This week in math was super interesting. We talked about geometry and spatial sense! In her book "Making Math Meaningful to Canadian Students K-8, 3rd Edition", Mariam Small describes this strand as, "one, but not the only, aspect of mathematics where visualization is important". She also discusses how teachers and students enjoy this particular strand because there are many opportunities for engaging hands-on activities that, in my opinion, can benefit both visual and kinesthetics learning styles. Essentially, Small (2016) states that geometry is the "study of shapes and spatial relationships". I will explore more of these notions below.
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The van Hieles Theoretical Model of Geometrical Thinking for Students
There are 5 sequential levels, starting with level 0 which begins at the primary level and progresses.
0. Visualization - This is the first phase for students when learning geometry. Here, they view different shapes and they have to recognize the differences and classify the shapes. Orientation and symmetry. Examples would be students can arranging basic shape blocks and using Tangrams.
1. Analysis - Here, students try to understand the different shapes. They must know the properties and attributes of shapes that define the shapes. Students study the shapes and analyze. Properties are now more important than the simple appearance of the shape. Examples students can do is sort shapes into groups like which triangles are obtuse and which triangles are symmetric. Students can also use Geoboards to make shapes themselves.
2. Informal deduction/Abstraction - Students now understand shapes and compare them to each other, they understand and recognize relationships. For example, questions you may ask students are 'how are squares connected to parallelograms?' and 'how are shapes different from one another and how are they similar?'
3. Deduction - This phase requires higher order thinking. Here, students can apply theories and use logical processes to understand concepts and definitions.
4. Rigor - Abstract thinking is involved and can extend to higher level thinking, for example it can be used and applied in University courses. Here, students must be able to use geometric reasoning to solve complex problems and issues that arise in the geometry strand.
Please review this great video for more clarification on this theory:
Activities that Help Students Understand Geometry
I found one activity in particular that my colleague presented to be very useful in presenting students. The activity was titled "Navigating Geo City" and it was composed of a cartesian plane grid with some coloured shapes distributed on it. The goal of the activity was to answer questions based on the shapes' locations and then practice doing reflections. The activity was very well presented because it was so colourful and easy to follow. This made the worksheet visually appealing and very engaging!
Ties to the Ontario Math Curriculum
We can tie this into the Ontario Math Curriculum. For the geometry and spatial sense strand, the overall expectation in Grade 4 states, "identify and describe the location of an object, using a grid map, and reflect two-dimensional
shapes". Grade 4 would be a good grade to implement the second level of van Hieles' theory of abstraction, where students recognize relationships between shapes and compare them to each other.
Other Resources to Teach Geometry
Before we end, I would like to share this video we watched in class. Using virtual story telling is a great way to get students engaged in the material, especially while setting up context and background for a math lesson during the Minds On activity. This video would be great for primary learners in stages 0 and 1 of van Hieles' theory.
That's it friends, enjoy exploring these resources and have a great week!
Teddy
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