"No, we are right, the data is wrong"

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.

The problem

Set up as a 3-act style task the video opposite was Act 1.  I filmed 20 rolls of a 30-sided dice but made sure they could not see what the last 5 of those rolls were.  They needed to figure out what the last five numbers rolled were.

In Act 2 they were first given the image below on the left (ACT 2A - click for a larger view) which was the numbers in roll order, numerical order and with the calculated values of mean, median and mode for the entire data set.

ACT 2A

ACT 2B (VIDEO)

ACT 2B (IMAGE)

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 task

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

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.