When angles co-exist, connections and relationships are formed.
I read through last year's angles planner to gain a sense of where I wanted the children this year to go with their understandings.
I caught myself thinking that and thought, should it be where I want them to go? Shouldn't it be where they want to go?
If we want the children in my class to develop a passion and authentic curiosity in maths, their wonderings have to drive the learning, not my preconceived ideas nor a maths scope & sequence document.
I remembered seeing a brilliant YouTube recently by one of my gurus, Kath Murdoch: What makes an inquiry-based teacher? and so thought rewatching that would help inspire me to rethink the approach my class and I could take this year.
I'm really glad I rewatched.
1. Because I always feel so inspired by Kath (to the point that a few years back I had the privilege of having a discussion with Kath and my team at Munich and found myself almost 'star-struck' by actually being with her that I found it unfortunately difficult to contribute to the discussion. Kath- if you happen to read this- forgive me; I'm not an imbecile- much!)
2. Because of what she said here: Inquiry teachers "know how to sit with the student and carefully scaffold. They know their curriculum so well that they can move with the students' interests in order to come to that curriculum. "
I looked at last year's planner and thought how I do know the curriculum related to angles and since I experienced the types of excitement and passion the children had last year, then I should trust myself and apply Kath's words.
We are a fair way into our angles unit now and the discoveries and sense of pride and wonderment the children are experiencing is fabulous buzz.
I didn't provide lines of inquiry like I did last year. We have kept the enquiry more open and far more student-driven.
An example is how last year we all cut corners off triangles and played around with them to find how they are connected to a straight angle. Whilst that is a really interesting connection for children to make, this time only the students who have chosen to enquire more deeply into triangles and their angles' connections made this investigation.
I listened and discussed with that small group about how they were playing around with the angles of a triangle and what they were discovering. They had just excitedly discovered that all the angles of every triangle add up to 180° and were trying to find out why that is. One student amongst them wasn't so interested in that why question so she was instead wanting to find out if there was a similar pattern with the exterior angles of triangles - brilliant! I thought of Kath's words and thought about how I could steer them to towards this connection with straight angles.
I asked what they had already thought of doing with the triangles and if they could see a connection with them adding to 180°. One student explained how she had just discovered how you can always fit at least two triangles into every quadrilateral so she felt there must be a relationship with their angles. She wanted to find out that connection more to which she suddenly thought: Circles are 360° and the other group just told me that all quadrilaterals' angles add to 360°. That wow moment grew more excited as she then exclaimed: Wait! Half of 360 is 180! There is another connection! And a straight angle is 180°!
I asked the group: Do we think there might be a connection between triangles adding to 180° and a straight angle being 180°?
They thought about this and felt it is plausible.
What could we do to find out?
No ideas yet.
Maybe we need to investigate more creatively. Let's see what is on our table that we could use to be creative with these angles.
Suddenly a student had an idea upon seeing the scissors he thought: We can cut the angles off the triangle!
So that's what they did. They started playing around with their cut out angles.
I brought over our display image of a straight angle and placed it on the table.
Light bulbs started flashing as they played more with the angles and suddenly could find the reason they were both 180°.
Later they excitedly shared this discovery with the other group exploring the angles of quadrilaterals. (I knew that group would benefit from this understanding and might further spark more wonderings for them to discover)
Instead of the whole class making this discovery, so far only a small amount of us have.
Does it matter with two-thirds of the class never make this discovery in this unit?
I don't think so.
What does matter the most is that each student's wonderings are being valued. They are all discovering different aspects of our central idea and that is what is the aim. Belongs each child gains an appreciation and understanding that angles have amazing connections with each other, it doesn't matter what particular concept they grasp.
What also matters for this unit is that each child gains a deeper sense of wonderment about maths patterns and connections.
The other group who are curious about the angle connections between quadrilaterals had discovered why quads add to 360° and a circle is 360° by being inspired by the thinking of the triangle enquirers!
This type of learning is what makes enquiry-based maths so great. Students feeding off and inspiring other students to deepen their learning.
We have discussed why we are enquiring into angles. I asked the children to show with their fingers (1 being low, 10 high) to how useful in real life having our new knowledge about angles is.
Many showed between 6 and 8 fingers. Others hovered between 1 and 4.
We listened to different theories why we thought so.
Have we ever seem our parents measuring angles?
- We might estimate angle sizes a lot in real life, but for the average person they don't ever measure them we theorised.
Our discussion came to the realisation that unless we plan to be a builder or an architect, the likelihood of using this knowledge is pretty slim.
So, why are we enquiring into them?
It's so important children know why we are learning something especially in maths so they can connect more deeply with their thinking.
We shared our ideas and came to the realisation that maths is about playing and that as mathematicians we can find joy and amazement in finding patterns and connections. Maths is like art. We enjoy learning how to paint, but is it likely that we will paint a picture for our living room when we grow up? So why learn to paint? How can we connect that reason with playing and exploring maths patterns and connections?
That discussion inspired quite a few of us and helped us gain another key understanding of what mathematical thinking is about. It's not about getting the right answer. It's about making explorations of a remarkable system created by people and all because some people decided we should make our number system based upon 10. We thought back to our first maths unit finding out about our base 10 number system and how it came to be. If we had a base 3 or base 12 number system, could we be making these same amazing discoveries about angle connections?
As mathematicians we can play and have fun making our own discoveries.
And that is a wonderment about maths!
When I look at our current angles planner, it has evolved a lot.
It began simply with the provocations, some pre-assessments and the central idea. The students have driven it completely with just that and the learning is amazing!!
It's no longer teacher-directed like last year's.
It's student driven and student owned.
And because of this, the sense of pride and confidence as mathematicians has soared even more than last year's learning.
Letting go is sometimes difficult to do as teachers, but as Kath points out, as inquiry teachers we simply need to believe in our knowledge of our curriculum and explore different strategies to help our students make their discoveries that leads to the curriculum rather than the curriculum leading the students.