A Mathematically Creative Four-Year-Old—What Do We Learn from Him?

Author(s)
Ruti Steinberg

ABSTRACT

A 4-year-old child, who is very interested and precocious in mathematics, was interviewed doing mathematical tasks in order to find out how advanced can a 4-year-old child be? His mathematical knowledge and ability are very high. Danny was able to count objects and add them, memorizing many of the addition facts. He was able to check if numbers are even or odd. He solved a variety of challenging addition, subtraction and multiplication problems. He could read and write large numbers in hundreds and thousands and could add large numbers. Analysis was done on the kind of problems the child was able to solve, their level of difficulty and the solution strategies the child used in light of what children usually do to solve these problems in ages 5-8 (Carpenter et al., 1999). Danny also showed creativity, including inventing problems for himself to solve and finding mathematical situations in his environment to attend to. An analysis was done on the creative components of his solutions and problem posing using the literature on creativity and creativity in mathematics learning (Leikin & Pitta-Pantazi, 2013). Interview with his mother showed that she supports Danny’s mathematical development by being responsive to his initiations and allowing him to explore his ideas autonomously. Her support was discussed in light of different support patterns of parents and what kind of support is especially beneficial to the child and encourages creativity (Leder, 1992). I discuss implications for education with an emphasis on what kindergarten and school can do to promote problem solving and creativity in mathematics.

KEYWORDS

Creativity in Mathematics; Mathematics Education; Early Childhood; Precocious Child in Mathematics

Creativity in Mathematics; Mathematics Education; Early Childhood; Precocious Child in Mathematics

Cite this paper

Steinberg, R. (2013). A Mathematically Creative Four-Year-Old—What Do We Learn from Him?.*Creative Education, 4,* 23-32. doi: 10.4236/ce.2013.47A1004.

Steinberg, R. (2013). A Mathematically Creative Four-Year-Old—What Do We Learn from Him?.

References

[1] Bahar, A. K., & Maker, C. J. (2011). Exploring the relationship be tween mathematical creativity and mathematical achievement. Asia Pacific Journal of Gifted and Talented Education, 3, 33-48.

[2] Bishop, A. J. (2002). Mathematical acculturation, cultural conflicts, and transition. In G. de Abreu, A. J. Bishop, & N. C. Presmeg (Eds.), Transitions between contexts of mathematical practices (pp. 193-212). Dordrecht: Kluwer Academic Press. doi:10.1007/0-306-47674-6_10

[3] Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179-202. doi:10.2307/748348

[4] Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten chil dren’s problem-solving processes. Journal for Research in Mathe matics Education, 24, 428-441. doi:10.2307/749152

[5] Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Ports mouth, NH: Heinemann.

[6] Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanc ed mathematical thinking (pp. 42-53). Dordrecht: Kluwer.

[7] Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use chil dren’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403-434. doi:10.2307/749875

[8] Franke, M. L. (2003). Fostering young children’s mathematical under standing. In C. Howes (Ed.), Teaching 4 to 8-year-olds: Literacy, math, multiculturalism, and classroom community. Baltimore, MD: Brookes.

[9] Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

[10] Hershkovitz, S., Peled, I., & Littler, G. (2009). Mathematical creativity and giftedness in elementary school: Task and teacher promoting creativity for all. In R. Leikin, A. Berman, & B. Koichu (Eds.), Crea tivity in mathematics and the education of gifted students (pp. 255-269). Rotterdam: Sense Publishers.

[11] Hirsh, R. A. (2010). Creativity: Cultural capital in the mathematics class room. Creative Education, 1, 154-161. doi:10.4236/ce.2010.13024

[12] Leder, G. C. (1992). Mathematics before formal schooling. Educational Studies in Mathematics, 23, 383-396. doi:10.1007/BF00302441

[13] Leikin, R. (2009a). Bridging research and theory in mathematics educa tion with research and theory in creativity and giftedness. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 383-409). Rotterdam: Sense Publishers.

[14] Leikin, R. (2009b). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129-145). Rotterdam: Sense Publishers.

[15] Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference? ZDM—The International Journal on Mathematics Edu cation, 45, 183-197.

[16] Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: The state of the art. ZDM Mathematics Education, 45, 159-166. doi:10.1007/s11858-012-0459-1

[17] Leikin, R., Berman, A., & Koichu, B. (2009). Creativity in mathematics and the education of gifted students. Rotterdam: Sense Publisher.

[18] Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solu tion tasks in developing knowledge and creativity in geometry. Jour nal of Mathematical Behavior, 31, 73-90. doi:10.1016/j.jmathb.2011.11.001

[19] Milgram, R., & Hong, E. (2009). Talent loss in mathematics: Causes and solutions. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativ ity in mathematics and the education of gifted students (pp. 149-163). Rotterdam: Sense Publishers.

[20] Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics, 13, 373-394. doi:10.1007/BF00366618

[21] Riley, M. S., Greeno, J. G., & Heller, J. (1983). Development of chil dren’s problem-solving ability in arithmetic. The Development of Mathematical Thinking (pp. 153-196). New York: Academic Press.

[22] Sak, U., & Maker, C. J. (2006). Developmental variations in children’s creative mathematical thinking as a function of schooling, age, and knowledge. Creativity Research Journal, 18, 279-291. doi:10.1207/s15326934crj1803_5

[23] Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown-ups cannot see as different? Early numerical thinking revisited. Cognition and Instruction, 23, 237-309. doi:10.1207/s1532690xci2302_3

[24] Sheffield, L. (2009). Developing mathematical creativity—Questions may be the answer. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 87-100). Rotterdam: Sense Publishers.

[25] Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM—The International Journal on Mathematics Education, 29, 75-80. doi:10.1007/s11858-997-0003-x

[26] Steinberg, R. (1985a). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, 337-355. doi:10.2307/749356

[27] Steinberg, R. (1985b). Keeping track processes in addition and sub traction. Paper Presented at the Annual Meeting of the American Educational Research Association, Chicago, IIlinois.

[28] Steinberg, R. M., Empson, S. B., & Carpenter, T. P. (2004). Inquiry into children’s mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 7, 237-267. doi:10.1023/B:JMTE.0000033083.04005.d3

[29] Tabach, M., & Friedlander, A. (2013). School mathematics and creativity at the elementary and middle grades level: How are they related? ZDM—The International Journal on Mathematics Education, 45, 227-238.

[30] Tiedemann, K., & Brandt, B. (2010). Parents’ Support in Mathematical Discourses. In U. Gellert, E. Jablonka, & C. Morgan (Eds.). Proceedings of the 6th International Conference on Mathematics Education and Society (pp. 428-437). Berlin: Freie Universitat Berlin.

[31] Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.

[32] Tsamir, P., Tirosh, D., Tabach, M., & Levenson, E. (2010). Multiple solution methods and multiple outcomes—Is it a task for kindergar ten children? Educational Studies in Mathematics, 73, 217-231. doi:10.1007/s10649-009-9215-z

[33] Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

[34] Warfield, J., & Yttri, M. J. (1999). Cognitively Guided Instruction in one kindergarten classroom. In J. V. Copley (Ed.). Mathematics in the early years. Reston, VA: NCTM.

[35] Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumenta tion, and autonomy in mathematics. Journal for Research in Mathe matics Education, 458-477. doi:10.2307/749877

[1] Bahar, A. K., & Maker, C. J. (2011). Exploring the relationship be tween mathematical creativity and mathematical achievement. Asia Pacific Journal of Gifted and Talented Education, 3, 33-48.

[2] Bishop, A. J. (2002). Mathematical acculturation, cultural conflicts, and transition. In G. de Abreu, A. J. Bishop, & N. C. Presmeg (Eds.), Transitions between contexts of mathematical practices (pp. 193-212). Dordrecht: Kluwer Academic Press. doi:10.1007/0-306-47674-6_10

[3] Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179-202. doi:10.2307/748348

[4] Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten chil dren’s problem-solving processes. Journal for Research in Mathe matics Education, 24, 428-441. doi:10.2307/749152

[5] Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Ports mouth, NH: Heinemann.

[6] Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanc ed mathematical thinking (pp. 42-53). Dordrecht: Kluwer.

[7] Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use chil dren’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403-434. doi:10.2307/749875

[8] Franke, M. L. (2003). Fostering young children’s mathematical under standing. In C. Howes (Ed.), Teaching 4 to 8-year-olds: Literacy, math, multiculturalism, and classroom community. Baltimore, MD: Brookes.

[9] Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

[10] Hershkovitz, S., Peled, I., & Littler, G. (2009). Mathematical creativity and giftedness in elementary school: Task and teacher promoting creativity for all. In R. Leikin, A. Berman, & B. Koichu (Eds.), Crea tivity in mathematics and the education of gifted students (pp. 255-269). Rotterdam: Sense Publishers.

[11] Hirsh, R. A. (2010). Creativity: Cultural capital in the mathematics class room. Creative Education, 1, 154-161. doi:10.4236/ce.2010.13024

[12] Leder, G. C. (1992). Mathematics before formal schooling. Educational Studies in Mathematics, 23, 383-396. doi:10.1007/BF00302441

[13] Leikin, R. (2009a). Bridging research and theory in mathematics educa tion with research and theory in creativity and giftedness. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 383-409). Rotterdam: Sense Publishers.

[14] Leikin, R. (2009b). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129-145). Rotterdam: Sense Publishers.

[15] Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference? ZDM—The International Journal on Mathematics Edu cation, 45, 183-197.

[16] Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: The state of the art. ZDM Mathematics Education, 45, 159-166. doi:10.1007/s11858-012-0459-1

[17] Leikin, R., Berman, A., & Koichu, B. (2009). Creativity in mathematics and the education of gifted students. Rotterdam: Sense Publisher.

[18] Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solu tion tasks in developing knowledge and creativity in geometry. Jour nal of Mathematical Behavior, 31, 73-90. doi:10.1016/j.jmathb.2011.11.001

[19] Milgram, R., & Hong, E. (2009). Talent loss in mathematics: Causes and solutions. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativ ity in mathematics and the education of gifted students (pp. 149-163). Rotterdam: Sense Publishers.

[20] Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics, 13, 373-394. doi:10.1007/BF00366618

[21] Riley, M. S., Greeno, J. G., & Heller, J. (1983). Development of chil dren’s problem-solving ability in arithmetic. The Development of Mathematical Thinking (pp. 153-196). New York: Academic Press.

[22] Sak, U., & Maker, C. J. (2006). Developmental variations in children’s creative mathematical thinking as a function of schooling, age, and knowledge. Creativity Research Journal, 18, 279-291. doi:10.1207/s15326934crj1803_5

[23] Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown-ups cannot see as different? Early numerical thinking revisited. Cognition and Instruction, 23, 237-309. doi:10.1207/s1532690xci2302_3

[24] Sheffield, L. (2009). Developing mathematical creativity—Questions may be the answer. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 87-100). Rotterdam: Sense Publishers.

[25] Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM—The International Journal on Mathematics Education, 29, 75-80. doi:10.1007/s11858-997-0003-x

[26] Steinberg, R. (1985a). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, 337-355. doi:10.2307/749356

[27] Steinberg, R. (1985b). Keeping track processes in addition and sub traction. Paper Presented at the Annual Meeting of the American Educational Research Association, Chicago, IIlinois.

[28] Steinberg, R. M., Empson, S. B., & Carpenter, T. P. (2004). Inquiry into children’s mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 7, 237-267. doi:10.1023/B:JMTE.0000033083.04005.d3

[29] Tabach, M., & Friedlander, A. (2013). School mathematics and creativity at the elementary and middle grades level: How are they related? ZDM—The International Journal on Mathematics Education, 45, 227-238.

[30] Tiedemann, K., & Brandt, B. (2010). Parents’ Support in Mathematical Discourses. In U. Gellert, E. Jablonka, & C. Morgan (Eds.). Proceedings of the 6th International Conference on Mathematics Education and Society (pp. 428-437). Berlin: Freie Universitat Berlin.

[31] Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.

[32] Tsamir, P., Tirosh, D., Tabach, M., & Levenson, E. (2010). Multiple solution methods and multiple outcomes—Is it a task for kindergar ten children? Educational Studies in Mathematics, 73, 217-231. doi:10.1007/s10649-009-9215-z

[33] Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

[34] Warfield, J., & Yttri, M. J. (1999). Cognitively Guided Instruction in one kindergarten classroom. In J. V. Copley (Ed.). Mathematics in the early years. Reston, VA: NCTM.

[35] Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumenta tion, and autonomy in mathematics. Journal for Research in Mathe matics Education, 458-477. doi:10.2307/749877