equations session 5 - eval

Final and most focused session yet.  We looked at calculations by exploring the link between rectangles and multiplication.
Far more homework appeared this week but I didn't really have an activity ready to show it to others and so missed that opportunity.  It's great that so much was done. I would really like to check what type of activity has happened.  Some ideas that seem to head in the right direction. Homework is posted on one of 3 or 4 walls depending on what type of activity they did or what type of question they explored.  This could raise the profile of exploration and facilitate a way to show off work and have others benefit from it.
Rectangular Number Islands was an activity that explorers were prewarned about.  I'm not sure if this had an impact or got enough from the video or watched it.  We needed to explain the activity quite a lot.  It was not necessarily a bad thing though as we got to see how explorers responded to the task and some of their assumptions.  It allowed the next activity to have clearer instructions.  Out of a group of 10 or so, it was only 3 or 4 who knew enough details to explore.  The rest had to ask or wait until we asked how they were doing.  The beauty of this as a creative activity, lots of explorers can get on with something on the map.  Also, there were a lot of examples of peer checking and support once people had put rectangles down.  Lots of types of conversation went on in the session.  They checked each others rectangles and helped with calculations.  Once it started to take shape they shared other aspects of the graph and drew pictures.
We had a few minutes at the end looking at involving rectangles in nested calculations.
First match calculations with rectangles                                                            
See if I can make new numbers by combining different types of calculation.
We didn't get on to explore the different calculators' responses to the some of the questions.
We had some interesting discoveries.  O was fantastic.  He tried a lot of different things with the rectangles to actually explore how the maths worked.  He had a subtraction and one of the adults helping the groups showed him how putting one shape on top of the other could help him see the difference.  O asked about minus numbers and said that he wasn't so sure of them but was confident enough to ask.  While initially reluctant to devote energy to understanding that, he tried a few things and then asked a few questions about it such as, how could we write it if it's negative ("Oh, it's the same as minus")  or "How do I know if it's minus or not?"
We had a quick plenary session where we talked about discoveries made.  All made possible by the power of the circle time I suspect.  It would be great to have a book of discoveries or some other kind of hall of fame type thing for explorers to have their achievements recorded.  A book that is regularly on display perhaps or a photo with a shield that can be written on and a picture can be taken with it.   The big work on display for explorers to look at at the end was also a good idea.

number relationships 4 - combining operations - eval

Our second session in our new home.  We have started trying to do our work in circles or at least communicate as a group in a circle while sitting on the floor.
This worked well.  As well as giving us the opportunity to get explorers to think about circles, it was a good way to get people's attention and work as a group.
We continued to begin the session with a recap of last week's activities by working through the booklet and arithmagons.
We focused on understanding multiplication represented as rectangles. We spent time finding rectangle numbers on a hundred square.  We had to choose a number and say what multiplication could have made it.
I set the challenge for explorers to find two rectangular numbers to make 17.  There was quite a mix of responses to this.  Some went with Rita and did more work on finding rectangular numbers first.  They worked well and had useful group discussions.  Many of those with me explored the topic well trying out different possibilities and finding some. There were a few who did not engage with it though. I'm not sure why they did not attempt this but it is going to be something that needs preparing for ready for next week.
One explorer chose to explore something else.  She solved it but it may have been better to encourage more participation at the beginning and moving on after a few tasks.

number relationships or equations session 3 - multiplicative - eval

 Our first session in our new home!!
We started by looking through the booklet of last week's session. We tried making number bonds to 9 sometimes with more than two rods and explored those ideas a little.  We introduced arithmagons that we didn't get a chance to do the previous week.  It is a good way to focus on the maths session after the games.
We were looking at multiplication. The first exploration was working with rectangles.  We tried to construct a numberline and match rectangles with their numbers.  There was opportunity to look at this in more detail and it was an activity that worked well for the explorers as a group.  We would have needed more blocks to spend more time on this but it would have been worth it.

From there, we went on to play the multiplication grid challenge.  The hope was to consolidate the ideas from the rectangles.  It was maybe a bit fast for that to happen.

We also looked at prime colours.  This was a bit bolted on and more could have been done with this if we had the chance to compare with the rectangles.

Plenty to explore but perhaps not enough time with each to make it possible for explorers to make something of the task.

equations - evaluation session 2 - number relationships - addition

Possibly a more directed session than the last but again with some successes in pure exploration.  We looked at number relationships based on addition (and subtraction).
Some things that I think need to be done in planning activities is to detect any particular outcomes that I am hoping explorers will find.  How many questions could they pose?  What models are currently available? How many are accessible and to who?  What else could be taken from these activities?  What meaning could there be to the explorers?  What contexts are relevant?
I wanted to get explorers to have a general appreciation of the relationships that exist between sum and difference.  I had three representations that I wanted to promote: equations, cuisenaire rods, number relationship triangles.

We started with dominoes and dice. Finding matching sums was ok for the group but not ideal as not everyone could actually get one example each time.  It was a useful activity but would be better with more dominoes or less people.

We split the group up after the introduction and explored cuisenaire rod number bonds.  It didn't work as well with the younger group as they identifying the rods as numbers was a challenging enough step for them and so it wasn't possible to add another task on top.  Working with dots on the dice was useful for them.  Rita ran their session and it was very successful.
We didn't get around to looking at the idea of difference much.  I started some of them on the towers puzzle but it didn't catch.

Explorers enjoy finding more examples, especially if they are in competition with others.  It is the next stage of exploring these examples that requires more thinking about.  Activities that encourage a decision can be a good start.  From there, explorers need to defend their decision or debate it.  This shouldn't be too difficult to set up once courses have been initially explored.  A discoveries board would be good with some incentive to post ideas there and multiple ways to contribute to the board.
The best thing is to get explorers to find a question beyond what has been presented.  I suppose that this can be incentivised or at least promoted as a goal that they can achieve.

Equations 1 eval

Numbers may be the most identifiable aspect of maths but calculations are the advance that created the discipline.
To focus on memorising or understanding is useful but how would it be to start with exploration.
Most explorers were able to evaluate the calculations we used but calculators were on hand for the others or for checking.
There were implied questions set up at each stage.  The first was what is the pattern?  What type of calculations produce negative numbers or decimals/fractions? Many explorers skipped this part of the activity and just found cards, brought them to the table and didn't think much more about it.  The activity would have benefited from more focus on the working out an answer and categorising.  I think that posting the calculations into a box of some kind might have helped raise the profile of that part of the activity and make it worth a child's mental activity. This was set up as an observation but it was still a useful thing to do at this stage in the course.

A recap of that activity and a new way to explore those patterns would be good at some point.  Perhaps revisit that activity in the last session.

 My aims were to get our explorers to see fairly specific things and I set activities up to support that aim.  This works to a degree but it is not really exploration.  It is far more like exploration training.
The model for this session was 1) Find calculations on cards, 2) find an answer for the calculation, 3) categorise the calculations by a type.
We started looking at ways of representing the calculations that we found.  Again, this was a bit too contrived and lost meaning or at least took on a personal meaning for me. It needed to be more about meaning that the explorers give it and discussing and sharing that.

There were some successes in terms of getting some of them used to the numbers on Cuisenaire rods and even build up to showing and understanding number relationships.

If you are set on exploration, you must accept that nothing may be found.  I don't know if I am ready to accept that or if I set the scene for an appropriate or manageable level of exploration. That would be a good thing to try to find out on this course.

Evaluation - session 5 - place value/number systems

We explored something behind the ideas of using place value as a number scheme and the effect of different place values or similarities and differences.
We started by listing all of the coins or notes that can be used in the UK. Explorers listed as many as they could on paper. I had examples of each of them and so I laid them out in a tabular form with different place values being in rows and the first digits being in columns (1, 2, 5 are the only first digits in current UK money.
I asked why and we looked at how easy or difficult it is to count in those numbers (1, 2, 5 and 10, 20, 50 etc.)  compared to other numbers.
From there we went on to see how the base 10 number square organised these tables and noticed that they were all in columns but non coin/note values were all over the place.
I moved on to getting explorers to create a little book of different place value based number squares.  Explorers went for it in a very exploratory manner. Quite a few parents were on board for this class and helped out. This was really useful and might be a good way to set up a session.  A particular family learning type session where I set up activities, resources and challenges and give some advice about how to let children explore and how to encourage them to go further.

It gave me a chance to work with some of the more reticent explorers which showed me the importance of a part like this in every session.  I would need to think about what I could do with those explorers who have no problem doing whatever comes into their head but don't check how they are doing.

The extension task was quite engaging and even parents seemed intrigued by it. They had to create one for base 12.  Explorer C61 did fantastically (with the help of R98) and managed to complete the table, refusing to give up until she did!

Evaluation Session 4 - Place value/Number systems

Did this session achieve its aims?  I'm not sure how well defined the aims of this session were. Much of the planning is something like a sculpture.  I start preparing activities without fully seeing the kind of experience I hope to create.  This continues into the sessions where I am always hoping for inspiration to help determine the next step.

We started by looking at the numbers that form the sequence of powers of 2.  Some of the explorers new about the game 2048 and so could say something about that and others could work out other interesting things to say about them.  Many others did not contribute.  I wonder what is best in these situations.  A think-pair-share routine might be a good thing to use for these parts.  Or think-pair-group-share. This could be linked with a badge acknowledgement system.  Contribution is not for everyone and it may be that allowing people not to contribute is just what they need.  Exploration must happen outside a comfort zone but by what means should someone get there.  And which part of the comfort zone should they be outside?  All of it?  unlikely but for the most hardened explorer and then that is probably just a different comfort zone.
I digress.
We went on to use the numbers in a slider game something like 2048.  This worked well and I noticed that girls and boys had very different approaches for this game.  Not surprisingly, boys tended to be competitive and girls collaborative. (or maybe that is surprising).
It worked well for all the groups and they managed to achieve at least up to 32 on each table.  The idea here was to get used to doubling these numbers and understanding their value in relation to each other.
We went on to look at exploding dots 2 to 1.  The challenge was set.  What is the maximum number of cubes that you can put in the system so that no cubes are pushed out side of the grid.  Google exploding dots to see how it works if this makes no sense.  There were still issues using this idea butt quite a few made good discoveries (including the set up discovery that any time there was just one cube, it corresponded to the numbers from the game.  That would have made a good challenge question.  What numbers end up with just one cube somewhere in the grid?
from there we looked briefly at the binary calculator and how to calculate to and from binary.  To give explorers a chance to use it, I set up some codes using the ascii grid.  Explorers tried to create some codes and two of them broke my code.
A varied and active session, with games, challenges and questioning
Almost a model maths explorers session.