Linear quantity models in US and Chinese elementary mathematics classrooms

References

• Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11, 25–61. doi:10.1207/S15327809JLS1101_2
   Web of Science ®Google Scholar
• Alibali, M. W., & DiRusso, A. A. (1999). The function of gesture in learning to count: More than keeping track. Cognitive Development, 14, 37–56. doi:10.1016/S0885-2014(99)80017-3
   Web of Science ®Google Scholar
• Alibali, M. W., & Nathan, M. J. (2012). Embodiment in mathematics teaching and learning: Evidence from students’ and teachers’ gestures. Journal of the Learning Sciences, 21, 247–286.
   Web of Science ®Google Scholar
• Beckman, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1), 42–46.
   Google Scholar
• Cai, J. (2005). U.S. and Chinese teachers’ knowing, evaluating, and constructing representations in mathematics instruction. Mathematical Thinking and Learning: an International Journal, 7, 135–169. doi:10.1207/s15327833mtl0702_3
   Google Scholar
• Cai, J., & Moyer, J. C. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In C. E. Greene & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 169–182). Reston, VA: NCTM.
   Google Scholar
• Chi, M. T. H. (1997). Quantifying qualitative analyses of verbal data: A practical guide. The Journal of the Learning Sciences, 6, 271–315. doi:10.1207/s15327809jls0603_1
   Web of Science ®Google Scholar
• Chi, M. T. H., & van Lehn, K. A. (2012). Seeing deep structure from the interactions of surface features. Educational Psychologist, 47(3), 177–188. doi:10.1080/00461520.2012.695709
   Web of Science ®Google Scholar
• Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/the-standards.
   Google Scholar
• Craig, S. D., Sullins, J., Witherspoon, A., & Gholson, B. (2006). The deep-level-reasoning-question effect: The role of dialogue and deep-level-reasoning questions during vicarious learning. Cognition and Instruction, 24, 565–591. doi:10.1207/s1532690xci2404_4
   Web of Science ®Google Scholar
• Ding, H. Y. (2010). Give students three-dimensional “mathematics”: The case of “number-shape combination.” (给学生一个立体的“数学”—例谈“数形结合”). People’s Education (人民教育), 7, 39–42.
   Google Scholar
• Ding, M. (2016). Opportunities to learn: Inverse operations in U.S. and Chinese elementary mathematics textbooks. Mathematical Thinking and Learning, 18(1), 45–68. doi:10.1080/10986065.2016.1107819
   Web of Science ®Google Scholar
• Ding, M. (2018). Modeling with tape diagrams. Teaching Children Mathematics. doi:10.5951/teacchilmath.25.3.0158
   Google Scholar
• Ding, M., & Li, X. (2014). Transition from concrete to abstract representations: The distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87, 103–121. doi:10.1007/s10649-014-9558-y
   Web of Science ®Google Scholar
• Dixon, J. K., Larson, M. R., Burger, E. B., Sandoval-Martinez, M. E., & Leinwand, S. J. (2012). Go math: Grades 1–4. Houghton: Mifflin Harcourt.
   Google Scholar
• English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Lawrence Erlbaum.
   Google Scholar
• Fruedenthal, H. (1991). Revisiting mathematics education. Dordrecht, The Netherlands: Kluwer Academic Publishers.
   Google Scholar
• Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: Making sense of her pedagogical moves. Educational Studies in Mathematics, 81, 325–345. doi:10.1007/s10649-012-9407-9
   Web of Science ®Google Scholar
• Goldin-Meadow, S., & Alibali, M. W. (2013). Gesture’s role in speaking, learning, and creating language. Annual Review of Psychology, 64, 257–283. doi:10.1146/annurev-psych-113011-143802
   PubMed Web of Science ®Google Scholar
• Goldstone, R. L., & Sakamoto, Y. (2003). The Transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414–466.
   PubMed Web of Science ®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, 69–110. doi:10.1207/s15327809jls1401_4
   Web of Science ®Google Scholar
• Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177. doi:10.1207/s15327833mtl0102_4
   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
• Huang, R., & Li, Y. (2017). Teaching and learning mathematics through variation: Confucian heritage meets western theories. Rotterdam, The Netherlands: Sense.
   Google Scholar
• Kintsch, W. (1986). Learning from text. Cognition and Instruction, 3, 87–108. doi:10.1207/s1532690xci0302_1
   Google Scholar
• Larkin, J., & Simon, H. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11, 65–99. doi:10.1111/j.1551-6708.1987.tb00863.x
   Web of Science ®Google Scholar
• Lesh, R., Cramer, K., Doerr, H., Post, T., & Zawojewski, J. (2003). Model development sequences. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 35–70). Mahwah, NJ: Lawrence Erlbaum.
   Google Scholar
• Li, Y., & Huang, R. (2013). How Chinese teach mathematics and improve teaching. New York, NY: Routledge.
   Google Scholar
• Ma, L. (1999). Knowing and teaching elementary mathematics: Understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical Thinking and Learning, 10, 374–406. doi:10.1080/10986060802291642
   Google Scholar
• Murata, A., & Kattubadi, S. (2012). Grade 3 students’ mathematization through modeling: Situation models and solution models with multi-digit subtraction problem solving. The Journal of Mathematical Behavior, 31, 15–28. doi:10.1016/j.jmathb.2011.07.004
   Google Scholar
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
   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.
   Web of Science ®Google Scholar
• Nunes, T., Bryant, P., & Watson, A. (2009). Key understandings in mathematics learning: A report to the Nuffield Foundation. London, UK: Nuffield Foundation.
   Google Scholar
• Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction, 22(1), 37–60. doi:10.1207/s1532690Xci2201_2
   Web of Science ®Google Scholar
• Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129. doi:10.1126/science.1142109
   PubMed Web of Science ®Google Scholar
• Rittle-Johnson, B., & Star, J. R. (2009). Comparing with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101, 529–544. doi:10.1037/a0014224
   Web of Science ®Google Scholar
• Stigler, J., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York, NY: Free Press.
   Google Scholar
• Sun, L., & Wang, L. (2012). Mathematics (Grade 1, volumes 1 and 2). Nanjing, China: Jiang Su Education Press.
   Google Scholar
• Sun, L., & Wang, L. (2013). Mathematics (Grade 2, volumes 1 and 2). Nanjing, China: Jiang Su Education Press.
   Google Scholar
• Sun, L., & Wang, L. (2014). Mathematics (Grade 3, volumes 1 and 2). Nanjing, China: Jiang Su Education Press.
   Google Scholar
• Sun, X. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76, 65–85. doi:10.1007/s10649-010-9263-4
   Web of Science ®Google Scholar
• Technical Education Research Centers. (2012). Investigations in number, data, and space. Upper Saddle River, NJ: Pearson.
   Google Scholar
• Vanduzer, K. (2016). Tape diagrams, big feelings and other predicaments of teaching. Retrieved from https://discoveryandconjecture.wordpress.com/2016/11/19/tape-diagrams-big-feelings-and-other-predicaments-of-teaching/.
   Google Scholar