Understanding Van Hiele's Geometric Levels

The Van Hiele theory describes the different levels of understanding students develop in geometry. It progresses from basic visual recognition to complex logical deduction. Let's look at the relevant levels:

• Level 0 (Visualisation): At this level, students recognise shapes based purely on their appearance. They might call a shape a "square" simply because it looks like one, without necessarily knowing its defining properties. They can sort shapes into broad categories (e.g., curves vs. straight lines, shapes with corners vs. without).
• Level 1 (Analysis): Students at this level move beyond appearance and start understanding shapes based on their properties. They can identify and define shapes by their characteristics, such as side lengths and angle measures. For example, they understand that a square has four equal sides and four right angles. They can describe the relationship between properties within a shape (e.g., opposite sides of a rectangle are equal).
• Level 2 (Relationship/Abstraction): This level involves understanding the connections and hierarchies between different geometric shapes. Students learn that some shapes are special types of others (e.g., a square is a type of rectangle, and also a type of rhombus). They can classify shapes based on multiple properties and understand that one definition can encompass others.

Analyzing Mohan's Folding Task

Mohan's Mathematics teacher gave him a rectangular page and asked him to make a square by folding it. Mohan successfully folded the paper from one side to create a square.

To perform this task, Mohan needed to understand the properties of a square. A square is defined as a quadrilateral with four equal sides and four right angles (each angle measuring $90^\circ$). A rectangle has opposite sides equal and four right angles.

When folding a rectangle to make a square, Mohan implicitly uses the properties of both shapes. He likely took the shorter side of the rectangle and folded the longer side over until it matched the shorter side, creating a crease. This action results in a square where the side length is equal to the shorter side of the original rectangle. For instance, if the rectangle has length '$l$' and width '$w$', and '$w \le l$', Mohan would fold it to create a square with side length '$s = w$'.

This process requires more than just visual recognition (Level 0). It involves:

• Knowing the defining property of a square: all sides must be equal.
• Analyzing the given rectangle's dimensions.
• Applying the property of equal sides to construct the square from the rectangle.

Identifying Mohan's Level of Understanding

Let's connect Mohan's actions to the Van Hiele levels:

• Level 0 (Visualisation): Mohan did more than just visually identify shapes. He actively manipulated the paper based on geometric properties.
• Level 1 (Analysis): This level fits Mohan's actions well. He needed to understand the property of "equal sides" for a square and apply it to the rectangular paper. He analyzed the shape and its properties to perform the transformation.
• Level 2 (Relationship): While Mohan implicitly uses the fact that a square is related to a rectangle, the core task involves constructing the square based on its properties, not necessarily explaining the relationship between squares and rectangles in a formal way.
• Level 3 (Deduction): This level involves logical proofs and formal reasoning, which is not required for simply folding a piece of paper into a square.

Therefore, Mohan's ability to fold the rectangle into a square demonstrates an understanding based on the properties of shapes.

Conclusion on Geometric Levels

Mohan's task of making a square from a rectangle by folding requires him to know and apply the properties of a square, specifically that all sides are equal. This aligns directly with Level 1 (Analysis) in the Van Hiele theory of geometric understanding, where students focus on the properties that define shapes.