I was stumped, where was I going wrong?  There I was standing in front of my class and I couldn't get the maths to work. There was nothing wrong with my thinking, there was nothing wrong with my calculation, but yet it still wasn't working.  It was then that it struck me, I looked at the question and realised that the maths wasn't working because the question I had put up wasn't a triangle. If the longest side was 9, then there was no way that sides of 3 and 4 would form a point above that side, they wouldn't even come close to joining. I felt like an idiot, I wanted to crawl under the desk and hide.

On that day, I had ran through my carefully prepared examples and they wanted more, so I made the questions up on the spot, I drew a triangle, put some numbers against them and tried to work it out. This is something I had done on other occasions without incident, but this time I picked wrong. It was a valuable learning experience for me as a young teacher and this moment still sticks with me, even 12 years later.

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Late last year I was doing some work on mathematical reasoning at the school, I gave the maths teachers the question above and gave them the time to figure it out. Not surprisingly they were all able to figure out that B and C were definitely not right angled, however they all used Pythagoras' Theorem to determine the answer, which was fine because I put no restrictions on it. ​​

So my next question to them was "If you knew nothing about Pythagoras' Theorem, how would you approach this question, how could you eliminate at least some of the possibilities?"  This was an interesting process, to look at at how they now approached the problem. It wasn't until I asked them to draw triangles B and C that they realised that they were impossible, they didn't need to draw them, it was just that stimulus that got them thinking about it in that way. 

I began to realise that knowing that the sum of the two smaller sides must be bigger than the largest side is not a frequently discussed property of triangles, at least from our high school perspective of shape, I really don't think I knew much about it until I was standing in front of that Year 10 class and trying to figure out why the maths wasn't working. We are often told that triangles have got three angles and three straight sides, we also are told sometimes that the sides must be connected, however I never remember exploring the conditions under which those sides would be connected and how this property extends to polygons with more than three sides.  We are told about angle sum in polygons, but don't explore ideas around the sum of sides.

This experience has taught me a few things 

1. ​We can draw a lot of things that can't physically exist, sometimes these impossible object can hinder what we are trying to achieve in a lesson, such as when the idea is being introduced for the first time,and sometimes they can enhance what is going on in the classroom by extending or consolidating their understanding.
   
   
2. My teaching back then was focused on 'rule-following' and 'answer-getting', this has limitations, it is not enough to just know how to do it when it is working, you also need to be able to figure out what is happening when it is not working. You need to reason through whether it is an error in your thinking or an error elsewhere.
   
   
3. Seemingly simple ideas such as a triangle or counting are often a lot more complex than we realise.  Taking the time to play with these ideas, to stay curious about them, is well worth our time and effort, it helps us to think beyond the obvious and undercover hidden depth and complexity.

For a number of years now a colleague and I have talked about teaching together. Over the last three years we have both taught year 8 maths, but in different years, never in the same year.  Over that time we have worked together extensively in developing, refining and adjusting our teaching program to make it more effective. We bounce ideas off of each other all of the time so it was exciting to find out that this year everything aligned so that we could teach together we also have another teacher and their class who is joining with us.  

The double class works across 3A:09 and 3A:10, and the triple class works across all of the spaces shown with tables, we did start off with the triple class in only 3A:09, 3A:10, and 3A:11 but found that they needed a bit more space.  When they are working on questions or when we are working on more open problem solving tasks then we always work in this way, one big open space. The are able to sit with anyone from any of the classes in that space, as long as they are working productively. The two or three teachers in that space work as a team to address student questions as they arise, we are not three separate classes, we are one big class. 

However having one big class does have it's limitations for some activities, when we are doing some specific teaching of key ideas it requires a smaller class. Often with this specific teaching you need to respond to what you see in the room, you can get a feel for the class in relation to whether it making sense to them and you need to respond to the questions raised this is too hard to do effectively with a class of 75 and therefore at those times we shut the walls and teach smaller groups.  However those smaller groups are not necessarily the people on the class list of that teacher.  Because each student in the large class has generally received help from each of us, they have come to understand which of us is able to help them best, so they are free to go into any of the three classes to get this specific teaching from the teacher they feel they will be able to understand the most. The examples we use in each class are always the same, we do that preparation in advance, but they just hear it from the teacher of their choice.

Why the change?
At the start of this post I talked about how we had wanted to do this for a number of years, part of this is because we get on really well, and we teach in similar ways, but from my perspective at least there were some really important reasons for students in why we trying this approach.

• We are asking students to work collaboratively and take risks in class.  In order to take risks you need to feel comfortable with the people you are working with.  You  have to be comfortable to say that "I have no idea about this" without fear of ridicule.  This allows them a larger selection of people to choose from in determining who they feel safe working with.  
• For many, having mathematical discussions is difficult, they are not sure of what it looks like, with multiple teachers in the class we are able to model those conversations and show how we build on or propose different ideas to others, we are able to model what it looks like to try and unpack or clarify another person's thinking.  We bounce ideas off of each other.
• If we are asking kids to work collaboratively they need to see the adults that work with them working collaboratively as well, they need to see us working together
• Our students get multiple perspectives on the ideas being presented.  I have found that sometimes some small differences in the phrasing of how an idea is presented can make all the difference to some students.  By having 2 or 3 teachers in the same room they can make sure they get the help they need from which ever teacher they feels works best for them.
• It allows kids to work in a space that works best for them, some prefer to be in the big open area where there is lots of people to bounce ideas off, some prefer to be tucked away in the 3A:04 classroom where they can focus their attention a bit more, this separation into distinct learning zones is not possible when you have one classroom that you are working out of.
• There is a large quantity of student ideas, it is not just the ideas of a few but the ideas of many that contribute to the discussions.

How is it working...so far
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It all seems to be working fairly well.  It has been a very new experience for us, and there were some teething issues such as how do you mark attendance when you have told 75 kids to sit with whoever, but we sorted that out.
we have learnt a lot along the way, but it seems to be working better than it normally would if we had been separate classes.  The structure seems to be working like we intended and seems to be addressing the things that we thought needed addressing. Kids are more comfortable in taking risks, they have greater agency, they are getting the help they need and they are producing some very good work. Is it perfect at the moment... no, but we need to be aware that this is still a very different for the adults and for the kids in the room and we have had enough success early on to see it is worth pursuing