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If you search Ashburton using the search box on our home page, you will find Ashburton Primary School has contributed significantly to our story telling in the past. As with all schools there has been staff change and this year at Years 4 and 5 they found themselves with six willing and able teachers of whom only two had experience with helping students learn to work like a mathematician. A good reason for running a PD day. One of the core components of their approach to learning to work like a mathematician has been regular use of tasks, so the first part of the day was planned to induct the 'newbies' into successfully introducing tasks into their classroom. The process was: 50 minutes as a workshop for the six staff which modelled (not described) how to introduce the tasks. This included significant time for pairs to work on their own task. 50 minutes debriefing the structure and purpose of the First Task Lesson and planning to enact it with a class. 50 minutes in two rooms enacting the First Task Lesson with 4 half classes of Year 4 students. One experienced teacher in each room to support. 50 minutes debriefing the four introductions, then learning about the principles of a task as the Tip of an Iceberg and the Three Lives of a Task, in particular converting the task to whole class investigation. Did it work? Seems so. The teachers felt they had learnt and were ready to continue using tasks regularly. How about the students? Well, here's one example of the journal work done in class. Click to enlarge the photo. Students had about 30 minutes on their task within the lesson. Does this seem like reasonable output for two students using tasks and maths journals for the first time? See Link List below for: An article titled Recording at Ashburton Primary School which was written a couple of years back to illustrate how initial recording such as this can be nurtured further. Task Centre Principles and Discussion Lessons.

The first task lesson starts with the statement of context. School mathematics is learning to work like a mathematician. That's followed by a listing of what the children think a mathematician's work is about. The discussion is summarised with the comment that a mathematician's work begins with an interesting problem... ...and over there on the table there's a whole bunch of interesting problems that the school has purchased to help us get started. One pair of students is invited to select a task. The teacher works with this pair to develop a preferred starting routine of: Count ... Read ... Try Then other pairs are invited, in an orderly way, to select a task. As the students work in pairs with their tasks, the teacher adds Record and Ask to the expected routine. The end of the lesson involves 5 minutes to recount and pack up and 5 minutes to share and reflect on one group's task. Detailed support for this Introductory Lesson is integrated into the workshop leader's notes of the eTask Package. (See Link List below.)

Professional learning continued in the afternoon with:

5. 50 minutes as a Discussion Lesson with a Year 5 class.
6. 50 minutes debriefing the Discussion Lesson and planning the next steps as individuals and as a teaching team.

The investigation used was based on Task 111, Square Numbers. This experience has since become the core of a detailed lesson plan designed to support self-directed professional development in a group of mathematics teachers. The plan has been added to the From The Classroom section of the Cameo for this task. The Cameo also has two other From The Classroom contributions which make it even richer.

That is why Square Numbers has been chosen as one of the cameos you are invited to get to know this month.
See Link List below.

My granddaughter finished school a bit 'off colour' the other day and then was sick in the night. Her temperature was still a bit high in the morning, although she seemed to be picking up, so guess who got the call to look after her at home while mummy went to work.

When I arrived, apart from being shown where the food and the tea bags were and how to work the new control-everything-from-the-one-panel remote, I was presented with a list of possible activities for the day if needed. Mother and daughter had apparently discussed them in preparation for my arrival. One of the things on the list was 4 times tables.

The young lady had just started Year 3, so a reasonable expectation that she learn this set of facts to automatic recall level. A mathematician's work is certainly about investigating problems, but those investigations are going to be more efficiently tackled if key tools are available for instant use. So, as the day progressed we tried a few times - in between lots of other nice stuff. (I would have been happy to watch Moana for the umpteenth time.)

The first try was while munching Vegemite toast - no butter. Just asking a few to see what she knew. If she knew them she knew them. No problem. If she didn't, she tried to work them out. The process of working out seemed to be to start from one you know and count forward by ones. Understandable, but not really group counting and definitely not automatic response. Fifteen minutes and we had only worked on three or four.

Later we had another try when I found a laminated 0 - 100 chart. We used a marking pen to circle any she knew automatically when asked. Then she tried to fill in the gaps. My phone rang after we started this and by the time I finished the call - 3 minutes or so - she had circled every counting by 4 number from 0 to 100 (with only one slip up). So pattern was a tool she could use. We looked closer and she started to see the last digit pattern. That was enough though - we had been at it at least another 15 minutes.

However, not long after she asked if we could use 'that phone game we did before'. She meant the calculator game we had played with the calculator app on the phone. It's called 6 Times, but of course it can be 4 Times. It involves two people and two calculators. Both press 4 X then just one secretly presses another digit and =.

Player A shows their screen to Player B. Then Player B presses a guessed digit and they compare screens. This kept us occupied for a while but it was like pulling teeth because there was no way she was going to guess unless she was sure she was right. Mind you, there was a point where she said, "No, it can't be 31, that's an odd number." This was followed by the guess of 32, which turned out to be the same as on my screen. At this point I suggested she close her eyes and say '4 times 8 = 32' to herself three or four times to see if that would help her remember it. We tried that with a couple of others as well.

It was too wet to play outside, so we lined up some chairs and cushions in the longeroom and took some of the dolls for a trip to Sydney. During lunch - scrambled eggs on toast which we made together; she was clearly feeling better - I was thinking we hadn't really come too far with the automatic response. The back of mind was wondering what I was missing.

After lunch we looked at the activity list again. Having been to Sydney and back it was okay to do that. She chose the one that involved using her pad device. She showed me that she could go on line and log in to a site to do some maths. The software took her to some part where she had finished last time and I was fascinated with what happened next. A voice told her how to do a very particular type of maths exercise (not times tables), the screen wrote on itself, then she was given one to do ... and another ... and another ... and then ... this became like pulling teeth too.

I am certain she would have stopped well before getting her score out of 10 if I hadn't been sitting beside her encouraging her to get it over with.

For me it was almost the same as the way I was taught maths when I was at school, except I could see the teacher's face and all ten on the board at the same time and the teacher could see when I was bored witless. Technology has made such advances since then.

Then I had a thought. If we could go on line to this site, then we could also go to Mathematics Centre. So I headed for the Picture Puzzles (see Link List below) and the 2 & 4 Times Tables Torture:

   

These puzzles are designed as a timed game involving two people, but it was easy to adapt for one. We were told there were going to be 20 challenges, so on paper I wrote 1 to 20 with space for answers. I told Little One that she could use the timer on my phone to set any time for herself to work out the 20 answers. I watched as she rolled the timer around to 20 minutes, paused, then rolled it on to 30.

Then it was up to her. The slides were not in 'times tables order'. She studied each one as she chose it, wrote her answer and moved on to the next one. She didn't lose concentration. When she reached the big STOP sign on the screen she was momentarily surprised, then slapped her finger onto the timer to stop it. Nine minutes and a few seconds!

• Apart from the trip to Sydney this was more involvement (self-driven involvement in fact) than I had seen all day.
• Why?
• What are the features offered by the Picture Puzzle that are likely to have captivated and absorbed her?

Any sensible grandparent would have stopped there, but that's not one of my characteristics. She seemed to know she had the 20 correct, but we hadn't really checked them. I encouraged her to show me how she did these ones:

   

The first had given trouble all day. She mostly worked out the 4 rows of 5 counting by ones, then sort of recognised that the 20 and 4 was like when she started at 4 x 5 because she knew it and then counted 4 more. Counting by ones was used to start with in the second one too, then all of a sudden, and with delight, she burst out with 'if that's 20 then it's 40'. An 'aha' moment. Now it was time to stop. Besides, we heard mummy's car roll in.

The young one is still seeking automatic response. And I am still seeking ways to help her.

Editor's Note: This Picture Puzzle derives from a Calculating Changes activity. Members have access to the whole class investigation form of Times Tables Torture which includes a slide for each times table from 1 x 1 to 10 x 10. Teachers select the ones they want to use and make their own slide show. The slide show is first shown to the whole class as a timed challenge, then the discussion moves to How do you know? and Can you check it another way?

Task 111, Square Numbers

Square numbers are developed in way that draws attention to them as the sum of a sequence of odd numbers beginning with 1. The challenge is to find a way of knowing the number of blocks needed to make any given square number. Mathematicians need to be able to check their work another way, so this also becomes part of the investigation. There may in fact be more than one other way. The cameo has been greatly enhanced by three From The Classroom contributions. One is the extended whole class lesson plan described above, one is an open-ended Investigation Guide shared by Damian Howison and one takes the problem into senior secondary school by using it as an example of the need for Proof by Mathematical Induction.

See Link List below for the Square Numbers cameo.

In the eTask Package, this task is in the 'Easy To Make' category as most schools will have wooden or plastic blocks or square tiles in six different colours. It doesn't matter which colours because the task card lists them as Colours A, B ... F.

Task 16, Octaflex

Visual reasoning is as important to a mathematician as symbolic reasoning. The card presents three challenges (getting the pieces back into the frame is the third) which encourage students to use and extend their spatial reasoning. The iceberg questions in the cameo take the task further and include content in perimeter, area and factors. See Link List below for the Octaflex cameo.

In the eTask Package, this task is in the 'Special Equipment' category because the equipment is not common is schools. If you happen to have one - you only need one - of the original foam Octaflex sets then the task immediately becomes easy to make. If not then notes provided with the Package (and in the cameo) suggest ways of making a set.