PROOF: What Do You See? How do You Know?

It was a great week.  As a teacher, you know that you have had a great week when one of your 14 yr old students starts a class discussion like this:

Mrs P, will you be my maths teacher forever?  I've always hated math, but with you I not only seem to like it, but I get it as well.  I am making connections that are really cool!

Even better, the rest of the class joins in on the discussion.  This is the same week that I returned the Year 9 exam, discussed as a whole class how to critically reflect on their exam results and use their exam as a point of growth for Year 10.  It was also after  teaching the skill of factorising by grouping in pairs, and then refactorising by finding the common algebraic expresson factor.  I am pretty stoked if a student came out of that lesson feeling proud!

I teach a whole range of ability students in years 8 and 9.  When I teach, we not only look at what we are learning, but how best to learn what we are learning.  A couple of years ago I took the online Coursera course called

 'Learning how to Learn - powerful mental tools to help you master tough subjects"

delivered by Dr Barbara Oakley, Professor of Engineering at UC San Diego.  Since taking this course, I make teaching the strategies for learning a part of my weekly lesson, revision, and student learning objective.  Many of the students I teach are surprised that there are actually techniques that they can apply to improve their learning.  Once they realise that they can actually learn, the growth mindset begins, and successful learning follows.

 

How YOU  have made my week of teaching so positive 

This blog post is dedicated to many of you - my online community - who share with me your best tools for learning.   The Coursera course is just one element of professional learning that has improved me as a teacher.  This week, I was able to use a number of other resources shared by YOU.

The Year 8 topic for the last two weeks was Geometry.  We started with just a mind map of what the students could remember from previous years' learning.  My students had just returned from a two week break, so their brains had not fully woken up.  So I gave them a hint:

     Objects.................Properties...................Relationships..............Language

It took a while, but in their groups, they started to recall prior learning.  And they appreciated having the first couple of lessons just to review the technical language for angles on parallel lines and triangle properties.  The learning was helped by this contribution from David Wees, Formative Assessment Specialist at New Visions for Public Schools who, with his team provided a simple visual  representation for proving the angle sum of a triangle.

This activity proved a nice challenge because it used algebraic representations of angles.  When trying to work out each of the missing angles, some of the students who had prior knowledge of the angle sum of a triangle tried to use that - but I clarified that they could not, as they had yet to 'prove' the angle sum of a triangle was 180 degrees.

The foundation notion of 'what is proof' was set.  Within the geometry unit, we went on to even more exciting lessons that involved extensive discussion, conjecture, and debate.

Scootle provided the next learning activity with its 'Quadrilateral Explorer' interactive. 
I like to use this interactive to generate small group sharing, followed by whole class discussion.  The Scootle 'Quadrilateral Explorer' activity has never failed to generate intense discussion, confusion, and then clarification on the properties that define specific quadrilaterals, and why a square is a rectangle but a rectangle is not a square!

With the concept of 'Proof - What do you see and How do you Know?  set in place, over the next few lessons my students continued to:

• proving the angle relationships on parallel lines by measure (protractor)
• Conjecture and Proof that the exterior angle sum of any polygon - by algebraic equations
• Prove the conditions for triangle congruence (SSS, RHS, SAS, AAS) by construction and comparing (students love a measurement cut and paste activity).  This activity also produced the counter example proof that SSA was NOT a condition for the congruence of two triangles.

And Finally,  an introduction to deductive proof using a statement-reason table.  Inclusion of a fun Ted Ed video

 

All in all, I think my students have had a range of learning activities during this 12 lesson module that allowed for learning maths using tools, concrete materials, small group sharing, whole class discussion, challenge, confusion, resolution, collaboration, and visual representation.   The learning focus of reasoning geometrically about properties of triangles and quadrilaterals using by connecting prior learning with precise mathematical language to build new learning was achieved -  and it was a lot more fun than standing at a chalkboard!