Addressing Diagram and Definition Misconceptions

The first chapter of Putting Essential Understanding of Geometry in Grades 9-12 is focused on student misconceptions with diagrams and definitions and how to address them. Student "difficulties and misconceptions include the naive or inexact use of formal terms definitions, naming conventions, or symbols; an inability to link verbal and pictorial representations, and an underdeveloped spatial reasoning." If someone asked me a month ago what mistakes I see students making repeatedly in geometry, I would have listed several of the misconceptions mentioned in this quote.  So, what do we do as geometry teachers to help better support our students in these and other areas.

The authors indicate "4 Stumbling Blocks" when it comes to how students learn geometry.

1. A lack of prerequisite knowledge and skills
2. A tendency to interpret diagrams as pictures, not as representations or relationships
3. An inability to link verbal and pictorial representations
4. A lack of precision in the use of terms, definitions, and symbols. 

To address these issues, the teacher must first identify where the students lie in the van Hiele levels of geometric reasoning. The majority of students in my geometry classes fall into levels 2 and 3. (The book refers to the lowest level as level 0.) The level of thought expected by a secondary geometry course would be level 4, which means more scaffolding will be needed to try to get my students to the level at which they're expected to be performing.

The authors then refer to van Hiele suggestions for how to help students progress through the levels.  The suggested process for helping a student progress from one level to the next has 5 phases.  Students start by working through an inquiry or investigation and make conjectures.  The students then investigate the topic further using materials designed by the teacher to help reveal relationships. A discussion follows in which the teacher and students restate conjectures using precise, accurate language. In the fourth phase, students complete more complex tasks with the aim of more thoroughly investigating the topic.  Lastly, students review and summarize what they've learned.  (This 5 step process is referred to throughout the text.) 

Things I noted from this chapter:

1. Students may see diagrams as static images instead of the dynamic relationships they represent. 
2. Students may also struggle with the multitude of symbols that are used to mark congruent parts or parallel/perpendicular lines, among others.
3. Students may encounter problems when naming angles or other objects.
4. Students may have problems translation between diagrams and definitions.

Several activities are included in the text to help address these issues.

Here's what I've decided to change in my classes:

• SCAFFOLD!  I am admittedly weak in this area. 
• Use the 5 step process described above to help students deal with particularly difficult concepts. 
• Allow students more time to develop definitions for figures.  Revising definitions needs to be shown as an essential part of the process, not criticism.
• When writing definitions, consider this:  A definition should exhibit "precision in terminology, isolation of the concept, essentiality, noncontradiction, and noncirculation." These means a definition should only draw upon previously defined terms or undefined terms in the most concise way possible. It should clearly separate examples from non-examples of the concept such that all properties mentioned in the definition can exist in the object being defined.  Lastly, and obviously, the definition should not include the term being defined. 
• Allow time for students to practice drawing diagrams given a definition (and vice versa).
• This is a BIG one!  Take the necessary time for students to practice a concept instead kowtowing to the belief that everything in the standards must be taught. 

This is all going to amount to a dramatic shift in my teaching of geometry.