Level 0 – Pre-recognition – inability to distinguish between figures
Level 1 – Visualization – recognize by appearance alone
Level 2 – Analysis – see shapes as collections of properties but don’t see relationships
Level 3 – Abstraction – see relationships between properties and between figures
Level 4 – Deduction – able to construct proofs, understand definitions, and know the meaning of conditions
Level 5 – Rigor – comparing mathematical systems
Geometric Understanding – By using van Hiele’s levels of geometric understanding, students progress through a level of understanding from 0 to 5. This understanding is dependent upon educational experiences and not age, grade level, or maturity. So, I understand this to mean that students must truly understand the material before they can progress to the next level. Moving on from memorizing information rather than understanding it will keep students from progressing.
Teaching Above a Student’s Level – If a teacher tries to teach content that is above a student’s level, they will not be able to master the information, will try to memorize but may easily forget the material, or be unable to apply the information to a given situation. So, the importance of knowing your students’ understanding level before proceeding with new material is crucial.
Instructional Practices – Students will be able to learn when they actively experience the objects and when they engage in discussion and reflection. So, this means that just lecture and reading will not help students learn. They must be afforded opportunities to investigate and experiment with the geometrical figures. Once investigations and exploration have taken place, then students can make use of information from lectures or books to aid in their learning. The final piece of this must be reflection and their understanding.
The Role of Language – Each level of geometric understanding has its own language associated with it and an interpretation of that language. It is important to discuss and have students talk about geometry using this language. As student understanding increases, so does the use of more specific language. I understand this to mean that initially students will understand that a triangle has 3 sides and talk about it in those terms. Later students will learn that triangles have 3 angles and they all add up to a certain measure. As understanding is gained, language will increase to improve and specify the learning.