Browse > Article
http://dx.doi.org/10.7468/mathedu.2022.61.1.109

An Analysis of Teachers' Knowledge on the Strategies for Understanding and Solving Equations by Fourth Graders  

Pang, JeongSuk (Department of Elementary Education (Mathematics Education), Korea National University of Education)
Lee, Yujin (Daejeon Heungryong Elementary School)
Publication Information
The Mathematical Education / v.61, no.1, 2022 , pp. 109-126 More about this Journal
Abstract
The purpose of this study is to explore how well teachers anticipate students to understand and solve equations. For this purpose, a questionnaire of the equal sign was developed, and 20 fourth-grade classes were selected as research participants. Teachers in each class were asked to predict various strategies on how their own students would respond to the questionnaire, and a total of 450 students from the 20 classes solved the questionnaire. As a result of the analysis, the teachers could predict students' computational strategies and relational strategies easily but did not fully understand that some students used both strategies or employed incorrect computational or relational strategies. The students tended to use relational strategies better than the teachers expected. They also employed operational strategies more often than the teachers expected. The teachers predicted that students' strategies would be influenced by the types of the problems such as equation-structure items and equation-solving items, whereas the students were more influenced by the forms of equations in the problems. Based on these results, several implications for the knowledge to which teachers needed to attend were discussed so that elementary school students could enhance the relational understanding of the equal sign.
Keywords
equal sign; equation; relational understanding; knowledge of student understanding; teacher knowledge;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317-326. https://doi.org/10.1007/BF00311062   DOI
2 Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G. (2015). A specific misconception of the equal sign acts as a barrier to children's learning of early algebra. Learning and Individual Differences, 38, 61-67. https://doi.org/10.1016/j.lindif.2015.01.001   DOI
3 Kim, J. W., Choi, J. Y., & Pang, J. S. (2016). How do elementary school students understand '='? Performance on various item types. Journal of Educational Research in Mathematics, 26(1), 79-101.
4 Matthews, P. G., & Fuchs, L. S. (2020). Keys to the gate? Equal sign knowledge at second grade predicts fourth-grade algebra competence. Child Development, 91(1), 14-28. https://doi.org/10.1111/cdev.13144   DOI
5 Lee, D. H. (2017). An analysis on the error types of elementary students and pre-service teachers in mixed calculations of natural number. Journal of the Korean School Mathematics Society, 20(2), 141-161.
6 Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Routledge.
7 Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221-247. https://doi.org/10.1080/10986060701360902   DOI
8 Blanton, M., Otalora, Y., Brizuela, B. M., Gardiner, A. M., Sawrey, K. B., Gibbins, A., & Kim, Y. (2018). Exploring kindergarten students' early understandings of the equal sign. Mathematical Thinking and Learning, 20(3), 167-201. https://doi.org/10.1080/10986065.2018.1474534   DOI
9 Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children's algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258-288. https://doi.org/10.2307/30034868   DOI
10 Kang, M. H., (2009). Study of elementary school students' concept of equal sign through the analysis of their corrent responses to nonstandard structure of equivalent equation problems. Open Education Practice Study, 14, 17-30.
11 Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children's understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316-350. https://doi.org/10.5951/jresematheduc.43.3.0316   DOI
12 McNeil, N. M., Fyfe, E. R., & Dunwiddie, A. E. (2015). Arithmetic practice can be modified to promote understanding of mathematical equivalence. Journal of Educational Psychology, 107(2), 423. https://doi.org/10.1037/a0037687   DOI
13 Molina, M., & Ambrose, R. (2008). From an operational to a relational conception of the equal sign: Third graders' developing algebraic thinking. Focus on Learning Problems in Mathematics, 30, 61-80.
14 Jung, H. J., & Choi, C. W. (2014). An analysis of elementary school teacher's knowledge of concept of equality. Journal of Elementary Mathematics Education in Korea, 18(2), 211-236.
15 Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children's understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26, 195-217. https://doi.org/10.1080/07370000801980845   DOI
16 McNeil, N. M., & Alibali, M. W. (2005). Why won't you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883-899. https://doi.org/10.1111/j.1467-8624.2005.00884.x   DOI
17 Stephens, A. C. (2006). Equivalence and relational thinking: Preservice elementary teachers' awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 9(3), 249-278. https://doi.org/10.1007/s10857-006-9000-1   DOI
18 Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers' knowledge of students' understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272. https://doi.org/10.1080/10986060701360910   DOI
19 Bajwa, N. P., & Perry, M. (2021). Features of a pan balance that may support students' developing understanding of mathematical equivalence. Mathematical Thinking and Learning, 23(1), 1-27. https://doi.org/10.1080/10986065.2020.1700587   DOI
20 Blanton, M., Isler-Baykal, I., Stroud, R., Stephens, A., Knuth, E., & Gardiner, A. M. (2019). Growth in children's understanding of generalizing and representing mathematical structure and relationships. Educational Studies in Mathematics, 102(2), 193-219. https://doi.org/10.1007/s10649-019-09894-7   DOI
21 Yim, J. H. (2013). Discrepancy between reading and writing equality number sentences in Korean language. Journal of Elementary Mathematics Education in Korea 17(2), 207-223.
22 Pang, J. S., & Choi, I. Y. (2016). An analysis of algebraic rhinking by third graders. Education of Primary School Mathematics, 19(3), 223-247. https://doi.org/10.7468/jksmec.2016.19.3.223   DOI
23 Prediger, S. (2010). How to develop mathematics-for-teaching and for understanding: The case of meanings of the equal sign. Journal of Mathematics Teacher Education, 13(1), 73-93. https://doi.org/10.1007/s10857-009-9119-y   DOI
24 Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology, 103(1), 85-104. https://doi.org/10.1037/a0021334   DOI
25 Simsek, E., Xenidou-Dervou, I., Hunter, J., Dowens, M. G., Pang, J., Lee, Y., McNeil, N., Kirkland, P., & Jones, I. (in press). Factors associated with children's understanding of mathematical equivalence: An investigation across six countries. Journal of Educational Psychology. https://doi.org/10.1037/edu0000747
26 Stephens, A., Sung, Y., Strachota, S., Torres, R. V., Morton, K., Gardiner, A. M., Blanton, M., & Stroud, R. (2020). The role of balance scales in supporting productive thinking about equations among diverse learners. Mathematical Thinking and Learning, 1-18. https://doi.org/10.1080/10986065.2020.1793055   DOI
27 Britt, M. S., & Irwin, K. C. (2011). Algebraic thinking with and without algebraic representation: A pathway for learning. In J. Cai, & E. Kunth (Eds.), Early algebraization (pp. 137-159). Springer.
28 Stephens, A. C., Knuth, E. J., Blanton, M. L., Isler, I., Gardiner, A. M., & Marum, T. (2013). Equation structure and the meaning of the equal sign: The impact of task selection in eliciting elementary students' understandings. The Journal of Mathematical Behavior, 32(2), 173-182. https://doi.org/10.1016/j.jmathb.2013.02.001   DOI
29 Vermeulen, C., & Meyer, B. (2017). The equal sign teachers' knowledge and students' misconceptions. African Journal of Research in Mathematics, Science and Technology Education, 21(2), 136-147. https://doi.org/10.1080/18117295.2017.1321343   DOI
30 Blanton, M., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5. National Council of Teachers of Mathematics.
31 Fischer, J. P., Sander, E., Sensevy, G., Vilette, B., & Richard, J. F. (2019). Can young students understand the mathematical concept of equality? A whole-year arithmetic teaching experiment in second grade. European Journal of Psychology of Education, 34(2), 439-456. https://doi.org/10.1007/s10212-018-0384-y   DOI
32 Capraro, R. M., Capraro, M. M., Yetkiner, Z. E., Ozel, S., Kim, H. G., & Kucuk, A. R. (2010). An international comparison of grade 6 students' understanding of the equal sign. Psychological Reports, 106(1), 49-53. https://doi.org/10.2466/PR0.106.1.49-53   DOI
33 Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.
34 Fauskanger, J., & Mosvold, R. (2013). Teachers' mathematical knowledge for teaching equality. In Lindmeier, A. M. & Heinze, A. (Eds.). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 289-296). PME.