Sunday, 23 October 2016

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:
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!

Sunday, 9 October 2016

Desmos Learning: The Inscribed Hexagon

Making connections with  Desmos…..The Inscribed Hexagon

I get so excited when other educators share their learning experiences on @desmos with me via @MtBos or other blogs!  It motivates me, excites me, and the first thing I want to do is suss it out myself! 

Here is one construction I created in response to a calendar problem solution posted in NCTM Mathematics Teacher (vol.110, Number 1, August 2016) presented by Scott Smith in the Reader Reflections   I loved the solution so much that I immediately wanted to re-create it in Desmos!

Inscribing a regular hexagon into a unit circle.

When I play/create/explore/learn using Desmos, I have to think not only from the perspective of myself as the learner, but also myself as the teacher of my Year 9 students who will be the learners.
  • What might my students be challenged by in this activity?  
  • What might they be able to do  based on their existing knowledge
  • What questions might they ask when stumped?
  • How might this activity extend their thinking?

Questions /ideas that came into my mind:

  1.      How do I graph a  unit circle ?
  2.     How might I locate the vertices of the hexagon as coordinates on the circle?
  3.     Should I use Cartesian or polar coordinates?
  4.     What are polar coordinates?  Do I know about them to use them?
  5.     Can I use Pythagoras or trigonometry to find the coordinates of the six points?
  6.     How can I find the coordinates using function notation?
  7.     How can I represent my circle as a function?
  8.     How can I use the point coordinates that I have graphed to draw a line?
  9.     How can I transform the lines that I draw using the point gradient formula?
  10.    How can I change the intervals for which the edges of my hexagon are graphed?
This simple geometric construction, and all of the thinking (as a year 9 student) took me about 1 hour to construct.  And the learning I did was phenomenal!  I started with something simple, started asking questions, explored, realised I did not have enough knowledge, when back to the knowledge I have, then practiced and applied the learning I have done over the past few years using basic functions and simple coordinate geometry!

After thoughts - could I have done this as a simple compass and straight edge exercise - how?  How might you use a simple construction activity like this one to teach other topics?   

How many different ways are there to inscribe a hexagon into a unit circle?  I would love to know your thoughts and ideas - share your experiences in the comments section.

This activity really excited me as a teacher - and I believe it will excite my students as well.  Can’t wait to use it!

Monday, 3 October 2016

Why now? Reflections and a call for your advice!

More on beginning this blog....

In teaching, I take my inspiration from all the connections I have in my online PLC.  My professional learning community helps me to understand and apply better instructional strategies to bring forth better learning.  My online Learning neighbourhood shares with me the best ways to  use technology as a learning tool (thank you Desmos) and not just as an informational resource.  My linkedIn connections (Daniel groenewald, Tom Barrett ) remind me that I have worked with some amazing, professional educators.  And at the end of a long week, if I am feeling frazzled, I know I need only to turn to Twitter or my weekly RSS feeds to find like thinkers, passionate educators who will put me back in that positive mindset about why I love teaching mathematics!

Reading tweets ( @aitsl, @d_groenewald,@edutopia) and blog posts (Desmos, Dan Meyer, Samjshaw) help me to connect and enrich both my own learning and that of my students!  It's a win win for me - I learn, I'm inspired, I teach better lessons, my students are inspired, grades improve..............and yet.........

Even with the great progress I've observed in the learning achievements of students as a result of implemented researched based instructional strategies, I feel that many teachers on the ground are still missing out! "Too much content to get through", they say, or "need to make sure they are ready for their end of year entrance exams...".

Sometimes I feel like I may not be making a difference.

So my questions are.....

     1.      How can I more effectively share my experiences, failures, triumphs, ideas, questions and curiosities with others?

     2.       What suggestions do you have to effectively share your learning about best practice?

     3.         What is the best way to generate more discussion within my school based teaching community about the best way to teach a particular concept, topic, or idea?

And this is why I am going to build this blog.  If I can connect with even one maths educator as a learning partner, then I will be happy😊.