NRICH has been invited to lead six face-to-face PD days for primary teachers in the London Borough of Tower Hamlets over the academic year 2018-19, focusing on the mathematical language of problem solving, reasoning and fluency.

Two teachers from each primary school will come together on each day with two members of the NRICH primary team to explore ways of embedding the three aims of the primary mathematics curriculum into everyday practice, with a particular focus on the role of mathematical talk and developing vocabulary.

Each session will offer opportunities to explore the role of NRICH tasks in nurturing confident, resourceful and enthusiastic teachers and learners of mathematics. Gap tasks will provide a focus for shared reflection and the extended nature of this programme will give delegates the chance to contribute to a wider mathematical community.

This page summarises the content of each day. If you would like anything uploading to this page, or have any queries about the project, please contact the NRICH primary team.

The six face-to-face days will take place at the Professional Development Centre in Tower Hamlets on the following dates from**9.30am to 3.30pm:**

**Day 1: Tuesday 13 November**

Here is a pdf of the PowerPoint slides we used on the day: 20181113THDay1.pdf

These are the points that were raised after we had tried the Quad Match task:

These are the suggestions we had for ways of personalising Quad Match for our individual settings:

As we were engaged with the task En-counters, Fran and Liz noted down some of their observations:

Here are the points that were raised as we reflected on our experience of En-counters:

We made the following observations having had a go at Stringy Quads:

We had a go at a 'Which One Doesn't Belong' task where we had cards that said 'octagon', 'triangle' and 'arrow head'. Here is a summary of our thoughts:

Fran referred to the Radio 4 programme 'Thinking Allowed', which she had presumed was called 'Thinking Aloud'. We discussed this play on words and wondered whether 'talking aloud' and 'talking allowed' were good principles for our classrooms.

Fran and Liz noted the following 'overheard' statements as we tried Factor Lines:

*Please refer to the Teacher Takeaway slide in the pdf for a reminder of the gap tasks from Day 1 to prepare for Day 2.*

**Day 2: Thursday 13 December**

Here is a pdf of the PowerPoint slides we used on the day: 20181213THDay2.pdf

In the first session, we reflected on our experiences of trying at least one task from day 1 with our learners. Fran and Liz picked out some common threads from the table discussions:

We tried One Big Triangle to focus on ways to get started on a problem. Here are the comments we shared specifically about this task:

And here are the strategies we discussed, which we felt helped us get started:

To exemplify the second stage of the problem-solving process (working on the problem), we had a go at Two-digit Targets. Fran invited us to consider the following points as we worked:

We discussed the phrase 'closest to' and that, mathematically speaking, the number closest to 50, for example, is 50 itself. We decided that '01' is not a two-digit number as the zero in the tens column is not a place holder (there could be an infinite number of zeros to the left of the 1).

We talked about different ways of judging whether one set of five numbers is better than another set. Two pairs of children might compare one answer at a time and whichever pair's number better matches the criteria gains a point. The pair who has the most points after all five numbers have been compared is the winner. Alternatively, learners could determine the 'ideal' number for each criteria and then find the difference between their answer and the ideal. The pair with the lowest score would then have the best solution. (The task Four-digit Targets could be used as a follow-up.)

Having tried Which Scripts?, Fran asked us to consider these questions, which are sufficiently generic to be useful for many tasks:

We were introduced to the Totality game as an example of a Low Threshold High Ceiling (LTHC) task. Having played it several times, we suggested ways to tweak the game:

The final task of the day was Possible Pieces, which illustrated some of the characteristics of a rich (but not LTHC) task. Possible Pieces was created for the National Young Mathematicians' Award, which is a joint venture between NRICH and Explore Learning. (More information can be found here.)

*Please refer to the Teacher Takeaway slide in the pdf for a reminder of the gap tasks from Day 2 to prepare for Day 3.*

**Day 3: Tuesday 29 January**

Here is a pdf of the PowerPoint slides we used on the day: 20190129THDay3.pdf

We began by reflecting on our experiences of trying out at least one task since day 2. The following key themes emerged:

All delegates then considered the differences between thinking and reasoning. Here are all the responses. The table below summarises the common points:

Having watched the silent Ken Ken video, we had the following observations:

We watched the video of the three girls having a go at a Ken Ken and we discussed examples of novice and more expert reasoning:

In session 2, we explored Poly Plug Rectangles using grids and counters, firstly one on one and then a pair vs another pair. We agreed that the following language/vocabulary had helped us to articulate our reasoning:

We explored the game further, using the new interactivity. This led into a discussion about whether rectangles place diagonally on the grid are allowed e.g.

Although the interactive itself does not allow these rectangles (as they do not form an array), it would be good to allow learners to decide for themselves whether or not they should be part of the task when playing using the grid and counters. The important point is that learners should be encouraged to give reasons as to why or why not.

Alice and Charlotte mentioned that they had found the downloadable recording sheets available in the Teachers' resources section of Poly Plug Rectangles very useful as they had asked children to highlight a plug if they guessed that position. This made it relatively easy to see quickly which learners had used a small number of guesses when trying to locate a rectangle.

After lunch, we took time to explore The Remainders Game using the iPads. We began to try to justify whether or not we could have used fewer clues to be certain of knowing the computer's chosen number. We will be following this up on day 4.

Our final task of the day was What's it Worth?, which will appear in a slightly different format as a primary live task (with a new interactivity) just before the February half term.

*Please refer to the Teacher Takeaway slide in the pdf for a reminder of the gap tasks from Day 3 to prepare for Day 4.*

**Day 4: Tuesday 26 February**

Here is a pdf of the PowerPoint slides we used on the day: 20190226 TH Day4.pdf

We reflected on what we had tried out at school since day 3. The following points were noted:

Some participants offered suggestions for ways to support KenKens in the classroom, using the template of NRICH's teachers' resources:

Fran then invited everyone to check whether the dominoes she had given out on each table were a full set. The following were overheard as delegates worked together to find out:

Having become more familiar with the structure of a set of dominoes, we had a go at the task Amy's Dominoes. Fran presented us with one of the published solutions (Bjorn's) but cut up so that the idea was to put it in the correct order. We discussed how this approach i.e. going from someone else's reasoning was helpful to improve our own reasoning skills. We also looked at the poster solutions sent in by Lyneham Primary to the same task, and compared them to each other, and to the five stages of reasoning.

In session 2, we revisited The Remainders Game which we'd played on day 3. We looked at copies of five different emergent reasoning journeys contributed last time and then each pair 'worked up' one of these so that, for example, the reasoning was more sophisticated and/or the language was improved. Here are the resulting annotations which all worked on the same starting point:

The following relate to the second starting point:

The following relate to the third starting point:

The following relate to the fourth starting point:

The following relate to the fifth starting point:

We spent some time considering how we might define some mathematical words taken from this curriculum glossary published by NCETM. In pairs, we then chose one of the words we were confident about and created a definition for it along with three 'non-definitions'. This pdf contains the full set.

In session 3 after lunch, we tried Hundred Square which involved looking for, and explaining, patterns. The points below were jotted down by Liz as she overheard delegates talking:

We discussed various ways of thinking about the problem and trying to generalise a way to find out what number is on the back of any number. We realised that the equation at the bottom of the sheet photographed below i.e. n + x = (2n - 10) + 1 did not work in fact, but the thinking helped us with a different way of considering the problem, illustrated by the image:

We looked at the published solutions to this task to get a sense of the fact that each one is relatively brief, it is the whole together that makes the rich 'tapestry'. Fran and Liz emphasised that it is worth sending in partial solutions to live tasks as they can be 'woven together' with other solutions that we receive.

We talked about getting the most out of rich mathematical tasks, whether they are from NRICH or elsewhere. We had a look at the freely available 100 ideas for using a hundred square, produced by the National Numeracy Strategy in Cumbria and discussed which of them might provide particularly fruitful opportunities for develping reasoning.

Fran offered a series of challenges on the large hundred square which we had taped to the back of the room, drawing out the reasoning in each case.

*(In the image above, the ideas highlighted in green were those we actually tried together.)*

Our last task of the day was Baravelle. Liz gave us chance to look at the image for a minute in silence, then talk to a neighbour about what we saw, before being able to talk and look at the same time. We were then challenged to recreate the image without being able to see it.

Fran and Liz drew attention to the Habits of Mind tasks available on NRICH. (Baravelle appears in the 'Being Resourceful' list, along with The Remainders Game.)

*Please refer to the Teacher Takeaway slide in the pdf for a reminder of the gap tasks from Day 4 to prepare for Day 5.*

**Day 5: Tuesday 2 April**

Here is a pdf of the PowerPoint slides we used on the day: 20190402THDay5.pdf

We began Day 5 by reflecting on staff meetings that delegates have led. This document includes all participants' observations, categorised as being related to before, during or after the meeting. We then discussed experiences back at school of working on NRICH tasks since last time. During the whole-group discussion, one delegate mentioned the work of an oracy project, Voice 21, of which his school is a part. We talked about improving children's conversational skills in general, which then impacts on their ability to reason mathematically. Fran recommended a TED talk given by Celeste Headlee in which she outlines ten ways to have a better conversation.

The first mathematical task of the day was School Fair Necklaces. We worked together to develop a system for finding all possible solutions, which is an example of*proof by exhaustion*.

Two-digit Targets and 6 Beads, both tasks we have worked on during previous days together, are also tasks which lend themselves to this kind of proof.

In session 2, we were introduced to Strike it Out. We watched the sequence of moves which simulated this two-player game (see the PowerPoint presentation in the teachers' resources section) and considered the following questions:

Card sorts of each proof were available for participants to use whenever they felt it helpful. This task offers an opportunity for*proof by logical reasoning*.

After lunch, Liz invited everyone to...

Our final task of the day was Dart Target, which was written for the National Young Mathematicians' Award. Fran and Liz also flagged up the new Maths Club Activities page and reminded everyone about the upcoming webinar for pupils on June 11.

*Please refer to the Teacher Takeaway slide in the pdf for a reminder of the gap tasks from Day 5 to prepare for Day 6.*

**Day 6: ****Tuesday 25 June **

Here is a pdf of the PowerPoint slides we used on the day: 20190625 TH Day6.pdf

We began our last day together having a go at Neighbourly Addition, the task which featured in the recent student webinar. We used the video footage from the webinar so we got a sense of how the session had been run and the kind of responses we had received from children around the world. Several delegates had participated themselves with their classes, or knew that other classes in their school had taken part.

As we worked on the task, we talked about the power of 'what else...?', in other words having a culture of continuing to ask questions and investigate mathematical ideas, even when the original question has been answered.

We discussed how the footage could be used in individual classrooms, and also the fact that most NRICH tasks have published children's solutions so a similar model could be used, in that children in the class could have a go at a task, then look at published solutions to compare with their own and build on others' ideas. The advantage of using the webinar footage itself might be that as the teacher, you would have more opportunities to stand back and observe your class. Alternatively, the webinar footage could also help get a sense of a possible way to structure the task, even if the video itself is not used in the classroom.

Here are some examples from one school of pupils' work during the webinar:

We took some time to reflect on the staff meetings led over the course of the project. Delegates were asked to pull out the three most helpful pieces of advice from the document we had collated last time we met. Six key themes emerged, which are represented on the image below, alongside some examples of the advice that was offered in relation to each theme. Here is a pdf version of the image which will be easier to read.

We looked at some examples of solutions that had been submitted to the live tasks from Tower Hamlets schools. We also shared a submission from a teacher who described how she had introduced Digit Addition to her class, and some of their responses. (It is now published on the solution page.) Fran pointed out that the teacher clearly enjoyed the process of reflecting on the children's learning. This kind of submission is particularly valuable for NRICH as it supplies not only some solutions from the children, but paints a fuller picture of the problem-solving process unfolding in the classroom. Fran acknowledged that a narrative like this takes time to communicate, but the teacher is clear about the benefit they experienced and users of the site benefit from reading her account.

We then moved on to considering what we understand by the term 'mathematical fluency' and common misconceptions surrounding it, firstly as individuals and then on tables.

We had a go at Dicey Addition with a pair playing against another pair, and talked about how we were making decisions about where to place each digit. We then tried Dicey Operations in Line, with everyone using the same nine digits, and being allowed to place them once all nine were known. The aim was to prove how the digits should be placed to get as close to 1000 as possible. Liz explained that one of the reasons for sharing these two linked tasks is that they are great examples of contexts that can develop fluency alongside reasoning.

Fran shared NRICH's view of fluency, which encompasses more than just fluency in the context of number and calculation. She explained that we will be revamping our Fluency feature to reflect this wider picture. She also shared a range of research, which has led NRICH to think of fluency as having five key aspects:

There was then an opportunity to talk about our experiences of working on NRICH tasks since we last met. Participants were invited to use the five aspects of fluency above as a 'reflective lens' through which to view the task/s they had tried.

One teacher brought in the 'class book' in which the class' mathematical journey over the year is recorded. Here are a few sample pages:

In order to consider fluency in the context of geometry, we worked on Inside Triangles together. Although this task is great for developing fluency in a spatial sense, it is also lends itself to discussions about working systematically being a key problem-solving skill and reasoning in the sense of proof. (Liz flagged up the interactive geoboards and the problem Nine-pin Triangles which would make a nice follow-up task.)

Fran then offered us four calculations and asked us to consider which order we would do them in and why. Here are the calculations:

121 - 78

121 - 59

121 - 4

121 - 20

This led to a discussion about how 'easy' each was to solve. We reflected on the fact that we had used different methods for each calculation and how we would like children to be able to do what we had done. We agreed that being able to select calculation methods appropriately was one element of being fluent in mathematics. Professor Ruth Merttens, from whom we magpied this idea (!), suggests that it is tasks like this (multiple calculations as part of the same question) that will help children develop fluency, rather than practising a single method over and over again, then moving on to the next method and so on.

Fran then gave us YOYO time (**y**ou **o**n **y**our **o**wn) to consider how we would each work out 5 x 18 mentally, and then write down how we did it. We were then given images showing various ways of representing the calculation and asked whether our way was depicted, and to try to make sense of the other ways. We watched a video where
students described their thinking as they calculated, and were asked to match the student with their representation. We discussed how visual representations of mathematics can help deepen understanding and enhance fluency. More information about the role of visual mathematics, including these examples, can be found in this article by Jo Boaler on YouCubed.

We moved on to a game, Spiralling Decimals, and again thought about which aspects of fluency might be developed through playing it. The game element means that children will be practising a very large number of decimal comparisons, whilst the more confident players can focus on trying to develop a winning strategy. We discussed the idea of a spiral number line more generally and talked about the possibility of a linked task in which childen tried to position numbers on a spiral number line of paper, then opened it out. Alternatively, they could accurately mark numbers on a paper number line, then fold it into a spiral. (The 'Possible extension' section of the teachers' resources contains some useful images.)

Our final task of the day was Shape Times Shape, which we introduced in a slightly different way to the suggestions in the teachers' resources section. We invited participants to each take one strip out of an envelope (each one depicting some images comprising a calculation) and compare with a partner. This was widened to include a discussion with their whole group (up to 6 people on a table) to decide what they (collectively) know. They were then asked to take the remaining strips out and were shown the full text of the problem. (This task is part of our What is the Question? feature, which encourages children to become more fluent with times tables, but starting from answers rather than questions.)

We rounded off the day, and indeed the project, by encouraging delegates to develop their positive mindset so they could continue the momentum already gained in schools over the course of the year. Rather than a 'yes, but...' attitude, we suggested a 'yes, and...' approach, which would harness the progress made and continue the fantastic achievements already evident.

**If you took part, please share your thoughts about the 2018-19 Tower Hamlets/NRICH programme by filling in a feedback form.**

Two teachers from each primary school will come together on each day with two members of the NRICH primary team to explore ways of embedding the three aims of the primary mathematics curriculum into everyday practice, with a particular focus on the role of mathematical talk and developing vocabulary.

Each session will offer opportunities to explore the role of NRICH tasks in nurturing confident, resourceful and enthusiastic teachers and learners of mathematics. Gap tasks will provide a focus for shared reflection and the extended nature of this programme will give delegates the chance to contribute to a wider mathematical community.

This page summarises the content of each day. If you would like anything uploading to this page, or have any queries about the project, please contact the NRICH primary team.

The six face-to-face days will take place at the Professional Development Centre in Tower Hamlets on the following dates from

Here is a pdf of the PowerPoint slides we used on the day: 20181113THDay1.pdf

These are the points that were raised after we had tried the Quad Match task:

These are the suggestions we had for ways of personalising Quad Match for our individual settings:

As we were engaged with the task En-counters, Fran and Liz noted down some of their observations:

Here are the points that were raised as we reflected on our experience of En-counters:

We made the following observations having had a go at Stringy Quads:

We had a go at a 'Which One Doesn't Belong' task where we had cards that said 'octagon', 'triangle' and 'arrow head'. Here is a summary of our thoughts:

Fran referred to the Radio 4 programme 'Thinking Allowed', which she had presumed was called 'Thinking Aloud'. We discussed this play on words and wondered whether 'talking aloud' and 'talking allowed' were good principles for our classrooms.

Fran and Liz noted the following 'overheard' statements as we tried Factor Lines:

Here is a pdf of the PowerPoint slides we used on the day: 20181213THDay2.pdf

In the first session, we reflected on our experiences of trying at least one task from day 1 with our learners. Fran and Liz picked out some common threads from the table discussions:

We tried One Big Triangle to focus on ways to get started on a problem. Here are the comments we shared specifically about this task:

And here are the strategies we discussed, which we felt helped us get started:

To exemplify the second stage of the problem-solving process (working on the problem), we had a go at Two-digit Targets. Fran invited us to consider the following points as we worked:

We discussed the phrase 'closest to' and that, mathematically speaking, the number closest to 50, for example, is 50 itself. We decided that '01' is not a two-digit number as the zero in the tens column is not a place holder (there could be an infinite number of zeros to the left of the 1).

We talked about different ways of judging whether one set of five numbers is better than another set. Two pairs of children might compare one answer at a time and whichever pair's number better matches the criteria gains a point. The pair who has the most points after all five numbers have been compared is the winner. Alternatively, learners could determine the 'ideal' number for each criteria and then find the difference between their answer and the ideal. The pair with the lowest score would then have the best solution. (The task Four-digit Targets could be used as a follow-up.)

Having tried Which Scripts?, Fran asked us to consider these questions, which are sufficiently generic to be useful for many tasks:

We were introduced to the Totality game as an example of a Low Threshold High Ceiling (LTHC) task. Having played it several times, we suggested ways to tweak the game:

The final task of the day was Possible Pieces, which illustrated some of the characteristics of a rich (but not LTHC) task. Possible Pieces was created for the National Young Mathematicians' Award, which is a joint venture between NRICH and Explore Learning. (More information can be found here.)

Here is a pdf of the PowerPoint slides we used on the day: 20190129THDay3.pdf

We began by reflecting on our experiences of trying out at least one task since day 2. The following key themes emerged:

All delegates then considered the differences between thinking and reasoning. Here are all the responses. The table below summarises the common points:

Having watched the silent Ken Ken video, we had the following observations:

We watched the video of the three girls having a go at a Ken Ken and we discussed examples of novice and more expert reasoning:

In session 2, we explored Poly Plug Rectangles using grids and counters, firstly one on one and then a pair vs another pair. We agreed that the following language/vocabulary had helped us to articulate our reasoning:

We explored the game further, using the new interactivity. This led into a discussion about whether rectangles place diagonally on the grid are allowed e.g.

Although the interactive itself does not allow these rectangles (as they do not form an array), it would be good to allow learners to decide for themselves whether or not they should be part of the task when playing using the grid and counters. The important point is that learners should be encouraged to give reasons as to why or why not.

Alice and Charlotte mentioned that they had found the downloadable recording sheets available in the Teachers' resources section of Poly Plug Rectangles very useful as they had asked children to highlight a plug if they guessed that position. This made it relatively easy to see quickly which learners had used a small number of guesses when trying to locate a rectangle.

After lunch, we took time to explore The Remainders Game using the iPads. We began to try to justify whether or not we could have used fewer clues to be certain of knowing the computer's chosen number. We will be following this up on day 4.

Our final task of the day was What's it Worth?, which will appear in a slightly different format as a primary live task (with a new interactivity) just before the February half term.

Here is a pdf of the PowerPoint slides we used on the day: 20190226 TH Day4.pdf

We reflected on what we had tried out at school since day 3. The following points were noted:

Some participants offered suggestions for ways to support KenKens in the classroom, using the template of NRICH's teachers' resources:

Fran then invited everyone to check whether the dominoes she had given out on each table were a full set. The following were overheard as delegates worked together to find out:

Having become more familiar with the structure of a set of dominoes, we had a go at the task Amy's Dominoes. Fran presented us with one of the published solutions (Bjorn's) but cut up so that the idea was to put it in the correct order. We discussed how this approach i.e. going from someone else's reasoning was helpful to improve our own reasoning skills. We also looked at the poster solutions sent in by Lyneham Primary to the same task, and compared them to each other, and to the five stages of reasoning.

In session 2, we revisited The Remainders Game which we'd played on day 3. We looked at copies of five different emergent reasoning journeys contributed last time and then each pair 'worked up' one of these so that, for example, the reasoning was more sophisticated and/or the language was improved. Here are the resulting annotations which all worked on the same starting point:

The following relate to the second starting point:

The following relate to the third starting point:

The following relate to the fourth starting point:

The following relate to the fifth starting point:

We spent some time considering how we might define some mathematical words taken from this curriculum glossary published by NCETM. In pairs, we then chose one of the words we were confident about and created a definition for it along with three 'non-definitions'. This pdf contains the full set.

In session 3 after lunch, we tried Hundred Square which involved looking for, and explaining, patterns. The points below were jotted down by Liz as she overheard delegates talking:

We discussed various ways of thinking about the problem and trying to generalise a way to find out what number is on the back of any number. We realised that the equation at the bottom of the sheet photographed below i.e. n + x = (2n - 10) + 1 did not work in fact, but the thinking helped us with a different way of considering the problem, illustrated by the image:

We looked at the published solutions to this task to get a sense of the fact that each one is relatively brief, it is the whole together that makes the rich 'tapestry'. Fran and Liz emphasised that it is worth sending in partial solutions to live tasks as they can be 'woven together' with other solutions that we receive.

We talked about getting the most out of rich mathematical tasks, whether they are from NRICH or elsewhere. We had a look at the freely available 100 ideas for using a hundred square, produced by the National Numeracy Strategy in Cumbria and discussed which of them might provide particularly fruitful opportunities for develping reasoning.

Fran offered a series of challenges on the large hundred square which we had taped to the back of the room, drawing out the reasoning in each case.

Our last task of the day was Baravelle. Liz gave us chance to look at the image for a minute in silence, then talk to a neighbour about what we saw, before being able to talk and look at the same time. We were then challenged to recreate the image without being able to see it.

Fran and Liz drew attention to the Habits of Mind tasks available on NRICH. (Baravelle appears in the 'Being Resourceful' list, along with The Remainders Game.)

Here is a pdf of the PowerPoint slides we used on the day: 20190402THDay5.pdf

We began Day 5 by reflecting on staff meetings that delegates have led. This document includes all participants' observations, categorised as being related to before, during or after the meeting. We then discussed experiences back at school of working on NRICH tasks since last time. During the whole-group discussion, one delegate mentioned the work of an oracy project, Voice 21, of which his school is a part. We talked about improving children's conversational skills in general, which then impacts on their ability to reason mathematically. Fran recommended a TED talk given by Celeste Headlee in which she outlines ten ways to have a better conversation.

The first mathematical task of the day was School Fair Necklaces. We worked together to develop a system for finding all possible solutions, which is an example of

Two-digit Targets and 6 Beads, both tasks we have worked on during previous days together, are also tasks which lend themselves to this kind of proof.

In session 2, we were introduced to Strike it Out. We watched the sequence of moves which simulated this two-player game (see the PowerPoint presentation in the teachers' resources section) and considered the following questions:

- How do you play?
- How do you win?
- What other questions do you have?

Is it possible to create a string of number sentences that uses all the numbers on the:

0-20 number line?

1-20 number line?

Any number line with a set of consecutive whole numbers?

0-20 number line?

1-20 number line?

Any number line with a set of consecutive whole numbers?

Card sorts of each proof were available for participants to use whenever they felt it helpful. This task offers an opportunity for

After lunch, Liz invited everyone to...

Jot down a number.

Jot down the next two consecutive numbers.

Add your numbers together.

Jot down the next two consecutive numbers.

Add your numbers together.

We did this seven times and talked about how that felt. We were asked what we noticed:

Participants were invited to choose one of the noticings and try to explain why it occurs, working up their explanations into a proof. Here is a visual proof of why the total is always a multiple of 3:

This task is one way of introducing Three Neighbours and offers chance for learners to engage with*generic proof*. We summarised the three types of proof we had focused on (see slide 16 in the PowerPoint presentation linked above), which are the most accessible to primary learners.

We had a go at a selection of Always, Sometimes or Never? tasks as another way into proof. We made a note of any statements we were unsure about and also had the option to re-draft statements so they would be categorised differently. We circulated round the room looking at each other's sortings.

Then there was an opportunity to share observations from the reasoning walks delegates had undertaken back at school. In pairs (where partners were from different schools), we shared artefacts/photos/adecdotes and having done this with three different colleagues, we returned to consider next steps in same-school pairs. The photos below illustrate one school's examples of reasoning:

Participants were invited to choose one of the noticings and try to explain why it occurs, working up their explanations into a proof. Here is a visual proof of why the total is always a multiple of 3:

This task is one way of introducing Three Neighbours and offers chance for learners to engage with

We had a go at a selection of Always, Sometimes or Never? tasks as another way into proof. We made a note of any statements we were unsure about and also had the option to re-draft statements so they would be categorised differently. We circulated round the room looking at each other's sortings.

Then there was an opportunity to share observations from the reasoning walks delegates had undertaken back at school. In pairs (where partners were from different schools), we shared artefacts/photos/adecdotes and having done this with three different colleagues, we returned to consider next steps in same-school pairs. The photos below illustrate one school's examples of reasoning:

Our final task of the day was Dart Target, which was written for the National Young Mathematicians' Award. Fran and Liz also flagged up the new Maths Club Activities page and reminded everyone about the upcoming webinar for pupils on June 11.

Here is a pdf of the PowerPoint slides we used on the day: 20190625 TH Day6.pdf

We began our last day together having a go at Neighbourly Addition, the task which featured in the recent student webinar. We used the video footage from the webinar so we got a sense of how the session had been run and the kind of responses we had received from children around the world. Several delegates had participated themselves with their classes, or knew that other classes in their school had taken part.

As we worked on the task, we talked about the power of 'what else...?', in other words having a culture of continuing to ask questions and investigate mathematical ideas, even when the original question has been answered.

We discussed how the footage could be used in individual classrooms, and also the fact that most NRICH tasks have published children's solutions so a similar model could be used, in that children in the class could have a go at a task, then look at published solutions to compare with their own and build on others' ideas. The advantage of using the webinar footage itself might be that as the teacher, you would have more opportunities to stand back and observe your class. Alternatively, the webinar footage could also help get a sense of a possible way to structure the task, even if the video itself is not used in the classroom.

Here are some examples from one school of pupils' work during the webinar:

We took some time to reflect on the staff meetings led over the course of the project. Delegates were asked to pull out the three most helpful pieces of advice from the document we had collated last time we met. Six key themes emerged, which are represented on the image below, alongside some examples of the advice that was offered in relation to each theme. Here is a pdf version of the image which will be easier to read.

We looked at some examples of solutions that had been submitted to the live tasks from Tower Hamlets schools. We also shared a submission from a teacher who described how she had introduced Digit Addition to her class, and some of their responses. (It is now published on the solution page.) Fran pointed out that the teacher clearly enjoyed the process of reflecting on the children's learning. This kind of submission is particularly valuable for NRICH as it supplies not only some solutions from the children, but paints a fuller picture of the problem-solving process unfolding in the classroom. Fran acknowledged that a narrative like this takes time to communicate, but the teacher is clear about the benefit they experienced and users of the site benefit from reading her account.

We then moved on to considering what we understand by the term 'mathematical fluency' and common misconceptions surrounding it, firstly as individuals and then on tables.

We had a go at Dicey Addition with a pair playing against another pair, and talked about how we were making decisions about where to place each digit. We then tried Dicey Operations in Line, with everyone using the same nine digits, and being allowed to place them once all nine were known. The aim was to prove how the digits should be placed to get as close to 1000 as possible. Liz explained that one of the reasons for sharing these two linked tasks is that they are great examples of contexts that can develop fluency alongside reasoning.

Fran shared NRICH's view of fluency, which encompasses more than just fluency in the context of number and calculation. She explained that we will be revamping our Fluency feature to reflect this wider picture. She also shared a range of research, which has led NRICH to think of fluency as having five key aspects:

- Accuracy
- Efficiency
- Flexibility
- Understanding
- Reasonableness

There was then an opportunity to talk about our experiences of working on NRICH tasks since we last met. Participants were invited to use the five aspects of fluency above as a 'reflective lens' through which to view the task/s they had tried.

One teacher brought in the 'class book' in which the class' mathematical journey over the year is recorded. Here are a few sample pages:

In order to consider fluency in the context of geometry, we worked on Inside Triangles together. Although this task is great for developing fluency in a spatial sense, it is also lends itself to discussions about working systematically being a key problem-solving skill and reasoning in the sense of proof. (Liz flagged up the interactive geoboards and the problem Nine-pin Triangles which would make a nice follow-up task.)

Fran then offered us four calculations and asked us to consider which order we would do them in and why. Here are the calculations:

121 - 78

121 - 59

121 - 4

121 - 20

This led to a discussion about how 'easy' each was to solve. We reflected on the fact that we had used different methods for each calculation and how we would like children to be able to do what we had done. We agreed that being able to select calculation methods appropriately was one element of being fluent in mathematics. Professor Ruth Merttens, from whom we magpied this idea (!), suggests that it is tasks like this (multiple calculations as part of the same question) that will help children develop fluency, rather than practising a single method over and over again, then moving on to the next method and so on.

Fran then gave us YOYO time (

We moved on to a game, Spiralling Decimals, and again thought about which aspects of fluency might be developed through playing it. The game element means that children will be practising a very large number of decimal comparisons, whilst the more confident players can focus on trying to develop a winning strategy. We discussed the idea of a spiral number line more generally and talked about the possibility of a linked task in which childen tried to position numbers on a spiral number line of paper, then opened it out. Alternatively, they could accurately mark numbers on a paper number line, then fold it into a spiral. (The 'Possible extension' section of the teachers' resources contains some useful images.)

Our final task of the day was Shape Times Shape, which we introduced in a slightly different way to the suggestions in the teachers' resources section. We invited participants to each take one strip out of an envelope (each one depicting some images comprising a calculation) and compare with a partner. This was widened to include a discussion with their whole group (up to 6 people on a table) to decide what they (collectively) know. They were then asked to take the remaining strips out and were shown the full text of the problem. (This task is part of our What is the Question? feature, which encourages children to become more fluent with times tables, but starting from answers rather than questions.)

We rounded off the day, and indeed the project, by encouraging delegates to develop their positive mindset so they could continue the momentum already gained in schools over the course of the year. Rather than a 'yes, but...' attitude, we suggested a 'yes, and...' approach, which would harness the progress made and continue the fantastic achievements already evident.