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

1. Despite the different names given to the types of assessment (e.g. tests, assignments, practice problems) they all looked the same, every sheet of paper that I gave to them over the course of the year pretty much looked the same apart from a few isolated exceptions.  For the entire year I was really only requiring them to do one type of thinking which was follow the example given and repeat n times.
   
   
2. The amount of work that I was giving them to do was excessive, there is no way they could have possibly got most of it done

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...

1. Better questions 
   The kids I teach still get practice problems, I feel they still have a place in the maths classroom, but these practice questions are now much better thought out. Previously I may have turned to a page in the text book, or printed out a worksheet, but all of those questions looked the same. They could do a lot of them because the thinking didn't change from question 1 to question 20.  But I am trying to really design question sets that tackle the thinking from a range of perspectives that challenge them to think about the concept in different ways, that encourage them to challenge themselves and to push that understanding further.  When question sets are designed in this frame of mind, they are not going to complete as many questions as they may have in the past.
   
   
2. More problems
   I see questions and problems as different, they are not interchangeable terms. In my own mind questions are those types of mathematical experience where you are practicing something you have already learnt as consolidation, the difference is that previously how they learnt it came from me, it was my procedure, but now how they learn it comes from them, it is built on a base of conceptual understanding, and that is where problems come in.  Again in my own head problems are those mathematical experiences for which you don't immediately have a way of solving it locked away in your head, it is new or unfamiliar, it feels a bit uncomfortable.  These types of questions were rarely approached early in my career but have become a massive part of my current teaching, and like the questions above, I think very carefully about what problems I put in front of them, they need to develop the underlying grounding and framework of the concepts they are looking at. Kids find such an enormous amount of satisfaction in working on these problems (although it still takes a lot of time to get them used to thinking about maths in this way), on moving from not knowing to having a solution that works for them.  This solution may not look the same as others in the class, but they know it works, they know how it works and they can apply it consistently. They have spend the time developing the concept in their own head through their interactions with others.
   
   
3. Greater Collaboration
   Again back in first year of teaching, getting students to collaborate on tasks was rare, they did the occasional group project.  But now that idea of collaboration is much more important, they are working more on problems, not questions, they need to bounce ideas around off others, they need to see how others think about it. When you are working on a difficult problem, one you are unfamiliar with you generally want to test your ideas against what others are thinking
   
   
4. More Reasoning
   Mathematical reasoning was also noticeably absent in my early teaching but again with the change in teaching and the move to more collaboration and more problems, reasoning has become much more important, they can't keep it all locked away in their head, they need to get it out as they are engaging with the task.  I rarely ever asked why but now the why is central to what we do.  What works does not matter if we don't understand why it works, and what works doesn't matter if we cannot communicate why it works to others.  If the aim of the problems is to develop conceptual understanding then unpacking how they think about the problem and helping them to connect their ideas with the ideas of others becomes vital, it is not the procedure that constitutes the conceptual understanding, it is the connections they make with other ideas to form new ideas. In this environment I want my students to be able to confidently and competently convince me and others of their line of thinking

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.

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.