Research And Developing Mathematics Knowledge Child Development Perspectives, 2022
DOI:
https://doi.org/10.61166/elm.v1i2.11Keywords:
conceptual knowledge, flexibility, procedural knowledgeAbstract
Proficiency in mathematics is critical to success academically, economically, and in life. Greater success in math is related to entering and completing college, earning more in adulthood, and making more optimal decisions concerning health. Knowledge of math begins to develop at a young age, and this early knowledge matters: Knowledge of math at or before school entry predicts outcomes in math and reading across primary and secondary school. More than one children struggle to learn math. For example, only 60% of fourth-grade and 55% of eighth-grade students in the United States performed at or above proficiency in math on the 2020 National Assessment of Educational Progress, and proficiency rates were even lower for African-American and Hispanic children and for children from low-income homes. More than one students do not master challenging math content. Developing strong knowledge about mathematics is important for success academically, economically, and in life, but more than one children fail to become proficient in math. Research on the developmental relations between conceptual and procedural knowledge of math provides insights into the development of knowledge about math. First, competency in math requires children to develop conceptual knowledge, procedural knowledge, and procedural flexibility. Second, conceptual and procedural knowledge often develop in a bidirectional, iterative fashion, with improvements in one type of knowledge supporting improvements in the other, as well as procedural flexibility. Third, learning techniques such as comparing, explaining, and exploring promote more than one type of knowledge about math, indicating that each is an important learning process. Researchers need to develop and validate measurement tools, devise more comprehensive theories of math development, and bridge more between research and educational practice.
Downloads
References
Canobi, K. H. (2009). Concept-procedure interactions in children's addition and subtraction.
Journal of Experimental Child Psychology, 102, 131-149. doi:10.1016/j.jecp.2008.07.008
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362. doi:10.1037//0022-0663.93.2.346
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: Constructing adaptive expertise (pp. 1-34). Mahwah, NJ: Erlbaum.
Star, J. R., & Newton, K. J. (2009). The nature and development of expert's strategy flexibility for solving equations. ZDM Mathematics Education, 41, 557-567. doi:10.1007/s11858- 009-0185-5
Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). London, UK: Psychology Press.
Rittle-Johnson, B., & Schneider, M. (2020). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1118-1134). Oxford, UK: Oxford University Press.
Rittle-Johnson, B., Schneider, M., & Star, J. R. (2020). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27, 587-597. doi:10.1007/s10648-015-9302-x
Xu, F., Spelke, E. S., & Goddard, S. (2005). Number sense in human infants. Developmental Science, 8, 88-101. doi:10.1111/j.1467-7687.2005.00395.x
Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843-859. doi:10.1037/a0019824
Cowan, R., Donlan, C., Shepherd, D.-L., Cole-Fletcher, R., Saxton, M., & Hurry, J. (2011).
Basic calculation proficiency and mathematics achievement in elementary school children. Journal of Educational Psychology, 103, 786-803. doi:10.1037/a0024556
Fuson, K. C. (1988). Children's counting and concept of number. New York, NY: Springer- Verlag.
Rittle-Johnson, B., & Koedinger, K. R. (2009). Iterating between lessons concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79, 483-500. doi:10.1348/000709908X398106
Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525-1538. doi:10.1037/a0024997
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175- 189. doi:10.1037//0022-0663.91.1.175
Siregar, T. (2023). A Penerapan Model Pembelajaran Problem-Based Learning Untuk Meningkatkan Prestasi Belajar Matematika Siswa Di SMA Negeri 1 Sinunukan. COMPETITIVE: Journal of Education, 2(2), 94–102. https://doi.org/10.58355/competitive.v2i2.9
McNeil, N. M., Fyfe, E. R., & Dunwiddie, A. E. (2014). Arithmetic practice can be modified to promote understanding of mathematical equivalence. Journal of Educational Psychology, 107, 423-436. doi:10.1037/a0037687
Rittle-Johnson, B. (2006). Promoting transfer: Effects of self-explanation and direct instruction. Child Development, 77, 1-15. doi:10.1111/j.1467-8624.2006.00852.x
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404-411. Retrieved [AU: Please add date retrieved] from http://www.jstor.org/stable/30034943
Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students' flexibility in solving two- digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93, 627-638. doi:10.1037//0022-0663.93.3.627
Loehr, A. M., Fyfe, E. R., & Rittle-Johnson, B. (2014). Wait for it… delaying instruction improves mathematics problem solving: A classroom study. The Journal of Problem Solving, 7, 36-49. doi:10.7771/1932-6246.1166
DeCaro, M. S., & Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn from instruction. Journal of Experimental Child Psychology, 113, 552- 568. doi:10.1016/j.jecp.2012.06.009
Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. Second Handbook of Research on Mathematics Teaching and Learning, 1, 371-
[AU: Is this a journal?] [AU: Please provide doi.]It is a book chapter - see below for citation
Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Charlotte, NC: Information Age Publishing.
Renkl, A. (2011). Instruction based on examples. In R. E. Mayer & P. A. Alexander (Eds.), Handbook of research on learning and instruction (pp. 272-295). New York, NY: Routledge.
Webb, N. M., Franke, M. L., Ing, M., Wong, J., Fernandez, C. H., Shin, N., & Turrou, A. C. (2014). Engaging with others’ mathematical ideas: Interrelationships among student participation, teachers’ instructional practices, and learning. International Journal of Educational Research, 63, 79-93. doi:10.1016/j.ijer.2013.02.001
Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J.,…Stigler,
J. W. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study (NCES 2003-013). Retrieved 10/1/2016 from http://nces.ed.gov/timss
Price, G. R., Palmer, D., Battista, C., & Ansari, D. (2012). Nonsymbolic numerical magnitude comparison: Reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults. Acta Psychologica, 140, 50-57. doi:http://dx.doi.org/10.1016/j.actpsy.2012.02.008
Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory, and math achievement in early elementary school. Journal of Cognition and Development, 14, 187-202. doi:10.1080/15248372.2012.664593
Star, J. R., & Rittle-Johnson, B. (2016). Toward an educational psychology of mathematics education. In L. Corno & E. Anderman (Eds.), Handbook of educational psychology (3rd ed., pp. 257-268). New York, NY: Routledge.
Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2020). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41-54. doi:10.1016/j.cedpsych.2014.05.005
Davenport, J., Kao, Y. S., & Schneider, S. A. (2013). Integrating cognitive science principles to redesign a middle school math curriculum. In M. Knauff, M. Pauen, N. Sebanz, & I. Wachsmuth (Eds.), Proceedings of the 35th annual conference of the cognitive science
Reyna, V. F., Nelson, W. L., Han, P. K., & Dieckmann, N. F. (2009). How numeracy influences risk comprehension and medical decision making. Psychological Bulletin, 135, 943-973. doi:10.1037/a0017327
Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, 24, 1301-1308. doi:10.1177/0956797612466268
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P.,
Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428-1446. doi:10.1037/0012-1649.43.6.1428
National Assessment of Educational Progress. (2020). 2020 mathematics results. Retrieved 10/1/2016 from http://www.nationsreportcard.gov/reading_math_2020/ - mathematics?grade=4
Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. In J. P. Mestre & B. H.
Ross (Eds.), Psychology of learning and motivation: Cognition in education (Vol. 55, pp. 199-222). Waltham, MA: Elsevier.
Day, S. B., & Goldstone, R. L. (2012). The import of knowledge export: Connecting findings and theories of transfer of learning. Educational Psychologist, 47, 153-176. doi:10.1080/00461520.2012.696438
Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24-34. doi:10.1016/j.learninstruc.2012.11.002
Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22, 206-214. doi:10.1016/j.learninstruc.2011.11.001
Ziegler, E., & Stern, E. (2016). Consistent advantages of contrasted comparisons: Algebra learning under direct instruction. Learning and Instruction, 41, 41-51. doi:10.1016/j.learninstruc.2020.09.006
Legare, C. H. (2014). The contributions of explanation and exploraton to scientific reasoning.
Child Development Perspectives, 8, 101-106. doi: 10.1111/cdep.12070
Lombrozo, T. (2006). The structure and function of explanations. TRENDS in Cognitive Science, 10, 464-470. doi:10.1016/j.tics.2006.08.004
McEldoon, K. L., Durkin, K. L., & Rittle-Johnson, B. (2013). Is self-explanation worth the time? A comparison to additional practice. British Journal of Educational Psychology, 83, 615-632. doi:10.1111/j.2044-8279.2012.02083.
Chi, M. T. H. (2000). Self-explaining: The dual processes of generating inference and repairing mental models Advances in Instructional Psychology: Educational Design and Cognitive Science (Vol. 5., pp. 161-238). Mahwah, NJ: Erlbaum.
Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education. (online first, so no volume or page numbers yet) doi.10.1007/s11858-017-0834-z
Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41, 75-86. doi:10.1207/s15326985ep4102_1
Schwartz, D. L., Chase, C. C., Chin, D. B., & Oppezzo, M. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psychology, 103, 759-775. doi:10.1037/a0025140
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Torang Siregar, Ahmad Arisman, Iskandarsyah, Risky Ardian, Awal Harahap
This work is licensed under a Creative Commons Attribution 4.0 International License.