Garthe Jones

I am one of the publishers here at Mathletics with 10+ years of experience teaching high school students.  I am look forward to hearing your thoughts and ideas on this and other topics in maths.

Constructing Reasoning Skills

There are many fantastic geometric software tools out there now that allow teachers to develop dynamic environments where simple interactive tools can be set up to change variables and allow the consequences of such changes to be observed immediately.  They can be extremely simple to use and many good teachers encourage their students to build their own great demonstrations to display their understanding of concepts. These wonderfully powerful packages are understandably becoming the norm in many classrooms.

Despite all this genuine praise for such software, I sometimes question whether their increasing use without the support of ‘archaic’ pencil and paper methods is leading to students losing a key step in their mathematical reasoning and spatial development.

Geometric software allows students to demonstrate advanced skills by building their own proofs of concept using the tools available in the package. These tools are generally based around the setting up of formulas and parameters using functions written into the program.  For example they might simply use a command that instructs the software on what they want such as  “bisect [a,b,c] ” to bisect an angle.   Important mathematical skills in terms of being methodical, sequencing steps and using algebraic techniques are all definitely getting well-oiled here.  However, using this method to build a geometric proof relies less on students actually understanding and experiencing geometric concepts or constructions and more on an algebraic approach.

The bisector of an angle construction is a simple example of what might be lost by moving too quickly to this approach. It connects nicely to the line forming the diagonal of a Rhombus. I have witnessed the ‘mental light bulb’ moment going off first hand with this exact example and as a teacher it was wonderful.

The thinking about the bisector from many good students was:

‘Ok, so  if I draw another two equal arcs that cross each other from the first one I have drawn I actually get a rhombus, and the diagonals of a rhombus bisect their angles’. This is a strong connection with the geometric theory.  Another great response from a student was along the lines of ‘I thought I could only draw circles using a compass’.

You can also show students through the steps with a geometric package but I am not convinced yet that the connection would be as strong. I am by no means implying that similar moments are not achievable using software tools however the ‘pay-off’ learning moment from personal experience, specifically in the topic of geometry, seems to be so much stronger for the students when achieved incorporating both methods together.

This following application I believe demonstrates how pencil and paper geometric constructions paired with a software package can lead to setting off an investigative spark in many students.

The Euler line is formed by a combination of three geometric constructions. Euler found that the orthocentre, centroid and circumcentre of a triangle are collinear.

The different techniques needed for each construction can really engage students when they get it to work. The question can be raised by students if this will always be the case for any triangle. This is where the software can be used to step in and continue this inquiry-based learning by allowing them to drag the vertices of the triangle and see what happens for all different shaped triangles. Furthermore a challenge can be then presented to the students to see if they can make all three points lie on top of each other. If they can, comment on the triangle that allows this to occur.

I hope teaching pencil and paper geometric construction skills will not be left behind to become a forgotten fossil of Mathematics. It appears to be increasingly fading out of syllabi around the globe yet to me has a real purpose in building the framework of good mathematical reasoning processes within students. Especially since the requirement to complete formal geometric proofs still sit firmly within many syllabi  The introduction of calculators so many years back did not mean performing mental calculations were entirely left behind. I hope the same happens for pencil and paper geometrical construction tools so they stay on as an important supplement to the great software demonstrations now available.

Garthe Jones
Secondary Mathematics Publisher, 3P Learning
B Eng Tech (Electronics & Communications) UNE
Grad. Dip Education (Secondary Mathematics) UNSW