Geometry is one mathematical domain that has opportunity for research to understand which instructional strategies work best when teaching geometry to elementary students. The National Council of Teachers of Mathematics (NCTM) does give suggestions on the types of concepts that should be a part of any geometry program. These concepts include:
Analyzing characteristics and properties of two-and three-dimensional geometric shapes and developing mathematical arguments about geometric relationships
Specifying locations and describing spatial relationships using coordinate geometry and other representational systems
Applying transformations and using symmetry to analyze mathematical situations
Using visualization, spatial reasoning, and geometric modeling to solve problems (NCTM, n.d.)
Growing up, I remember learning about these concepts through behaviorist learning approaches such as memorization, repetition, and a lot of listening to and mimicking the teacher, making it easy to disconnect from the subject of geometry altogether.
Behaviorist learning theories continued to dominate in high school with the introduction of writing geometric proofs. My friends loved this approach to geometry while I thought it was nothing more than an exercise in futility. I could never understand why we were spending 40+ minutes proving that a right triangle is a triangle, for example, when everyone in the room already agreed at the beginning of the class period that the shape in question represented a right triangle. Furthermore, there were no contextual representations so while in high school I never understood how important these geometric concepts are in the real world.
In undergrad, I welcomed the separation from Geometry courses as none were not required for my degree in elementary education. And then when I began my teaching career as a middle school science teacher, my deep thinking centered around habitats and ecosystems, not comparing and contrasting geometric shapes.
Teaching elementary-age students forced me to confront my contrarian disposition toward geometry and to research alternative teaching and learning approaches.
We as elementary teachers are laying the groundwork for student success with geometry courses in high school and beyond with how we teach children to interact and realize aspects of the mathematical practices, such as how to construct viable arguments and how to critique the reasoning of others.
When I started teaching 3rd grade, I always made sure my students knew the names of 2D and 3D shapes, including remembering that a rhombus is a shape, and that parallel lines never touched, but for years I struggled to push student thinking deeper into student-centered activities requiring students to create and convince within the geometry domain. I was teaching the way I was taught by treating geometry as a domain reserved for low level thinking involving memorizing vocabulary and identifying shapes.
After joining local and national mathematics organizations, I started learning about other teaching approaches that focus on placing the learner at the center to elicit authentic student thinking about geometry, such as Van de Walle et a. (2014).
According to Van de Walle et al. (2014)-
Geometry is a 'network of concepts, ways of reasoning and representation systems' (Battista, 2007, p. 843) used to explore and analyze shape and space. This critical area of mathematics appears in everything from global positioning systems to computer animation.
(Van de Walle, et al., 2014, p. 344)