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Transition from Textbook to Classroom Instruction in Mathematics: The Case of an Expert Chinese Teacher

  • Research Article
  • Published:
Frontiers of Education in China

Abstract

This study reports how an expert Chinese teacher implements mathematics textbook lessons in enacted instruction. Our video analysis indicates that both textbook and enacted teaching included only one worked example; however, the teacher engaged students in unpacking the example in great depth. Both the textbook and the enacted teaching showed “concreteness fading” in students’ use of representations. However, the Chinese teacher incorporated students’ self-generated representations and facilitated students’ active modeling of quantitative relationships. Finally, the Chinese teacher asked a greater number of deep questions than were suggested by the textbook. These deep questions often occurred as clusters of follow-up questions that were either concept-specific or promoted comparisons which facilitated connection-making between multiple representations and solutions.

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References

  • Baroody, A. J. (1999). Children’s relational knowledge of addition and subtraction. Cognition and Instruction, 17(2), 137–175. https://doi.org/10.1207/S1532690XCI170201

    Article  Google Scholar 

  • Bi, H. L, & Wan, Y. Q. (2013). 课堂教学中教科书使用模型的构建与应用 [Construction and application of textbook usage models in classroom teaching]. 当代教育科学 [Contemporary Education Sciences], 12, 14–17.

    Google Scholar 

  • Brousseau, G. (2002). Theory of didactical situations in mathematics. New York, NY: Kluwer Academic Publishers.

    Google Scholar 

  • Cai, J. F. (1995). A cognitive analysis of U.S. and Chinese students’ mathematical performance on tasks involving computation, simple problem solving, and complex problem solving. Journal for Research in Mathematics Education, 7, i–151. doi: 10.2307/749940

    Google Scholar 

  • Cai, J. F. (2000). Mathematical thinking involved in U.S. and Chinese students’ solving process-constrained and process-open problems. Mathematical Thinking and Learning, 2(4), 309–340. https://doi.org/10.1207/S15327833MTL0204_4

    Article  Google Scholar 

  • Cai, J. F. (2005). U.S. and Chinese teachers’ knowing, evaluating, and constructing representations in mathematics instruction. Mathematical Thinking and Learning, 7(2), 135–169. https://doi.org/10.1207/s15327833mtl0702_3

    Article  Google Scholar 

  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Chi, M. T. H. (2000). Self-explaining: The dual processes of generating interference and repairing mental models. In R. Glaser (Ed.), Advances in instructional psychology: Educational design and cognitive science (Vol. V, pp. 161–238). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Common Core State Standards Initiative. (2010). Mathematics standards. Retrieved November 10, 2018, from http://www.corestandards.org/Math/

    Google Scholar 

  • Craig, S. D., Sullins, J., Witherspoon, A., & Gholson, B. (2006). The deep-levelreasoning-question effect: The role of dialogue and deep-level-reasoning questions during vicarious learning. Cognition and Instruction, 24(4), 565–591.

    Article  Google Scholar 

  • Creswell, J. W. (2014). Research design: Qualitative, quantitative, and mixed methods approaches (4th ed.). Thousand Oaks, CA: Sage.

    Google Scholar 

  • Ding, M. X. (2016). Opportunities to learn: Inverse relations in U.S. and Chinese textbooks. Mathematical Thinking and Learning, 18(1), 45–68. https://doi.org/10.1080/10986065.2016.1107819

    Article  Google Scholar 

  • Ding, M. X., & Carlson, M. A. (2013). Elementary teachers’ learning to construct high-quality mathematics lesson plans: A use of IES recommendations. The Elementary School Journal, 113(3), 359–385. doi: 10.1086/668505

    Article  Google Scholar 

  • Ding, M. X., Chen, W., & Hassler, R. (in press). Linear quantity models in the US and Chinese elementary mathematics classroom. Mathematical Thinking and Learning.

  • Ding, M. X., & Li, X. B. (2010). A comparative analysis of the distributive property in the US and Chinese elementary mathematics textbooks. Cognition and Instruction, 28(2), 146–180. https://doi.org/10.1080/07370001003638553

    Article  Google Scholar 

  • Ding, M. X., & Li, X. B. (2014). Transition from concrete to abstract representations: The distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87(1), 103–121. https://doi.org/10.1007/s10649-014-9558-y

    Article  Google Scholar 

  • Ding, M. X., Li, Y. P., Li, X, B., & Gu, J. (2013). Knowing and understanding instructional mathematics content through intensive studies of textbooks. In Y. P. Li & R. J. Huang (Eds.), How Chinese teach mathematics and improve teaching (pp. 66–82). New York, NY: Routledge.

    Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103–131. https://doi.org/10.1007/s10649-006-0400-z

    Article  Google Scholar 

  • English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Franke, M. L., Webb, N. M., Chan, A. G., Ing, M., Freund, D., & Battey, D. (2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education, 60(4), 380–392. https://doi.org/10.1177/0022487109339906

    Article  Google Scholar 

  • Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children’s mathematics understanding. Learning and Instruction, 35, 104–120. https://doi.org/10.1016/j.learninstruc.2014.10.004

    Article  Google Scholar 

  • Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. The Journal of the Learning Sciences, 14(1), 69–110. https://doi.org/10.1207/s15327809jls1401_4

    Article  Google Scholar 

  • Greeno, J., & Riley, M. (1987). Processes and development of understanding. In R. E. Weinert & R. H. Kluwe (Eds.), Metacognition, motivation, and understanding (pp. 289–313). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320(5875), 454–455. doi: 10.1126/science.1154659

    Article  Google Scholar 

  • Kieran, C. (Ed.). (2018). Teaching and learning algebraic thinking with 5-to 12-year-olds: The global evolution of an emerging field of research and practice. Cham, Switzerland: Springer.

    Book  Google Scholar 

  • 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(2), 75–86. https://doi.org/10.1207/s15326985ep4102_1

    Article  Google Scholar 

  • Koedinger, K., & Nathan, M. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of the Learning Sciences, 13(2), 129–164. https://doi.org/10.1207/s15327809jls1302_1

    Article  Google Scholar 

  • Li, X. B., Ding, M. X., 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 in the United States. Cognition and Instruction, 26(2), 195–217. https://doi.org/10.1080/07370000801980845

    Article  Google Scholar 

  • Li, Y. P., & Huang, R. J. (Eds.). (2013). How Chinese teach mathematics and improve teaching. New York, NY: Routledge.

    Google Scholar 

  • Ma, L. P. (1999). Knowing and teaching elementary mathematics: Understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • McNeil, N. M., & Fyfe, E. R. (2012). “Concreteness fading” promotes transfer of mathematical knowledge. Learning and Instruction, 22(6), 440–448. https://doi.org/10.1016/j.learninstruc.2012.05.001

    Article  Google Scholar 

  • Ministry of Education. (2001). 九年制义务教育全日制小学数学教学大纲 (试用). [Elementary mathematics teaching and learning standards for nine-year compulsory and full-time schooling (experimental edition)]. 北京, 中国: 北京师范大学出版社 [Beijing, China: Beijing Normal University Press].

    Google Scholar 

  • Mok, I. A. C., Cai, J. F., & Fung, A. T. F. (2008). Missing learning opportunities in classroom instruction: Evidence from an analysis of a well-structured lesson on comparing fractions. The Mathematics Educator, 11(1/2), 111–126.

    Google Scholar 

  • Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical Thinking and Learning, 10(4), 374–406. https://doi.org/10.1080/10986060802291642

    Article  Google Scholar 

  • Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40, 282–313.

    Google Scholar 

  • Nicol, C. C., & Crespo, S. M. (2006). Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials. Educational Studies in Mathematics, 62(3), 331–355. https://doi.org/0.1007/s10649-006-5423-y

    Article  Google Scholar 

  • Pashler, H., Bain, P. M., Bottge, B. A., Graesser, A., Koedinger, K., McDaniel, M., & Metcalfe, J. (2007). Organizing instruction and study to improve student learning (NCER 2007–2004). Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education.

    Google Scholar 

  • Perry, M., VanderStoep, S. W., & Yu, S. L. (1993). Asking questions in first-grade mathematics classes: Potential influences on mathematical thought. Journal of Educational Psychology, 85(1), 31–40. http://dx.doi.org/10.1037/0022-0663.85.1.31

    Article  Google Scholar 

  • Programme for International Student Assessment (PISA). (2012). PISA 2012 results in focus: What 15-year-olds know and what they can do with what they know. Retrieved November 3, 2018, from http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

    Google Scholar 

  • Radford, L., & Roth, W.-M. (2011). Intercorporeality and ethical commitment: An activity perspective on classroom interaction. Educational Studies in Mathematics, 77(2/3), 227–245. https://doi.org/10.1007/s10649-010-9282-1

    Article  Google Scholar 

  • Radford, L., Schubring, G., & Seeger, F. (2011). Signifying and meaning-making in mathematical thinking, teaching, and learning. Educational Studies in Mathematics, 77(2/3), 149–156. https://doi.org/10.1007/s10649-011-9322-5

    Article  Google Scholar 

  • Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. https://doi.org/10.3102/00346543075002211

    Article  Google Scholar 

  • Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35(5), 352–388. doi: 10.2307/30034820

    Article  Google Scholar 

  • Renkl, A., Atkinson, R. K., & Grobe, C. S. (2004). How fading worked solution steps works: A cognitive load perspective. Instructional Science, 32(1/2), 59–82. https://doi.org/10.1023/B:TRUC.0000021815.74806.f6

    Article  Google Scholar 

  • Resnick, L. B., Cauzinille-Marmeche, E., & Mathieu, J. (1987). Understanding algebra. In J. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics (pp. 169–203). Oxford, England: Clarendon.

    Google Scholar 

  • Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316(5828), 1128–1129. doi: 10.1126/science.1142103

    Article  Google Scholar 

  • Rittle-Johnson, B., & Star, J. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.

    Article  Google Scholar 

  • Schank, R. (2011). Teaching minds: How cognitive science can save our schools. New York, NY: Teachers College Press.

    Google Scholar 

  • Schifter, D. (2018). Early algebra as analysis of structure: A focus on operations. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5-to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 309–328). Cham, Switzerland: Springer.

    Chapter  Google Scholar 

  • Schleppenbach, M., Flevares, L. M., Sims, L. M., & Perry, M. (2007). Teachers’ responses to student mistakes in Chinese and U.S. mathematics classrooms. The Elementary School Journal, 108(2), 131–147.

    Article  Google Scholar 

  • Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage.

    Google Scholar 

  • Stein, M. K., Remillard, J. T., & Smith, M. S., (2007). How curriculum influences student learning. In F. K. Lester, Jr. (Ed.). Second handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York, NY: The Free Press.

    Google Scholar 

  • Su, L., & Wang, N. (2011). 小学数学 [Elementary mathematics textbook]. 南京, 中国: 江 苏教育出版社 [Nanjing, China: Jiangsu Education wblishing House].

    Google Scholar 

  • Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89.

    Article  Google Scholar 

  • Wang, S. W. (2011). 调适教科书: 使用教科书的实然与应然取向之间的中庸之道 [Adapting textbooks: The doctrine of the mean between the real and ideal implementation of textbooks]. 教师教育研究 [Teacher Education Research], 23(5), 43–48.

    Google Scholar 

  • Zhu, X. M., & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition and Instruction, 4(3), 137–166. https://doi.org/10.1207/s1532690xci0403_1

    Article  Google Scholar 

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Acknowledgements

This study is supported by the National Science Foundation CAREER program under Grant No. DRL-1350068 at Temple University. Any opinions, findings, and conclusions in this study are those of the authors and do not necessarily reflect the views of the National Science Foundation. We are grateful to the editors and three anonymous reviewers’ insightful feedback as well as Laurie Shirley Esposito’s editorial assistance with the manuscript revision. Both authors contributed equally to the article.

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  1. College of Teacher Education, Nanjing Xiaozhuang University, Nanjing, 210017, China

    Wei Chen

  2. College of Education, Temple University, Philadelphia, PA, 19122, USA

    Meixia Ding

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  1. Wei Chen
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  2. Meixia Ding
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Correspondence to Meixia Ding.

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Chen, W., Ding, M. Transition from Textbook to Classroom Instruction in Mathematics: The Case of an Expert Chinese Teacher. Front Educ China 13, 601–632 (2018). https://doi.org/10.1007/s11516-018-0031-z

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