Comparing and Discussing Multiple Strategies: an Instructional Approach to Promote Algebra Learning
äºImplementation and Replication Studies in Mathematics EducationSearch for other papers by Haoyi Wang in
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Abstract
This study replicated Durkin et al. (2021) to examine the effects of the Comparing and Discussing Multiple Strategies (CDMS) instructional approach on studentsâ algebra learning. To test the generalizability of prior findings across cultural and curricular contexts, we implemented an isomorphic intervention in Grade 10 classrooms in China. Within a quasi-experimental pretestâposttest design, 251 students from 12 classes and 6 teachers participated. Bayesian multilevel analyses confirmed robust positive effects on studentsâ potential flexibility and positive directional trends for procedural flexibility, conceptual knowledge, and procedural knowledge, while effects on spontaneous flexibility were negligible. These findings suggest that CDMS may support repertoire expansion before enabling strategic selection. Unlike in the previous sample, our teachersâ self-efficacy and outcome expectancy beliefs remained stable. Nonetheless, variations in instructional quality continued to shape the CDMS intervention effects on student algebra learning. Implications for culturally responsive replication, instructional design, and professional development for algebra teaching are discussed.
The impact sheet to this article can be accessed at https://doi.org/10.1163/26670127-bja10035 under Supplementary Materials.
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-
Aguilar, M. S. (2020). Replication studies in mathematics education: What kind of questions would be productive to explore? International Journal of Science and Mathematics Education, 18(1, Suppl.), 37â50. http://doi.org/10.1007/s10763-020-10069-7.
-
Bandura, A. (1997). Self-efficacy: The exercise of control. New York: WH Freeman.
-
Bandura, A. (2000). Exercise of human agency through collective efficacy. Current Directions in Psychological Science, 9(3), 75â78. https://doi.org/10.1111/1467-8721.00064.
-
Boerst, T., Sleep, L., Ball, D., & Bass, H. (2011). Preparing teachers to lead mathematics discussions. Teachers College Record, 113(12), 2844â2877. https://doi.org/10.1177/016146811111301207.
-
Bonett, D. G. (2012). Replication-extension studies. Current Directions in Psychological Science, 21(6), 409â412. https://doi.org/10.1177/0963721412459512.
-
Bürkner, P.-C. (2017). brms: An R Package for Bayesian Multilevel Models Using Stan. Journal of Statistical Software, 80(1), 1â28. https://doi.org/10.18637/jss.v080.i01.
-
Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., & Hiebert, J. (2018). The Role of Replication Studies in Educational Research. Journal for Research in Mathematics Education, 49(1), 2â8. https://doi.org/10.5951/jresematheduc.49.1.0002.
-
Cai, J., Moyer, J. C., Wang, C., Wang, N., & Nie, B. (2024). Student learning and instructional tasks in different curricular contexts: A longitudinal study. International Journal of Educational Research, 125, 102329. https://doi.org/10.1016/j.ijer.2024.102329.
-
Cai, J., & Nie, B. (2007). Problem solving in Chinese mathematics education: Research and practice. ZDM, 39(5), 459â473. https://doi.org/10.1007/s11858-007-0042-3.
-
Charles, R., Hall, B., Kennedy, D., Bellman, A., Bragg, S., Handlin, W., & Murphy, S. (2015). Algebra 1, common core (Student ed.). Pearson.
-
Chen, Q. (2024). Exploring urban and rural studentsâ procedural flexibility in solving linear equations: An investigation of seventh-grade students in Jiangsu Province (Unpublished masterâs thesis). East China Normal University.
-
Durkin, K., Rittle-Johnson, B., Star, J. R., & Loehr, A. (2021). Comparing and Discussing Multiple Strategies: An Approach to Improving Algebra Instruction. The Journal of Experimental Education, 91(1), 1â19. https://doi.org/10.1080/00220973.2021.1903377.
-
Ebby, C. B., Hess, B., Pecora, L., & Valerio, J. (2024). Facilitating collaborative inquiry into practice around artifacts of mathematics teaching. Journal of Mathematics Teacher Education, 1â24. https://doi.org/10.1007/s10857-024-09649-z.
-
Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science, 7(2), 155â170. https://doi.org/10.1207/s15516709cog0702_3.
-
Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95, 393â405. https://doi.org/10.1037/0022-0663.95.2.393.
-
Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive psychology, 15(1), 1â38. https://doi.org/10.1016/0010-0285(83)90002-6.
-
Hammack, R., Yeter, I. H., Pavlovich, C., & Boz, T. (2024). Pre-service elementary teachersâ science and engineering teaching self-efficacy and outcome expectancy: Exploring the impacts of efficacy source experiences through varying course modalities. International Journal of STEM Education, 11(1), 4. https://doi.org/10.1186/s40594-024-00464-9.
-
Hattikudur, S., Sidney, P. G., & Alibali, M. W. (2016). Does comparing informal and formal procedures promote mathematics learning? The benefits of bridging depend on attitudes toward mathematics. The Journal of Problem Solving, 9(1), 2. https://doi.org/10.7771/1932-6246.1180.
-
Jacobs, J., Scornavacco, K., Harty, C., Suresh, A., Lai, V., & Sumner, T. (2022). Promoting rich discussions in mathematics classrooms: Using personalized, automated feedback to support reflection and instructional change. Teaching and Teacher Education, 112, 103631. https://doi.org/10.1016/j.tate.2022.103631.
-
Jankvist, U. T., Aguilar, M. S., Misfeldt, M., & Koichu, B. (2021a). Launching Implementation and Replication Studies in Mathematics Education (IRME). Implementation and Replication Studies in Mathematics Education, 1(1), 1â19. https://doi.org/10.1163/26670127-01010001.
-
Jankvist, U. T., Sánchez Aguilar, M., Misfeldt, M., & Koichu, B. (2021b). What to Replicate?. Implementation and Replication Studies in Mathematics Education, 1(2), 141â153. https://doi.org/10.1163/26670127-01010015.
-
Jiang, R., Liu, R., Star, J., Zhen, R., Wang, J., Hong, W., Jiang, S., Sun, Y., & Fu, X. (2021). How mathematics anxiety affects studentsâ inflexible perseverance in mathematics problemâsolving: Examining the mediating role of cognitive reflection. British Journal of Educational Psychology, 91(1), 237â260. https://doi.org/10.1111/bjep.12364.
-
Jiang, R., Star, J. R., Hästö, P., Li, L., Liu, R. D., Tuomela, D., Prieto, N. J., Palkki, R., Abánades, M. Ã., & Pejlare, J. (2023). Which one is the âbestâ: A cross-national comparative study of studentsâ strategy evaluation in equation solving. International Journal of Science and Mathematics Education, 21(4), 1127â1151. https://doi.org/10.1007/s10763-022-10282-6.
-
Ke, X., & Newton, K. J. (2024). English learners learn from worked example comparison in algebra. Instructional Science, 52(5), 831â858. https://doi.org/10.1007/s11251-024-09668-6.
-
Kruschke, J. K. (2015). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan (2nd ed.). Academic Press.
-
Lauermann, F., & Butler, R. (2021). The elusive links between teachersâ teaching-related emotions, motivations, and self-regulation and studentsâ educational outcomes. Educational Psychologist, 56(4), 243â249. https://doi.org/10.1080/00461520.2021.1991800.
-
Li, J. (2021). Research on the flexibility teaching of math problem solving program based on metacognitive strategy [Masterâs thesis, Shandong Normal University]. Wanfang Data.
-
Liu, R. D., Zhen, R., Ding, Y., Liu, Y., Wang, J., Jiang, R., & Xu, L. (2018). Teacher support and math engagement: roles of academic self-efficacy and positive emotions. Educational Psychology, 38(1), 3â16. https://doi.org/10.1080/01443410.2017.1359238.
-
Lynch, K., & Star, J. R. (2014a). Teachersâ views about multiple strategies in middle and high school mathematics. Mathematical Thinking and Learning, 16(2), 85â108. https://doi.org/10.1080/10986065.2014.889501.
-
Lynch, K., & Star, J. R. (2014b). Views of struggling students on instruction incorporating multiple strategies in Algebra I: An exploratory study. Journal for Research in Mathematics Education, 45(1), 6â18. https://doi.org/10.5951/jresematheduc.45.1.0006.
-
Lynch, K., & Star, J. R. (2014c). Exploring teachersâ implementation of comparison in Algebra I. The Journal of Mathematical Behavior, 35, 144â163. https://doi.org/10.1016/j.jmathb.2014.07.003.
-
Mata-Pereira, J., & da Ponte, J. P. (2017). Enhancing studentsâ mathematical reasoning in the classroom: teacher actions facilitating generalization and justification. Educational Studies in Mathematics, 96(2), 169â186. https://doi.org/10.1007/s10649-017-9773-4.
-
McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9780429029608.
-
Ministry of Education of the Peopleâs Republic of China (2022). Mathematics curriculum standards for compulsory education.
-
Mok, S. Y., Hämmerle, C. S., Rüede, C., & Staub, F. C. (2022). How do professional development programs on comparing solution methods and classroom discourse affect studentsâ achievement in mathematics? The mediating role of studentsâ subject matter justifications. Learning and Instruction, 82, 101668. https://doi.org/10.1016/j.learninstruc.2022.101668.
-
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics.
-
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
-
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics.
-
Newton, K. J., & Star, J. R. (2013). Exploring the nature and impact of model teaching with worked example pairs. Mathematics Teacher Educator, 2(1), 86â102. https://doi.org/10.5951/mathteaceduc.2.1.0086.
-
Ngu, B. H., & Phan, H. P. (2024). Instructional approach and acquisition of mathematical proficiency: Theoretical insights from learning by comparison and cognitive load theory. Asian Journal for Mathematics Education, 3(3), 357â379. https://doi.org/10.1177/27527263241266765.
-
Paul, J. F., Dreher, A., Wang, T., Hsieh, F., & Lindmeier, A. (2024). Culture-specific norms regarding high-quality use of task potential for mathematical learning â Contrasting researchersâ perspectives from Germany and Taiwan. Journal for Didactics of Mathematics, 44(14). https://doi.org/10.1007/s13138-024-00237-5.
-
Peopleâs Education Press. (2019). Grade 10 Mathematics: Semester 1. Beijing, China: Peopleâs Education Press.
-
Pirie, S. E., & Schwarzenberger, R. L. E. (1988). Mathematical discussion and mathematical understanding. Educational Studies in mathematics, 19(4), 459â470. https://doi.org/10.1007/BF00578694.
-
Riggs, I. M., Fischman, D. D., Riggs, M. L., Jetter, M. E., & Jesunathadas, J. (2018). Measuring teachersâ beliefs in relation to teaching mathematics with mathematical practices in mind. School Science and Mathematics, 118(8), 385â395. https://doi.org/10.1111/ssm.12303.
-
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561â574. https://doi.org/10.1037/0022-0663.99.3.561.
-
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 (Vol. 55, pp. 199â225). Academic Press. https://doi.org/10.1016/B978-0-12-387691-1.00007-7.
-
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836â852. https://doi.org/10.1037/a0016026.
-
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2017). The power of comparison in mathematics instruction: Experimental evidence from classrooms. In Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 273â295). Academic Press. https://doi.org/10.1016/B978-0-12-805086-6.00012-6.
-
RittleâJohnson, B., Star, J. R., & Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures?. British Journal of Educational Psychology, 82(3), 436â455. https://doi.org/10.1111/j.2044-8279.2011.02037.x.
-
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and instruction, 16(4), 475â5223. https://doi.org/10.1207/s1532690xci1604_4.
-
Shavelson, R. J., & Towne, L. (Eds.). (2002). Scientific research in education: Committee on scientific principles for education research. Washington, DC: National Academy Press.
-
Silver, E. A., & Cai, J. (1996). An Analysis of Arithmetic Problem Posing by Middle School Students. Journal for Research in Mathematics Education, 27(5), 521â539. https://doi.org/10.5951/jresematheduc.27.5.0521.
-
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24(3), 287â301. https://psycnet.apa.org/doi/10.1016/j.jmathb.2005.09.009.
-
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404â411. https://doi.org/10.2307/30034943.
-
Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565â579. https://doi.org/10.1016/j.learninstruc.2007.09.018.
-
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102(4), 408â426. https://doi.org/10.1016/j.jecp.2008.11.004.
-
Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31(3), 280â300. https://doi.org/10.1016/j.cedpsych.2005.08.001.
-
Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41â54. https://doi.org/10.1016/j.cedpsych.2014.05.005.
-
Star, J. R., Tuomela, D., Joglar-Prieto, N., Hästö, P., Palkki, R., Abánades, M. Ã., ⦠& Liu, R. D. (2022). Exploring studentsâ procedural flexibility in three countries. International Journal of STEM Education, 9(1), 1â18. https://doi.org/10.1186/s40594-021-00322-y.
-
Star, J., Rittle-Johnson, B., Durkin, K., Shero, M., & Sommer, J. (2020). Teaching for Improved Procedural Flexibility in Mathematics. In Gresalfi, M. and Horn, I. S. (Eds.), The Interdisciplinarity of the Learning Sciences, 14th International Conference of the Learning Sciences (ICLS) 2020 (Vol. 4, pp. 2285â2288). Nashville, Tennessee: International Society of the Learning Sciences.
-
Tschannen-Moran, M., & Woolfolk Hoy, A. (2001). Teacher efficacy: Capturing an elusive construct. Teaching and Teacher Education, 17(7), 783â805. https://doi.org/10.1016/S0742-051X(01)00036-1.
-
Usiskin, Z. (1999). Conceptions of school algebra and uses of variables. In B. Moses (Ed.), Algebraic thinking, grades Kâ12: Readings from NCTMâs school-based journals and other publications (pp. 7â13). Reston, VA: National Council of Teachers of Mathematics.
-
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335â359. https://doi.org/10.1007/BF03174765.
-
Wang, H., & Star, J. R. (2023). Investigating algorithm-oriented flexibility and structure-informed flexibility in mathematics learning. Asian Journal for Mathematics Education, 2(1), 16â41. https://doi.org/10.1177/27527263231163593.
-
Xie, S., Fan, L., Chen, Q., & Star, J. R. (under review). Investigating studentsâ opportunity to learn procedural flexibility: A comparative study of mathematics textbooks between the US and China.
-
Xu, C., Wells, E., LeFevre, J. A., & Imbo, I. (2014). Strategic flexibility in computational estimation for Chinese-and Canadian-educated adults. Journal of Experimental Psychology: Learning, Memory, and Cognition, 40(5), 1481â1497.
-
Xu, L., Liu, R. D., Star, J. R., Wang, J., Liu, Y., & Zhen, R. (2017). Measures of potential flexibility and practical flexibility in equation solving. Frontiers in Psychology, 8, 1368. https://doi.org/10.3389/fpsyg.2017.01368.
-
Zohar, A., & Peled, B. (2008). The effects of explicit teaching of metastrategic knowledge on low-and high-achieving students. Learning and instruction, 18(4), 337â353. https://doi.org/10.1016/j.learninstruc.2007.07.001.
Supplementary Materials
| å ¨é¨æé´ | è¿å»ä¸å¹´ | è¿å»30天 | |
|---|---|---|---|
| æè¦æµè§æ¬¡æ° | 170 | 170 | 97 |
| å ¨ææµè§æ¬¡æ° | 27 | 27 | 18 |
| PDFä¸è½½æ¬¡æ° | 58 | 58 | 41 |
Comparing and Discussing Multiple Strategies: an Instructional Approach to Promote Algebra Learning
äºImplementation and Replication Studies in Mathematics EducationSearch for other papers by Haoyi Wang in
Current site
Google Scholar
PubMed
Search for other papers by Qiuyu Chen in
Current site
Google Scholar
PubMed
Search for other papers by Jon R. Star in
Current site
Google Scholar
PubMed
Abstract
This study replicated Durkin et al. (2021) to examine the effects of the Comparing and Discussing Multiple Strategies (CDMS) instructional approach on studentsâ algebra learning. To test the generalizability of prior findings across cultural and curricular contexts, we implemented an isomorphic intervention in Grade 10 classrooms in China. Within a quasi-experimental pretestâposttest design, 251 students from 12 classes and 6 teachers participated. Bayesian multilevel analyses confirmed robust positive effects on studentsâ potential flexibility and positive directional trends for procedural flexibility, conceptual knowledge, and procedural knowledge, while effects on spontaneous flexibility were negligible. These findings suggest that CDMS may support repertoire expansion before enabling strategic selection. Unlike in the previous sample, our teachersâ self-efficacy and outcome expectancy beliefs remained stable. Nonetheless, variations in instructional quality continued to shape the CDMS intervention effects on student algebra learning. Implications for culturally responsive replication, instructional design, and professional development for algebra teaching are discussed.
The impact sheet to this article can be accessed at https://doi.org/10.1163/26670127-bja10035 under Supplementary Materials.
| å ¨é¨æé´ | è¿å»ä¸å¹´ | è¿å»30天 | |
|---|---|---|---|
| æè¦æµè§æ¬¡æ° | 170 | 170 | 97 |
| å ¨ææµè§æ¬¡æ° | 27 | 27 | 18 |
| PDFä¸è½½æ¬¡æ° | 58 | 58 | 41 |
