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.

After only Act 2A

• The were really clear about their reasoning for the highest unknown number. They recognised it had to be 23 or 24, but realised if it was 24 there would be two modes, 23 and 24, so the number couldn't be 24 it had to be 23.
• They were really firm on the idea that the second highest unknown number had to be 15. They recognised that this number and the number 17 made up the numbers required for the median. As 16 was the median they were clear that 16 was the centre of 15 and 17, hence choosing 15 for the other number.
• Many figured out that the remaining three unknown numbers had to have a sum of 27 to give the required mean.  They came up with multiple sets of numbers that would be able to work in those places.

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

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.

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.

• 10 blocks put into a box
• I took the box around to students, 5 took out a block and showed it to the class, this was recorded on board.  Students had to guess at what the last 5 blocks were
• Discussed guesses, first 5 blocks were all red and blue so all the guesses were also red and blue
• drew out another 2 blocks (green and another blue) and asked them to guess again, told them that two of the blocks are colours that they have already seen
• Did the reveal on the other blocks

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

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.

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

• Cost per minute
  
• Cost every 5 minutes
  
• Where they took the taxi from
  
• What their destination is
  
• Time taken to the arrive to the destination
  

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.

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.