Using Manipulatives in Solving and Posing Mathematical Problems

Affiliation(s)

^{1}
Faculty of Education, The National University of Malaysia, Bangi, Malaysia.

^{2}
Texas A&M University, College Station, USA.

ABSTRACT

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.

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.

Cite this paper

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.

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

References

[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

[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