by NCTM President J. Michael Shaughnessy
NCTM Summing Up, October 2011
Is it just me, or does geometry seem to be in the background these days? Slogans like “Algebra for all” and policies designed to ensure that all students complete algebra by the end of eighth grade have caught the national attention in recent years. When states or national organizations develop sample assessment tasks, they usually begin the process with tasks that involve arithmetic operations or algebraic concepts and procedures. Geometry tasks are often lower on their priority list. The new Common Core State Standards for Mathematics (CCSSM) appear to highlight number and operations, algebra, and functions, putting geometry a tad off to the side. Even in some of NCTM’s own series, the first books to emerge have often been on algebra or number and operations. Geometry has always been well represented in curriculum, instruction, and assessment, but it seems to come a bit later to the party.
If algebra is the language of mathematics, geometry is the glue that connects it. Geometry offers topics that provide mathematical challenges for students as they progress from kindergarten to college. For example, suppose we consider the geometry of tessellations—tilings of the plane with geometric shapes. Students at the primary level, pre-K–grade 2, can begin to build shapes with pattern blocks or other collections of polygons. They can use the polygons to form tiling patterns and to solve shape puzzles by placing polygons together to cover a picture or to make a shape. Students at the intermediate level, grades 3–5, can investigate which collections of regular polygons they can use to tessellate a tabletop—to cover the plane with no gaps or no overlaps. They soon discover that only three regular polygons (triangle, hexagon, square) will tile the plane all by themselves (forming regular tessellations), without any other type of regular polygon. Students can also begin to create tessellations using collections of more than one regular polygon, such as combinations of equilateral triangles and squares.
Students in middle school can search for all the semi-regular tessellations, those tessellations that consist of sets of regular polygons where the same configuration of polygons occurs at every vertex. The concept of symmetry (e.g., translational, rotational, reflection symmetry) can be introduced in a tessellation environment. By the time students are in high school, tessellations can provide an excellent medium for investigating and characterizing symmetry patterns by using geometric transformations. Geometric transformations play a pivotal role in the development of geometric concepts and processes in the new Common Core State Standards. In college mathematics, this same tessellation environment can provide an opportunity for investigating and characterizing symmetry groups in abstract algebra. Students can find such symmetry groups in frieze patterns or wall-paper designs.
Geometry provides an incredibly rich medium for both mathematical content and mathematical reasoning progressions. In fact, a variety of geometries can model aspects of the universe. Not only do we have the “flat” geometry handed down from Euclid, but we also have spherical and hyperbolic geometry, which provide models of the universe on curved surfaces, and taxicab geometry, which provides an environment for exploring space in which the meaning of distance has altered. Furthermore, in any of these geometric systems, we can put things in motion and consider issues like symmetry or invariants under certain geometric transformations. One could make a good case for beginning the study of mathematics with geometry, and making geometry the priority content, rather than number.
Geometry provides the yin to the yang of algebra and number. We should give geometry as much consideration in our curriculum development, professional development for teachers, assessments, and standards implementation, as we do algebra and number. Geometry, algebra, and number should be equal, but not separate, because the connections between algebra and geometry are important and powerful for our students. The connection I mentioned between the geometry of tessellations and the algebra of symmetry groups is just one example. Centuries ago, Descartes provided us with one of the most powerful connections in the history of mathematics when he linked algebra and geometry with his invention of the coordinate plane. Consider that the geometry of graphs of functions brings a visual life to the algebra of their equations. The geometry of functions is easily accessible through the dynamic environments now available to us for teaching with graphing tools in applications on computers and calculators. More recently, we have seen the beauty of the connection between the algebra of recursive computations and the geometry of fractals.
Perhaps in the curriculum of school mathematics, or in standards, or in assessment, it must be the case that some mathematical strands take priority over others. But we must never forget geometry or miss the opportunities to connect it with the other content areas—particularly, algebra and probability. Geometry is a crucial part of the mathematical education of our students and of our citizens.
Missing out on geometry would be a tragedy for many students, since so many blossom in mathematics and enjoy it for the very first time when they encounter geometry. “I finally like mathematics,” they say. “This is really different!”
Geometry opens doors and minds for some students that other parts of mathematics leave closed. We must continue to provide a variety of options for our students so that they can tap the mathematician that resides within. For many of our students, it is geometry that first opens that door, providing a breath of fresh mathematical air.