As I move toward the teaching profession, thinking about how and what I want to present to my students, I naturally think about the developmental levels of the students that greatly impact how they are able to learn in the hopes that I can attend to the skills that my students have, while strengthening the skills and qualities that students of their age traditionally are coming into. Looking forward to teaching fifth graders this upcoming school year, I am excited about the math that I will be able to do with the students. I love getting past knowing the basic shapes, but getting into classifying shapes based on shared properties. So as I plan for this material I have to be conscious of what will likely work, and what likely will not work as I teach these students. Luckily, the research done on Learning Theory by Dina van Hiele can largely assist and influence the teaching methods I will bring to my classroom.

One of the more respected voices in the mathematics world, perhaps Dina van Hiele’s greatest contribution to the subject was his theory of the development of young students’ geometric thinking. As students progress in their mathematical understanding, van Hiele suggested that they go through four distinct levels of thinking in dealing with geometric properties. As a teacher thinks about teaching certain geometric concepts to students, then, there is an optimal order of skills the teacher should be emphasizing as they work through the particular unit.

van Hiele’s first level of geometric comprehension is called “Visualization.” In this level, students are able to look at shapes and name them according to what shape prototype they look most similar to. They are not quite able to describe which exact properties they notice that make them give the shape the name they do, but they do know that a given shape looks familiar to a shape they have seen with that same name in the past. For example, a student would be able to tell us from experience that the classroom door is a similar shape to the whiteboard, and those shapes are rectangles. Conversely, when a student hears the word rectangle the image that comes to mind will be that of the door or the whiteboard. This skill can be enhanced by doing such an activity, where students are provided with the names of objects around the room and are challenged to categorize them by their shape, or they could be given the categories of shapes, and then told to find objects around the room that would fall into those categories.

After Visualization, van Hiele says that students’ next level of understanding is “Descriptive/Analytic.” Once in this level, students are able to identify a shape based on multiple properties that, in conjunction with other properties, are exclusive to a given shape. A square, for example, could be classified as always having four right angles and four congruent sides. This goes beyond just looking at a shape and giving it a name, such as in the Visualization stage, as it includes knowing the specific qualifications of shapes. Using the same activities from the Visualization activity, students could be asked to extend their work by defending why their answers were correct, describing the characteristics of the classroom objects that qualify them as the shape they are claimed to be.

The next stage of geometric understanding is “Abstract/Relational.” In this stage students are challenged to create abstract definitions of shapes, and note what characteristics are necessary for a shape to be named as it is. Not only that, but it is key in this stage to think of these characteristics as more than just descriptive facts, but ways in which to organize shapes hierarchically. So once students can see, identify and describe an object as a shape, they need to be able to say what makes that shape unique. Saying a shape has “four sides and four right angles” does not specify if the shape is a rectangle in general, or more specifically a square. The student would have to be able to say that either all of the sides are congruent, or the opposite sides are congruent and the adjacent sides are not, thus demonstrating their knowledge of the difference between a square and a traditional rectangle (traditional = opposite sides congruent, adjacent sides not congruent). My activity that could take part in this stage is having students create a hierarchy of shapes, that is, a hierarchy showing how shapes are related to each other, and whether one shape always qualifies as another shape, but is slightly more specific, such as the square-to-rectangle relationship.

The fourth level of van Hiele’s geometric understanding is called “Formal Deduction and Proof.” In this level, students are able to understand a sequence of statements and can formulate a conclusion from the givens they are presented. An activity that could be used for this level would be to give students a list of qualities that a shape must have, and challenge them to make a shape that fits those guidelines. Ultimatley students are trying to justify a conclusion as a result of the facts presented to them, much like famous mathematicians have been doing for millennia!

Sometimes there’s a fifth level to include some post-secondary & professional geometry: axiomatic reasoning. Typically people credit she and her then-husband Pierre.

C’s: 4/5

consolidation –

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