Constructivist Learning Design for Advanced-Level Mathematics in Singapore classrooms

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.

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Notes

  1. 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.

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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.

figurea
  • 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.

figureb

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.

figurec

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.

figured

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.

figuree

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.

  1. (i)

    How would you describe this solid of revolution?

  2. (ii)

    How do you find/estimate the volume of this solid?

figuref

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?

figureg

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.

figureh

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

figurei

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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

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Keywords

  • Constructivist learning
  • Contrasting cases
  • Volume of solid of revolution
  • Productive failure