Victory through the Van Hiele Levels of Geometric Reasoning

Improving the ways we teach young children geometry is something I feel very passionately about. Although this area of mathematics is introduced at a very young age in many of our nation’s schools, countless students never claim victory with a firm grasp of spatial reasoning. Instead, these unfortunate students find solace in constructing a shallow understanding of geometry that’s riddled with holes and dead-ends. Regrettably, I was one of these deficient students. Instead of achieving mastery in geometric reasoning, I claimed only failure and incomprehension. When my classmates were climbing toward a rock-solid understanding of central geometric concepts with past experiences serving as a dependable compass, I was left clinging to the outer boundaries led only by a frail and desperate grasp of the subject matter. When others were filling themselves with the fruitful mathematical skills of spatial concepts, I continually found myself starved, drained, and dreadfully confused.

Geometry and I have never experienced anything but a critically strained relationship. As a result of my past frustrations and failures, I want to do all I can to shield my future students from the evils of a hollow understanding of geometry. I want them to leap over the same hurdles in their mathematical journey that hindered me for so long and then experience geometry in ways that I could have never dreamed possible in my own youth.

With a mastery of geometric understanding, students are equipped with a lifetime of possibilities. Understanding the hierarchical levels of geometric reasoning that every student progresses through is the key to unlocking these possibilities and setting every young boy and girl on a course toward victory. IMAGES, found at http://images.rbs.org , is a website devoted to the connection between understanding geometry and improving the strategies teachers use to instruct, and it is a valuable resource for understanding the process every child must go through to attain geometric mastery.

According to the instrumental work of Dutch educators Pierre van Hiele and Dina van Hiele-Geldof, children progress through 5 levels of special understanding that advance them toward geometric thinking. These 5 levels, labeled the Van Hiele levels of geometric reasoning, are not age-dependent, but hinge upon rich geometric experiences that are developmentally appropriate.
The first level in the Van Hiele model is visualization. Children at this stage are able to recognize and name shapes solely based upon their appearance but cannot pinpoint the properties of the shapes. Likewise, students may see some of the characteristics of the shapes, but do not necessarily use them for sorting or recognition. This level is based upon perception rather than reasoning of geometric shapes.

Level two is analysis. It is at this level that students grow to understand the properties of shapes and some of the related vocabulary that describes the properties. However, students are not yet capable of understanding the relationships between these properties. For example, when a child describes a geometric object, he or she may be able to list properties of the shape, but will not necessarily know which properties are important enough to use.
The third level is based upon informal deduction. Students at this level of maturity are able to recognize relationships between properties and are capable of creating meaningful definitions while forming logical arguments to justify their reasoning.

Next is the deduction level of the Van Hiele model of geometric reasoning. Students passing through this phase do not just identify the characteristics of geometric shapes, but are also equipped to construct proofs that use both postulates and definitions together.

The highest level in the Van Hiele hierarchy is labeled the rigor stage and students who have reached this final stop along their way to geometric reasoning understand the formal aspects of deduction, such as creating and comparing mathematical systems. A college-level geometry course functions at this level of thought and students who have reached the rigor level are capable of using proof by contrapositive as well as indirect proof. Furthermore, students at this level can also understand non-Euclidean systems.

These five levels (visualization, analysis, informal deduction, deduction, and rigor) must be experienced in sequential order if a student is to gain mastery of geometric concepts. Failure to progress through each of the levels will assuredly lead to gaps in spatial comprehension. When teachers understand each of these necessary levels and how to effectively mold their classroom experiences in geometry around each progressive level, students will claim the prize that I relentlessly sought in my younger years: a fruitful and rock-solid relationship with geometry. Victory will be theirs.