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I notice that geometry students frequently have difficulty with representations of 3-dimensional objects in 2 dimensions. Today, we worked with physical manipulatives in order to help visualize where right triangles can occur in 3 dimensions in both pyramids and rectangular prisms (the focus is on fluency with the Pythagorean Theorem and noting its application in many contexts.) I chose to create physical manipulatives instead of finding online 3d manipulatives because I felt as if the physical manipulatives would provide more insight than simply seeing a draggable, yet still 2d, projection.

For clarity:

Physical manipulative in conjunction with 2d drawing: Photo of maipulative

Some online examples of "virtual manipulatives": 1 2

My question is: Is there research supporting my intuition? Are students who have difficulty translating between 2D and 3D more benefited by a physical model than a virtual manipulative, or the other way around?

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    $\begingroup$ My own experience is that weaker student benefit greatly from a physical model and get lost with the 2d virtual manipulatives. Note that I taught gifted students but some of them were weak in visualization and only understood after handling the 3D model. $\endgroup$ – Amy B Mar 9 '16 at 13:14
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This is not what you seek, because it compares two different physical manipulatives, rather than physical vs. virtual. But I find it interesting partly because my own research involves studying nets of polyhedra.

Scott, Jacqui, Anton Selvaratnam, and Lynden Rogers. "Using Bendable and Rigid Manipulatives in Primary Mathematics: Is One More Effective Than the Other in Conceptualising 3D Objects from Their 2D Nets?." TEACH Journal of Christian Education 6.1 (2012): 10. (Article link)


          Manipulatives


Abstract excerpts. The purpose of this study was to compare the effectiveness of two different types of manipulatives, bendable and rigid, as aids for the conceptualisation of 3D solids from 2D nets...Contrary to initial expectations, the bendable nets, although more attractive to pupils, did not prove superior to the rigid variety. In fact, the most noticeable advances in conceptualisation followed teaching experiences using the rigid nets.

They cite an article I couldn't find:

Shaw, J. M. "Manipulatives enhance the learning of mathematics." Houghton Mifflin Mathematics. (2002).

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Here's a piece comparing virtual manipulatives to traditional teaching without manipulatives, in the context of community college remedial courses:

Violeta Menil and Eric Fuchs, "Teaching Pre-Algebra and Algebra Concepts to Community College Students through the Use of Virtual Manipulatives", Improving Undergraduate Mathematics Learning, CUNY Office of Academic Affairs, 2012. (Link)

Statistically significant improvements were found regarding exam performance and student attitude for prealgebra/arithmetic classes. None are found for either in elementary algebra classes.

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