# On mathematical understanding: perspectives of experienced Chinese mathematics teachers

## Abstract

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.

## Keywords

Mathematical understanding Achieving mathematical understanding Experienced teachers Chinese teachers Perspective of classroom practitioners## Notes

### Acknowledgments

The research reported in this paper was supported by a grant from the Spencer Foundation and a grant from the University of Delaware’s Center for Global and Area Studies and the Center for the Study of Diversity. Any opinions expressed herein are those of the authors and do not necessarily represent the views of the Spencer Foundation and the University of Delaware. We gratefully acknowledge the feedback and editorial assistance from Stephen Hwang, Margaret Walshaw, and reviewers, which contributed to the paper’s improvement. Of course, any errors remain solely the responsibility of the authors.

## References

- Anderson, J. R., Reder, L. M., & Simon, H. A. (2000). Applications and misapplications of cognitive psychology to mathematics education.
*Texas Educational Review.*http://actr.psy.cmu.edu/wordpress/wp-content/uploads/2012/12/146Applic.MisApp.pdf. - Baroody, J. (1987).
*Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers*. New York, NY: Teachers College Press.Google Scholar - Baroody, A. J. (1999). Children’s relational knowledge of addition and subtraction.
*Cognition and Instruction,**17*, 137–175.CrossRefGoogle Scholar - Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge.
*Journal for Research in Mathematics Education,**38*, 115–131.Google Scholar - Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotna (Ed.),
*Proceedings of the 30th conference of the international group for the psychology of mathematics education*. Prague, Czech Republic: PME.Google Scholar - 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.CrossRefGoogle Scholar - Borgen, K. L., & Manu, S. S. (2002). What do students really understand?
*Journal of Mathematical Behavior,**21*, 151–165.CrossRefGoogle Scholar - Brownell, W. A. (1935). Psychological considerations in the learning and the teaching of arithmetic. In W. D. Reeve (Ed.),
*The teaching of arithmetic*(Tenth Yearbook of the National Council of Teachers of Mathematics) (pp. 1–31). New York: Columbia University.Google Scholar - Bruner, J. S. (1966).
*Toward a theory of instruction*. Cambridge: Harvard University Press.Google Scholar - Cai, J. (2000). Mathematical thinking involved in U.S. and Chinese students' solving process-constrained and process-open problems.
*Mathematical Thinking and Learning: An International Journal, 2,*309–340. CrossRefGoogle Scholar - Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving? Exploring the impact of early algebra learning and teachers’ beliefs.
*Journal of Mathematical Behavior, 23,*135–167.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.CrossRefGoogle Scholar - Cai, J., Ding, M., & Wang, T. (2014). How do exemplary Chinese and U.S. mathematics teachers view instructional coherence?
*Educational Studies in Mathematics, 85,*265–280.CrossRefGoogle Scholar - Cai, J., & Knuth, E. (2011).
*Early algebraization: A global dialogue from multiple perspectives.*New York: Springer.CrossRefGoogle Scholar - Chi, M. T. H., de Leeuw, N., Chiu, M., & LaVancher, C. (1994). Eliciting self-explanations improves understanding.
*Cognitive Science,**18*, 439–477.Google Scholar - Chi, T. H., & VanLehn, K. A. (2012). Seeing deep structure from the interactions of surface features.
*Educational Psychologist,**47*(3), 177–188.CrossRefGoogle Scholar - Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development.
*Educational Researcher,**23*(7), 13–20.CrossRefGoogle Scholar - Darling-Hammond, L. (1994). Performance-based assessment and educational equity.
*Harvard Education Review,**64*(1), 5–30.CrossRefGoogle Scholar - Davis, R. B. (1992). Understanding of “Understanding”.
*Journal of Mathematical Behavior,**11*(3), 225–242.Google Scholar - Ding, M., & Li, X. (2010). A comparative analysis of the distributive property in the US and Chinese elementary mathematics textbooks.
*Cognition and Instruction, 28*, 146–180.CrossRefGoogle Scholar - Ding, M., Li, X., Piccolo, D., & Kulm, G. (2007). Teacher interventions in cooperative-learning mathematics classes.
*Journal of Educational Research, 100,*162–175.CrossRefGoogle Scholar - Ding, M., Li, Y., Li., X, & Gu, J. (2013). Knowing and understanding instructional mathematics content through intensive studies of textbooks. In Y. Li, & R. Huang (Eds.),
*How Chinese teach mathematics and improve teaching*(pp. 66–82). New York: Routledge.Google Scholar - Ellis, A. B. (2007). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations.
*The Journal of The Learning Sciences,**16*, 221–262.CrossRefGoogle Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education. China lectures*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Gay, L. R., & Airasian, P. (2000).
*Educational research: Competencies for analysis and application*(6th ed.). Upper Saddle River: Merrill.Google Scholar - Gravemeijer, K. P. E. (1994).
*Developing realistic mathematics education*. Utrecht: CD Beta Press.Google Scholar - Greeno, J. G. (1978). Understanding and procedural knowledge in mathematics instruction.
*Educational Psychologist,**12*, 262–283.CrossRefGoogle 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: Lawrence Erlbaum Associates.Google Scholar - Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.),
*How Chinese learn mathematics: Perspectives from insiders*(pp. 309–347). Singapore: World Scientific.CrossRefGoogle Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York: Mcmillan.Google Scholar - Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997).
*Making sense: Teaching and learning mathematics with understanding*. Portsmouth: Heinemann.Google Scholar - Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),
*Conceptual and procedural knowledge: The case of mathematics*(pp. 1–27). Hillsdale: Lawrence Erlbaum Associates.Google Scholar - Huang, R., & Li, Y. (2009). Pursuing excellence in mathematics classroom instruction through exemplary lesson development in China: A case study.
*ZDM-International Journal of Mathematics Education,**41*, 297–309.CrossRefGoogle 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.CrossRefGoogle Scholar - Lester, F. A. (2003).
*Research and issues in teaching mathematics through problem solving*. Reston: National Council of Teachers of Mathematics.Google Scholar - Lewis, C. (1988). Why and how to learn why: Analysis-based generalization of procedures.
*Cognitive Science,**12*, 211–256.CrossRefGoogle Scholar - 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 in the United States.
*Cognition and Instruction, 26,*195–217.CrossRefGoogle Scholar - Lobato, J. (2006). Alternative perspectives on the transfer of learning: History, issues, and challenges for future research.
*The Journal of The Learning Sciences,**15*, 431–449.CrossRefGoogle Scholar - Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling.
*Journal for Research in Mathematics Education,**36*, 101–136.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Understanding of fundamental mathematics in China and the United States*. Mahwah: Lawrence Erlbaum Associates.Google Scholar - Marton, F., Tse, L. K., & dall’Alba, G. (1996). Memorizing and understanding: The keys to the paradox? In D. A. Watkins & J. B. Biggs (Eds.),
*The Chinese learner: Cultural, psychological and contextual influences*(pp. 69–83). Hong Kong: Comparative Education Research Centre.Google Scholar - Mason, J. (1989). Mathematical abstraction seen as a delicate shift of attention.
*For the Learning of Mathematics,**9*(2), 2–8.Google Scholar - Meel, D. E. (2003). Models and theories of mathematical understanding: Comparing Pirie and Kieren’s model of the growth of mathematical understanding and APOS theory. In A. Selden, E. Dubinsky, G. Harel & F. Hitt (Eds.)
*Research in collegiate mathematics education. V. Issues in mathematics education*(pp. 132-181). Providence, RI: American Mathematical Society.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: Author.Google Scholar - National Research Council. (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions.
*American Educational Research Journal,**45*(4), 1080–1110.CrossRefGoogle Scholar - Nisbett, R. E., Peng, K., Choi, I., & Norenzayan, A. (2001). Culture and systems of thought: Holistic versus analytic cognition.
*Psychological Review,**108*, 291–310.CrossRefGoogle Scholar - Piaget, J. (1952).
*The child’s conception of number*. London: Routledge and Kegan Paul.Google Scholar - Pirie, S., & Kieren, T. (1992). Creating constructivist environments and constructing creative mathematics.
*Educational Studies in Mathematics,**23*, 505–528.CrossRefGoogle Scholar - Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it?
*Educational Studies in Mathematics,**26*, 165–190.CrossRefGoogle Scholar - Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics.
*Educational Psychologist,**47*, 189–203.CrossRefGoogle Scholar - Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom.
*Science,**316*, 1128–1129.CrossRefGoogle Scholar - Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem solving.
*Cognition and Instruction,**23*, 313–349.CrossRefGoogle Scholar - 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.CrossRefGoogle 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*, 561–574.CrossRefGoogle Scholar - Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals.
*Mathematical Thinking and Learning,**8*(4), 359–371.CrossRefGoogle Scholar - Skemp, R. (1976). Relational understanding and instrumental understanding.
*Mathematics Teaching,**77*(1), 20–26.Google Scholar - Star, J. (2005). Reconceptualizing procedural knowledge.
*Journal for Research in Mathematics Education,**36*, 404–411.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: The Free 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.CrossRefGoogle Scholar - Sweller, J. (2006). The worked example effect and human cognition.
*Learning and Instruction,**16*, 165–169.CrossRefGoogle Scholar - Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P., & Verschaffel, L. (2009). Solving subtraction problems by means of indirect addition.
*Mathematical Thinking and Learning,**11*, 79–91.CrossRefGoogle Scholar - Trends in International Mathematics and Science Study (2003).
*Highlights from the Trends in International Mathematics and Science Study (TIMSS) 2003.*Retrieved from http://nces.ed.gov/pubs2005/2005005.pdf. - Trends in International Mathematics and Science Study (2007).
*Highlights from TIMSS 2007: Mathematics and Science Achievement of US Fourth and Eighth*-*Grade Students in an International Context.*Retrieved from http://nces.ed.gov/pubs2009/2009001.pdf. - Wearne, D., & Hiebert, J. (1988). A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers.
*Journal for Research in Mathematics Education,**19*(5), 371–384.CrossRefGoogle Scholar - Weiss, I. R., Smith, S., & O’Kelley, S. K. (2009). The Presidential award for excellence in mathematics teaching: Setting the standard. In J. Cai, G. Kaiser, R. Perry, & N.-Y. Wong (Eds.),
*Effective mathematics teaching from teachers’ perspectives: National and international studies*(pp. 281–304). Rotterdam: Sense Publisher.Google Scholar - Wertheimer, M. (1959).
*Productive thinking*(Enlarged ed.). New York: Harper and Row**(Original work published 1945)**.Google Scholar