Mathematics always amazes me in terms of how something simple can produce something seemingly magical. Andrew Blair over at inquirymaths.co.uk has been a real inspiration in this. He states that:
Inquiry Maths is a model of teaching that encourages students to regulate their own activity while exploring a mathematical statement (called a prompt). Inquiries can involve a class on diverse paths of exploration or in listening to a teacher’s explanation. In Inquiry Maths, students take responsibility for directing the lesson with the teacher acting as the arbiter of legitimate mathematical activity.
I think this idea of self-regulation is a really key aspect. Now this doesn’t mean the absence of teaching or the absence of curriculum content. I have found many times that teachers have a belief that mathematical investigations/inquiries are somehow separate from the day to day mathematics and the curriculum objectives that they need to cover.
But actually, let us take a step back and consider what maths is in the first place. I happen to really like this definition:
Mathematics is a collection of extended, collaborative
games of ‘what if’, played by mathematicians who make up sets of rules
(axioms) and then explore the consequences (theorems) of following
It is this game of ‘what if’ that has been the source of many lesson ideas for me. To give an example of that, let me show you how one lesson came to be.
My Year 6 class were studying angles, I asked myself, what if I take a rectangle and draw a line at the same angle from each corner?
Immediately, some interesting things crop up. I have a rectangle in the middle and triangles and quadrilaterals surrounding it. I wonder if that is always the case. What if I wanted to know all the angles in the intersections between the lines and the lines and the edge of my bounding rectangle? Are they determinable?
This is essentially what my class set out to investigate. As mentioned before though, self-regulation is the key. Borrowed from Andrew’s excellent site is the use of regulatory cards. Working in a primary school, I use a set of cards that I have feel work well for me but do look at various different options here. Here are my set:
In keeping with the definition of mathematics above, I have a ‘What if…?’ card.
That’s because I want them to take this initial idea or prompt and go in their own direction. An obvious one would be, what if I used a different bounding shape? Or what if I used different angles – a, a + 10, a + 20, a + 30. This can be so empowering. The children are able to challenge themselves at an appropriate level, everyone capable of making different discoveries. A lot of the fear of maths evaporates because they became the expert on their line of inquiry.
But embedded into this lesson are some really useful curriculum objectives:
- draw given angles, and measure them in degrees (°)
- angles at a point and 1 whole turn (total 360°)
- angles at a point on a straight line and half a turn (total 180°)
- other multiples of 90°
- compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
- recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles
And who knows where else it will lead us.
This is the power of playing what if. Just trying out things, seeing what the consequences are can produce astonishing results. Sometimes little experiments and doodles produce great mathematical discoveries. The Ulam spiral is a very well known example of this.
Doodle yourself, try a what if, and see what you discover.
P.S. It has been a long long time since my last post. Hopefully not as long until my next one!