I am in my 14th year of teaching, when I look at how my own teaching it is sometimes hard to know just how much it has changed. But sometimes some small moments that bring that into real clarity, for me this moment was when I was clearing off my hard drive to make room for new stuff as it was getting too full. As I was going through my hard drive I sorted it by age, I figured that if stuff was going to go, it was probably the oldest stuff that hasn't been accessed or edited in a long time. I realised that I still had on there all the stuff I developed in my first year of teaching over 14 years ago. I was curious about the work I gave to my students back then, I was curious about how far I had come since then, so I had a look at it. When I looked at it I noticed a few things
When I reflect on that work now, I realise that at that time I didn't know any better, I didn't intended to do this to my students, I didn't think i was doing a bad job at the time as it was how I learned maths, it was what I learned at university in my preservice teaching program and it was what was expected when I was a student teacher. All of the models and examples around me for maths teaching that I was using to establish myself as a teacher were all saying the same thing. So out of interest I printed out just the practice problems that i am going to give to my students in the coming term of 10 weeks and then printed out all of the practice problems that I gave to my students on the same topic all the way back in my first year of teaching and the gif at the top of this post is the result of that. 14 years ago, for my 10 week unit of work, I was giving them 70 pages of practice problems. When I then looked at what I am planning on getting them to do the page count came out at 8 pages of practice problems across the 10 weeks. Just this little bit of data has made me reflect on what has changed in relation what I value now that I didn't value then. I am not expecting any less learning from them, so how am I spending the time that those extra 62 pages would have taken up. This additional 62 pages morphed into...
So 2 stacks of paper and 14 years later I have realised I am now asking them to do less busy work, so they can engage in a lot more, and a lot deeper thinking.
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The last 2 weeks of my life has been monopolised by one question, a question for which I am still not confident in any answer, a question asked by my 5 year old daughter who just started school this year. One day just after I had got home from work and I was beginning to make dinner she asked me "Daddy, how do you count to half?". The question immediately intrigued me, because as a maths teacher I felt I should be able to answer that question confidently, after all it is just about counting, but I found my self unsure of how to answer her. I really wasn't sure if the answer was " you can't count to half, one is the first counting number" or whether to tell her "you can count to half by counting by numbers smaller than one", I just wasn't sure if a number less than one could be denoted as a count. So I didn't answer her at that time, I told her what a great question it was, it was such a good question that she was able to trick me, even though I teach maths. But I promised her that I would also speak to some other maths teachers to see what they thought about the question.
I went to school the next day and asked every maths teacher at my school the same question, and they all gave me the same look, kinda puzzled, but also very intrigued, it was clear that, like me, they had never thought about counting by anything less than 1's. Most of them were willing to give an answer, but they did not give the same answer, and when I spoke to them about the other thinking I had on the question they also became unsure and reconsidered their initial position. I also asked a lot of primary colleagues in different schools as I figured that they ultimately had the responsibility around teaching counting so I felt they would have a better understanding than me. But again, they could see both sides of it,. I went to my resources on how concepts develop and could not find anything there, I even went to Google and could not find the answer anywhere on there. When I told my daughter all of this she was really amused that all of these teachers could not answer it and even Google and Siri couldn't answer it. So as a final ditch effort I took to twitter and tried to tap into the collective knowledge of the MTBOS, a global tribe of mathematics teachers.
But again I had no luck, lots were intrigued by the question, and lots told me how I could talk to my daughter about it, but that is not what I was after, I know how to talk to her about these things, I wanted to know whether they felt that half was a number you could count to.
The discussions have continued for quite a while now, and are still going, the more I think about it the more I think i have settled on an answer, but as one of the teachers I have been working on this closely with put it "even if 90% of the people I talked to disagreed with my position, this is one question where I don't think I would be convinced by that enough to change my mind".
I still feel like this question only has one answer, I feel that if I was asked to count what was on the plate and asked 100 other people then we should all get the same answer, but that has just not been my experience with this one innocent question from my daughter. I think if she asked most people, it would be so easy for them to give an answer without giving the answer much thought, they wouldn't even think twice about the complexity of the question, so I am really glad she asked me first. It made it really clear to me about about how closely we need to listen to these questions, even from learners who are just embarking on this learning journey. What was clear from this question was that she had been thinking a lot about counting and also the number half, she had figured out that half was a number, but had never used it in counting. She had identified what was a gap in the concept for her, but interestingly enough it also made me discover that it was a conceptual gap for me to as I had never considered it. 
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
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