experienced. But how do MTEs know if their work meets these educational goals? This article describes a drawing task designed to elicit the ways preservice teachers (PSTs) envision the teaching and learning of mathematics ( Lotan, Lythcott, & Hammerness, 2003

### John W. Bradshaw

The College Entrance Examination Board in its Document No. 108, containing a definition of the requirements in geometry adopted by the board on April 21, 1923, has included an appendix on drawing which opens with the following paragraphs:

“An important aid to the visualization of space figures is the ability to draw these figures on paper. The pupil should be trained to make neat free-hand drawings, inserting whatever construction lines are needed and supplementing the representation of the space figure by independent drawings of plane sections, whenever such contribute to clearness. Ruler and compasses may often be used with advantage, but no technical knowledge of descriptive geometry or the niceties of mechanical drawing forms any part of the requirement. The use of ruler and compasses in the examination is permitted and to a moderate degree, desirable but it is not prescribed.

### John W. Bradshaw

In earlier parts of this discussion^{1} we have been concerned with the picturing in oblique parallel projection of certain simple solids bounded by plane surfaces. In this part we shall continue the study with a consideration of the simplest curved surfaces, the right circular cylinder and the sphere. The Appendix on “Drawing” (pages 35- 39) of Document No. 108 of the College Entrance Examination Board on Geometry furnishes the outline; the following quotations are taken from this Document. References to figures are to the figures in that document, some of which are reproduced in this article.

### Margaret Joseph

Every teacher of solid geometry has no doubt experienced having a student fail to prove an original exercise just because his drawing of the figure required was so much out of proportion that he failed to establish the proper relationship between lines, angles, triangles, or planes necessary in the proof. If, for example, a student would draw a square for the base of an oblique prism or a circle for the base of a cone, instead of the parallelogram and ellipse as they appear in perspective, then the whole figure is distorted. Just recently I had a student come to me for help in deriving the formula for the altitude of a regular tetrahedron in terms of its edge *e*. As soon as I saw the rough sketch he was using, where the faces were scalene triangles, I knew why he had two unknowns in the equation that he was trying to solve. The altitude did not meet the base of the figure at its center. I made a better drawing for him and he immediately recognized his error and proceeded with the solution without further help from me.

### William F. Finzer and Dan S. Bennett

Students have many reasons for making a sketch with The Geometer“s Sketchpad. Their purpose may be to explore the behavior of a particular geometric figure, such as a rhombus, or to model a New users must understand the difference between a drawing and a construction physical situation, such as a ladder leaning against a wall. They may want to make a beautiful pattern inspired by Navajo rug designs, or their goal may be an animation—perhaps a Ferris wheel or a merry-go-round. No matter what the purpose, they want the finished sketch to behave well when they drag basic elements around in it. If investigating properties of rhombuses, they want the figure to remain a rhombus while they drag to explore its different sizes and shapes. Students want to end up with a *construction*, not merely a *drawing*.

### Sandra M. Crespo and Andreas O. Kyriakides

Drawing a picture is a problem-solving strategy widely encouraged by elementary mathematics textbooks and teachers. Indeed, drawing can be a powerful way of engaging many students, especially young ones, in representing and communicating their mathematical ideas. Children develop the ability to draw long before they learn to write, and the act and product of drawing are accessible to children of diverse cognitive, academic, cultural, and language backgrounds (children with visual impairments are an obvious exception). The power of drawing as a problem-solving strategy can be observed as young children draw solutions to problems that involve mathematical concepts beyond the level of the mathematics they have studied. The shortcoming of drawing as a problemsolving strategy is that some students favor drawing even when that strategy is the least efficient or viable for finding a solution. Although there are good reasons for asking students to draw when solving a mathematical problem, teachers must also consider what they themselves know and do not know about children's drawings and what sense they make of such representations.

### Keiichi Shigematsu and Larry Sowder

The problem-solving strategy “Make a drawing” is used in both the United States and Japan but with different degrees and varying amounts of success in each country. The purposes of this article are to compare what seem to be Japanese and United States practices in promoting the use of drawings in solving story problems and to use that comparison to suggest some teaching approaches.

### Janet A. Kelly

While working with third-, fourth-, and fifth-grade teachers in a National Science Foundation–sponsored project designed to enhance the mathematics and science teaching of in-service elementary teachers, we recognized that teaching mathematics problem solving was one of their greatest challenges. Discussions with the teachers revealed that most were using an algorithmic approach to problem solving with an emphasis on facts, rules, and procedures. Their students were being taught to solve word problems in a systematic, single-mode manner. We found that the teachers were most comfortable with the algorithmic approach because that is how they were taught mathematics when they were in school. As one teacher commented, “I was stunned to find out that not everyone worked math problems the same way.”

### Beverly Gimmestad Baartmans and Sheryl A. Sorby

Students should study the “geometry of one, two, and three dimensions” and through such study will develop the spatial skills needed in a variety of careers (NCTM 1989, 112). As instructors of geometry, we need to be able to demonstrate to our students the uses of mathematics and its connections to other disciplines. The purpose of this article is to demonstrate the kinds of spatial skills needed by engineers for their work and to suggest activities for the geometry classroom that we have used to help build these skills.