Abstract
Whilst Productive Failure (Kapur, Cognition and Instruction, 26:379–424, 2008, Instructional Science, 38:523–550, 2010; Kapur & Bielaczyc, Journal of the Learning Sciences 21:45–83, 2012) has been shown to be efficacious in deepening students’ learning of mathematics in Singapore classrooms, an action research has been undertaken to adapt such constructivist learning design for it to be useful to teachers in the context of the Advanced-Level mathematics instructional goals. Findings revealed the adapted learning design to be more effective in learning the targeted concept of finding volume of solid of revolution than the Lecture and Practice mode of instruction. Insights were also gleaned to inform further adjustments to the learning design.
This is a preview of subscription content, access via your institution.
Notes
- 1.
Informally speaking, a step function is a piecewise constant function that may have infinitely many pieces, but in this case, we are showing a step-function with two pieces.
References
Becker, J. P., & Shimada, S. (1997). The open-ended approach: a new proposal for teaching mathematics. Reston: National Council of Teachers of Mathematics.
Bruner, J. S. (1960). The process of education. Cambridge: Harvard University Press.
diSessa, A. A., Hammer, D., Sherin, B. L., & Kolpakowski, T. (1991). Inventing graphing: metarepresentational expertise in children. The Journal of Mathematical Behavior, 10(2), 117–160.
Fosnot, C. T. (Ed.). (1996). Constructivism: theory, perspective, and practice. New York: Teachers College Press.
Gibson, J. J., & Gibson, E. J. (1955). Perceptual learning: differentiation or enrichment. Psychological Review, 62, 32–41. https://doi.org/10.1037/h0048826.
Gick, M. L., & Paterson, K. (1992). Do contrasting examples facilitate schema acquisition and analogical transfer? Canadian Journal of Psychology, 46, 539–550. https://doi.org/10.1037/h0084333.
Kapur, M. (2008). Productive failure. Cognition and Instruction, 26(3), 379–424.
Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523–550.
Kapur, M. (2011). A further study of productive failure in mathematical problem solving: unpacking the design components. Instructional Science, 39, 561–579. https://doi.org/10.1007/s11251-010-9144-3.
Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45–83.
Kemmis, S. & Mctaggart, R. & Nixon, R.. (2014). The action research planner: doing critical participatory action research. https://doi.org/10.1007/978-981-4560-67-2.
Lee, N. H., Chua, L. C., Kapur, M., Lam, R. J., Zhu, Y., & Yeo, D. (2018). Translating productive failure in the Singapore A-level statistics curriculum. (NIE Research Brief Series No. 18-024). Singapore: National Institute of Education http://hdl.handle.net/10497/20736. Accessed 28 Sept 2020.
Lesh, R. A., & Doerr, H. M. (Eds.). (2003). Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching. Mahwah: Erlbaum.
Ministry of Education. (2019). Pre-university H2 mathematics syllabus. Singapore: Ministry of Education https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/2020-pre-university-h2-mathematics.pdf. Accessed 5 January 2020.
National Research Council. (1999). How people learn: bridging research and practice. Washington: The National Academies Press.
Noffke, S. E., & Somekh, B. (Eds.). (2009). The SAGE handbook of educational action research. London: SAGE.
Rittle-Johnson, B., & Star, J. R. (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. https://doi.org/10.1037/0022-0663.99.3.561.
Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101, 529–544. https://doi.org/10.1037/a0014224.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475–522. https://doi.org/10.1207/s1532690xci1604_4.
Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: the effects of telling first on learning and transfer. Journal of Educational Psychology, 103(4), 759–775.
Steffe, L. P., & Gale, J. (Eds.). (1995). Constructivism in education. Hillsdale: Earlbaum.
Tan, C. (2013). Learning from Shanghai: lessons on achieving educational success. New York: Springer.
Author information
Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendices
Appendix A1: ICC task 1
-
Section A (for students to discern the essential cross-sectional shape features that distinguished the different solids of revolution):
The 2-D planes on the right hand side when rotated about x- or y-axis will form a 3-D objects on the left hand side. Match each 3-D object on the left to its corresponding 2-D plane on the right. The first one has been done for you.
-
Section B (for students to discern the essential cross-sectional shape features that distinguished the different solids of revolution, regions are bounded by function curve and axes, to note angle of rotation is 2 π ):
Sketch the resulting complete 3-D solid to its corresponding shaded region when rotated about the given axis.
What is the angle of rotation to form a complete solid in Section B? Explain your answer.
-
Section C (for students to discern the essential cross-sectional shape features that distinguished the different solids of revolution, regions are bounded by function curve and axes, angle of rotation is π):
Sketch the resulting complete 3-D solid to its corresponding shaded region when rotated about the given axis.
Is there a difference in the angle of rotation to form a complete solid in Section B and Section C? Explain your answer.
-
Section D (for students to discern the essential cross-sectional shape features that distinguished the different solids of revolution, regions are bounded by two function curves, angle of rotation is 2π)
Sketch or describe the resulting complete solids of the required region when rotated about the given axis.
Appendix A2: ICC task 2
In your own group, come up with as many ways as possible to estimate/find the volume of each of the following abstract objects.
Consider the solid generated by rotating the region under the line y = b from x = 0 to x = a about the x-axis through 2π radians.
-
(i)
How would you describe this solid of revolution?
-
(ii)
How do you find/estimate the volume of this solid?
Consider the solid generated by rotating the region under y = f(x) from x = 0 to x = b about the x-axis through 2π radians, where
\( \mathrm{f}(x)=\left\{\begin{array}{cc}m,& 0\le x<a\\ {}n,& a\le x<b\end{array}\right. \)
y = f(x) is a step functionFootnote 1.
How would you describe this solid of revolution?
How do you find/estimate the volume of this solid?
Consider the solid generated by rotating the region under y = g(x) from x = 0 to x = b about the x-axis through 2π radians, where
\( \mathrm{g}(x)=\left\{\begin{array}{c}m,\kern0.75em 0\le x<a\ \\ {}n,\kern0.75em a\le x<b\\ {}p,\kern0.75em b\le x<c\\ {}q,\kern0.75em c\le x<d.\end{array}\right. \)
How would you describe this solid of revolution?
Find/estimate the volume of this solid.
Consider the solid generated by rotating the region under the curve y = f(x) from x = 0 to x = a about the x-axis through 2π radians.
How would you describe this solid of revolution?
How do you estimate/find the volume of this solid?
How then do you think these ideas translate to a general formula to calculate volume of such solids of revolution?
Post-instruction quiz questions
Rights and permissions
About this article
Cite this article
Tan, L.S., Bek, A.C.A. & Kok, Y.H. Constructivist Learning Design for Advanced-Level Mathematics in Singapore classrooms. Math Ed Res J (2021). https://doi.org/10.1007/s13394-020-00363-6
Received:
Revised:
Accepted:
Published:
Keywords
- Constructivist learning
- Contrasting cases
- Volume of solid of revolution
- Productive failure