The van Hiele Levels of Gemetric Understanding

 

Here’s some notes I took from the PowerPoint and PDF article:

• Pierre van Hiele and Dina van Hiele-Geldof (husband and wife team of Dutch educators)
• The theory explains why many students have difficulties with geometry, especially with formal proofs… writing proofs requires thinking at a comparatively high level, and students need to have more experiences in thinking at lower levels before learning formal geometric concepts
• Most geometry teachers are on a 3^rd or 4^th level
• Students should progress through level 2 by end of 8^th grade
• The levels are numbered differently in the PowerPoint and the article.  Listed below is the way that the van Hieles originally numbered them, while Americans started numbering them from 1-5 instead of 0-4.  Using the 1-5 scale, also allowed Clements and Battista to add level 0 as “pre-recognition”.
• Level 0/ Visualization: child may think a square is not a square, simply because it’s been rotated and has a point on the top (for example), teachers should engage students in activities that let them sort different shapes and identify the shapes in front of them, building, drawing, and putting together shapes
• Level 1/ Analysis: students can make generalizations about specific shapes, but cannot make generalizations about how the properties of different shapes relate to one another, students can list properties of the different shapes but are unable to see that they are sub-classes of one another (for example, squares are rectangles, and all rectangles are parallelograms), students can think of shapes in a class, instead of just what’s in front of them, activities should allow the students to identify the properties of the shapes, not just the shape itself, use physical models, make property lists, and discuss specific conditions that define a shape
•  Level 2/ Abstraction or Informal Deduction: able to understand abstract definitions and can meaningfully classify shapes into hierarchies, for instance, a child would know that if it’s a square, it’s also a rectangle because a square has all of the properties of a rectangle, activities should include tasks where properties of shapes is an important component, use informal deductive language that includes the terms: all, some, none, if… then, what if, etc., the relationship of polygons should also be investigated to establish if the converse is also valid, exp: if a quadrilateral is a rectangle, then it must have four right angles, but if a quadrilateral has four right angles, must it also be a rectangle?
• Level 3/ Deduction: this is where instruction should be at in our high school level classes, students can begin to construct proofs using postulates, axioms, & definitions
• Level 4/ Rigor: instruction at this level typically starts at the college level, students can work in different geometric or axiomatic systems
• What’s My Shape? Activity: I was doing fine on this activity until the very last question on the third example.  I was down to two different shapes, and the question was “Does the shape look like a kite?”.  The answer was no, but I thought the correct answer actually did look like a kite.  Did anybody else think this?  Granted it’s been a while since I actually flew a kite, but I guess it goes to show you how particular you have to pay attention to the questions you ask, because they can be perceived differently than you had intended… which I think is a lesson we all learnt when studying about Data and coming up with a workable survey question.
• There is a definite relationship between the NCTM Curriculum and Evaluation Standards for School Mathematics to the van Hiele theory, which is shown on page 7 of the article Dr. Higgins posted.
• Having viewed the PowerPoint and read the article, I have to say that I’m a little more at ease about teaching geometry to my students now.  This may sound crazy, but maybe I’ll even do well teaching geometry to elementary students, since I think I’m only a level 0-2 myself!  Furthermore, I really appreciated the part of the article that stated: “The van Hiele theory indicates that effective learning takes place when students actively experience the objects of study in appropriate contexts, and when they engage in discussion and reflection. According to the theory, using lecture and memorization as the main methods of instruction will not lead to effective learning” (page 7).  Once again, stressing the point that students learn math the best when it’s hands on learning and discovering the solutions through their own observations!