Gotta Love Math!

Familiarity with both the van Hiele levels of geometric thought and with Bloom’s Taxonomy is vital for the geometry teacher.  The two models seem to be parallel.  In Blooms, the first level is knowledge; in van Hiele, it is concrete.  Both deal with the knowledge of basic facts.  What is lacking is anything beyond the knowledge – that is, little or no experience in manipulating their knowledge.

In Blooms, the second level is comprehension, earmarked by being able to summarize information.  In van Hiele, the second level is Analysis, where students are able to determine relationships between ideas. These correlate because you have to be able to analyze what is important before you can comprehend it. The next level in Bloom’s is Comprehension.  In this level, students can work with a new, abstract idea by connecting it with past knowledge.  In van Heiles next level, Informal Deduction, students have a working, common sense knowledge of geometry.  They may not be able to formalize their learning, demonstrating their understanding with common rather than specific terminology. They are, in effect, using what they know to understand more complex ideas.

Bloom’s describes a level called “Analysis”, in which students are able to break down the components of a concept or idea and show how the parts of the idea are related.  This would correspond to van Hiele’s level of Deduction, where students are proving ideas deductively. Finally, Bloom’s lists the evaluation level, where students demonstrate an ability to make informed evaluations based on self-developed criteria. The top van Hiele level is Rigor, where students are able to maximize their geometric understanding by comparing and analyzing the systems used to describe the processes used. The correlation here is that students are understanding the parts of the concept and are able to analyze and justify their decisions.

Teachers of geometry would want to be highly familiar with both systems in order to guide their students to a high level of competence and facility with geometric concepts.  They can use the van Hiele levels in helping students of different ability succeed.  Keys to this are encouraging students to work together in problem solving; designing questions to help students understand the next level; and most importantly, design educational tasks that move students through the levels.  Lessons plans should include the concrete stage, move into determining relationships, to having a working knowledge of geometry, to being able to express their ideas in formal geometric language, to studying geometry with rigor, and finally seeking to apply and prove theorem.

It is important for a teacher to have predetermined questions to encourage student growth.  After reading the Perimeter and Area Using the van Hiele Model, by Carol E. Malloy, the following  questions were developed that will assist students in moving to the higher levels of van Hiele:  Can you compare the different strategies for adding to the perimeter?  What is your analysis of the affect of adding a block to a corner?  To a side?  What criteria would you use to evaluate your process through this activity?  (could be asked before or after the activity.)  How can you verify your response?