Mathematics Education in Singapore pp 241-268 | Cite as

# Metacognition in the Teaching and Learning of Mathematics

## Abstract

This chapter first presents the evolving conceptualisation of metacognition since it was first coined by Flavell in 1976. In particular, the issue of awareness, monitoring, and regulation of both cognitive and affective resources was examined. The role that metacognition plays in mathematical problem-solving was also examined, leading to a discussion of the role of metacognition in the Singapore School Mathematics Curriculum which has mathematical problem-solving as its central aim. In view of this, the conceptualisation of metacognition as well as the how’s of addressing metacognition in the Singapore mathematics classrooms were discussed from the intended curriculum point of view. Some of the local postgraduate works on metacognition and teaching, and learning of mathematics was also presented to provide an overview of the landscape of the work in this area that has been undertaken thus far. In addition, examples of ongoing works on metacognitive approaches, which have made some inroads in some local schools, were shared to give the reader a glimpse of how research in this area has impacted school practices locally. The chapter concludes with implications for addressing metacognition in the Singapore Mathematics classrooms from the perspective of teachers’ professional development.

## Keywords

Singapore School Mathematics Curriculum Cognition Metacognition Offline metacognition Online metacognition Metacognitive instructional strategies Mathematical problem-solving Teaching and learning of mathematics Reflection Reflective practice model Meta-metacognition Theory of mind Social metacognition## References

- Baker, L. (1991). Metacognition, reading, and science education. In C. Santa & D. Alvermann (Eds.),
*Science learning: Processes and applications*(pp. 2–13). Newark, DE: International Reading Association.Google Scholar - Bangert-Drowns, R., & Bankert, E. (1990).
*Meta-analysis of effects of explicit instruction for critical thinking.*Paper presented at the annual meeting of the American Educational Research Association Boston, MA.Google Scholar - Barkatsas, A. N., & Hunting, R. (1996). A review of recent research on cognitive, metacognitive and affective aspects of problem solving.
*Nordic Studies in Mathematics Education,**4*(4), 7–30.Google Scholar - Biggs, J. B. (1987).
*Student approaches to learning and studying*. Melbourne: Australian Council for Educational Research.Google Scholar - Brinol, P., & DeMarree, K. G. (2012). Social metacognition: Thinking about thinking in social psychology. In P. Brinol & K. G. DeMarree (Eds.),
*Social metacognition*(pp. 1–20). New York: Psychology Press.CrossRefGoogle Scholar - Brown, A. L. (1980). Metacognitive development and reading. In R. J. Sprio, B. C. Bruce, & W. F. Brewer (Eds.),
*Theoretical issues in reading comprehension*(pp. 453–481). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Brown, A. L., Bransford, J. D., Ferrara, R. A., & Campione, J. C. (1983). Learning, remembering, and understanding. In J. H. Flavell & M. E. Markman (Eds.),
*Handbook of child psychology*(4th ed.),*Vol. III, Cognitive development*(pp. 77–166). New York: Wiley.Google Scholar - Chang, S. C. (1989, March).
*A study of learning strategies employed by Secondary 4 Express and Normal pupils.*Paper presented at the Sixth ASEAN Forum on Child and Adolescent Psychiatry, Singapore.Google Scholar - Chang, S. C. A. & Ang, W. H. (1999, July).
*Emotions, values, good thinking.*Paper presented at the 8th international conference on thinking, Edmonton, Canada.Google Scholar - Chang, S. C., Yeap, B. H., & Lee, N. H. (2001). Infusing thinking skills through the use of graphic organisers in primary mathematics to enhance weak students’ learning. In J. Ee, B. Kaur, N. H. Lee, & B. H. Yeap (Eds.),
*New ‘Literacies’: Educational response to a knowledge-based society*(pp. 642–649). Singapore: Educational Research Association.Google Scholar - Chapin, S. H., O’Conner, C., & Anderson, N. C. (2013).
*Classroom discussions in Math: A Teacher’s guide for using talk moves to support the common core and more, Grades K-6: A multimedia professional learning resource*(3rd ed.). Sausalito, CA: Math Solutions Publications.Google Scholar - Cheng, P. (1999). Cognition, metacognition, and metacognitive theory: A critical analysis.
*The Korean Journal of Thinking and Problem Solving, 9*(i), 85–103.Google Scholar - Chiu, M. M., & Kuo, S. W. (2009). From metacognition to social metacognition: Similarities, differences, and learning.
*Journal of Educational Research,**3*(4), 1–19.Google Scholar - Costa, A. L. (2001). Habits of mind. In A. L. Costa (Ed.),
*Developing minds—A resource book for teaching thinking*(3rd ed., pp. 80–86). Alexandria, VA: Association for Supervision and Curriculum Development.Google Scholar - Costa, A. L., & Kallick, B. (Eds.). (2000).
*Activating & engaging habits of mind*. Alexandria, VA: Association for Supervision and Curriculum Development.Google Scholar - Costa, A. L., & Kallick, B. (2009) (Ed.).
*Habits of mind across the curriculum—Practical and creative strategies for teachers*. Alexandria, VA: Association for Supervision and Curriculum Development.Google Scholar - Cotton, K. (1991).
*Close-up #11: Teaching thinking skills*. Retrieved from Northwest Regional Educational Laboratory’s School Improvement Research Series Website. http://educationnorthwest.org/sites/default/files/TeachingThinkingSkills.pdf. - Davidson, J. E., Deuser, R., & Sternberg, R. J. (1994). The role of metacognition in problem solving. In J. Metcalfe & A. P. Shinmamura (Eds.),
*Metacognition: Knowing about knowing*(pp. 207–226). Cambridge, MA: Massachusetts Institute of Technology.Google Scholar - Dweck, C. S. (2002). Beliefs that make smart people dumb. In R. J. Sternberg (Ed.),
*Why smart people can be so stupid*. New Haven, CT: Yale University Press.Google Scholar - Dweck, C. S. (2006).
*Mindset: The new psychology of success*. New York: Random House.Google Scholar - Dweck, C. S. (2012).
*Mindset: How you can fulfil your potential*. London: Robinson.Google Scholar - Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.),
*The nature of intelligence*(pp. 231–235). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Flavell, J. H. (1979). Metacognition and cognitive monitoring.
*American Psychologist,**34,*906–911.CrossRefGoogle Scholar - Flavell, J. H. (1987). Speculations about the nature and development of metacognition. In F. E. Weinert & R. H. Kluwe (Eds.),
*Metacognition, motivation, and understanding*(pp. 21–29). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Flavell, J. H. (2002). Development of children’s knowledge about the mental world. In W. W. Hartup & R. K. Silbereisen (Eds.),
*Growing points in developmental science: An introduction*(pp. 102–122). Hove, UK: Psychology Press.Google Scholar - Flavell, J. H., Miller, P. H., & Miller, S. A. (2002).
*Cognitive development*(4th ed.). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Foong, P. Y. (1990).
*A metacognitive-heuristic approach to mathematical problem solving*. Unpublished doctoral thesis, Monash University, Australia.Google Scholar - Foong, P. Y. (1993). Development of a framework for analyzing mathematical problem solving behaviours.
*Singapore Journal of Education,**13*(1), 61–75.CrossRefGoogle Scholar - Freeman, W. (1995).
*Societies of brains*. Hillsdale, NJ: Lawrence Erlbaum and Associates.Google Scholar - Garofalo, J., & Lester, F. K., Jr. (1985). Metacognition, cognitive monitoring and mathematical performance.
*Journal for Research in Mathematics Education,**16*(3), 163–176.CrossRefGoogle Scholar - Halpern, D. F. (1998). Teaching critical thinking for transfer across domains: Dispositions, skills, structure training, and metacognitive monitoring.
*American Psychologist,**53,*449–455.CrossRefGoogle Scholar - Halpern, D. F. (2003).
*Thought and knowledge: An introduction to critical thinking*(4th ed.). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Hong, S. E., Lee, N. H., & Yeo, J. S. D. (2012). A metacognitive approach in kick-starting the understanding and planning phases of mathematical problem solving. In ICME-12 (Ed.),
*ICME-12 Pre-proceedings (Electronic)*(pp. 4615–4623). Seoul, Korea: ICME-12.Google Scholar - Jonid, S. H., Kwek, D., Hogan, D., & Towndrow, P. (2014).
*Report on metacognition*. Centre of Research in Pedagogy and Practice, Office of Educational Research, National Institute of Education, Singapore.Google Scholar - Khun, D. (2000). Theory of mind, metacognition, and reasoning: A life-span perspective. In P. Metchell & K. J. Riggs (Eds.),
*Children’s reasoning and the mind*(pp. 301–326). Hove, UK: Psychology Press.Google Scholar - Kuhn, D. (1999). Metacognitive development. In L. Balter & C. S. Tamis-LeMonda (Eds.),
*Child psychology: A handbook of contemporary issues*(pp. 259–286). Philadelphia, PA: Psychology Press.Google Scholar - Kuhn, D. (2000). Metacognitive development.
*Current Directions in Psychological Science,**9,*178–181.CrossRefGoogle Scholar - Leder, G. C. (1993). Reconciling affective and cognitive aspects of mathematical learning: Reality or a pious hope? In I. Hirabayashi, N. Nohda, K. Shigemathu, & F.-L. Lin (Eds.),
*Proceedings of the Seventeenth PME Conference*(pp. 46–65). Tsukuba, Ibaraki, Japan: University of Tsukuba.Google Scholar - Lee, N. H. (2003).
*A model for the practicing reflective Mathematics teacher.*Paper presented at the international conference on Thinking XI, Arizona, United States.Google Scholar - Lee, N. H. (2008).
*Enhancing mathematical learning and achievement of secondary one normal (academic) students using metacognitive strategies.*Unpublished doctoral thesis, Nanyang Technological University, Singapore.Google Scholar - Lee, N. H. (2010, August).
*Reflective practice in the teaching & learning of Mathematics.*Keynote Address, The Academy of Singapore Teachers’ Second LT-ST Network Meeting.Google Scholar - Lee, N. H. (2015, July).
*Metacognition in the teaching and learning of Mathematics*. Keynote Address, 2nd Mathematics Chapter Meeting, Singapore, Singapore.Google Scholar - Lee, N. H. (2016a, April).
*Using the problem wheel to Metacognitively Kickstart students’ problem solving.*Singapore.Google Scholar - Lee, N. H. (2016b, April).
*Addressing metacognition in the Singapore Mathematics Curriculum—Issues and approaches*. Paper presented at the Secondary Mathematics HOD Meeting, Ministry of Education, Singapore.Google Scholar - Lee, N. H. (2017a, May).
*Nurturing cognitive and metacognitive thinkers in the 21st century.*Keynote Address, 4th Anglican Character, Thinking and Service (ACTS) Seminar, Singapore.Google Scholar - Lee, N. H. (2017b, June & July).
*Promoting metacognition in primary school children*. In-Service Course for Primary Mathematics Teachers, National Institute of Education, Singapore.Google Scholar - Lee, N. H. (2017c, November).
*Metacognition in the Mathematics classroom*. In-Service Course for Secondary Mathematics Teachers, National Institute of Education, Singapore.Google Scholar - Lee, N. H., Lee, Y. Y. G., & Koo, C. C. (2013). Teachers’ promotion of students’ metacognition in mathematical modelling lessons. In M. Inprasitha (Ed.),
*Innovations and Exemplary Practices in Mathematics Education—Proceedings of the 6th East Asia Regional Conference on Mathematics Education*(Vol. 2, pp. 74–84). Khon Kaen University, Thailand: Center for Research in Mathematics Education (CRME).Google Scholar - Lee, N. H., Ng, K. E. D., Seto, C., Loh, M. Y., & Chen, S. (2016, September).
*Programmatic influence on Mathematics teachers’ metacognitive instructional strategies: A Singapore case study.*Paper presented at British Educational Research Association (BERA) annual conference 2016, Leeds, United Kingdom.Google Scholar - Lee, N. H., Poh, C. L., Chan, Y. L., Lye, W. L., Chan, Y. Y., & Leong, S. L. (1998). Critical thinking in the Mathematics class. In M. L. Quah & W. K. Ho (Eds.),
*Thinking processes—Going beyond the surface curriculum*(pp. 163–178). Singapore: Prentice Hall.Google Scholar - Lee, N. H., Yeo, D. J. S., & Hong, S. E. (2014). A metacognitive-based instruction for Primary Four students to approach non-routine mathematical word problems.
*ZDM—The International Journal on Mathematics Education,**46*(3), 465–480.CrossRefGoogle Scholar - Lo, C. L. (1995).
*Metacognitive strategy in solving Mathematics problems, learning approach and Mathematics achievement of students from a Junior College*. Unpublished master’s thesis, Nanyang Technology University, Singapore.Google Scholar - Lockl, K., & Schneider, W. (2006). Precursors of metamemory in young children: The role of theory of mind and metacognitive vocabulary.
*Metacognition and Learning,**1,*15–31.CrossRefGoogle Scholar - Loh, M. Y. (2015).
*Metacognitive strategies secondary one students employed while solving Mathematics problems.*Unpublished doctoral thesis, Nanyang Technological University, Singapore.Google Scholar - Loh, M. Y., & Lee, N. H. (2017). Empowering Mathematics learners with metacognitive strategies in problem solving. In B. Kaur & N. H. Lee (Eds.),
*Empowering Mathematics learners*(pp. 1–8). Singapore: World Scientific.Google Scholar - Marin, L. M., & Halpern, D. F. (2011). Pedagogy for developing critical thinking in adolescents: Explicitly instruction produces greatest gains.
*Thinking Skills and Creativity,**6,*1–13.CrossRefGoogle Scholar - Markman, E. M. (1977). Realizing that you don’t understand: A preliminary investigation.
*Child Development,**48,*643–655.CrossRefGoogle Scholar - McLeod, D. B. (1992). Research on affect in Mathematics education: A reconceptualisation. In D. Grouws (Ed.),
*Handbook of research on Mathematics teaching and learning*(pp. 575–596). New York: MacMillan.Google Scholar - Mevarech, Z. R., Tabuk, A., & Sinai, O. (2006). Meta-cognitive instruction in Mathematics classrooms: Effects on the solution of different kinds of problems. In A. Desoete & M. Veenman (Eds.),
*Metacognition in Mathematics education*(pp. 83–101). New York: Nova Science Publishers.Google Scholar - Ministry of Education. (1990a).
*Mathematics syllabus (Primary)*. Singapore: Author.Google Scholar - Ministry of Education. (1990b).
*Mathematics syllabus (Lower Secondary)*. Singapore: Author.Google Scholar - Ministry of Education. (2000a).
*Mathematics syllabus—Primary*. Singapore: Author.Google Scholar - Ministry of Education. (2000b).
*Mathematics syllabus—Lower Secondary*. Singapore: Author.Google Scholar - Ministry of Education. (2006a).
*Primary Mathematics syllabus*. Singapore: Author.Google Scholar - Ministry of Education. (2006b).
*Secondary Mathematics syllabus*. Singapore: Author.Google Scholar - Ministry of Education. (2012a).
*Primary Mathematics teaching and learning syllabus*. Singapore: Author.Google Scholar - Ministry of Education. (2012b).
*Ordinary-level and normal (academic)—Level Mathematics teaching and learning syllabus*. Singapore: Author.Google Scholar - Ministry of Education. (2012c).
*Normal (technical)-level Mathematics teaching and learning Syllabus*. Singapore: Author.Google Scholar - Misailidi, P. (2010). Children’s metacognition and theory of mind: Bridging the gap. In A. Efklides & P. Misailidi (Eds.),
*Trends and prospects in metacognition research*(pp. 279–291). Boston, MA: Springer.CrossRefGoogle Scholar - National Research Council. (2010).
*Educating teachers of Science, Mathematics, and Technology: New practices for the New Millennium*. Washing, DC: National Academy Press.Google Scholar - Ng, S. F. (2009). The Singapore Primary Mathematics Curriculum. In P. Y. Lee & N. H. Lee (Eds.),
*Teaching Primary School Mathematics—A resource book*(2nd ed., pp. 15–34). Singapore: McGraw-Hill.Google Scholar - Ng, K. E. D., Lee, N. H., Seto, C., Loh, M. Y., & Chen, S. (2016, September).
*Teachers’ conceptions of metacognition: Some preliminary findings from Singapore primary schools.*Paper presented at British Educational Research Association (BERA) annual conference 2016, Leeds, United Kingdom.Google Scholar - OECD. (2010).
*PISA 2009 results: Learning to learn—Student engagement, strategies and practices*. Author.Google Scholar - Paul, R. W. (1993).
*Critical thinking: What every person needs to survive in a rapidly changing world*(3rd ed.). Tomales, CA: Foundation for Critical Thinking.Google Scholar - Perkins, D. N., Simmons, R., & Tishman, S. (1990). Teaching cognitive and metacognitive strategies.
*Journal of Structural Learning,**10,*285–303.Google Scholar - Pólya, G. (1957).
*How to solve it*. Princeton: Princeton University Press.Google Scholar - Schmitt, M. C., & Newby, T. J. (1986). Metacognition: Relevance to instructional design.
*Journal of Instructional Development,**9*(4), 29–33.CrossRefGoogle Scholar - Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in Mathematics. In D. Grouws (Ed.),
*Handbook of research on Mathematics teaching and learning*(pp. 334–370). New York: MacMillan.Google Scholar - Silver, E. A. (1987). Foundations of cognitive theory and research for Mathematics problem solving instruction. In A. H. Schoenfeld (Ed.),
*Cognitive science and Mathematics education*(pp. 33–60). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Stillman, G. (2007). Applying metacognitive knowledge and strategies in applications and modelling tasks at secondary school. In W. Blum, P. L. Galbraith, H. Henn, & M. Niss (Eds.),
*Modelling and applications in Mathematics education (The 14th ICMI Study)*(pp. 165–180). NY: Springer.Google Scholar - Swartz, R. J., & Perkins, D. N. (1990).
*Teaching thinking: Issues & approaches*(Rev ed.). Pacific Grove, CA: Critical Thinking Press & Software.Google Scholar - Sweller, J., & Low, R. (1992). Some cognitive factors relevant to Mathematics instruction.
*Mathematics Education Research Journal,**4,*83–94.CrossRefGoogle Scholar - Tarricone, P. (2011).
*The taxonomy of metacognition*. New York: Psychology Press.Google Scholar - Teo, O. M. (2006).
*A small-scale study on the effects of metacognition and beliefs on students in A-level sequences and series problem solving.*Unpublished master’s thesis, Nanyang Technological University, Singapore.Google Scholar - The Discoveries of Reflective Practice. (2015, September).
*SingTeach.*Retrieved http://singteach.nie.edu.sg/issue54-classroom03/. - Wong, P. (1989, November).
*Students’ metacognition in Mathematical problem solving*. Paper presented at the annual meeting of the Australian Association for Research in Education.Google Scholar - Wong, P. (1992). Metacognition in Mathematical problem solving.
*Singapore Journal of Education,**12*(2), 48–58.CrossRefGoogle Scholar - Yap, Q. H. J. (2016).
*A metacognitive-heuristic approach to help low attainers in ratio word problems.*Unpublished master’s thesis, Nanyang Technology University, Singapore.Google Scholar - Yeap, B. H. (1997).
*Mathematical problem solving: A focus on metacognition*. Unpublished master’s thesis, Nanyang Technology University, Singapore.Google Scholar - Yeo, B. W. J. (2013).
*The nature and development of processes in mathematical investigation.*Unpublished doctoral thesis, Nanyang Technological University, Singapore.Google Scholar