The van Hiele Levels of Geometric Thinking

Cathcart, et al. Learning Mathematics in Elementary and Middle Schools. p.282-3

Dina and Pierre van Hiele are two Dutch educators who were concerned about the difficulties that their students were having in geometry.  This concern motivated their research aimed at understanding students’ levels of geometric thinking to determine the kinds of instruction that can best help students.

The five levels that are described below are not age-dependent, but, instead, are related more to the experiences students have had.  The levels are sequential; that is, students must pass through the levels in order as their understanding increases.  The descriptions of the levels are in terms of “students” – and remember that we are all students in some sense.

# Level 0 – Visualization

Students recognize shapes by their global, holistic appearance.

Students at level 0 think about shapes in terms of what they resemble and are able to sort shapes into groups that “seem to be alike.”  For example, a student at this level might describe a triangle as a “clown’s hat.”  The student, however, might not recognize the same triangle if it is rotated so that it “stands on its point.”

Level 1 – Analysis

Students observe the component parts of figures (e.g., a parallelogram has opposite sides that are parallel) but are unable to explain the relationships between properties within a shape or among shapes.

Student at level 1 are capable of describing the properties of shapes.  Thus, they are able to understand that all shapes in a group such as parallelograms have the same properties, and they can describe those properties.

• Students can separate shapes into groups.

• Students can describe the properties of given groups.

ACTIVITIES

1. List two exploring activities that your student could do to let them master level 1.
2. List two activities that your students could do with your help to master level 1?
3. List five activities that your students could do without your help to show they have mastered level 1?

Level 2 – Informal deduction (relationships)

## Students deduce properties of figures and express interrelationships both within and between figures.

Students at level 2 are able to notice relationships between properties and to understand informal deductive discussions about shapes and their properties.  This is when the students begin to understand the relationships between shapes and their different characteristics.

Students can see relationships between properties.

• Students can understand informal deductive discussion concerning shapes and their properties.

ACTIVITIES

1. List two exploring activities that your student could do to let them master level 2.
2. List two activities that your students could do with your help to master level 2?
3. List five activities that your students could do without your help to show they have mastered level 2?

# Level 3 – Formal deduction

Students can create formal deductive proofs.

Students at level 3 think about relationships between properties of shapes and also understand relationships between axioms, definitions, theorems, corollaries, and postulates.  At this level, students are able to “work with abstract statements about geometric properties and make conclusions based more on logic than intuition  (Van de Walle).  Students can create formal deductive proofs.  This is when the students understand axioms to solve problems.

• Students can think about relationships between properties of shapes

• Students can distinguish between axioms, definitions, theorems, corollaries, and postulates.

• Students can work with geometric abstracts and make conclusions based on logic.

ACTIVITIES

1. List two exploring activities that your student could do to let them master level 3.
2. List two activities that your students could do with your help to master level 3?
3. List five activities that your students could do without your help to show they have mastered level 3?

# Level 4 – Rigor

Students rigorously compare different axiomatic systems.

During this level the students are using different premises while developing different shapes.  Students at this level think about deductive axiomatic systems of geometry.  This is the level that college mathematics majors think about Geometry.

• Students think about deductive axiomatic systems of geometry.

ACTIVITIES

1. List two exploring activities that your student could do to let them master level 4.
2. List two activities that your students could do with your help to master level 4?
3. List five activities that your students could do without your help to show they have mastered level 4?

In general, most elementary school students are at levels 0 or 1; some middle school students are at level 2.  State standards are written to begin the transition from levels 0 and 1 to level 2 as early as 5th grade “Students identify, describe, draw and classify properties of, and relationships between, plane and solid geometric figures.”  (5th grade, standard 2 under Geometry and Measurement)  This emphasis on relationships is magnified in the 6th and 7th grade standards.

Interestingly, the sixth National Assessment of Educational Progress report (1997) reported that “most of the students at all three grade levels (fourth, eight, and twelfth) appear to be performing at the ‘holistic’ level (level 0) of the van Heile levels of geometric thought.”

Bibliography

Fuys, David, Dorothy Geddes, and Rosamond Tischler. "The van Hiele Model of Thinking in Geometry among Adolescents." Journal for Research in Mathematics Education Monograph No. 3. Reston, VA.: National Council of Teachers of Mathematics, 1988.

Malloy‚Carol E., and Susan Friel. “Perimeter and Area through the van Heile Model.” Math Teaching in the Middle School. Volume 5, Issue 2, October,1999.

van Hiele, P.M., van Hiele Geldof, D.(1958). A Method of initiation into geometry at secondary school. Report on Methods of Initiation into Geometry, J. B. Wolters

van Hiele, P.M.(1959). La penséede l' enfant la. géométrie. Bulletin de l'Association des Professorurs de Mathematiqe de. l'Emseignement Public no.198

van Hiele, P.M.(1986). Structure and Insight, Academic Press

Gutiérrez, A., Jaime, A., & Fortuny, J. M.(1991). An alternative paradigm to evaluate the acquisition of the van Hiele levels. Journal for Research in Mathematics Education, vol.22

Hoffer, A. (1983). van Hiele Based Research. Acquisition of Mathematical Concepts and Processes, Academic Press

Mayberry, J. (1983). The van Hiele levels of geometric thought in undergraduate preservice teachers. Journal of Research in Mathematics Education, vol.14

Sfard, A. (1987). Two conceptions of mathematical notions. Proceedings of PME11

Sfard, A. (1991). On the dual nature of mathematical conceptions. Educational Study in Mathematics, vol 22