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.