Observation Checklist: van Hiele Levels

According to a theory known as the van Hiele levels, students learn geometry at several cognitive stages, from recognizing objects to forming abstract arguments regarding geometric qualities. Utilizing an observation checklist based on the van Hiele levels can be highly beneficial for teachers in evaluating their students’ geometric reasoning skills. The signals on this checklist generally correspond to the five processes of the van Hiele model: Analysis, Visualisation, Deduction, Rigour, and Informal Deduction. Teachers can determine the precise level at which each student functions by closely examining how each student engages with geometric concepts. This enables more individualized education, pushing students to advance through the stages. Teachers can also complete an observation checklist using the van Hiele levels as a guide.

Visual Level 

1. Identifies a shape.
2. Constructs draws or copies a shape.
3. Names or labels shapes and other geometric configurations appropriately.
4. Compares and sorts shapes.
5. Verbally describes shapes.
6. Solves routine problems by operating on shapes.
7. Identifies parts of a figure.

Analysis Level

1. Identifies and tests relationships.
2. Recalls and uses appropriate vocabulary.
3. Compares two shapes.
4. Interprets and uses verbal descriptions of a figure to draw/construct a figure.
5. Interprets verbal or symbolic statements of rules and applies them.
6. Discovers properties of specific figures and generalizes properties.
7. Describes a class of figures in terms of its properties.
8. Given certain properties, identifies a shape.
9. Connect properties between classes of figures to compare.
10. Discovers properties of an unfamiliar class of figures.
11. Solves geometric problems.
12. Formulates and uses generalizations about the properties of figures.

Abstract Level

1. Identifies different sets of properties that characterize a class of figures and tests that these are sufficient.
2. Identifies minimum sets of properties that can characterize a figure.
3. Formulates and uses a definition for a class of figures.
4. Gives informal arguments.
5. Gives informal deductive arguments.
6. Gives more than one explanation to prove something and justifies these explanations.
7. Informally recognizes the difference between a statement and its converse.
8. Identifies and uses strategies or insightful reasoning to solve problems.
9. Recognizes the role of a deductive argument and approaches problems in a deductive manner.

Deductive Level

1. Recognizes the need for undefined terms, definitions, and basic assumptions (postulates).
2. Recognizes characteristics of a formal definition and equivalence of definitions.
3. Proves relationships in axiomatic settings.
4. Proves relationships between a theorem and related statements.
5. Establishes interrelationships among networks of theorems.
6. Compares and contrasts different proofs of theorems.
7. Examines effects of changing an initial definition or postulate in a logical sequence.
8. Establishes a general principle that unifies several different theorems.
9. Creates proofs from simple sets of axioms frequently using a model to support arguments.
10.  Gives formal deductive arguments.

Rigor

1. Rigorously establishes theorems in different axiomatic systems.
2. Compares axiomatic systems (e.g., Euclidean and non-Euclidean geometries); spontaneously explores how axiom changes affect the resulting geometry.
3. Establishes consistency of a set of axioms, independence of a hypothesis, and equivalency of different sets of axioms; creates an axiomatic system.
4. Invents generalized methods for solving classes of problems.
5. Searches for the broadest context in which a mathematical theorem/principle will apply.
6. Does in-depth study of the subject logic to develop new insights and approaches to logical inferences.