Introduction to Geometry
Mhairi (Vi) Maeers
Faculty of Education
University of Regina

As I think about geometry I cannot but help consider the following question:

 Is geometry a 'man'-made mathematical invention to describe the way the world is organized or is the world in itself geometrical?

As one's view of the world changes (from a 'flatland' two-dimensional space, to a parallel view, to a view of bent 'lines' -- e. g., light waves as opposed to straight lines) so does one's way of interpreting the world geometrically (i. e., from a Euclidean perspective to a non-Euclidean perspective). All present-day geometry can do for us is to help us in 1997 describe geometrically the shapes around us. New knowledge about outer space, the underwater world, etc. is changing our view of 'earth' and, consequently, our geometrical thinking.
The NCTM (1989) Curriculum and Evaluation Standards promote the development of spatial sense. The Standards state that the K-4 math curriculum should include two- and three-dimensional geometry so that students can:
 describe, model, draw, and classify shapes investigate and predict the results of combining, subdividing, and changing shapes develop spatial sense relate geometric ideas to number and measurement ideas recognize and appreciate geometry in their world

Similarly in grades 5-8, it is stated that the mathematics curriculum should include the study of the geometry of one, two, and three dimensions in a variety of situations so that students can:
 identify, describe, compare, and classify geometric figures visualize and represent geometric figures with special attention to developing spatial sense explore transformations of geometric figures represent and solve problems using geometric models understand and apply geometric properties and relationships develop an appreciation of geometry as a means of describing the physical world

Finally at the secondary level we are reminded that High school geometry should build on the strong conceptual foundation students develop in the new K-8 programs. Students should have opportunities to visualize and work with three dimensional figures in order to develop spatial skills fundamental to everyday life and to many careers. Physical models and other real-world objects should be used to provide a strong base for the development of students' geometric intuition so that they can draw on these experiences in their work with abstract ideas.
Geometry components dealt with at the elementary level include:
 A. 3-dimensional geometry--the geometry of solid objects such as cones, rectangular prisms, etc. B. 2-dimensional geometry--the geometry of the plane (flatland geometry)--topics like angles, rays, lines; common shapes such as squares, rectangles etc. C. Motion or transformational geometry--the geometry of motion, such as translations (slides), reflections (flips), rotations (turns). A transformation of a plane is any one-to-one correspondence between a plane and itself. A translation is a transformation of a plane that moves every point of the plane a specified distance in a specific direction along a straight line. Any transformation that preserves distance is called an isometry or a Îrigid motion.â Thus a translation is an isometry.

In addition to A, B, and C the study of geometry is about developing spatial sense, that is the ability to mentally picture objects and to maintain accurate perception of the objects under different orientations (Owens, 1990, cited in Cathcart, Pothier & Vance, 1997, p. 171). In elementary school, motion geometry introduces children to the concepts and language of congruency and similarity; they may initially use informal language to describe their motion geometry experiences, and gradually, by further immersion in the social and mathematical world of geometry, their language will become more precise.

Geometry Learning

Geometry learning in the early grades should be informal, involving explorations, discovery, guessing, and problem solving. The new Saskatchewan elementary mathematics curriculum advocates that problem solving be the central focus of the curriculum. Children can learn about problem solving (e. g., some strategies) by being involved in geometric problem solving explorations; they can learn geometric concepts via problem solving (problem solving is used here as a means to teach geometry--the problem would be carefully selected to address specific geometric principles); they can work together to solve geometry-type problems mainly to get experience in solving problems (this is teaching/learning for problem solving). Baroody (1993) suggests we overlap the three approaches in an integrated way--in other words, children get the opportunity to experience solving problems, by being immersed in a high-impact problem (intended to address particular geometric concepts), while at the same time being taught some strategies and negotiating others.
In keeping with the Professional Standards, the classroom should be a mathematical community, where children can work in groups, cooperate, negotiate, collaborate, relate mathematical ideas to each other, make mathematical connections, conjecture, invent, problem solve and engage in mathematical discussion, reasoning, and argument. The main emphasis from professional directives and other literature [e.g., topics like the history of the nature of mathematical ideas; social constructivism (Ernest, 1991)], is that mathematical ideas originated in peopleâs minds, were tested against reality, were experimented with, discussed with other people, written down, held up for public scrutiny, argued and validated, and finally became the clear polished finished products we see today. Children need also to have the experience of behaving like mathematicians (Fellows, 1991), of testing and discussing their ideas and their constructions against the constructions and ideas of others, of writing about their ideas, and being involved in justifying their ideas to others. This kind of mathematical community could be the norm in all classrooms.
Given, then, that this kind of mathematical community can exist, what is the teacher's role? The teacher could have four major functions during an 'instructional cycle': (1) determining the students' geometric thinking level; (2) structuring high-impact geometric learning experiences and engaging the students in 'open-ended' geometric explorations; (3) interacting by asking 'open-ended' questions and responding by validation (i. e., not giving Îrightâ answers, but instead showing appreciation for the contribution and prompting the child to be convinced, to share with others, to justify, 'What did you notice?â' 'How would you describe this?' 'Explain how you arrived at your answer.' 'Can you make a different one?' etc.); (4) assessing using alternative assessment strategies (conferences, open-ended performance tasks, observational records, portfolios, etc.).
An example of an open-ended task might be (from Cathcart, Pothier, & Vance, 1997, p. 172):
 Draw as many different four-sided figures as you can. Write about how they are different and how they are the same. Choose a block from the collection. Use the sticks and marshmallows to make a skeleton model of the shape. How many different ways can you connect six square tiles with at least one side attached?

A task such as described above will enable all students to participate, making the mathematics accessible to each child. The common experience will result in many different solutions, each child engaging in the experience and bringing to it his/her particular mathematical/historical background.
Perhaps more than any other mathematics topic, geometry can be integrated with art, social studies, science, and with other math topics. Three-dimensional objects can be collected and displayed in class, children can go on a geometry walk and study the symmetry of butterflies, fibonacci numbers in nature, architectural forms, tessellations, and so on.
This issue of Ideas & Resource for Teachers of Mathematics focuses on geometry, on geometric thinking and on powerful geometry activities at different grade levels. We invite you to try some of these ideas in your classroom. There are literally thousands of good ideas for classroom geometry activities. It is vital that these activities are connected to the curriculum and are used to enable children to learn the big ideas of geometry expressed in the curriculum.

References
 Baroody, A. (1993). Problem Solving, Reasoning, and Communicating, K-8. Helping Children Think Mathematically. Toronto, ON: Maxwell Macmillan Canada. Cathcart, W. G., Pothier, Y., and Vance, J. H. (1997). Learning Mathematics in Elementary School. Scarborough, ON: Prentice Hall Allyn and Bacon Canada. Ernest, P. (1991). The Philosophy of Mathematics Education. New York: The Falmer Press. Fellows, M. R. (1991). Computer science and mathematics in the elementary schools. (Report given to the Computer Science Dept.) University of Victoria, BC. National Council of Teachers of Mathematics (1989). The Curriculum and Evaluation Standards. Reston, VA: NCTM. Owens, D. T. (1990). Research into practice: Spatial abilities. Arithmetic Teacher, 37, 6, 48-51. 