2,793
Views
4
CrossRef citations to date
0
Altmetric
Articles

Evaluating the suitability of mathematical thinking problems for senior high-school students by including mathematical sense making and global planning

Pages 313-329 | Received 29 Dec 2015, Accepted 30 Mar 2016, Published online: 06 May 2016

References

  • Anderson, J. (2009, October). Mathematics curriculum development and the role of problem solving. Paper presented at the National Biennial Conference of Australian Curriculum Studies Association, Canberra.
  • Anthony, G., & Walshaw, M. (2009). Characteristics of effective teaching of mathematics: A view from the west. Journal of Mathematics Education, 2(2), 147–164.
  • Antonietti, A., Ignazi, S., & Perego, P. (2000). Metacognitive knowledge about problem-solving methods. British Journal of Educational Psychology, 70(1), 1–16.
  • Arslan, S. (2010). Do students really understand what an ordinal differential equation is? International Journal of Mathematical Education in Science and Technology, 41(7), 873–888.
  • Artzt, A. F., & Armour-Thomas, E. (1992). Development of a cognitive-metacognitive framework for protocol analysis of mathematical problem solving in small groups. Cognition and Instruction, 9(2), 137–175.
  • Briars, D. J. (2014). Curriculum materials matter: Evaluating the evaluation process. National Council of Teachers of Mathematics. Retrieved from http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Curriculum-Materials-Matter_-Evaluating-the-Evaluation-Process/.
  • Brown, A. L. (1987). Metacognition, executive control, self-regulation, and other more mysterious mechanisms. In F. E. Weinert, & R. H. Kluwe (Eds.), Metacognition, motivation, and understanding (pp. 65–116). Hillsdale, NJ: LEA.
  • Burkhardt, H., & Bell, A. (2007). Problem solving in the United Kingdom. ZDM: The International Journal on Mathematics Education, 39(5–6), 395–403.
  • Clark-Carter, D. (2010). Quantitative psychological research: The complete student's companion. East Sussex: Psychology Press.
  • De Corte, E. (1990). Towards powerful learning environments for the acquisition of problem solving skills. European Journal of Psychology of Education, 5(1), 5–19.
  • De Corte, E., & Verschaffel, L. (2006). Mathematical thinking and learning. In K. A. Renninger, I. E. Sigel, W. Damon, & R. M. Lerner (Eds.), Handbook of child psychology: Child psychology in practice (pp. 103–152). Hoboken, NJ: Wiley & Sons.
  • Doorman, M., Drijvers, P., Dekker, T., Van Heuvel-Panhuizen, T., De lange, J., & Wijers, M. (2007). Problem solving as a challenge for mathematics education in the Netherlands. ZDM: The International Journal on Mathematics Learning, 39(5–6), 405–418.
  • Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg, & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 253–284). Mahwah. NJ: LEA.
  • Efklides, A., & Misailidi, P. (2011). Introduction: The present and the future in metacognition. In A. Efklides, & P. Misailidi (Eds.), Trends and prospects in metacognition research (pp. 1–18). New York, NY: Springer.
  • English, L. D., & Kirshner, D. (2015). Changing agendas in international research in mathematics education. In L. D. English, & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 3–18). New York, NY: Taylor and Francis.
  • English, L. D., Lesh, R., & Fennewald, T. (2008, July). Future directions and perspectives for problem solving research and curriculum development. Paper presented at the 11th International Congress on Mathematics Education, Monterrey.
  • Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive-developmental inquiry. American Psychologist, 34(10), 906–911.
  • Francisco, J. M. (2013). Learning in collaborative settings: Students building on each other's ideas to promote their mathematical understanding. Educational Studies in Mathematics: An International Journal, 82(3), 417–438.
  • Freitag, M. (1994). Reading and writing in the mathematics classroom. The Mathematics Educator, 8(1), 16–21.
  • Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176.
  • Garofalo, J., & Trinter, C. P. (2012). Tasks that make connections through representations. Mathematics Teacher, 106(4), 302–307.
  • Ge, X., & Land, S. M. (2003). Scaffolding students’ problem-solving processes in an ill-structured task using question prompts and peer interactions. Educational Technology Research and Development, 51(1), 21–38.
  • Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49(2), 193–223.
  • Gravemeijer, K. (2004). Local theories as means support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128.
  • Grawe, N. D. (2011.) Beyond math skills: Measuring quantitative reasoning in context. New Directions for Institutional Research, 149, 41–52.
  • Guberman, R., & Leikin, R. (2013). Interesting and difficult mathematical problems: Changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teacher Education, 16(1), 33–56.
  • Halpern, D. F. (1998). Teaching critical thinking for transfer across domains. American Psychologist, 53(4), 449–455.
  • Hamilton, E. (2007). What changes are needed in the kind of problem-solving situations where mathematical thinking is needed beyond school?. In R. Lesh, E. Hamilton, & J. Kaput (Eds.), Foundations for the future in mathematics education (pp. 1–6). Mahwah, NJ: LEA.
  • Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its development. Mathematical Thinking and Learning, 7(1), 27–50.
  • Hatano, G., & Inagaki, K. (1986). Two courses for expertise. In H. Stevenson, H. Azuma, & K. Hakuta (Eds.), Child development and education in Japan (pp. 263–272). New York, NY: Freeman.
  • Heinze, A. (2005). Differences in problem solving strategies of mathematically gifted and non-gifted elementary students. International Education Journal, 6(2), 175–183.
  • Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288.
  • Hughes, R., Monaghan, J., Shingadia, E., & Vaughan, S. (2006). Revisiting routine questions. Teaching Mathematics and its Applications: International Journal of the IMA, 25(2), 90–96.
  • Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38(5), 1008–1022.
  • Lampert, M. (1990). When the problem is not the question and the solution is not the answer. American Educational Research Journal, 27(1), 29–63.
  • Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modelling. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte, NC: Information Age Publishing.
  • Lester, F. K., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. A. Lesh, & H. M. Doerr (Eds.), Beyond constructivism: Models and modelling perspectives on mathematics problem solving, learning, and teaching (pp. 501–518). Mahwah, NJ: LEA.
  • Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32(2), 251–265.
  • Mamona-Downs, J. (2007). Synopsis of the activities of working group 14, CERME-5, on the theme of ‘advanced mathematical thinking’. In D. Pitta-Pantazi & G. Philippou (Eds.), European research in mathematics education V: Proceedings of the 5th Congress of the European Society for Research in Mathematics Education. Retrieved from http://www.mathematik.uni-dortmund.de/-erme/CERME5b
  • National Council of Teachers of Mathematics. (2009). Guiding principles for mathematics curriculum and assessment. Retrieved from http:www.nctm.org/standard/content.aspx?id=23273
  • National Research Council. (2000). How people learn: Brain, mind, experience, and school (Expanded ed.). Washington, DC: National Academic Press.
  • Pantziara, M., Gagatsis, A., & Elia, I. (2004). Using diagrams as tools for the solution of non-routine mathematical problems. Educational Studies in Mathematics, 72(1), 39–60.
  • Polya, G. (1954). How to solve it. Princeton, NJ: Princeton University Press.
  • Pugalee, D. K. (1998). Promoting mathematical learning through writing. Mathematics in School, 27(1), 20–22.
  • Pugalee, D. K. (2001). Writing, mathematics, and metacognition: Looking for connections through students’ work in mathematical problem solving. School Science and Mathematics, 101(5), 236–245.
  • Quinnell, L. (2010). Why are mathematical investigations important?. Australian Mathematics Teacher, 66(3), 35–40.
  • Santel-Parke, C., & Cai, J. (1997). Does the task truly measure what was intended?. Mathematics Teaching in the Middle School, 3(1), 74–82.
  • Schneider, W. (2008). The development of metacognitive knowledge in children and adolescents: Major trends and implications for education. Mind, Brain, and Education, 2(3), 114–121.
  • Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM: The International Journal on Mathematics Education, 42(2), 149–161.
  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 334–370). New York, NY: Macmillan.
  • Schoenfeld, A. H. (2013). Reflections on problem solving theory and practice. The Mathematics Enthusiast, 10(1–2), 9–34.
  • Schukajlow, S., Kolter, J., & Blum, W. (2015). Scaffolding mathematical modelling with a solution plan. ZDM: Mathematics Education. 47(7), 1241–1254.
  • Shield, M., & Galbraith, P. (1998). The analysis of student expository writing in mathematics. Educational Studies in Mathematics, 36(1), 29–52.
  • Sierpinska, A. (2004). Understanding in mathematics. London: The Falmer Press.
  • Stigler, J. W., & Hiebert, J. (1999). Understanding and improving classroom mathematics instruction: An overview of the TIMSS video study. In B. Jaworski & D. Phillips (Eds), Comparing standards internationally: Research and practice in mathematics and beyond (pp. 119–134). Oxford, UK: Symposium Books.
  • Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with real world connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157–194.
  • Stylianou, D. A., & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Mathematical Thinking and Learning, 6(4), 353–387.
  • Sullivan, P., Clarke, D., & Clarke, B. (Eds.). (2013). Teaching with tasks for effective mathematics learning. In A. Peter-Koop, & P. Wilson (Eds.), Mathematics teacher education (Vol. 9). New York, NY: Springer.
  • Taylor, J. A., & McDonald, C. (2007). Writing in groups as a tool for non-routine problem solving in first year university mathematics. International Journal of Mathematical Education in Science and Technology, 38(5), 639–655.
  • Van den Heuvel-Panhuizen, M. (2001). Realistic Mathematics Education in the Netherlands. In J. Anghileri (Ed.), Principles and practice in arithmetic teaching (pp. 49–63). Buckingham: Open University Press.
  • Van Velzen, J. H. (2012). Teaching metacognitive knowledge and developing expertise. Teachers and Teaching: Theory and Practice, 18(3), 365--380.
  • Van Velzen, J. H. (2015). Eleventh-grade high-school students' accounts of mathematical metacognitive knowledge: Explicitness and systematicity. International Journal of Science and Mathematics Education, 14(2), 319--333.
  • Van Velzen, J. H. (submitted a). Senior high-school students' global planning of difficult non-routine mathematics problems.
  • Van Velzen, J. H. (submitted b). Experienced mathematics teachers' expected student difficulties regarding difficult non-routine mathematics problems.
  • Ward, J. P. (2006). Experimental mathematics in the curriculum (part 1). Teaching Mathematics and its Applications: International Journal of the IMA, 25(4), 205–215.
  • Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.
  • Wilkerson-Jerde, M. H., & Wilensky, U. J. (2011). How do mathematicians learn math? Resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78(1), 21–43.
  • Wismath, S., Orr, D., & Zhong, M. (2014). Student perception of problem solving skills. Transformative Dialogues: Teaching & Learning Journal, 7(3), 1–18.