Where I live in my part of Australia we are down to our last two weeks of school for the year before their long break over Christmas. Over the year I have been really trying to build their confidence in their own understanding and in the answer they are presenting as being a correct solution. A favorite phrase of both myself and Angela, the teacher I am team teaching with, is "convince me". When they come to us to check an answer, we don't want them to come to us for a tick or a cross. They know we are trying to build a culture where we examine not the answer, but the thinking that led to that answer. It is not their job to get the right answer, but to convince us that their answer makes sense. If the logic leading to the answer is sound and makes sense, then it also follows that the answer should also make sense. What I have found through this process is often is pretty easy to derail them. This is not something I do on purpose but often happens as I critique their reasoning. When I ask them a question about their solution they automatically assume that the answer is wrong rather than seeing it as me wanting to know more about their solution. Early on in the year I found myself needing to tell them it is correct but also let them know that I was not clear on how they got there, that there were gaps in the reasoning, knowing it is correct gave them the confidence to justify it further. As the year has progressed I have stopped telling them whether it is correct or in correct and have had them determine that through their justification. Their confidence with this is still developing, it is hard to really push your position firmly on whether it is correct if you have the uncertainty as to whether that really is the case. But I want that for them, I want them to get to the point of being confident in their mathematics and being confident in their reasoning even if they are not sure about whether they are correct. Today this culture was really strong in class, the confidence of our students in their answers and in their reasoning was firm, there was no way of derailing it, not today.
They first needed to use this information to figure out as much as they could about the missing numbers. This first part of Act 2 is not enough to figure out all the mystery numbers so it was only then that I showed them the video in the middle above (ACT 2B) which was how the mean, median and mode changed as the mystery numbers were added. The many ways that my students rocked this taskAfter only Act 2A
After Act 2B This one stumped them a bit at first, they didn't initially see how it could be useful, but in drawing them together and getting them to determine that they could use the mean effectively in that situation to help, many were away and gave it a good shot. The story I want to tell here is of two students who were able to use this Act 2B to find the 5 mystery numbers. They were really excited about it and went to check their reasoning with Angela. Angela had found a flaw in this Act 2B in relation to how Excel handles these calculations. I am not going to give away too much on this as I didn't pick it up and it is a nice one to think about, but I would be keen to hear about it in the comments. So when these students went to Angela to check and she asked for them to convince her, she said that she was not convinced as there was a part that did not seem to follow the pattern. When they noticed the inconsistency they thought they were wrong and went back and checked it all. As they began check it something interesting happened, rather than thinking there was something they were missing they started to get more and more convinced that they were correct and they said No, we are right, the data is wrong. This was amazing to me. They had developed so much confidence in their answer that they were happy to say that it was not them that was wrong, it was the way that the Excel was calculating the answer that was wrong, they were doubting the calculator instead of their own thinking. They were right, it was wrong, they had found a flaw that I knew Excel had, but I had failed to consider in designing this task, and I now owe them a chocolate which I am more than happy to do. I couldn't be prouder of my class today and it makes me really sad that my time with them this year is coming to an end, but it also makes me really clear of how far they have come this year.
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A lot of the activities I have talked about on this blog, or the ideas I talk about, are generally ones I have put significant time into. The activity I am talking about in this post has been a very successful one for me over the last few days. I would like to say that it was because of the careful and deliberate planning I did on the task prior to the lesson, but that would be a lie. Sometimes as the lesson is unfolding you see an opportunity present itself, and by following it through, sometimes planning the next step on the run, you can have a really good lesson. Planning the next step on the run was not a result of being disorganised, but a result of identifying an emerging need and recognising the need to follow up on it before moving any further.
They then had a conversation with their table groups about what they thought would be the colour of the other 5 blocks. Not surprisingly all the responses were some combination of red and blue blocks, however despite knowing that there were lots of other colours of blocks in the storage tub, they had not considered they could be part of the final five.
Next I wanted to introduce the idea of using probabilities to describe exactly what is in the box so I gave them the information listed below and then gave them the time in groups to have the discussions required to figure it out I have filled the box now with blocks according to the criteria below, can you tell me how many blocks of each colour are in the box? What was exciting about seeing them work on this was that I finally saw the classroom culture that I have been trying to build all year with them. In working together on the problem they were talking about the problem, they were critiquing each other's reasoning, they were asking questions of each other, they were willing to tell the group when the explanation still didn't make sense to them, forcing the person giving the explanation to justify their thinking more strongly. I think one of the most important aspects of their work though was their confidence with their answer. With these sorts of questions when they tell me they have the answer I try to ask a few questions to head their thinking down a line that creates some doubt that they have found the answer. This isn't done to trip them up, but is more designed to see if they have got to a point where they feel the have considered everything and have come to the only answer that works, to gauge their confidence in their own thinking. Normally when I ask students a question about their answer they take this as an indication that their answer is wrong, but this time, no matter what question I asked them about their answer they had confidence with it as they had determined that 12 was the only possible number of blocks as you could not have parts of blocks. The next day the aim was to move towards students being able to determine how to calculate the probability of pulling a block of a certain colour out of the box, to help facilitate this i made up a simulation of the box compostion from the previous day using excel. The excel file and a screen shot is shown below.
Like the day before i got students drawing the blocks from the box and this time we tracked the probabilities. when the first green block was drawing I asked them to predict what would happen to the probabilities of each colour.
Overall what started as a sidetrack, developed into some really great thinking, on some really important concepts and I couldn't be prouder of that lot today.
in looking at all of the aspects of maths that are expected to be covered over the course of a year I feel that statistics offers some really great opportunities. It allows a really authentic way for students to get into some real maths with data that matters to them and to others. It is not that there is not authentic ways to engage with the other aspects of the course and get into some real maths, it just sometimes seems like it doesn't seem like an authentic situation for them, there is not the emotional buy in. With statistics you have the opportunity to introduce some very provocative data sets or ones that directly speak about them. You can have some really great discussions about how statistics are used to make decisions, how others use them to make decisions that effect you, to look at how you or other people can use them to build convincing arguments (even if the stats are quite deceiving). I am at the very start of my unit on statistics, but I wanted to start it with them having a discussion about some data, their own diagnostic testing data. I chose to use this data for a because sometimes I feel they don't see the use of doing the testing, they don't always treat it properly and that skews our data. I also chose it as I feel that they do not always understand the results of the data, I wanted them to have much more awareness of what the data tells them and of how we use the data as eduators.
The last person to talk picks the next person to talk (from those with a counter out) by throwing them a ball or similar, only the one with the ball can talk. Once a person has used all of their counters they then cannot contribute any more to the discussion, they can sit and listen, but that is it. The teacher in this process is a facilitator, but tries to stay out of the discussion as much as possible. They provide the provocation to start off with, they pick the first person to talk and they will ask questions to stimulate the discussion if it dies off completely. In this role you need to avoid jumping in to help with that I tend to give myself the same number of talking stones that everyone else gets, if they only have three opportunities to add to the discussion then I only have three to stimulate it, it forces me to be strategic as well. This is hard when you completely disagree with what is being said, but you need to let students be the ones to respond them and to challenge that thinking.
I gave them two prompts to start their contribution with they could start with
I am so proud of the kids I teach today. Since the start of the year we have had a much stronger focus on patient problem solving than they might have been used to in the past and until today there has been a lot of resistance to attempt some of the tasks. But today they were superstars, it is the first time where almost all of them had persisted in solving a problem, even when they found it difficult. There had been pockets of this in the past, but to see most of the class doing it today was fantastic.
The lesson that we decided to look at was based on an ad produced by the motor accident commission in South Australia. The idea behind the lesson, which they came up with at the start, was to find out how long they were actually in the back of that taxi. For the sake of trying something a bit different we combined three maths classes with three maths teachers to see how it would go. Students were then asked to generate some ideas about what is the sort of information that they might need to know to answer the question and they came up with
From here they need to have some information in order to be able to progress forward firstly was how they calculate the taxi fares. Students were given the formula below, but were not shown how to use it at all. Cost of the fare = Distance cost + Waiting Time Cost + Flagfall It was great to see how students had made the decision that they needed to compile information from several sources realising that they needed to use the time of departure to find out what tariff they would use and then using the tariff information to relate back to the distance traveled. This continuous movement between different sources of information was really great to see. The path they took from this point was pretty different with each group. some were able to work through it themselves and some needed some further prompts, but very few actually gave up on the problem. Even when they did not have much success initially they really took on the prompt and tried to incorporate it into their thinking and that is why I am so proud of them today.

Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
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