Articles from NCTM Journals

NCTM publishes a variety of journals whose content addresses a multitude of mathematics educational concerns, topics, and trends. Below is a list of some of the articles, and their foci, which specifically addresses three-dimensional geometry, particularly in the middle and high school levels.

Creating Three-Dimensional Scenes

by Norm Krumpe, edited by Hollylynne Stohl and Suzanne R. Harper

Focus: From the “Technology Tips” section of the Mathematics Teacher, discusses a free program that teachers and students can use to create three-dimensional scenes. Author likens the POV-Ray language necessary to creating the scenes to that of Mathematica or Maple.

“I have used the program to help teach such concepts as transformational geometry; analytic geometry; and calculus (particularly volume of solids using shells and washers).”

Mathematics Teacher, Feb. 2005, p. 394

Developing Spatial Sense: Comparing Appearance with Reality

by Gwen Kelly, Tim Ewers, and Lanna Proctor

Focus: Activities to help students gain a working understanding of perspective.

“In our work with students, we have discovered that their background experiences with perspective have been insufficient. Their perception of the appearance of a block construction when pictured on a two-dimensional plane was often incorrect.” (

Mathematics Teacher, Dec. 2002, p. 702

Developing Spatial Understanding Through Building Polyhedrons

by Rebecca C. Ambrose and Karen Falkner

Focus: Article discusses children’s spatial sense as they build polyhedrons. As a result of building their own unique polyhedrons, some students were able to mentally dissect their polyhedron into smaller three-dimensional objects. For one activity, the students described their unique polyhedron and a few days later reconstruct their polyhedron using only their descriptions and memory. For another activity, the students built a “secret” polyhedron the teacher contained hidden in a bag. The students had to ask questions about the features/characteristics of the polyhedron.

“We were concerned that our students were not given opportunities to develop their spatial abilities. Developing spatial thinking has been neglected in school environments to the point that students who have exceptional ability in this domain reported disenchantment with school and were far less likely to pursue advanced academic degrees than their peers who have comparable abilities in other domains (Gohm, Humphreys, and Yao 1998).”

Teaching Children Mathematics, Apr. 2002, p. 442

Elastic Geometry and Storyknifing: A Yup’ik Eskimo Example

by Jerry Lipka, Sandra Wildfeuer, Nastasia Wahlberg, Mary George, and Dafna R. Ezran

Focus: Activities to help students get an introductory understanding of elastic geometry.

“In elastic geometry, we look at those properties of a shape that remain the same under conditions of stretching, pulling, or shrinking . . . Moving symbols from one perspective to another or creating two-dimensional symbols of three-dimensional objects encourages students to perform and imagine spatial transformations of objects.”

Teaching Children Mathematics, Feb. 2001, p. 340

From Tessellations to Polyhedra: Big Polyhedra

by Blake E. Peterson

Focus: Activities to help students understand the connections between tessellations to polyhedra presented in the context of needing to grass an indoor stadium for a game of soccer.

“The underlying theme in all these activities and explorations is the measure of the interior angles of regular polygons. By not having a specific algorithm for computing these angle measures, students are motivated to ask, ‘What are the angle measures?’ and ‘Why are they important?’ . . . From these connections, students gain mathematical understanding, which is, after all, our goal.”

Mathematics Teaching in the Middle School, Feb. 2000, p. 356

Hands-On Perspective

by Marc Frantz, Annalisa Crannell, Dan Maki, and Ted Hodgson

Focus: Activities aimed to help students master the basic concepts of perspective art.

“Some students are convinced they cannot draw three-dimensional solids at all, much less in perspective . . . The algebra underlying perspective art arises from the use of three-dimensional coordinates. Middle school and high school students routinely consider three-dimensional solids, yet the concept of three-dimensional coordinates is rarely taught.”

Mathematics Teacher, Apr. 2006, p. 554

Painting the Pyramid

by Christopher M. Kribs-Zaleta

Focus: Activity that extends the classic painting the cube activity.

“Painting the Pyramid, like Painting the Cube, brings together ideas from many different parts of mathematics. The geometric context provides opportunities for modeling and data gathering to identify patterns and the measurement notions by which to interpret them. The values of triangular and tetrahedral numbers are classic basic combinatorial problems, and indeed appear as diagonals of Pascal’s triangle. The patterns in formulae, especially the cubic ones, provide a nice illustration of the notion of translation, as we can see the shift of the expression for the nth tetrahedral number by 1, 4, and 5, both in the data (and models) and in the arguments of the expression.”

Mathematics Teacher, Nov. 2006, p. 281

Rotations of the Regular Polyhedra

by MaryClara Jones and Hortensia Soto-Johnson

Focus: A discussion on three-dimensional transformations.

“In high school, students use The Geometer’s Sketchpad to further their understanding of algebraic and geometric transformations. While students are exposed to two-dimensional transformations, i.e., mappings from R^2 to R^2, they rarely learn about three-dimensional transformations, which are mappings from R^3 to R^3. In this article we discuss the rotational symmetries of the regular polyhedra, which can be incorporated into a high school geometry classroom where students will have an opportunity to further their knowledge about two-dimensional rotations through the study of three-dimensional rotations.”

Mathematics Teacher, May 2006, p. 606

Three by Three Systems: More than Just a Point

by Joseph Ordinans

Focus: Activities to visually illustrate solving a three by three system.

“Most algebra 2 textbooks have considerable material for teaching consistent, independent 3 x 3 systems . . . Some textbooks briefly discuss a few of the other seven possibilities, but an in-depth look at these other possibilities leads to algebra that requires a considerable amount of insight and gives students a different look at the possibilities of a 3 x 3 system. Students are required to make a connection between geometric figures and the algebraic equations that generate those figures.”

Mathematics Teacher, Feb. 2006, p. 419

Volumes and Cube Dissections

by Ioana Mihaila

Focus: Discussion propelled by an intriguing problem involving ratios of volumes of platonic solids.

“Of course, the answer is not difficult to computer, but I kept thinking that some way should exist to decompose the pictures and find the answer without any ‘work’ . . . And the moral of the story is that a picture is worth a thousand words, or in this case, a half-page of algebra.”

Mathematics Teacher, Dec. 2002, p. 724

What Happens to Geometry on a Sphere?

by Janet M. Sharp and Corrine Heimer

Focus: Activity focused on students using plastic balls to learn about spherical geometry and the role of definitions in mathematics.

“We knew that the students would have to be able to visualize lines, segments, angles, and polygons in a plane and easily identify properties and definitions of these concepts before moving their thinking onto the sphere . . . We strongly encourage teachers to use the plastic balls in this discussion because, as our student, Alicia, said, ‘It was easier to learn about spheres’ [figures] on a ball than on paper.’”

Mathematics Teaching in the Middle School, Dec. 2002, p. 182

Why Not Three Dimensions?

by Steve Lege

Focus: Activities for higher level high school mathematics students to enhance their visualization of three dimensions. One activity required students to build an elliptical template for a variety of stovepipes through roofs with various pitches on a scaled-down measurement. Another activity required students to build models of quadric surfaces. While another had students build models of a variety of solids of revolution.

“When I reviewed our high school curriculum, I realized that my students did not have enough practice with situations that required analysis in three dimensions. Few situations that applied the three-dimensional ideas studied in geometry were found in second-year algebra and precalculus, and calculus students faced with rotations that produced solids had very little experience to help them with the visualizations required . . . Several students in this class confided to me that after they had actually made one model that demonstrated a solid formed by revolution, they found it much easier to visualize other problems, even those where the type of revolution was quite different . . . Students’ skill in working with three-dimensional situations can be enhanced through hands-on model building. We hope that all students develop these skills, yet the skills are often ignored orunderutilized by the current textbook curriculum.”

Mathematics Teacher, Oct. 1999, p. 560

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