Many learners struggle with Euclidean Geometry from Grade 10 because they say they cannot see it or they cannot write it down. Geometry has been a troublesome section in the curriculum for all of my classes, through each year. This is not only from the perspective of the learners trying to understand it, but also from my perspective of trying to teach it. I have been practising Euclidean Geometry for years and NOW I can SEE it! How can I teach my classes to weave through the maze that is Euclidean Geometry and learn to SEE it, but in a time period far shorter than what I have had to learn to see it?!!!
Some learners really struggle to SEE Euclidean Geometry! No matter what they try, they keep hitting a wall and are not able to achieve what they want. I am sure they wish that they could just download an app from their app store and install it into their brains to give them the “Geometric eye” (Godfrey (1910), as cited in Fujita & Jones (2002), p. 16). Unfortunately, this is not possible yet – but fortunately, this is not necessary!
The PROOF OF CONCEPT…
One of the authors of the Mind Action Series mathematics textbooks had a workshop that I attended. In this workshop, he explained his methods and ideas for teaching geometry. His ideas seemed so logical and obvious, yet I had not been using them! After implementing his methods with my Grade 11 class, I found that my learners were more responsive and had a significantly better understanding (and more importantly RECALL) of the work I had taught them. Even the following year, when those learners were in Grade 12, many commented to me that they remembered their Grade 11 geometry theorems far better than their friends in other classes. Now, I can share these world-changing methods with you! After reading this article, you will have a whole new perspective on what it takes to get that “Geometric eye“.
The THEORY behind teaching Euclidean Geometry…
There is a theoretical framework called The Van Hiele Levels that can be used in the learning and teaching of Euclidean Geometry. It lays out five tiers or levels of geometric thought. A learner would have to master each level before progressing to the next one. Let us have a look at each level:
Level 1 is called Recognition or Visualisation
This is where a learner can learn names of figures and recognises a shape as a whole, e.g. squares and rectangles seem to be different. It is important to remember that this is a purely visual skill without any deductive or inductive skills.
Level 2 is called Analysis
Once a learner progresses from Level 1 to Level 2, they will be able to identify properties of figures, e.g. rectangles have four right angles, circles have no right angles, etc. We must note that a learner has the ideas of the properties; however, they are in isolation.
Level 3 is Order or Informal Deduction
This means that simple deduction can be followed, but formal proof is not understood. This is a very important stage since it is the beginning of seeing proofs; however, this is informal deduction. Informal deduction means a learner may be able to follow a given proof, but they will not be able to write and structure it themselves.
The learner should have attained Level 3 by the time they finish Grade 9.
Level 4 is Deduction
This is where formal deduction takes place and the learner can write proofs with understanding. This means that the learner must be able to structure and write up formal proofs in the Statement-Reason format.
The learner should have attained Level 4 by the time they finish Grade 12.
Level 5 is Rigor
A learner could challenge axioms in different systems and determine if they would still be valid or not, since in the system in which the axioms are used, these axioms may break down. An example of this would be the axiom about the sum of angles in a triangle equalling 180 degrees. This holds if we are looking at normal geometry (planar); however, it does not hold if we are using a rounded surface (spherical).
This level is for post-matric courses generally.
It is important to see the order of these levels since they will clearly demonstrate the problem with how Euclidean Geometry is taught in most textbooks and classrooms.
The CURRENT PRACTICE of teaching Euclidean Geometry…
Euclidean Geometry is normally taught by starting with the statement of the theorem, then its proof (which includes the diagram, given and RTP – Required To Prove), then a few numerical examples and finally, some non-numerical examples. Have you seen the problem yet??
The PROOF is given at the beginning! It is given before the examples that help you understand how the theorem works. Therefore, a learner is being shown Level 4 (Deduction) before Level 3 (Informal Deduction)! This can cause confusion (as it did with former pupils in my class) where they use the method from proving the theorem instead of just using the theorem.
An example of this was when a former learner of mine used congruency to solve a problem examining Theorem 1 (since that is the method of proving Theorem 1 – chord meeting a perpendicular line from the centre of the circle). This meant that the solution they wrote was 6 lines long, instead of being 1 line (using the theorem). The idea of what a theorem was and how to use it was lost!
The NEW PRACTICE of teaching Euclidean Geometry…
The learning of the theorem needs to follow The Van Hiele Levels!
Level 1 (Recognition or Visualisation)
This means that a learner needs to learn the diagram of the theorem first. In my opinion, this is the most important step! This is the step that is very often brushed over very quickly, but it is the step that develops the “Geometric eye“. It is the step that will help a learner SEE the geometry since they know visually what they are looking for. If the learner does not know what it could look like when the theorem is applicable, how on Earth are they going to be able to see when to use it?!!
has an awesome animation that explains the diagrams (and subsequently statements) of the circle theorems (Click here).
Level 2 (Analysis)
This is where the learner will learn about properties, i.e. learning the statement of the theorem and the reason to be written next. Since the learner has already learnt the diagram, the statement (and reason) that they will use makes more sense since the statement refers to what happens in the diagram! This will mean that the linking in the brain of the learner will be easier and therefore, remembering it will be easier.
There are many ways that can be used to remember the statements of the theorems. Below, there is an example of using a song to remember them:
(Note: the theorem numbering in the video may be different to the numbering you know.)
Level 3 (Order or Informal Deduction)
This is where another crucial difference in this method appears! The examples start here. The learner will start with the simple numerical examples. The purpose of this is that numerical examples can be done via simple deduction or informal deduction. They are generally not multi-step calculations (and if they are, they are still short). This helps the learner to be able to practise their “Geometric eye” by identifying the necessary theorem using the diagram they have learnt. Then they can further practise the application of that theorem.
This can be seen through an example on (Click here).
Level 4 (Deduction)
The proof of the theorem should only be introduced here! This way the learner has already learnt the visual aspect of the theorem, properties of the theorem and practised using it. Now, the learner can delve deeper into WHY they can use the theorem. It will also set up the idea of how to set up a formal proof. The learner will then be able to extend their knowledge of numerical examples to non-numerical examples and show/prove questions. This will invoke more abstract thinking (which will not be a problem since their thinking progression has been correct).
has an example that shows this progression (Click here).
Level 5 (Rigor)
The learner does not need to reach this level for high school.
Learners will be able to process and assimilate the new information being taught far more easily if learning it can follow these levels and will be able to structure it in their minds far better! With such a perfectly neat structure, Euclidean Geometry will never be a worry again!