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Note: This blog post is part 3 of a three part series discussing the Van Hiele model which has significantly influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics (NCTM) and the new proposed Common Core Standards. Our math game called “Sunken Shapes” also follows this model to help kids learn geometry for Pre-K, Kindergarten and Grade 1. The game will be launched in the App store in the next few weeks. Please don’t forget to give us your feedback. Part 1 of the series can be found here. Part 2 of the series can be found here.

Level 2: Abstraction

At level 2, the objects of thought are the geometric properties of shapes and the relationships between those properties. So, kids start understanding that if one set of properties is true for a shape, then this may imply another property to be true. They understand necessary and sufficient properties. Because of their deeper understanding of the properties of the shapes, they can start recognizing that a square is also a rectangle. At this level of understanding of geometric shapes, kids can logically reason and answer simple “what if” scenarios. What if I stretch a square from the two opposite sides? Kids at level 2 are, however, not ready to understand or apply formal geometric proofs.

Kids at level 2 gain experience by identifying and experimenting with the “necessary” and “sufficient” properties of shapes. Initially they will list all the properties that they associate with a shape as necessary and sufficient. But with guidance, they will minimize them.

Note: This blog post is part 2 of a three part series discussing the Van Hiele model which has significantly influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics (NCTM) and the new proposed Common Core Standards. Our math game called “Sunken Shapes” also follows this model to help kids learn geometry for Pre-K, Kindergarten and Grade 1. The game will be launched in the App store in the next few weeks. Please don’t forget to give us your feedback. Part 1 of the series can be found here

Level 1: Analysis

At level 1, the objects of thought are classes of shapes rather than individual shapes. They start recognizing that these classes of shapes have certain properties. For example, any square (not just “this” square) has “4 equal sides, 4 equal angles, parallel opposite sides, etc.” At this level, they also start recognizing that for identifying shapes, features like size and orientation are irrelevant. However, they may still not to understand the concept of overlap in categories – to them, a square is not a rectangle, and a rectangle is not a parallelogram. A significant difference between level 0 and level 1 is that kids start to see specific shapes or drawings as representatives of classes of shapes based on the properties-match that they can find in the drawings.

Kids at level 1 gain initial experience by working on one class of shapes at a time. They can be given 5 different rectangle shapes, for example, and then they can list down the common properties they see in all those shapes. After some experience, they can start categorizing those properties under headings like sides, angles, diagonals, symmetries etc. At an advanced stage of this level, kids will be able to group and differentiate shapes based on multiple properties. And they will understand that if a property applies to a few shapes available, then it will apply to all shapes in the category.

Continue reading: Part 3 of the series

Note: This blog post is part 1 of a three part series discussing the Van Hiele model which has significantly influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics (NCTM) and the new proposed Common Core Standards. Our math game called “Sunken Shapes” also follows this model to help kids learn geometry for Pre-K, Kindergarten and Grade 1. The game will be launched in the App store in the next few weeks. Please don’t forget to give us your feedback.

Introduction

We all think about geometry slightly differently. However, we are all capable of thinking and reasoning in geometric contexts. There are five levels in which kids learn to reason in geometry. These levels do not tell us how much knowledge of geometry we have. Instead, they describe how we think and what types of geometric ideas we think about. Kids start at the lowest level (0), and with experience and instruction, progress to higher levels of sophistication. A well experienced elementary school student can reach up to level 2. However, without such experiences, even an adult will remain only in level 1. With our focus on elementary math education, we will only discuss the first three levels (Level 0 to Level 2).

Level 0 (Visualization)

At this visualization level, kids’ focus is on the overall appearance of the shape, with little or no attention to the properties of the shape. A square is a square because it matches with the “square prototype” that the kid has in mind. So, to the kid’s mind, a square that has been rotated at a 45-degree angle will not be a square anymore. It may become a diamond, or an unrecognizable shape. For kids in this level, the shapes can be changed by rotation or rearrangement. The goal at this level is to explore how shapes are alike and different and to use these ideas to create categories of shapes, like squares, triangles, circles, cylinders, etc. The kids may start by grouping a triangle and a square together (they both have corners). But with more experience, they will start recognizing the differences between the more pointy corners of a triangle and the right angle corners of the square.

Kids at level 0 gain most experience by evaluating common shapes and discussing interesting features that they see in the shapes. They can later compare two shapes and discuss features in the shapes that are alike and that are different. A kid at an advanced stage of this level should be able to sort multiple shapes based on a feature based sorting mechanism chosen by the kid, and should start noticing some traditional geometric properties.

Continue reading: Part 2 of the series