If Only Clairaut had Dynamic Geometric Tools

If Only Clairaut had Dynamic Geometric Tools

Sue Pawula

Summary

      This discussion centered on Clairaut’s historic-genetic approach and how it can be combined with technology through using dynamic geometry software to better afford student development of conceptual foundation based on exploration, discovery, and explanation before they are exposed to formal geometry proof.  CCSSM expects middle school students to use informal debate to establish facts about the angle sum and exterior angle of triangles, angles created when parallel lines are cut by a transversal, and angle-angle criterion for similarity of triangles.  Clairaut believed that students new to geometry should focus on big ideas instead of propositions whose truth may be discovered by concentrating on the details. The three ideas focused on were 1) ”in any triangle, the greater side subtends the great angle”, 2) “a circle does not cut another circle at more than two points”, and 3) “if two circles touch one another then they will not have the same center.”  Teachers could pose questions to challenge students such as “it has been said that the sum of the angles of a triangle (any triangle) is equal to the sum of the angles of any other triangle…is this true?”  Following that students should be expected to use the dynamic geometry tools to help them visualize that a particular theorem holds even when factors change. The software tools will allow them to recognize, explain, and generalize geometric properties through interaction and exploration on the computer where they can drag the vertices of a triangle.  Teachers should coordinate and organize the flow of student thought, prompting students to recognize invariants. Student discussion and reflection should follow as to what they observed and rationally decided about geometric properties. Students are better able to become dynamic learners that discriminate between variants and invariants when they are allowed to explore and identify geometric ideas through the use of the dynamic software.  All of this requires the teacher to guide student reflection and encourage them to think back on the geometric meaning they have garnered from this exercise.

Reflection

Students can begin building a much better foundational knowledge base through hands on activities such as this.  The new geometric software is amazing and can really allow students to experience the changes that occur and yet factors that remain the same, through the simple movement of a cursor.  Reading the article and looking at the diagrams made me understand more fully what they were requiring students to experience and the understanding they would walk away with after the exercise.  As always, the teacher has to be mindful of any misconceptions that may have taken place and guide the discussion to make sure student understanding is complete and accurate.  This is a great way for students to construct knowledge on which to scaffold.

Chang, H., & Reys, B. (2013/2014). If only clairaut had dynamic geometric tools. Mathematics Teaching in the Middle School, 19(5), 280-287.