### A Response Indicating Geometric Thought on a Higher Level

When I was both a doctoral student at the University of Louisville and also teaching mathematics for elementary teachers in Indiana, I worked with others to develop some interview tasks for three-, four-, and five-year-olds about numbers and shapes. One of our favorites, and one that was also a favorite of the children we interviewed, concerned identifying some basic shapes. In this task, the interviewer would place within the child’s reach a collection of flat shapes, including circles, triangles, squares, and rectangles. One request that was part of our task protocol was, “I want you to place on the mat all of the triangles and only the triangles.”

The first few times the preschool children were asked to identify triangles, I was surprised at how quickly and discerningly most of them were able to do so. Then, the interviewer would ask, “How do you know these are triangles?” Some would respond, “Because my mother told me” or “Because my teacher told me,” or very confidently, “Because I’m smart.” However, a “power-answer” to the question that many of the children gave was, “Because it has three sides.”

I believe a child’s recognition that every triangle must have three sides is an example of a geometry milestone, not unlike the Cardinality Principle for counting. The answer is powerful because it indicates that the child is moving to a higher level of geometric thinking — not making their shape identification based on just the appearance of the shape but on its properties. This response elicits higher-level questions such as

- How many corners must it have?
- Are all the sides the same length?
- Are the corners pointy or wide?
- Are there any square corners?

Then, when children move to a higher level and begin to think about the properties of shapes, they can classify shapes as three-sided (triangles); four-sided (quadrilaterals, squares, rectangles, etc.); five-sided (pentagons); and six-sided (hexagons). The following poster can be used for conversations with children about differences and similarities in properties of two-dimensional shapes.

### The Van Hiele Levels of Geometric Thought: A Learning Trajectory for Geometry

A theoretical learning trajectory for thinking about shapes was developed in the 1950s by two Dutch high school teachers, Pierre van Hiele and Dina van Hiele-Geldof. They became troubled by the poor performance of their high school students in geometry. When studying Piaget’s interviews, they noticed that some of his tasks required knowledge of math principles and vocabulary that were beyond the child’s current understandings. Their research demonstrated how a mismatch between the level of teaching and the level of student understanding could mean new ideas would not be assimilated properly into long-term memory.[1] Hence, for their companion doctoral dissertations, Pierre and Dina van Hiele proposed that students moved through levels of thought in understanding geometry in a similar manner to levels we move through in thinking about quantity. Then, beginning in preschool a progression of geometric activities can be used raise student thinking levels and prepare high school geometry students for writing a deductive argument in the Euclidean tradition. Here are the five Van Hiele Levels of Geometric Thought.[2]

**Level 0:**Recognize geometric shapes by their appearance. The student can name them but not explicitly describe their properties**Level 1:**Identify, describe, and analyze a shape’s properties. A rectangle has four right angles and a rhombus has four equal sides. A square has both four 4 right angles and 4 equal sides.**Level 2:**Classify shapes into sets and subsets according to their properties. Venn diagrams are useful for Level 2 geometric thinking. Squares can be classified as rectangles because they have 4 right angles. Thus, squares are a sub-class of rectangles because they have the additional property of all of sides having equal length. A square is a diamond (or rhombus) because it has four sides of equal length. A squares are in a sub-class of diamonds because they have the additional property of 4 right angles. Even so, students thinking on this level are not yet ready to organize sequences of statements to justify observations.**Level 3:**Students develop sequences of statements to deduce one statement from another in Euclid’s deductive system. In a traditional high school geometry course, a student might be asked to prove that the sum of the measures of the three angles of a triangle is 180^{o }or to prove the Pythagorean Theorem. However, at this level, even though a student understands how to write a proof using Euclidean definitions, postulates, and theorems, he does do not understand relationships between one deductive system and another.**Level 4:**Students recognize that the world of mathematics includes a wide array of deductive systems. In modern times, mathematicians have created other systems of geometry that are non-Euclidean.[3]

For preschool and elementary school students conversations will remain on Levels 0, 1, and 2 — identifying a shape by its appearance; considering the properties of shapes; and then classifying shapes into different categories based on their properties. [4] Furthermore, children with geometry experiences on these three lower levels have a greater probability of success with a high school geometry course which is typically taught on Level 3.

___________________________________________

[1] I like to connect this van Hiele idea with Vygotsky’s belief that teaching above the scope of a student’s current Zone of Proximal Development (ZPD) was usually unsuccessful.

[2] My resource for information about the Van Hiele levels was a chapter written by Alan Hoffer in Lesh & Landau’s 1983 publication, *Acquisition of Mathematics Concepts and Processes* (p 205-227).

[3] Two examples of non-Euclidean geometries are hyperbolic and elliptical.

[4] You can find van Hiele-based geometry activities on Levels 0, 1, and 2 in Chapter 20 of *Elementary and Middle School Mathematics: Teaching Developmentally, Seventh Edition* (Van de Walle, Karp, and Bay-Williams, 2010).