Teaching Geometry for Deeper Understanding Using the 5 Van Hiele Levels

What Needs to Happen at Each Level and How to Incorporate Activities that Lead Students from one Level to the Next

What makes geometry concepts so challenging for some students?  There have been many teachers who have attempted to make the formal understanding easier with new programs and methods.

​Sometimes they are successful, sometimes not so much. A huge challenge can be tackling two-column proofs, which are seemingly detached from practical life applications, for the first time.  It can seem like a grueling task to teach and to learn.
 
Here is some insight connecting to the Van Hiele levels for learning Geometry. 

​If you know what makes this subject difficult and what level your students are currently at, then you can use these suggestions to figure out what you can do to help your students learn.

PART 1:  So, Why is Teaching (& Learning!) Geometry So Difficult? 

​Not enough proving in early education
 
The foundation for learning geometry doesn’t start in high school. It starts in elementary and middle school. Students need to learn how to start proving and explaining why things occur before they hit high school level classes. This helps lay the groundwork for geometry as well as other subjects in high school. If students were introduced to simple informal proofs and required to reasonably justify statements, they would be far more prepared for the formal proofs to come.
 
Of course the specific geometry concepts wouldn’t be on the same level, but introducing the pattern of thoughts earlier is better. Students need to know how to explain, prove, and show why long before they are in high school geometry.
 
Low Levels of Conceptual Understanding
 
We all know that a semester flies by. Teachers aren’t able to spend the amount of time needed to truly cover all levels of understanding needed to be successful. More than that, because geometry concepts aren’t being introduced early, there are always some very confused students. While some students can pick it up easily, others are left grasping at straws of understanding.
 
The Van Hiele levels help us get a better grasp on this lack of understanding that too many students experience. The Van Hiele Model is a theory which describes how students learn geometry. It is based on the doctoral work of husband and wife, Dina van Hiele-Geldof and Pierre van Hiele.
 
According to their model and other research, students enter geometry with a low Van Hiele level of understanding. Learning complex concepts like proofs require a much higher level (level 4 of 5) and most students enter the class at a level 2 or 3. Students without basic knowledge or the ability to back up statements with reason are easily set up to fail in class.
 
General Cognitive Development
 
Even when notions of geometry are introduced early, some students just aren’t ready on a developmental level. Jean Piaget and his theories on cognitive development help shed some light on this. An individual needs to achieve the formal operational stage in order to understand, formally reason and build proofs.
 
The problem with this is that some students haven’t reached that stage yet, since it can span from adolescence into early adulthood (ages 11-20). That means that if a student hasn’t reached that level of cognitive development, then it will be pretty hard for them to understand geometry. Even though there isn’t much we can do about this, knowing that it is a factor is important. 

Part 2:  About the Van Hiele Levels for Geometry Learners

​The levels are based around the idea that students can understand geometry visually at a young age, then from there develop the concepts behind the properties to the point that they can first just identify, then think more abstractly about the principles.  They get to the point of analyzing a figure or shape.  Eventually, they then progress to the point of making deductions. Students at the higher level know how axioms, theorems, and converses operate enough to extend the properties.  At the highest level, students are able to think like a mathematician.  They realize that axioms are more arbitrary than concrete, and therefore can finally extend their thinking to accept non-Euclidean geometry.
 
A student’s level is based on his/her experiences, not age.  Adding a series of experiences that allows students to interact with geometric figures helps them to move on to the next level. 
 
Another thing to keep in mind is that all the language and vocabulary that you use has to match the particular student level.  Always use correct terminology, but be aware that if you do not support your students with explicit vocabulary instruction to be sure you are all understanding the terms as you use them, you’ll lose some students.  Those who fall behind will only be able to memorize and scrape by.  They will not fully understand, and this can prevent them from ever achieving the next phase / level.
 
Even students of two different levels working together may have a hard time communicating about the geometric properties at play.  Partners will need to be using the same level of vocabulary to work together effectively.

Part 3:  Ways to Teach Geometry for Deeper Understanding Using the Van Hiele Levels

As it goes with most learning, the earlier the better.  A head start in the early grades is the only way to make sure students have a better van Hiele level when they enter high school geometry. It’s imperative that we improve basics of learning in elementary and middle school for abstract and relational levels. How can we incorporate this?
 

• Visual recognition in elementary school (grades 2-5)
• Drawing practice (for accuracy)
• Practice the relationships of different shapes (grades 6-8)
• Hands-on activities (with manipulatives), ideally with some level of inquiry / exploration
• Teaching students to explain, prove and show why

 
Geometry is a subject that builds upon itself.  A student who is lost at the beginning  is likely to be lost for the duration of their school career. That’s why it’s imperative to have underlying concepts taught throughout all education levels.
 
Here are some examples of what you’ll see at each level, and the types of activities that will help students solidify their learning in order to pass on to the next level.

​Level 0: VISUALIZATION
 
Understanding:  The child can recognize and classify individual shapes based on classic examples.
 
Thought Process:  The shape is a rectangle because it looks like a box.
 
Challenges:  When figures are turned a different way or do not fit the “typical” visual presentation, the child may not think it qualifies (examples: The child may think that a very thin scalene triangle is not a triangle since it’s not the classic equilateral shape they learned.  He/she may think that a square that is turned is not actually a square, but instead is a “diamond.”
 
Activities for Middle / High Schoolers stuck at Level 0:

• Practice transformations (to see that shapes are congruent when turned, etc.)
• Use geometry software to explore triangles, parallel and perpendicular lines, angles, quadrilaterals, and circles.  When students can drag vertices and display measures, they can really begin to grasp how the properties of each figure work.

​Level 1: ANALYSIS
 
Understanding: The student can identify figures, or components of figures.  He/she accepts basic, separate properties of geometry.
 
Thought Process:  This set of angle measures is from an isosceles triangle because two of the measures are the same, and the sum is 180 degrees.
 
Challenges: The student may have difficulty progressing from one property to another.  The sequence of steps may be foggy, and the ability to “see” where to begin in a more complex figure or a problem that requires two properties may be lacking.
 
Activities for Middle / High Schoolers at Level 1:

• Play games that allow students to convert words into diagrams and figures into words, like this "Pass It On" relay game.
• Use hands-on inquiry activities as much as possible.  Here is a sample set for circle theorems and another for triangle sum theorem.

• Allow students to practice just identifying situations where a certain property may be at play.  They need a lot of practice with recognizing figures and reinforcement of the properties before they can begin to apply the more abstract concepts.  This "twisted fingers" game is like tabletop twister for practicing special angle pairs along a transversal.
• Take time for clear, concise notes.  During lecture time, be sure to focus in on new vocabulary terms.  Build up the basics.  Try "doodle notes" that blend graphics and text to help students remember big ideas and new terms. 

​Level 2: ABSTRACTION
 
Understanding:  A student can draw conclusions and is able to order and connect properties.  He/she understands conditions and can begin to USE deductive logic, but does not yet understand the meaning of deduction or how to formally show a proof.
 
Thought Process:  This figure has two pairs of congruent sides, so it could be a rectangle or maybe a parallelogram…. Ah, but there are no right angles, and opposite pairs of angles are congruent, so it must be a parallelogram.
 
Challenges:  The student may have trouble extending a definition beyond its most basic interpretation.  It’s difficult to explain or follow a deductive reasoning sequence that represents a formal proof with many steps.
 
Activities for Students at Level 2:

•  Incorporate challenges in which students have to use multiple properties layered upon one another to find missing measures or solve problems.  Angle puzzles offer a fun and challenging task that fits this level of learner well.   This set reviews  for properties of angles in triangles, and here is another for circle theorems.​

• Try card sorts that require students to use a variety of theorems across a diverse mixture of input (words, diagrams, measures...) so students make connections between geometric properties and have to sometimes take multiple steps before they can successfully classify a figure.  This set combines properties for triangles, while this one reviews quadrilaterals​

• For additional practice, use games where students have to make decisions based on the properties of a figure.  For example, in this game where the goal is to cross off four squares in a row, the student can decide which way to classify each triangle (by sides or by angle measures).  This requires them to determine both before they play if they want to use any strategy at all for what they cross off next.  

Again, be sure to include a variety of input information.  Use some cards with diagrams, some with word explanations and definitions, and some with measurements so that students get a blend of practice with all forms and can build mental connections.

​Level 3: DEDUCTION
 
Understanding:  The student has a grasp of reasoning & logic and can communicate this to others.  He/she is able to construct formal proofs.
 
Thought Process:  Since a implies b, and b implies c, then a implies c.  In this case a is true, so I can conclude that c must be true as well.
 
Challenges:  Students in phase 3 understand theorems, undefined terms, definitions, and axioms, but have a fixed view of them.  They see axioms as concrete and have difficulty comprehending non-Euclidean geometry.
 
Activities for High School Students at Level 3:
 

• Explicitly teach your students logic, including conditional statements, conjectures, laws of syllogism and detachment, ​etc.  Focus on the skills and understanding behind deductive reasoning.  In your practice activities, be sure to include examples in all formats: some in logical language (p --> q, some in day-to-day examples about students / animals, etc, some in geometry terms, and some in algebra terms as a review.  The variety of examples will help students connect how logic works in geometry to how it works in other areas.

• When you dive into proofs, be sure that your students have the basics of logic and deduction down.  Then, start with the postulates that they will need for their first proofs.  Instead of jumping right into geometric proofs, check out this approach that offers more scaffolding and fills the gaps that often lead students to struggle with the first few weeks of proofs.

Level 4: RIGOR
 
Understanding:  The grasp of geometry has been extended to the level of a mathematician.  The student has an understanding non-euclidian geometry and is able to compare the study of geometry to other areas.
 
(Most of our high school students won’t get to this level yet.)

---

​You can’t just throw a baby into water and expect them to know how to swim. They need to see the water, touch the water, practice and experiment with and experience it. Then they slowly learn the more complex aspects of swimming such as floating, manipulating limbs to move, holding your breath under water and so on.
 
That is much like geometry. Students who have never had an actual understanding of the basics in geometry are going to be lost. They can easily drown in the onslaught of information that is geometry. No matter what level of math is taught, at some point a lesson that includes basic geometry is not only needed, it’s crucial. Be sure to give your students the time, space, and solid practice that allows them to progress from one level to another!

For more tips on teaching math, updates, and resources, enter your email here:

To Read Next: