Question

TPcaion of 1. Describe in your own words the first three van Hiele levels of geometric thought (levels 0, 1, and 2), includ- ing the object of thought and product of thought. How would activities aimed at levels 0, 1, and 2 differ?

Answer 1

The first van Hiele level focuses on the visual attributes of a figure.

Level 0 Visualization (Basic visualization or Recognition) At this level people use visual perception and nonverbal thinking.

They recognize geometric figures by their shape as “a whole” and compare the figures with their prototypes or everyday things (“it looks like door”), categorize them (“it is / it is not a…”).

They use simple language. They do not identify the properties of geometric figures.

Level 1 Analysis (Description) At this level people (students) start analyzing and naming properties of geometric figures.

They do not see relationships between properties, they think all properties are important (= there is no difference between necessary and sufficient properties).

They do not see a need for proof of facts discovered empirically. They can measure, fold and cut paper, use geometric software etc.​​​​​​​

Level 2 Abstraction (Informal deduction or Ordering or Relational) At this level pupils or students perceive relationships between properties and figures. They create meaningful definitions.

They are able to give simple arguments to justify their reasoning.

They can draw logical maps and diagrams. They use sketches, grid paper, geometric SW.

Students recognize figures based on their visual observations which also dominate how students at this level sort and classify shapes.

The product of thought for students at the first level in turn focuses on students grouping shapes that are visually alike.

For the second van Hiele level, students are able to focus thought on classes of shapes versus one single figure.

Students at this level can focus on classes of shapes and consider the properties of the classes.

Therefore, the products of thought at this level pertain to students defining properties of shapes as they gain understanding of how shapes are grouped or classed together.

Level 2 of the van Hiele theory pertains to an object of thought focused on properties of shapes.

Students at this level are able to think about the properties of shapes to develop relationships within the properties.

The thinking of shapes within this levels begin to exhibit logical thinking in terms of the properties allowing students to question proofs of the shape properties.

Noting this, the product of thought at level 2 promotes student logical reasoning to determine relationships between properties of shapes.

Activities in level 0, 1, and 2 differ based on the visual components as well as the though process provoked in the differing activities.

In the first level, students work with physical shapes that they can observe and group according to visual similarities.

The next level activities the thinking moves beyond visual observation to also include properties and their connection to the objects grouped by the students.

This idea is also carried over into level 2 thinking as students begin to logically reason the relationships of shapes within groups with a stronger emphasis on definitions.