Advertisement

Mathematics Education Research Journal

, Volume 26, Issue 2, pp 353–376 | Cite as

Facilitating and direct guidance in student-centered classrooms: addressing “lines or pieces” difficulty

  • Meixia Ding
  • Xiaobao Li
Original Article
First Online:

Abstract

This study explores, from both constructivist and cognitive perspectives, teacher guidance in student-centered classrooms when addressing a common learning difficulty with equivalent fractions—lines or pieces—based on number line models. Findings from three contrasting cases reveal differences in teachers’ facilitating and direct guidance in terms of anticipating and responding to student difficulties, which leads to differences in students’ exploration opportunity and quality. These findings demonstrate the plausibility and benefit of integrating facilitating and direct guidance in student-centered classrooms. Findings also suggest two key components of effective teacher guidance including (a) using pretraining through worked examples and (b) focusing on the relevant information and explanations of concepts. Implementations are discussed.

Keywords

Teacher guidance Facilitating guidance Direct guidance Learning difficulty Equivalent fractions Student-centered classroom 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We wish to thank Kelley Marshall for helpful assistance. Special thanks to Dr. Xiaobao Li at Widener University for serving as the second coder in this study.

References

  1. Anderson, J. R., Reder, L. M., & Simon, H. A. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review, Summer, 29–49.Google Scholar
  2. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389–407.CrossRefGoogle Scholar
  3. 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. PME: Prague.Google Scholar
  4. Bright, W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19, 215–232.CrossRefGoogle Scholar
  5. Bruner, J. S. (1961). The art of discovery. Harvard Educational Review, 31, 21–32.Google Scholar
  6. Cai, J. (2003). What research tells us about teaching mathematics through problem solving. In F. Lester (Ed.), Research and issues in teaching mathematics through problem solving (pp. 241–254). Reston: National Council of Teachers of Mathematics.Google Scholar
  7. Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97, 3–20.CrossRefGoogle Scholar
  8. Chapin, S., O’Connor, M. C., & Anderson, N. (2009). Classroom discussions: Using math talk to help students learn, Grades K-6 (2nd ed.). Sausalito: Math Solutions.Google Scholar
  9. Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64, 293–316.CrossRefGoogle Scholar
  10. Chazan, D., & Ball, D. L. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19(2), 2–10.Google Scholar
  11. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20.CrossRefGoogle Scholar
  12. Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., et al. (1991). Assessment of a problem-centered second grade mathematics project. Journal for Research in Mathematics Education, 22, 3–29.CrossRefGoogle Scholar
  13. Dean, D., & Kuhn, D. (2006). Direct instruction vs. discovery: The long view. Science Education, 91, 384–397.CrossRefGoogle Scholar
  14. Ding, M. (2007). Knowing mathematics for teaching: Case studies of teachers’ responses to students’ errors and difficulties in teaching equivalent fractions. Unpublished dissertation. Texas A&M University. College Station, TX.Google Scholar
  15. 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
  16. Ding, M., Li, X., Capraro, M. M., & Kulm, G. (2011). A case study of teacher responses to a doubling error and difficulty in learning equivalent fractions. Investigations in Mathematics Learning, 4(2), 42–73.Google Scholar
  17. Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17, 283–342.CrossRefGoogle Scholar
  18. Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth: Heinemann.Google Scholar
  19. Forman, E., & Ansell, E. (2001). The multiple voices of a mathematics classroom community. Educational Studies in Mathematics, 46, 115–142.CrossRefGoogle Scholar
  20. Franke, M. L., & Kazemi, E. (2001). Learning to teach mathematics: Developing a focus on students’ mathematical thinking. Theory into Practice, 40, 102–109.CrossRefGoogle Scholar
  21. Franke, M. L., Kazemi, E., & Battey, D. (2007). Understanding teaching and classroom practice in mathematics. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 225–256). Greenwich: Information Age.Google Scholar
  22. Gagné, R. M., & Paradise, N. E. (1961). Abilities and learning sets in knowledge acquisition. Psychological Monographs, 75, 518.CrossRefGoogle Scholar
  23. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30, 393–425.CrossRefGoogle Scholar
  24. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., et al. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12–21.CrossRefGoogle Scholar
  25. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth: Heinemann.Google Scholar
  26. 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 Psychologists, 41(2), 75–86.CrossRefGoogle Scholar
  27. Klahr, D., & Nigam, M. (2004). The equivalence of learning paths in early science instruction: Effects of direct instruction and discovery learning. Psychological Sciences, 15, 661–667.CrossRefGoogle Scholar
  28. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (Eds.). (1998). Connected mathematics: Bits and pieces 1. Understanding rational numbers. Teacher’s Edition. Menlo Park: Dale Seymour.Google Scholar
  29. Leinhardt, G., & Smith, D. A. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 99, 247–271.CrossRefGoogle Scholar
  30. Mayer, R. E. (2005). Principles for managing essential processing in multimedia learning: segmenting, pretraining, and modality principles. The Cambridge handbook of multimedia learning (pp. 169–182). New York: Cambridge University Press.Google Scholar
  31. Meloth, M. S., & Deering, P. D. (1999). The role of the teacher in promoting cognitive processing during collaborative learning. In A. M. O’Donnell & A. King (Eds.), Cognitive perspectives on peer learning (pp. 235–255). Mahwah: Erlbaum.Google Scholar
  32. Mitchell, A., & Horne, M. (2008). Fraction number line tasks and the additivity concept of length measurement. In M. Goos, R. Brown, & K. Makar (Eds.), Proceedings of the 31st annual conference of the mathematics education research group of Australasia (pp. 353–360). Adelaide: Merga.Google Scholar
  33. Moreno, R. (2004). Decreasing cognitive load in novice students: Effects of explanatory versus corrective feedback in discovery-based multimedia. Instructional Science, 32, 99–113.CrossRefGoogle Scholar
  34. Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30, 122–147.CrossRefGoogle Scholar
  35. Ni, Y. (2000). How valid is it to use number line to measure children’s conceptual knowledge about rational number? Educational Psychology, 20, 139–152.CrossRefGoogle Scholar
  36. O’Connor, M. C., & Michaels, S. (1993). Aligning academic task and participation status through revoicing: Analysis of a classroom discourse strategy. Anthropology & Education Quarterly, 24, 318–335.CrossRefGoogle Scholar
  37. Piaget, J. (1978). Success and understanding. Cambridge: Harvard University Press.Google Scholar
  38. Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of Mathematics. Educational Psychologist, 47, 189–203.CrossRefGoogle Scholar
  39. Santagata, R. (2004). “Are you joking or are you sleeping?” Cultural beliefs and practices in Italian and U.S. teachers’ mistake-handling strategies. Linguistics and Education, 15, 141–164.CrossRefGoogle Scholar
  40. Sfard, A. (2001). There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning. Educational Studies in Mathematics, 46, 13–57.CrossRefGoogle Scholar
  41. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Helping teachers learn to better incorporate student thinking. Mathematical Thinking and Learning, 10, 313–340.CrossRefGoogle Scholar
  42. 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
  43. Sweller, J. (2005). Implications of cognitive load theory for multimedia learning. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 19–30). New York: Cambridge University Press.CrossRefGoogle Scholar
  44. Sweller, J. (2006). The worked example effect and human cognition. Learning and Instruction, 16, 165–169.CrossRefGoogle Scholar
  45. Sweller, J., van Merriënboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251–296.CrossRefGoogle Scholar
  46. Webb, N. M., Franke, M. L., De, T., Chan, A. G., Freund, D., Shein, P., et al. (2009). ‘Explain to your partner’: teachers’ instructional practices and students’ dialogue in small groups. Cambridge Journal of Education, 39, 49–70.CrossRefGoogle Scholar
  47. Wu, H. (2009). What’s sophisticated about elementary mathematics? American Educator, 33(3), 4–14.Google Scholar
  48. Zack, V., & Graves, B. (2001). Making mathematical meaning through dialogue: “Once you think of it, the Z minus three seems pretty weird. Educational Studies in Mathematics, 46, 229–271.CrossRefGoogle Scholar

Copyright information

© Mathematics Education Research Group of Australasia, Inc. 2014

Authors and Affiliations

  1. 1.Department of Teaching and LearningTemple UniversityPhiladelphiaUSA
  2. 2.Center for EducationWidener UniversityChesterUSA

Personalised recommendations

Cite article

Buy options