Prime Colours - Session 3

Explorers are reminded of the system.
Elicit anything that they know about prime colours.

Our goal is to learn to tell numbers as prime colours and vice versa.

First in two groups attempt to put them in numerical order with colours facing up.  Rules, touch one card at a time.  You can look but must be put down colours up.

Discuss which numbers are easy to remember and which ones harder.

Create a web of prime colours.  link each card with other cards that are just one disk away from them.  Start with numbers up to 20.  If those are completed, they can be given more cards to add to their web.

When finished explorers can turn the numbers over to reveal the numbers and a number web.  They can then talk about the connections and how one could travel on the web. What's the quickest way to get from one number to a different one

Session 1 - Number sense - planning

Introductions - say name and describe an activity that you've done this week.  the rest of us have to find maths in their description.

What would a world without numbers be like?

Activity 1 - What are numbers? - research activities that others have done on this idea

- imagine a world where there was no counting.  What would be different?  What would be better?  What would be more difficult.

Task one - teach Jason and Rita how to count.  We will act like pre numerate people, we will be taught to count but get it wrong in as many ways as we can think of (counting the same one twice, missing some out, counting blank space, not understanding that there is one thing separate from another)

Task them with becoming prenumerate

Tell stories about your day so far and you cannot include any idea of a number.
Role play - act out a situation (given situation or make up themselves) where the people cannot use numbers

Task - How do you tell which has more without counting - Stones or blocks of different colours

Task start to develop a number system either using sticks or stones.  Produce numbers 1 - 5.  See how quickly people can identify the number.
Maybe we could introduce number words as clicks (3 clicks means 3 etc.) also and they can identify the cards that way.

Plenary discussion - What are the most difficult aspects of life for non numerate people?  What has made things easier for you today?  What other developments would you need?

My entry into maths

This week, I've had a few reasons to recall a time in primary school at around age 8 or 9, when I discovered maths.  It was a discovery much like those medieval explorers who upon arriving in populated lands claimed (or had claimed on their behalf) discovery.

And so it should be.  I had travelled roads that were new to me using maps that warned me of dragons and other terrible monsters lurking in wait.  I continued going down those roads, finding dead ends, trying other routes and generally finding out new things about this maths at every turn.  I was an explorer and a discoverer!

So anyway how did I discover all of this maths that had already been discovered?

I was in year 4, which was called year 5 I think in those days and possibly even year 2 a few years before me.  I do not know how I came to be sitting in the library while the rest of my class was elsewhere but I was there.  I'm also not sure what I was doing but I imagine some generic form of work.  My class teacher, Miss Hutchins I think, was doing number investigations or surveys with children in my class.  So it was that I got to observe these tests for an afternoon.

I only remember one of the puzzles.  There were a number of light blue blocks and a number of orange blocks and we were asked if we could tell which colour had more blocks without counting.

I must have become interested in this process at some point and started paying attention to what was happening.  I guess Miss Hutchins noticed this and so started including me in the discussions.  No one was able to do the no counting puzzle.

I came up with a way to do this perhaps 4 or 5 children into the survey.  I had been included in a few children's attempts at it and had had a lot more time to think about it than any of the other children.  I remember solving that puzzle to this day though.  I don't remember much else about the other tasks or anything else I did in that year of school.  I do remember that research had shown that preschool children were more likely to be able to work that puzzle out.  I remember my teacher commented to my mum that since that day I seemed to like her more than I had previously.  I did not at the time have the impression that I was good at maths or smart.  At some point after that, I did but I don't know how much of an impact that test had in creating this impression for me. But it seems like a turning point to me now.

I have put together a story something like the following to make sense of this event.  I learnt something fundamental about numbers.  So fundamental that it was too easy to teach. But an idea that with out some degree of meditation on it, without the process of coming up with it yourself to solve a problem, you might not be able to discover as an important idea.  Because you have far more sophisticated tools to work with numbers, you could take the foundation upon which they are build for granted and lose much of the ability to create something new in mathematics.  I really am thankful to Miss Hutchins.  I can't imagine that I was aware of that at the time but must have shown it on some level.  Maybe just being aware that someone had taken an interest in me and I had been able to reward their faith in me.  I don't know but I remember her favourably to this day.

The idea has been with me ever since as well as the appreciation for understanding the fundamentals of whatever it is that I'm pursuing.  This is an aspect of maths education that I feel is not as well supported.  Fairly easy to understand why.  How would you get students to value for example, exploring why 2 + 2 equals 4?  It takes real insight to see the importance of this kind of investigation unless you have a problem you are interested in that requires that understanding.

Whilst preparing a programme on number sense and doing some consultancy with a young lady who is having trouble finding and activating her mathematical tools, I thought more about this and discussed it with a few people.  I have a sense of the power that I was given at that time though I'm sure that it was not a single event as in my story nor can it be used to solve everyone's problems in maths.  It holds some important lessons that I intend to delve into further.  I'd love to hear if anyone uses activities or has a programme that focusses on the real fundamentals of maths also.  Maybe even a discussion of what the fundamentals of maths are.

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.

Place value session 3 - evaluation

What did I expect on going into this session? Some of my concerns were the amount of instructions needed for Explorers to be able to benefit from the tasks. As a group, they aren't the most attentive. I realise that M28, P79and R2D are far more likely to stop and listen attentively. Cat and joy pick things up excellently, as they did today if they're in the mood for it. 007 is on and off. If c61 is there, she's always involved. The littler ones need a more instruction drill, drill type of approach which may mean either that they should be doing less involved tasks or that they are not ready to explore. Our founding principal is that everyone is ready to explore from birth but not all ready to be influenced in this direction.
The games session started with draughts and different forms of chess. There was limited choice but everyone was engaged to a degree. The little ones used the pieces as play things and fought each other. I wanted to influence their actions but I was unable able to.  For me it would have been good to get them using the moves if not strategies. Maybe a knight's move challenge or chess solitaire would have been good.  Suicide chess may have worked also. It was great that the older ones tried all the games even though the rules changed a bit. Should have brought and set up "Go".

The place value session had a healthy dose of chaos added to the mix. We started with finger signed numbers which would be good as a regular feature.
We looked at the sequence of powers of 2 which were well known to those who played 2048.
Moved on to Exploding dots. The task was to find out how many would end up in the 4th 5th and sixth positions. Most had big problems sticking to the method. Gq8 and m28 did it really well. And discovered everything that had been set up to discover. But what next for them?
It was a useful way for the others to try the drill to learn the system. I can make it optional next time with others using  the binary converters. I'll set it as homework.
My aim is a comparable place value system. Ideally we would be able to compare them using our abacuses or something. It would ready them for Base 12.
Next week, converting to and from Binary and Base 5 to Base 10. Set up a spy system using ascii so that they can decode a message.

Points, lines and shapes 1 evaluation

Perpendicular!

The Brixton sessions have a very different character to Tulse hill. We are in our own room so no one can see what we do and we can use the space much more as we like.
My plan for floor space has on occasion led to other ideas for activities much like today's block and roll idea.

It was very much a workshop style session today.  We had some nice elements such as the parallel/perpendicular call which should be a feature.  One explorer called out regularly and others started getting into it at the end.

Finding parallel and perpendicular examples in the room was ok but would have been much better in the playground.

The triangle investigation was not as thorough as I had thought it might be but I misjudged the time it took.  We didn't get many doubles ( I think two explorers did a different 7).  But it would be good to mock up a few straws on 7, 8, and 9 and see if they can get more than one triangle by moving them around.

We talked a bit about the history of right angles and historic ways of creating them.  I hoped to get on to creating a grid on the floor using parallel and perpendicular lines but didn't get a chance to do that.  Instead we played block and roll using paper.  This was quite a good way to do it though as it was a better place to introduce the game.  They can practise online and we can try to create the grid next week.

We didn't get to use rulers but we started to see the use of set squares and we can use set squares that we make ourselves.

Games focused session evaluation

One of the challenges I face in this and perhaps all of the sessions I facilitate is to figure out my role in the growth process. Both of the sessions this week gave me an opportunity to reflect on this aspect. These are some of the roles I picked out.

Challenger
Ideally I would like my main role to be to set challenges that take care of themselves. An Explorer gets a vision and embarks on the mission. This was most evident in the straw challenge where the initial challenge was set and then they went off and tried different things to achieve the tallest tower possible.

Setting the challenge with the rubik's snakes to create a square worked well in this regard.  On Saturday E and J discussed ideas about why it could or might not be possible

Ideas board

Our twin explorers are fantastic and so advanced for their age.  And so different. Chatting about ideas with Will was very cool.  None of them were very mathematical but they were linked to the activities at least initially.  But it gave me a chance to see more about his way of thinking and I assume it gave him more of a chance to be heard and find out something that was relevant to the moment.

Encouragement
this is mainly in the planning of the process of the session.  I am trying to gamify the workshop to the extent that explorers try a range of activities and review what they have done.  I am attempting to do that by a levelling up scheme where comments are reviewed and both reviewing and having well reviewed comments gets explorers points that lead to levelling up.  Still in progress.

Authority
We had problems this week with a couple of explorers going a bit wild before I got there.  It is fairly clear that it is getting close to the time when I'll have to move this session on to another venue as there we are quite limited there and it's clear that it is becoming a strained relationship but that's another point.



This is a strange role and one I'm exploring as I need to be able to influence some of their decisions about exploration but many of them clearly would not accept if it went against a decision that they have already made.  This is a good thing up to a point and I need to make sure that I distinguish that line.

To Provide examples
We started by playing split 3 ways on paper.  This was not a problem for many of the children but some needed help getting some numbers to start playing with.  Once they had the idea that I could help them to get numbers to play with, those that needed this were comfortable asking for this help and would then get on with playing the rest of the game.  I gave examples some times as a demonstration with blocks, sometimes working out the answers and sometimes just checking

Games Focus
I think that focussing on games development is a really useful way for me to see these sessions.  I don't need it to be the explicit focus for explorers, just to be there as a way of framing my planning of the session.  It will be useful to see how it works for this term.

Multiplication session 5/4 - Evaluation

This was a key session in the course and really the pinnacle of the programme. The idea of multiplication being a relationship between numbers is a difficult one to see and so not easy to explore. 

I wanted to make an activity that forced them to think in terms of multiplying numbers and that also might lead to the project idea of representing primes as parts and composite numbers as completed products.  

To encourage this idea, I tried a few things:

We looked at characteristics of different multiples and how tables of multiples are symmetrical.  We looked at other tables that linked characteristics and asked about why the central diagonal didn't have symmetrical doubles.

We explored the patterns made by times tables in a 100 square.

I restricted the numbers to only the "friendly" primes.  We explored the numbers created using 2, 3, 5, and 11 only.  I called these the friendly primes because they have easily discoverable patterns in a 100 square.

I recommended the playing the app Alchemy that involves mixing base elements (Air, fire, water, earth) to make all kinds of compounds.

We started the session in Brixton using cards that had prime factors in the form of coloured dots on the back.  I used this as a way to remind everyone about the colours introduced last week and to get questions and observations.  The first two observations were about the card with 3 red dots (8 as it is 2x2x2) and three green dots (27 - 3x3x3) on first encounter, everyone assumed that they should be 6 and 9 respectively.  I half explained half left them to figure it out.  It would have been good to have some strategies for that.  Even if I had just explained the homework rule that they can only use the numbers from the dots and the times button.  That may have still caused them to think 2 x 3 instead of 2x2x2.
I don't think it was a big problem anyway.  E and N went on to get the concept very well as did the rest of them I think.  One or two didn't really get past the times tables on 100 square that we did last week but reinforcement of that is a good thing.  I think the way C20 understood it afterwards was one of the best moments as he often does not get our activities or draw his focus away from whatever it was on but today he did a lot of the prime map and also asked questions about what he was doing and checked with me regularly.
An activity that was very helpful for setting this one up was to create number system using the 100 square.  We added blocks corresponding to the colours for that number.  I started off with this one just letting them do what they wanted but had to work more strictly to get a good outcome.  I took the board away and asked if they thought they could find some cards and  then build a tower with the correct colours and number of blocks.  They all got on well with that.  transferring it to the paper copy was a little more difficult.
I tried a 2048 style game at the end but I haven't thought the rules through sufficiently yet so I'll try that another day.  One thing is that I should leave out the miss a go part as too many goes were missed and that made the game too slow.

Session 4/3 evaluation

 It came to me about an hour before the session and proved to be a really useful resource for working with a hundred square. I could see it being used for all kinds of number square investigations.
The basis for this week's exploration was understanding how the multiples of certain primes are distributed over the hundred square.  As has become a feature of this programme, I wonder how much meaning explorers are able to fix onto maths like this. Nevertheless, we are committed to providing experiences that allow children to explore ideas in as many ways as possible.
We started with colour coded cards and wanted to see if our explorers could break the code.  Those who were there succeeded in saying lots of things about the possible links with colours and numbers.  Many noticed times tables and combined times tables.
We discussed prime numbers and what I called friendly primes. From that we worked on colouring a 100 square using the same code.  This is a way to start explorers seeing the patterns.  I should have started with 11 with this one though as many of them got the idea from the 2 times table that they all just go down.  and without any more thought proceeded to colour the numbers in, in columns (rather than the diagonal line of squares.
This was on top of the templates that revealed the answers if help was needed.
This session was also an opportunity to get to know some of the boys a bit more.  One of them had been asked to try colouring in the 2 times table.  He just said that he didn't want to do it.  His mum checked in with him and told me that he didn't know his two times table.  This was a bit of a revelation (as he is very able mathematically) but now I think back, I have excused playing up as boredom rather than inability.  This will be an important aspect of the sessions to master.  When to offer support. I imagine it will be whenever someone doesn't want to do an activity.  It works very differently with others of course.

 We had quite a good game to finish up.  We started with all the colour code cards colour down.  I threw the 2, 3, 5 and 11 cards up and asked explorers to find cards that had the same colours as those that had landed colour up.  It was a bit of a dash with explorers turning up every card that they could. rather than figuring it out from the numbers.  Little by little I suppose.

 I don't take in homework in maths explorers sessions but this week, one of our explorers had really looked at the missing multiples game on Nrich.  She went through lots of levels and really figured out how it worked.  I should find a way to get more homework in or talked about or recognised.  Hopefully with the project this time round.  I've set homework to find all the numbers under 100 that are built using friendly primes.  I must remember to send out a reminder.