Scientific Research

An Academic Publisher

**Using Manipulatives in Solving and Posing Mathematical Problems** ()

In mathematics classrooms, teachers use multiple representations to help students explore and develop abstract concepts. Students are engaged in problem solving as they manipulate objects as they search for a solution. They also can enhance their profound knowledge when posing a scenario problem that matches to the appropriate manipulatives. The integration of manipulatives during teaching and learning can conceptually support students’ acquisition of symbols and mathematical language.

Share and Cite:

Rosli, R. , Goldsby, D. and Capraro, M. (2015) Using Manipulatives in Solving and Posing Mathematical Problems.

*Creative Education*,**6**, 1718-1725. doi: 10.4236/ce.2015.616173.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Barnett-Clarke, C., Fisher, W., Marks, R., & Ross, S. (2010). Developing Essential Understanding of Rational Number: Grades 3-5. Reston, VA: The National Council of Teachers of Mathematics. |

[2] | Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational-Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes (pp. 91-126). New York, NY: Academic Press. |

[3] |
Bruner, J. S. (1973). Organization of Early Skilled Action. Child Development, 44, 1-11. http://dx.doi.org/10.2307/1127671 |

[4] | Burns, M. (2007). About Teaching Mathematics: A K-8 Resource (3rd ed.). Sausalito, CA: Math Solutions. |

[5] |
Cass, M., Cates, D., Smith, M., & Jackson, C. (2003). Effects of Manipulative Instruction on Solving Area and Perimeter Problems by Students with Learning Disabilities. Learning Disabilities Research & Practice, 18, 112-120.
http://dx.doi.org/10.1111/1540-5826.00067 |

[6] |
Clements, D. H. (1999). “Concrete” Manipulatives, Concrete Ideas. Contemporary Issues in Early Childhood, 1, 45.
http://dx.doi.org/10.2304/ciec.2000.1.1.7 |

[7] |
Cramer, K. A., Post, T. R., & del Mas, R. C. (2002). Initial Fraction Learning by Fourth- and Fifth-Grade Students: A Comparison of the Effects of Using Commercial Curricula with the Effects of Using the Rational Number Project Curriculum. Journal for Research in Mathematics Education, 33, 111-144. http://dx.doi.org/10.2307/749646 |

[8] | Cramer, K., & Henry, A. (2002). Using Manipulative Models to Build Number Sense for Addition of Fractions. In B. Litwiller & G. Bright (Eds.), Making Sense of Fractions, Ratios, and Proportions: 2002 Yearbook (pp. 41-48). Reston, VA: National Council of Teachers of Mathematics. |

[9] | De George, B., & Santoro, M. A. (2004). Manipulatives: A Hands-On Approach to Math. Principal, 84, 2. |

[10] | Dewey, J. (1997). Experience and Education. New York, NY: Simon & Schuster (Original Work Published in 1938). |

[11] |
Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual Knowledge Falls through the Cracks: Complexities of Learning to Teach Mathematics for Understanding. Journal for Research in Mathematics Education, 24, 8-40. http://dx.doi.org/10.2307/749384 |

[12] | Empson, S. B. (2002). Organizing Diversity in Early Fraction Thinking. In B. Litwiller, & G. Bright (Eds.), Making Sense of Fractions, Ratios and Proportions: 2002 Yearbook (pp. 29-40). Reston, VA: National Council of Teachers of Mathematics. |

[13] | Even, R., & Tirosh, D. (2002). Teacher Knowledge and Understanding of Students’ Mathematical Learning. In L. English (Ed.), Handbook of International Research in Mathematics Education (pp. 219-240). Mahwah, NJ: Erlbaum. |

[14] | Fuson, K. C., Kalchman, M., & Bransford, J. D. (2005). Mathematical Understanding: An Introduction. In M. S. Donovan, & J. Bransford (Eds.), How Students Learn Mathematics in the Classroom (pp. 217-256). Washington DC: National Research Council. |

[15] | Hatfield, M. M., Edwards, N. T., Bitter, G. G., & Morrow, J. (2003). Mathematics Methods for Elementary and Middle School Teachers (4th ed.). New York: John Wiley & Sons. |

[16] | Hunt, A. W., Nipper, K. L., & Nash, L. E. (2011). Virtual vs. Concrete Manipulatives in Mathematics Teacher Education: Is One Type More Effective than the Other? Current Issues in Middle Level Education, 16, 1-6. |

[17] |
Karshmer, A. I., & Farsi, D. (2008). Manipulatives in the History of Teaching: Fast Forward to Auto Mathic Blocks for the Blind. In K. Miesenberger, J. Klaus, W. Zagler, & A. Karshmer (Eds.), Computers Helping People with Special Needs (vol. 5105, pp. 915-918). Lecture Notes in Computer Science, Berlin: Springer.
http://dx.doi.org/10.1007/978-3-540-70540-6_137 |

[18] | Kelly, C. A. (2006). Using Manipulatives in Mathematical Problem Solving: A Performance Based Analysis. The Montana Mathematics Enthusiast, 3, 184-193. |

[19] | Marsh, L. G., & Cooke, N. L. (1996). The Effects of Using Manipulatives in Teaching Math Problem Solving to Students with Learning Disabilities. Learning Disabilities Research & Practice, 11, 58-65. |

[20] |
Mathematics Science and Technology Education University of Illinois (2011). MSTE Online Resource Catalog.
http://mste.illinois.edu/resources/ |

[21] |
McNeil, N. M., & Jarvin, L. (2007). When Theories Don’t Add up: Disentangling the Manipulatives Debate. Theory into Practice, 46, 309-316. http://dx.doi.org/10.1080/00405840701593899 |

[22] | Montessori, M. (1964). The Montessori Method (A. E. George, Trans.). New York: Schocken. (Original Work Published in 1912) |

[23] |
Moss, J., & Case, R. (1999). Developing Children’s Understanding of the Rational Numbers: A New Model and an Experimental Curriculum. Journal for Research in Mathematics Education, 30, 122-147. http://dx.doi.org/10.2307/749607 |

[24] |
Moyer, P. (2001). Are We Having Fun Yet? How Teachers Use Manipulatives to Teach Mathematics. Educational Studies in Mathematics, 47, 175-197. http://dx.doi.org/10.1023/A:1014596316942 |

[25] | Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What Are Virtual Manipulatives? Teaching Children Mathematics, 8, 372-377. |

[26] | Moyer-Packenham, P. S., & Westenskow, A. (2011). An Initial Examination of Effect Sizes for Virtual Manipulatives and Other Instructional Treatments. In L. Paditz, & A. Rogerson (Eds.), Proceedings of the 11th International Conference of the Mathematics Education into the 21st Century Project—MEC 21: On Turning Dreams into Reality. Transformations and Paradigm Shifts in Mathematics Education, (Vol. 1, pp. 236-241). Rhodes University, Grahamstown: Oxford University Press. |

[27] |
Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of Virtual Manipulatives on Student Achievement and Mathematics Learning. International Journal of Virtual and Personal Learning Environments, 4, 35-50.
http://dx.doi.org/10.4018/jvple.2013070103 |

[28] | Moyer-Packenham, P. S., Salkind, G., & Bolyard, J. J. (2008). Virtual Manipulatives Used by K-8 Teachers for Mathematics Instruction: Considering Mathematical, Cognitive, and Pedagogical Fidelity. Contemporary Issues in Technology and Teacher Education, 8, 202-218. |

[29] | National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author. |

[30] |
National Library of Virtual Manipulatives (2010). Interactive Online Math Lessons.
http://enlvm.usu.edu/ma/nav/doc/intro.jsp |

[31] |
Piaget, J. (1964). Part I: Cognitive Development in Children: Piaget Development and Learning. Journal of Research in Science Teaching, 2, 176-186. http://dx.doi.org/10.1002/tea.3660020306 |

[32] |
Puchner, L., Taylor, A., O’Donnell, B., & Fick, K. (2008). Teacher Learning and Mathematics Manipulatives: A Collective Case Study about Teacher Use of Manipulatives in Elementary and Middle School Mathematics Lessons. School Science and Mathematics, 108, 313-325. http://dx.doi.org/10.1111/j.1949-8594.2008.tb17844.x |

[33] |
Rosli, R., Capraro, M. M., Goldsby, D., Gonzalez y Gonzalez, E., Onwuegbuzie, A. J., & Capraro, R. M. (2015). Middle Grade Preservice Teachers’ Mathematical Problem Solving and Problem Posing. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp. 333-354). New York: Springer.
http://dx.doi.org/10.1007/978-1-4614-6258-3_16 |

[34] |
Shodor Education Foundation (2011). Interactivate. http://www.shodor.org/interactivate/ |

[35] |
Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15, 4-14.
http://dx.doi.org/10.3102/0013189X015002004 |

[36] | Sowder, J. T., Philipp, R. A., Armstrong, B. E., & Schappelle, B. P. (1998). Middle-Grade Teachers’ Mathematical Knowledge and Its Relations to Instruction: A Research Monograph. Albany, NY: State University of New York. |

[37] | Steen, K., Brooks, D., & Lyon, T. (2006). The Impact of Virtual Manipulatives on First Grade Geometry Instruction and Learning. Journal of Computers in Mathematics and Science Teaching, 25, 373-391. |

[38] | Suh, J., Moyer, P. S., & Heo, H. J. (2005). Examining Technology Uses in the Classroom: Developing Fraction Sense Using Virtual Manipulative Concept Tutorials. Journal of Interactive Online Learning, 3, 1-20. |

[39] | Suzuka, K., Sleep, L., Ball, D. L., Bass, H., Lewis, J. M., & Thames, M. K. (2009). Designing and Using Tasks to Teach Mathematical Knowledge for Teaching. Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers ATME Monograph, 6, 7-23. |

[40] | Swan, P., & Marshall, L. (2010). Revisiting Mathematics Manipulative Materials. Australian Primary Mathematics Classroom, 15, 13-19. |

[41] |
Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. (2009). Dual Representation and the Linking of Concrete and Symbolic Representations. Child Development Perspectives, 3, 156-159.
http://dx.doi.org/10.1111/j.1750-8606.2009.00097.x |

[42] | Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2009). Elementary and Middle School Mathematics: Teaching Developmentally (7th ed.). Boston, MA: Allyn& Bacon/Merill. |

[43] | von Glasersfeld, E. (1989). Constructivism in Education. In T. Husen, & T. N. Postlethwaite (Eds.), The International Encyclopedia of Education (supplementary vol., pp. 162-163). Oxford: Pergamon. |

[44] | Vygotsky, L. (2009). Interaction between Learning and Development. In M. Gauvain, & M. Cole (Eds.), Readings on the Development of Children (5th ed., pp. 42-48). New York: Worth. (Reprinted from M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds., Mind in Society: The Development of Higher Psychological Processes (pp. 71-91). 1978, Cambridge, MA: Harvard University.) |

[45] |
Zuckman, O., Arida, S., & Resnick, M. (2005). Extending Tangible Interfaces for Education: Digital Montessori-Inspired Manipulatives. http://guzdial.cc.gatech.edu/hci-seminar/uploads/29/p859-zuckerman.pdf |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.