Geometric Thinking

In mathematics education, the Van Hiele model by Pierre van Hiele and Dina van Hiele-Geldof (husband and wife) is a theory that describes how students learn geometry.


The best known part of the theory are the five levels which the van Hieles postulated to describe how children learn to reason in geometry. These levels are often understood to describe how children understand shape classification, but they actually describe the way that children reason about shapes. And progress from one level to the next is more dependent on mathematical experiences than chronological age. 


In Level 0 (Visualization), children can name and recognize shapes by their appearance, but cannot specifically identify properties of shapes. Although they may be able to recognize characteristics, they do not use them for recognition and sorting.


In Level 1 (Analysis), children begin to identify properties of shapes but do not make connections between different shapes and their properties. Irrelevant features, such as size or orientation, become less important, as students are able to focus on all shapes within a class. They are able to think about what properties make a rectangle.


In Level 2 (Informal Deduction), children are able to recognize relationships between and among properties of shapes or classes of shapes and are able to follow logical arguments using such properties.
In Level 3 (Deduction), children go beyond just identifying characteristics of shapes and are able to construct proofs using postulates or axioms and definitions.


In Level 4 (Rigor), children have entered the highest level of thought in the van Hiele hierarchy and can work in different geometric or axiomatic systems.





My coursemates and I were asked to find the interior angles of a pentagon. Most of us immediately gave the answer, "360 degrees" which was incorrect. After some thoughts and discussion, the sum of interior angles is 540 degrees instead!


How do we problem solve and find the interior angles then?


We were given a square piece of paper, and I got my answer by adding 5 right angles and 2 angles of 45 degrees together. Other coursemates divided the pentagon into three triangles, then multiply 180 degrees by 3 because the total sum of a triangle is always 180 degrees.


Somehow, I find that we are actually at Level 3 of the Van Hiele model, and is doing problem solving in which properties of shapes are important components.




And here's something to share, a MindMap of the Van Hiele Model of Geometric Thought.

Developing Number Sense

Even before children starts school, most of them would have already been exposed to basic concepts of numbers and counting by their parents through counting the number of sweets, fingers, or even educational television programs and charts.

Numbers can be found everywhere, from the telephone to the clock, and even the lift. Young children who started counting may be able to count numbers verbally in the correct order. However, they may not be able to connect the numbers in a one-to-one manner where each item only gets one count.

Thus, it is important to support the children in gaining full understanding of the concepts by using a variety of activities and experiences.

Next, I will talk about what is suggested about teaching of number sense in the textbook and what is being practised in pre-school teaching.

 = already in practice

= not a common practice

On Page 128 of the text, it is suggested that any games or activity that involves counts and comparison can be used to develop children's understanding of counting, such as rolling a die and collecting the indicated number of counters. When children are able to count forward in number sequence, they also learn to count on and count back, and this will lead to the concept of number sense.

In my preschool, children play similar games such as 'snakes and ladders' and 'aeroplane chess'. Instead of buying, the teachers will get the children to be involved in making the games, exposing the little ones to numbers. 'Snakes and ladders' involves the children in count and comparison when the die is rolled, as well as counting on and counting back

               when the counter lands on a ladder or a snake.

In the section on 'one and two more, one and two less', the book suggests using materials to play the 'more or less' game to learn about the relationships between numbers and reflecting on them.

In my preschool, the teachers do use materials to introduce the concept too. By exploring

               with concrete materials, children are able to better visualize, understand, and reflect on the
               relationship of 'more and less'.

Calculators can also be used for the same concept of introducing 'more and less' in the section on 'one and two more, one and two less'. In my preschool, it is not a common practice to use such technology as the teachers are not exposed to and are unaware of the benefits of using it as a teaching tool.

The next section on 'anchoring numbers to 5 and 10' seems quite unfamiliar to me. After reading, I am able to understand why it is useful to develop relationships for the numbers 1 to 10 due to the important anchors of 5 and 10. However, it is not a common practice for the teachers to introduce the relationships among numbers using this approach, or to
               use the five-frames or ten-frames tool as an activity.