# Using Spatial Imagery in Geometric Reasoning

Michael T. Battista
Kent State University
Kent, OH 44242

Douglas H. Clements
State University of New York at Buffalo
Buffalo, NY 14260

Edited by Grayson H. Wheatley
Florida State University
Tallahassee, FL 32306-3032

Two second-grade students are investigating geometry with Logo. They are working in a TEACH environment in which the turtle-movement commands they enter are not only executed but simultaneously recorded in a procedure (Battista and Clements, in press; Clements 1983-84). During earlier lessons, they had successfully maneuvered the turtle to draw squares of various sizes. For each of these squares, they had used 90-degree turns. When asked how they knew whether a figure was a square, however, neither student had mentioned 90-degree turns.

## Classroom Episodes

These two students are then individually asked to use the turtle to draw a tilted square (they are shown a picture of a square whose sides are not vertical or horizontal). Each begins by turning the turtle and making it go forward. Megan uses a trial-and-error approach to find the turns, turning the turtle by increments until the heading looks right. She then makes the turtle go forward and examines the results, erasing and reissuing the turn command if necessary. As she completes her figure, she announces that it is a square, even though her turns were not of 90 degrees and though she has used forward inputs of 30, 5, and 5 for the last side instead of 30 as for the first three sides. (She had said earlier that squares have all their sides equal.)

 Teacher: How do you know it's a square for sure? Megan: It's in a tilt. But it's a square because if you turned it this way it would be a square.

Megan's reasoning is clearly at the visual level in the van Hiele hierarchy (Battista and Clements 1988; Clements and Battista, in press). She thinks of shapes as visual wholes rather than as geometric objects possessing properties. According to van Hiele, for students at this level, "There is no way, one just sees it" (van Hiele 1986, 83). As her remarks indicate, however, for Megan there is a "way." This student has reasoned that the figure in front of her is a square by using spatial imagery. That is, she has observed that squares that are tilted often do not look quite like squares. So perhaps because her intention had been to draw a square, she visualizes that this figure can be turned to look like a square in standard orientation. She does so even though the figure drawn does not have all the properties that she previously named for squares. The students' next task is to use the turtle to draw a tilted rectangle. Coletta, who has discovered that she needs 90-degree turns to draw a square, uses 90s on her first attempt at making the tilted rectangle, reasoning as follows:

 Colletta: Because a rectangle is just like a square but just longer, and all the sides are straight. Well, not straight, but not tilted like that (makes an acute angle with her hands). They're all like that (shows a right angle with her hands), and so are the squares. Teacher: And that's 90 (showing hands put together at a 90 degree angle)? Coletta: Yes.

In other discussions, she has also stated that a square is a rectangle.

Teacher: Does that make sense to you?
Coletta: It wouldn't to my [4-year-old] sister, but it sort of does to me.
Teacher: How would you explain it to her?
Coletta: We have these stretchy square bathroom things. And I'd tell her to stretch it out and it would be a rectangle.

Here again, we see how a student uses visual imagery to reason in a geometric context. First, she reasons that rectangles have 90-degree turns because of their visual similarity to squares. Second, it "sort of made sense" that a square is a rectangle because a square could be stretched into a rectangle. This response may be more sophisticated than one might initially think, for Coletta has just demonstrated her knowledge that squares and rectangles are similar in having angles made by 90-degree turns. Thus, she may have understood at an intuitive level that all rectangles could be generated from one another by certain legal transformations, ones that preserve 90-degree angles. That is, her conceptual knowledge may have influenced her imagery. Contrast Coletta's use of imagery with that of Megan, whose visual transformation contrasted with her verbal statement. We have additional evidence that Coletta's thinking about shapes was based more on properties than Megan's. So Coletta's imagery was more likely to be influenced by property-based considerations.

Another example of visual reasoning occurs as fifth grader works with Logo Geometry (Battista and Clements, in press). Ryan is deciding whether he can draw various figures with a rectangle procedure that takes the lengths of the sides as input. He has successfully used the procedure to draw a tilted rectangle (labeled 4 in fig. 1) and is now reflecting on his unsuccessful attempt to make a nonrectangular parallelogram (labeled 7).

Teacher: Could you use different inputs, or is it just impossible?
Ryan: Maybe if you used different inputs. (Ryan types in a new initial turn. He stares at the picture of the parallelogram on the activity sheet.) No, you can't. Because the lines are slanted, instead of a rectangle going like that. (He traces a rectangle over the parallelogram.)
Teacher: Yes, but this one's slanted (indicating the tilted rectangle, labeled 4, that Ryan had successfully drawn with the Logo procedure).
Ryan: Yeah, but the lines are slanted (meaning nonperpendicular). This one's still in the size (shape) of a rectangle. This one (parallelogram) -- the thing's slanted. This thing (rectangle) isn't slanted. It looks slanted, but if you put it back (shows a turn by gesturing, meaning to turn it so that the sides are vertical and horizontal), it wouldn't be slanted. Any way you move this (the parallelogram), it wouldn't be a rectangle (shaking his head). So there's no way.

Ryan has used three of the four classes of image processes defined by Kosslyn -- generating an image, inspecting an image, and transforming an image (1983). First, after making the initial turn and trying to choose the inputs, he recognizes that the relationship between adjacent sides is not consistent with the implicit definition of a rectangle in the Logo procedure. He then generates an image of a rectangle that is consistent with this definition, tracing it on the activity sheet. Second, he inspects this image and compares it with the parallelogram, noting the differences.

Third, to elaborate the difference in the "slantiness" of the figures, Ryan mentally transforms his images of the rectangle and the parallelogram. His assertion that only the rectangle could be rotated to look like a rectangle is the capstone of his argument. His emerging knowledge of the properties of figures is supported by his visual reasoning.

In sum, we see a reciprocal effect. Work with Logo involving thinking about properties of figures has refined Ryan's visual reasoning. In turn, his visual reasoning has supported his analysis of the properties of figures.

Finally, an episode with a kindergarten student illustrates another natural role that imagery plays in geometric problem solving. John is asked to use Logo to draw an open path that has three bends in it. He draws a "box" in which the beginning and ending points do not touch. He explains that he "just thought of all the paths we made. One of the paths had the right number." John visually reviews solutions to similar problems, analyzing each to see if it meets the constraints for the new problem.

These episodes are consistent with the hypothesis (Battista 1990) that spatial visualization may be a more important factor in geometric problem solving for students who are at the visual, rather than higher, levels of geometric thinking. That is not to say, however, that visual imagery is unimportant at higher levels of thought. Many students that we have classified at the descriptive-analytic level (the level immediately following the visual level) are inhibited in their reasoning and problem solving because their spatial imagery is not coordinated with their knowledge of properties. For instance, many of these students know the properties of a rectangle but have difficulty recognizing whether a figure in a nonstandard spatial orientation (such as shape 7 in fig. 1) possesses these properties. Moreover, because students at all levels often reason by considering specific images, their reasoning can go astray if they make a mistake in generating, inspecting, or transforming those images. In sum, visual imagery -- when properly developed -- can make a substantial contribution at all levels of geometric thinking. Thus, in teaching geometry, we should not focus solely on properties of figures and relationships among them. We should also help students develop vivid images and coordinate these images with their conceptual knowledge.

## Instruction Implications

The foregoing discussion suggest two recommendations for teaching:

1. Recognize that a great many students use visual imagery to reason. Respect this mode of thinking. Help students use it to analyze and convince themselves of the truth of geometric ideas.

2. Discuss visual reasoning with students. Ask questions that might help students incorporate conceptual knowledge into their visual-reasoning processes. For instance, after Coletta gave her bathroom tile justification that squares are rectangles, the teacher might have drawn a trapezoid and asked, "I think I can stretch this figure into a rectangle. Is this geometric figure a rectangle?" This question would have prompted Coletta to clarify the nature of her "stretch" transformation. Also, when discussing a concept such as that of a triangle, make sure to use a variety of different examples, such as many different sizes and shapes of triangle in nonstandard orientations. This approach will help students evoke dynamic images when reasoning about concepts. When students have become familiar with mathematical transformations, they can be encouraged to describe their mental transformations using mathematical terms.

Many of the tasks that we have presented to students in our research and curriculum-development projects can promote students' coordination of visual imagery and conceptual knowledge. Some examples follow. (In our project, students in grades K-1 used a single-key version of Logo, students in grades 2-4 used TEACH, and students in grades 5-6 used an enhanced version of regular Logo [Battista and Clements, in press].)

Activity 1. Ask students to draw a square in Logo. Then ask them to draw a tilted square. Do the same for rectangles. (Grades 1-4)

Activity 2. Give students a procedure RECT that draws a rectangle when the two dimensions for the rectangle are given as input. Ask them which figures (as in fig. 1) they can draw with RECT. (Grades 2-6)

Activity 3. Show students the three figures in figure 2, two at a time. Ask how figure A is different from figure B. (The majority of students will give a global response, such as, "It's bigger or longer," not referring specifically to side lengths.) Give the students a Logo procedure that draws figure A and let them test it on a computer. Then ask them how they can alter their procedure to draw figure B. Repeat with figures B and C. (Grades 2-6)

Note: If you do not have access to a computer, you can still use Logo-like commands. Have students work in pairs, one reading the commands and one acting as the turtle, following the commands with a metric ruler (FORWARD 50 means forward 50mm) and a protractor. Students take turns acting as the turtle on paper.

### References

Battista, Michael T. "Spatial Visualization and Gender Differences in High School Geometry." Journal for Research in Mathematics Education 21 (January 1990):47-60.

Battista, Michael T. and Douglas H. Clements. "A Case for a Logo-based Elementary School Geometry Curriculum." Arithmetic Teacher 36 (November 1988):11-17.

-----. Logo Geometry. Morristown, N.J.: Silver Burdett & Ginn. In Press.

Clements, Douglas H. "Supporting Young Children's Logo Programming." Computing Teacher 11 (December 1983 - January 1984):24-30.

Clements, Douglas H. and Michael T. Battista. "Geometry and Spatial Reasoning." In Handbook of Research on Mathematics Teaching, edited by D. A. Grouws. Reston, VA: National Council of Teachers of Mathematics and Macmillan. In press.

Kosslyn, Stephen M. Ghosts in the Mind's Machine. New York: W. W. Norton & Co., 1983.

van Hiele, Pierre M. Structure and Insight. Orlando, Fla.: Academic Press, 1986.