar 

  • Siegler, R. S., & Svetina, M. (2006). What leads children to adopt new strategies? A microgenetic/cross sectional study of class inclusion. Child Development, 77, 997–1015.

    Article  Google Scholar 

  • Swanson, H., & Beebe-Frankenberger, M. (2004). The relationship between working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471–491.

    Article  Google Scholar 

  • Van Luit, J. E. H., & Naglieri, J. A. (1999). Effectiveness of the MASTER program for teaching special children multiplication and division. Journal of Learning Disabilities, 32, 98–107.

    Article  Google Scholar 

  • Zhang, D. (2011). The Effects of Strategic Training on Improving Multiplication Strategic Development in Children with Math Difficulties. Unpublished dissertation. West Lafayette, IN: Purdue University.

  • Zhang, D., Xin, Y. P., & Si, L. (2013). Transition from intuitive to advanced strategies in multiplicative reasoning for students with math disabilities. The Journal of Special Education, 47(1), 50–64.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Educational Psychology, Rutgers University, 10 Seminary Place, Room 312, New Brunswick, NJ, 08901, USA

    Dake Zhang

  2. Fordham University, 113 West 60th Street, Room 1012F, New York, NY, 10023, USA

    Yi Ding

  3. Clemson University, Clemson, SC, USA

    Dave E. Barrett

  4. Purdue University, West Lafayette, IN, USA

    Yan Ping Xin

  5. Beijing Normal University, Beijing, China

    Ru-de Liu

Authors
  1. Dake Zhang
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  2. Yi Ding
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  3. Dave E. Barrett
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  4. Yan Ping Xin
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  5. Ru-de Liu
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Corresponding author

Correspondence to Ru-de Liu.

Additional information

Dake Zhang. Assistant professor, Department of Educational Psychology at Rutgers University, 10 Seminary Place, Room 312, New Brunswick, NJ 08901 USA. E-mail: dake.zhang@gse.rutgers.edu; Website: http://gse.rutgers.edu/dake_zhang

Current themes of research:

Her research interests include assessment and interventions for students with mathematics learning difficulties.

Most relevant publications in the field of Psychology of Education:

Zhang, D., Ding, Y. Stegall, J., & Mo, L. (in press). The effects of visual chunking on improving visual imagery skills for solving geometry problems in students with math disabilities. Learning Disability Research and Practice.

Zhang, D., Xin, Y. P., & Si, L. (in press). Transition from intuitive to advanced strategies in multiplicative reasoning for students with math disabilities. The Journal of Special Education.

Xin, Y. P., Hord, C., Zhang, D., Cetintas, S, Park. J., & Si, L. (in press). The effects of computer-assisted instruction in teaching conceptual model-based problem solving. Learning Disabilities: A Multidisciplinary Journal.

Ding, Y., Yang, L.-Y., Guo, J.-P., Zhang, D., Ning, H., & Richman, L. C. (in press). Rapid automatized naming skills and immediate memory functions in Chinese elementary readers who read English as a second language. The Journal of Learning Disabilities.

Zhang, D., & Xin, Y. P. (2012). A follow-up meta-analysis of word problem solving interventions for students with math learning problems. The Journal of Educational Research, 105 (5), 303-318.

Yi Ding. Assistant professor of School Psychology Program at Fordham University. , 113 West 60th Street, Room 1012F. New York, New York 10023 USA.E-mail: yding4@fordham.edu; Website: http://www.fordham.edu/academics/colleges__graduate_s/graduate__profession/education/divisions/psychological__educa/school_psychology_pr/school_psychology_fa/ding/index.asp

Current themes of research:

Dr. Ding’s research interests include assessment for children with reading and mathematics learning disabilities.

Most relevant publications in the field of Psychology of Education:

Ding, Y., Yang, L.-Y., Guo, J.-P., Zhang, D., Ning, H., & Richman, L. C. (accepted). Rapid automatized naming skills and immediate memory functions in Chinese elementary readers who read English as a second language. Journal of Learning Disabilities.

Taylor, R., Ding, Y., Felt, D., & Zhang, D. (2011). Effects of tier-I intervention on nonsense word fluency in first graders. School Psychology Forum: Research in Practice, 5(2), 54-73.

Ding, Y., Yang, L.-Y., Salyers, K., Harper, H., Guo, J.-P., Liu, H., & Feng, Y.-H. (2010). Assessing needs and challenges perceived by caregivers and teachers of children with autism spectrum disorder in China. Journal of International Association of Special Education. 11(1), 4-14.

Cochrane, W. C., Salyers, K., & Ding, Y. (2010). An examination of the preparation, supervisor’s theoretical model, and university support for supervisors of school psychology interns. Trainer’s Forum: Journal of Trainer’s of School Psychology, 29 (1), 6-23.

Ding, Y., Richman, L. C., Yang, L.-Y., & Guo, J.-P. (2010). Rapid automatized naming skills and immediate memory functions in Chinese Mandarin speaking elementary readers. Journal of Learning Disabilities, 43(1), 48-61.

Dave E. Barrett. Alumni Distinguished Professor at School of Education at Clemson University. Website: http://www.clemson.edu/hehd/departments/education/faculty-staff/profile.html?userid=bdavid

Current themes of research:

His research interests include cognitive and emotional development for exceptional children.

Most relevant publications in the field of Psychology of Education:

Katsiyannis, A., Barrett, D.E. & Losinski. M. (in press). Miranda rights: Implications for juveniles with disabilities. Intervention in School and Clinic.

Katsiyannis, A., Barrett, D.E. & Zhang, D. (in press). Juvenile Offenders with Disabilities: Challenges and Promises. In E.L. Grigorenko (Ed.) Handbook of Juvenile Forensic Psychology and Psychiatry. New York: Springer.

Zhang, D., Hsu, H., Katsiyannis, A., Barrett, D.E. & Ju, S (2011). Adolescents with disabilities in the juvenile justice system: Patterns of recidivism. Exceptional Children, 77, 283-298.

Zhang, D., Barrett, D.E., Katsiyannis, A., & Yoon, M. (2011). Juvenile offenders with and without disabilities: Risks and patterns of recidivism. Learning and Individual Differences, 21, 12-18.

Barrett, D.E., Katsiyannis, A. & Zhang, D. (2010). Predictors of offense severity, adjudication, incarceration and repeat referrals for juvenile offenders: A multi-cohort replication study. Remedial and Special Education, 31, 261-275.

Yan Ping Xin. Associate Professor in Special Education at Purdue University. Website: https://collaborate.education.purdue.edu/edst/xin/default.aspx

Current themes of research:

Dr. Xin’s research includes effective instructional strategies in mathematics problem solving with students with learning disabilities or problems, algebra readiness, computer-assisted differentiated instructional system, cross-culture curriculum comparison, and meta-analysis. Recently, she proposed Conceptual Model-based Problem Solving that emphasizes algebraic expression of mathematical relations in word problem solving.

Most relevant publications in the field of Psychology of Education:

Xin, Y. P., Liu, J., & Zheng, X., (2011). A Cross-Cultural Lesson Comparison on Teaching the Connection between Multiplication and Division. School Science and Mathematics, 111(7), 354-367

Xin, Y. P., Zhang, D., Park, J. Y., Tom, K., Whipple, A., & Si, L. (2011). A Comparison of Two Mathematics Problem-Solving Strategies: Facilitate Algebra-Readiness. The Journal of Educational Research, 104, 1-15.

Xin, Y. P. & Zhang, D. (2009). Exploring a conceptual model-based approach to teaching situated word problems. The Journal of Educational Research, 102(6), 427-441.

Xin, Y. P. (2008). The effect of schema-based instruction in solving word problems: An emphasis on pre-algebraic conceptualization of multiplicative relations. Journal for Research in Mathematics Education. 39, 526-551.

Xin, Y. P., Wiles, B., & Lin, Y. (2008). Teaching conceptual model-based word-problem story grammar to enhance mathematics problem solving. The Journal of Special Education, 42, 163-178.

Ru-de Liu. Professor in School of Psychology at Beijing Normal University. Website: http://psych.bnu.edu.cn/www/index.php/Web/shizi_de_de/tid/132/id/16

Current themes of research:

Dr. Liu’s research interests include students’ cognitive processing in solving mathematics problems, and the evaluation of educational technologies in mathematics instruction in elementary and secondary schools.

Most relevant publications in the field of Psychology of Education:

Liu, R. (2010). Psychological research in educational technology in China. British Journal of Educational Technology, 41(4), 593-606.

Hung, W., Jonassen, D., & Liu, R (2008). Problem-Based Learning. In J. Michael Spector, M. David Merrill, Jeroen Van Merrienboer, & Marcy P. Driscoll (Eds), Handbook of Research on Educational Communications and Technology. NY: Lawrence Erlbaum Associates, 185-506.

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Zhang, D., Ding, Y., Barrett, D.E. et al. A comparison of strategic development for multiplication problem solving in low-, average-, and high-achieving students. Eur J Psychol Educ 29, 195–214 (2014). https://doi.org/10.1007/s10212-013-0194-1

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