Numeracy for All
Thinking Out of the Box
Summary
The project aimed to introduce teachers to the Dutch Realistic Mathematics
Education (RME) approach to teaching in order to raise the confidence and motivation
of low achieving pupils. This approach sets Mathematics in a real context and in order
to allow teachers to help their pupils form a deep understanding of number
processes, pupils are expected to explore patterns, put forward explanations and
conjectures, and be confident and adept at convincing one another of their thinking.
The schools involved were Dalbeattie High and its main feeder school Dalbeattie
Primary. Both schools are situated in the small town of Dalbeattie in rural Dumfries
and Galloway Two teachers from each sector took part in the project and the pupils
involved were from the lowest sets in P7, S1 and S2.
The main resource used as valuable background information for the RME approach
was the ‘Young Mathematicians at Work’ series of books and CD-ROMS which
described ‘Mathematics in the City’, an innovative American project based on the
RME philosophy. The teachers based their lessons on examples within the books.
The project focussed initially on the use of the ‘array’ as a thinking tool for
developing understanding of the process of multiplication. An array, in this context, is
where objects are laid out in rows and columns.
Midway through the project it was clear that the S2 pupils were
not engaging as was hoped and that some ‘thinking out of the box’
was needed.
An ‘enterprising type’ project was set up where the S2 pupils would
provide resource materials for a group of children at a local Primary
school. The pupils began to engage with their learning and
motivation was high. This successful mini project culminated in a
visit from the local Bank Manager who discussed the skills that the pupils had gained
and how they were relevant in the work place.
Overall, this project proved to be a
worthwhile learning experience for
teachers and pupils alike. It was an
opportunity for cross- sector working
and engaging in creative pedagogy.
Lessons were learned and further
development was identified; lifelong
learning in action!
Numeracy for All
Thinking Out of the Box
1. Introduction
‘Establishing good numeracy skills is necessary for successful learning across the
curriculum and developing these skills needs to be of high priority for all children,
young people and their teachers.’
ACfE Building the Curriculum 1
Acquiring ‘good numeracy skills’ means more than just regurgitating number bonds or
multiplication tables, but being able to discuss, manipulate, apply and feel confident in
using mathematics in real life situations.
According to international assessment programmes, the Netherlands has a higher
level of attainment in acquiring numeracy skills than Scotland (TIMMSS 2003, PISA
2001). The approach to teaching in the Netherlands, called Realistic Mathematics
Education (RME), is based primarily on the work of the renowned mathematician Hans
Freudenthal who claimed that people learn mathematics by actively investigating
realistic problems. It is different to the approaches used in Scotland in the following
ways:
 ‘Realistic situations are used as a means of developing pupils’ mathematics as
opposed to contexts as an introduction to mathematics or as an application of
mathematics.
 There is less emphasis on the use of algorithms and more on understanding
and refinement of strategies.
 There is less emphasis on linking single lessons to direct content acquisition
and more on gradual development over a longer period of time.
 There is greater emphasis on research into learning and teaching.
 There is use of guided reinvention rather than discovery learning or teachers
explaining.’
( Eade, F et al, 2006)
Maarten Dolk, researcher and developer of mathematics education at the
Freudenthal Institute and Catherine Twomey Fosnot, Professor of Education at the
City College of New York have written a series of professional books for teachers.
This series of books, Young Mathematicians at Work, describe Mathematics in the
City, an innovative project where teachers helped children construct a deep
understanding of number and operations in a maths-workshop environment. The
rationale behind Mathematics in the City is in accordance with the four capacities of
A Curriculum for Excellence (successful learners, confident individuals, responsible
citizens and effective contributors) and therefore was used as a model during this
project.
Schools
Two schools were involved in the project: Dalbeattie High with a roll of 380 and
Dalbeattie Primary with a roll of 280. Both schools are situated in Dalbeattie, a small,
picturesque town near the centre of rural Dumfries and Galloway.
In Dalbeattie Primary, the main feeder school for Dalbeattie High, 36% of the pupils
have support needs and are in levels 2, 3 and 4 of EPSEM. The children are set for
maths according to ability and for the purpose of this project it was decided to work
with the lowest ability group of 27 pupils in Primary 7. The school is already heavily
involved with the Critical Skills Programme.
In Dalbeattie High, it was decided to work with the 17 pupils in the lowest S1 set and
the 10 pupils in the lowest S2 set. The teacher of the S2 set was particularly anxious
that her pupils were not engaging with their maths and motivation was low.
2. Aims/Outcomes
2.1 Aims of the Project
•
•
To raise pupils’ confidence and motivation in their ability to solve problems by
using realistic contexts.
To improve the group’s understanding of the Dutch Realistic Mathematics
curricula.
2.2 Questions to be addressed
•
•
•
Can the use of the ‘array’ as a model of thinking help deepen pupils’
understanding of number?
Does learning in a real context raise motivation and have an impact on learning?
Does spending time reflecting on strategies and developing thinking improve
pupils’ confidence and understanding?
2.3 Pupil Outcomes
By the end of the project pupils will:
• demonstrate increased understanding of a particular area of numeracy
• display positive attitudes towards numeracy
• show willingness to solve problems, communicate and work as a team.
2.4 Teacher Outcomes
By the end of the project teachers will have:
• gained an understanding of the methodology underpinning the Realistic
Mathematics Programme
• had the opportunity to share their experiences of using new methodology with
colleagues from another sector.
3.Process
An informal initial meeting was held with the teachers involved in August 2006 to
outline the project and hand out copies of ‘Young Mathematicians at Work,
Constructing Multiplication and Division’,
(Twomey, Fosnot, and Dolk,2001) This text would act as their ‘bible’ during the
project as it focuses on how to develop an understanding of multiplication and
division by, among other things:
 ‘describing and illustrating what it means to do and learn mathematics
 providing strategies to help teachers turn their classrooms into maths
workshops that encourage and reflect mathematizing
 examining several ways to engage and support children as they construct
important strategies and big ideas related to multiplication.’
In September 2006, the teachers met for a Training Day to discuss the above text
and to study the accompanying interactive CD, ‘Working with the Ratio Table,
Mathematical Models’.
The interactive CD allowed the teachers to watch and interact with a video that
depicts a classroom teacher as he listens to, questions, and interprets students’
thinking; develops connections between mathematical ideas and strategies; and
ultimately develops a vibrant mathematical community in the classroom.
The teachers made the following observations from the video:
 The students were very comfortable with mathematical language.
 There was a very positive classroom ethos with no ‘put downs’.
 The students were confident about discussing their strategies.
 The students were very patient towards each other.
 The students and the teacher displayed active listening by the way they attend
to each other’s ideas by paraphrasing or questioning to ensure real
understanding.
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The students learned from one another by the use of effective questioning
that developed their thinking.
In order to reinforce understanding, the teacher frequently employed the
technique of asking the students to share their thinking with their shoulder
partner.
The teacher accepted all answers without the use of praise, or comments on
whether the answers were right or wrong.
The students were given thinking time.
The following points were also discussed:
 The teacher had obviously had training in this methodology and acted as a
‘model’ for the students.
 All the pupils appeared to always be on task.
 It appeared that the pupils had been learning in this way for quite some time.
3.1Baseline Assessment
The Primary 7 pupils completed reflection logs and the S1/2 constructed mind maps.
The following was noted:

The primary pupils find all maths difficult and it is important to them how they
feel about the teacher and her response to requests for help. (see Appendix 1)

The S1/2 made the following comments:
 ‘I like the teachers.’
 ‘Sometimes maths is hard because it is not what I have done.’
 ‘I like games.’
 ‘We think maths is important.’
 ‘I do not like writing in jotters.’
 ‘Make maths better by making it easier to understand.’
 ‘I feel more confident about talking out in class.’
3.2 Learning Contexts
The children were set the following challenges:
Primary 7
1. Mr Griggs the grocer delivers the healthy fruit to our school. He needs to
count out the fruit quickly because he delivers to lots of schools. How could we
help him?
The pupils worked in pairs and were given pictures of fruit displayed in arrays that
subtly suggested skip counting, but pupils who needed to count by ones could.
Strategies offered by the children were then shared as a class. (See appendix 2)
2. Mrs Burn has just bought 186 pencils that she wants to give out fairly to each
of the 6 tables in our class, how many should each table receive?
The pupils worked in pairs on this sharing problem that develops the big idea that
dealing fairly produces fair groups. (See Appendix 2 for the pupils’ strategies)
3. Mrs Ingham has to sort out a big meeting in the hall. She wants us to work out
how many tables she needs Thomas to put out for 81 people. Each table has
room for 6 people.
This is a grouping problem as opposed to the previous sharing one. As the teacher
presented the problem, she drew a picture of one table showing each chair and then
presented a second table with the numeral 6. By doing this she suggests some
possible strategies, but does not complete the picture and does not try to lead the
children toward the use of one image over the other. (See Appendix 2 for the pupils’
comments)
4. Mrs Ingham wants to give everyone at the meeting one cup of coffee each and
each of the coffee pots holds 7 cups. How many coffee pots do we need?
This problem does not lend itself to counting in the hopes of moving pupils away from
counting one at a time. Coffee pots are not as easy for the pupils to draw as the
tables were, nor is it easy to demarcate each cup. The teacher hopes that the pupils
will use symbolisation and/or the distributive property of multiplication.
Secondary Sector
To set the scene in both classes for the purpose of the project the pupils were asked
what jobs they hoped to do when they left school. The pupils wrote the jobs on postits which they placed on the wall for discussion. They were then asked what skills
they would need to have to help them with these jobs when they left school. The
teacher talked about her son who was the local Bank Manager.
The skills elicited were:
 Communication
 Problem solving
 Being a team player.
The pupils were then set the following challenge which was adapted from M.
Askew’s ‘Teaching mental strategies’.
S1/S2 The DIY Problem
Steven, the local Bank Manager has to be very quick with number calculations
when interviewing customers. He has asked for our help. Can we come up with
strategies for him for multiplication?
1. The pupils worked in pairs and were given a grid. (See Appendix 3) They
decided to work with the 7 times table as this, they felt, was the hardest
table. They were asked to complete the grid by filling in what they felt
were the easiest ones and discuss how they might work out the harder ones.
Throughout the lesson, the pupils were reminded of the learning challenges:
 Be a team player
 Share ideas and strategies
 Find smart ways to multiply numbers
The strategies were shared as a class.
2. Using the strategies discussed in 1, the pupils tackled a grid with
multiplication by 8 and then by 13. The teacher introduced the idea of a
blank array as a thinking model when tackling multiplication by 13.
3. Using the blank array as a thinking model for the distributive property of
multiplication, the pupils tackled multiplication of 2 digit numbers by 2
digits.
S2
In order to introduce fair sharing where the pupils can generate and model for
themselves mathematical ideas related to fractions the teacher began with the
following context:
The school had been organising various trips recently and the canteen staff had made
up large baguettes for the pupils to share at lunch time. Some of the pupils had been
complaining that the system wasn’t fair. What do you think?
Threave Gardens 10 pupils 3 baguettes
Hill Walking
5 pupils 2 baguettes
Carlisle
5 pupils 3 baguettes
Swimming
10 pupils 5 baguettes
If each baguette costs £1 how much should each pupil pay?
3.3 Thinking out of the Box
Although one or two of the S2 pupils were confident with sharing strategies, it was
proving very difficult to engage the others in discussion. The teacher was concerned
that, due to the pupils’ general lack of confidence and especially their reticence to
discuss their mathematics, learning wasn’t being internalised. It was time for thinking
‘out of the box’!
It was decided that the S2 pupils would produce a set of materials for a group of
children at the primary school next door.
Plan
1. The primary and S2 teacher met and decided that ‘time’ would be the focus for
the project.
2. To enable them to decide on a context for the worksheets and games, the S2
pupils prepared questionnaires for the primary children.
3. The S2 pupils visited the primary school in order to both meet their ‘clients’
and complete the questionnaires.
4. The S2 pupils produced sets of card games, worksheets and answer sheets.
5. The S2 pupils took the materials to the primary and explained to the pupils how
to play the games.
6. Steven, the local Bank Manager visited the S2 pupils in order to see the
materials and discuss the skills that the pupils needed to use during the
project. (See video clip)
4.Evaluation
S1
The pupils found the DIY problem challenging yet the teacher was impressed by the
high level of engagement during the first lesson, especially by one boy who even after
the end of period bell had rung, still persisted on sharing more strategies! This boy
was eventually moved to a higher set. More pupils were moved up later.
The use of the array helped pupils to tackle simple multiplication problems eg 8x13,
but initially 18x13 proved a step too far for some pupils. As the pupils had not had
earlier experience of working with the array, the teacher decided to go back to first
principles and use pictures of trays of fruit etc followed by the use of square blocks
and square paper. As understanding improved, the teacher was then able to
reintroduce the blank array with more success.
S1 Pupils’ Reflections
Positive
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‘I love maths very much.’
‘Maths is not so hard.’
‘Maths is sometimes fun.’
‘I am confident in maths.’
‘It is cool.’
‘I feel more confident.’
‘I feel better in maths now.’
Negative
 ‘I find it boring and easy.’
S2 The Baguette Problem
The teacher was amazed at how deeply the pupils had to think when tackling the
‘baguette’ problem and how difficult it was for them to articulate their strategies for
solving the problem and the small group made discussion limited. She herself found it
challenging to be non-committal with her comments, allowing her pupils to develop
their thinking skills. She was anxious that she might ‘get it wrong’ during the project,
as she was aware of the importance of both the context and the numbers chosen. The
numbers were carefully chosen in order to enable the pupils to develop an
understanding of tenths and hundredths for later percentages work.
The pupils found the ‘baguette’ problem challenging and needed more practice with
concrete materials to develop a real understanding of equivalent fractions therefore
an equivalent fraction chart with ideas for practical activities was purchased.
The DIY Problem
As a result of working with arrays during the DIY problem, the pupils had a better
understanding of square numbers. The array work will be developed later when
tackling area of rectangles.
The Times Problem
The ‘time’ project was very worthwhile as it
equipped the pupils with a variety of skills. The
teacher is keen to develop other enterprising
approaches to sustain the pupils’ motivation and
self-esteem.
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The pupils were very keen to engage with the ‘time’ project.
The pupils were given real, meaningful opportunities to engage in
questioning, discussion and problem solving.
Cross curricular links were made as the pupils were engaged in Literacy
and IT activities.
The pupils were working collaboratively.
The pupils were involved in an enterprising approach to their learning.
The teacher reported that the pupils were more motivated towards
their learning and were more confident during discussions.
During S2 Parents Evening, various parents reported on how pleased they
were with their children’s response to Steven’s visit. They felt that
Steven’s visit had helped to raise their children’s self esteem, as he told
them that during their project they had displayed skills of team building,
communication and problem solving and these were the skills that he
would look for in a prospective employee!
S2 Pupils’ Reflections
Positive
 ‘Group work was good fun and gave
me confidence because it was
working in a different way.’
 ‘Visiting the Primary was something
different.’
 ‘Using the computers was fun
because we were still learning but
still having fun.’
 ‘Making your own worksheets was
great because again it was
different.’
 ‘Visit from Bank Manager was great
because it is good getting some
advice from a different person.’
Negative
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‘Didn’t like making worksheets.’
‘Don’t like going to the Primary.’
‘Don’t like group work.’
‘Don’t like the visit from the
Bank Manager.’
Table of S2’s comments
Activity
Group work
Visiting the Primary
Using the computers
Making the worksheets
Visit from the Bank Manager
Liked
4
4
6
3
4
Disliked No Comment
1
2
3
1
4
2
1
S1/S2 Teachers’ Reflections
Generally, teachers’ perceptions were that:
 Pupils were making connections within different areas of mathematics.
 It is important that the specifics of the context are right in order for the
learning to be within the pupil’s ‘zone of proximal development’.
 Motivation was high when pupils were engaged in real and meaningful learning
experiences.
 Because the pupils were set for mathematics from S1 and these were the
lowest sets, there often wasn’t a more able pupil to ‘spark off’ ideas.
 Early in the year, especially when the teacher is new, lack of confidence can
be a major issue in lower ability sets.
P7
Pupils’ Reflections
The children were posed the following question:
‘What did you think about helping to solve the problems?’
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‘You got to work with different people because then you would be able to find
out how different people did things ‘cause you would be able to find out if they
did it in a different way from you.’
‘I enjoyed it because problem solving is fun and I liked who I worked with
because Sam helps and doesn’t just leave everything to me.’
‘The problem about fruit was fun because it was based on something.’
‘I really enjoyed it and it was good for team work.’
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‘I liked it because it gave me a challenge and I liked working out the problems.
At first it was hard then further on it was easy.’
‘I really enjoyed it because of the amount of different ways to solve a
problem.’
‘I discovered new ideas and methods to help me with this area of maths.’
‘I enjoyed working with a partner to solve the problems.’
‘It was hard but my partner made it easy, with his help and support it was
easy.’
‘I learned different ways of finding solutions.’
‘Drawing the questions was a bit easier.’
‘One thing I didn’t like was that you never got to choose who you worked with.’
Teachers’ Reflections
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‘Children learn better and are more motivated when the context is
meaningful.’
‘The use of the array helped the children to understand multiplication.’
‘Children enjoy sharing their strategies and this has helped their
confidence.’
‘We thoroughly enjoyed the whole experience.’
All teachers involved in the project agreed that they had enjoyed both working with
colleagues from another sector and sharing their experiences. It was agreed that
although they had gained a basic understanding of the methodology underpinning the
Realistic Mathematics Programme, there was much more to learn in order to
implement it fully in all areas of mathematics.
5. Implications
Along with equal groupings, sharing, jumps on a number line and scalings, the use of
the array is a model of multiplication and division that pupils need to experience as it:
 provides a visual picture of multiplication and the commutative law.
 helps to develop the long multiplication algorithm.
 provides opportunities for realistic contexts.
While during the project, the use of the array did help to deepen pupils’
understanding of multiplication, perhaps if these pupils had had experience of all the
multiplication models earlier on in their learning development then their
understanding and achievement would have been better at the outset. Therefore,
more relevant in-depth CPD opportunities need to be made available for all staff in
both sectors to discuss and share not only how pupils learn but the learning processes
that they follow.
Frank Eade and others, from the Centre of Maths Education in Manchester
Metropolitan University, have been developing a programme for teachers in order to
introduce the Realistic Mathematics Education approach. It would be advantageous to
make contact with the University.
For most pupils, learning in some sort of context is necessary for understanding and
making links between other curricular areas. It is also more likely to be meaningful,
relevant and stimulating.
For some pupils however, especially those who have continually been failed by an
outmoded system that has not been geared towards them as neither personalities nor
learners, only an ‘enterprising type’ context led by educators who are willing to
challenge their own beliefs about pedagogy and change their classroom behaviour to
engage successfully with their pupils, will suffice to promote true intrinsic motivation.
This ideal contains all the elements or ‘drivers’ of McLean’s motivation model and will
be a necessary step towards empowerment, (McLean, 2003)
In planning for this ‘enterprising type’ experience it will be necessary for the teacher
to build in ‘challenges’ that necessitate learning taking place. It is during this time
that pupils will be given the opportunity to discuss and reflect on various strategies in
order to solve problems and develop their thinking skills.
‘Content, judiciously selected for its rich contributions to thinking and learning,
becomes the vehicle to carry the learning processes.’ ( Costa, 2001)
A vital element of the philosophy of RME is that the pupils engage in discussion and
questioning about their mathematics. During the project this proved to be difficult at
times due to the fact that the pupils were all from the lowest sets and there were no
more-able pupils to ‘spark off’ ideas, model learning behaviour or be catalysts for
developing thinking skills for their peers.
It was apparent in the classrooms involved in the ‘Mathematics in the City’ project
that the pupils were of mixed ability. A real community of learners had been
developed where all pupils were expected to be able to access and engage with the
mathematics. There was real collaboration between the pupils and all were
empowered.
Professor Brian Boyd of Strathclyde University states that, ‘Perhaps the strongest
argument against setting is that it is irrelevant. Formative assessment and learning
and teaching approaches that stress co-operation, creativity and pupil understanding
should make setting a thing of the past.’ (TESS August 4, 2006)
6. Conclusions
In order to access the Dutch Realistic Mathematics curricula, in-depth professional
development of staff is vital, to allow teachers to challenge and adapt their own
beliefs about how pupils learn and to gain knowledge of the learning processes and
methodology entailed. Evidence appears to suggest that in order to have optimum
engagement with this model;
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it needs to be introduced much earlier than Primary7/S1 before the pupils
have acquired any negative self-belief patterns
pupils need to be placed in mixed ability classes in order to maximise the
cognitive development of all children.
The project provides evidence that many pupils will engage willingly in solving
problems that are set in realistic contexts, but for others who have become
disenchanted by years of curriculum architecture that failed to address their
particular needs, a more ‘enterprising’ approach is required to achieve success. It is
hoped that A Curriculum for Excellence will be the long-awaited impetus for changes
that are needed to take pedagogy and learning into the 21st Century to empower and
enable all pupils and educators alike to become successful learners, confident
individuals, responsible citizens and effective contributors.
7. References/Resources
Arthur, L. (2001) Developing minds: a resource book for teaching thinking, ASCD.
Askew, M., Robinson, D. and Mosley, F. (2001) Teaching mental
strategies: number calculations in years 5 and 6. BEAM Education.
Hersch, S., Cameron, A. and Dolk, M. (2006) Young mathematicians at work, sharing
submarine sandwiches, grades 5-8: a context for fractions. New Hampshire,
Heinemann.
This facilitator’s package includes:
 Overview Manual
 Facilitator’s Guide
 Interactive CD
McLean, A. (2003) The motivated school. Paul Chapman Publishing.
Twomey, Fosnot, C. and Dolk, M. (2001) Young mathematicians at work, constructing
multiplication and division. New Hampshire, Heinemann.
Twomey, Fosnot, C. and Dolk, M. (2002) Young mathematicians at work, constructing
fractions, decimals and percentages. New Hampshire, Heinemann.
Twomey, Fosnot, C. and Dolk, M. (2006) Young mathematicians at work, working with
the ratio table, grades 5-8. New Hampshire, Heinemann.
This facilitator’s package includes:
 Overview Manual
 Facilitator’s Guide
 Interactive CD
Appendix 1
Primary 7 Initial Reflection Log Comments
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‘I’m not good at maths but I’m never worried because I’m not scared to ask for
help.’
‘Mrs. Burn explains it very well. ‘
‘Mental maths is my least favourite.’
‘Mrs. Burn always helps and makes maths fun.’
‘Maths is fun but I’m not always confident.’
‘If you finish your work you can play a maths game.’
‘Computer suite is fun.’
‘I’m better at mental maths now.’
‘Mrs. Burn is nice and kind.’
‘Maths is hard.’
Appendix 2
Comments noted as P7 pupils were on task
1.Help Mr Griggs count his fruit more quickly:
Rory and Sean
“It’s tables!”
Zoe and Amy
“Your way will take longer and it has to be quicker.”
Shona and Edith
“We can count in 2’s or it’s 4x3.”
Sam and Ashley
“1,2,3, that’s just normal counting and it has to be quicker.”
“We’ve already got that, see 6, 12, 18, 24 etc”
“That’s good thinking Sam.”
“It’s the 4 times table.”
“We could count in 4’s.”
Jack and Christopher
“4 fives are 20 and another 20 is 40.
It’s easier that way.”
Plums:
“6+6=12 12+12=24 24+24=48 48+12=54”
“How did we get that again?”
“Right count the row and the side then times it, 9x6=54”
“This is hard!”
Daniel and Grant
We counted 6 rows of 9= 54, then we halved it and then doubled it.”
[They’ve been doing doubling and halves in class work.]
Brandon and Shane
“We counted rows of 9 and 6 down the side.’
Sophie and Shannon
“We just did our tables 2,3,4,!”
Ewan and Cameron
“Miss out the top row of 9 then count 9x5=45 and the other 9 makes 54.
Ewan and Cameron were interesting because they did 9x5 because they knew the
answer then added the 9 on.
2. Help Mrs Burn decide how many pencils go on each table.
Ashley and Edith
Drew 6 tables, then put 10 pencils on each, then another 10, then another 10, we’ve
got 6 left so we’ll put 1 pencil on each table.
Shannon C + Daniel P
A/A but counted in tens as they were going along till there were 6 left,
“So that leaves 1 more for each table.”
Ewan + Rory
Tried dividing 186 by 2, then 3, then 6.
They used practical materials [tens + units] to divide out the pencils into 6 piles.
“If you divide 186 by 6 you get 31 so now we’ll just draw it.”
Cameron + Sean
Started from 6x10=60, then another 6x10 and go from there.
They took the 60 away from 186, then another 60 etc
Cameron was drawing out the tables with pencils and Sean was counting out the tens
and units.
Sam + Zöe
Started with 6x12 because there were 12 pencils in Sam’s packet of pencils then took
it away from 186, 186-72, leaving 114 then took another 72 away leaving 42, they then
divided what was left by 6, 42 divided by 6=7 [by counting the stations of the 6 times
tables till they got to 42.
12+12+7=31
Ewan + Sophie
We knew 10x10 =100 so we gave 10 to each table, then another 10 which was 21 so we
had 66 left so we gave them another 10 each, then we gave them out 1 at a time.
Shane + Emma
Took a long time to decide on strategy and finally after hearing another group talking
about tens, started putting 10’s on each table till they got to 180, then they put 1 on
each table.
Shane was more focused on drawing the 6 tables than solving the problem.
Brian, Shona + Grant
Put 10 on each table then decided to put another 10 and another 10 till only 6 were
left so we put 1 on each table.
Jack H + Jade
Practical materials first then they did the sum.
“First we put 10 on each table which was 60, then another 10 which was 120, then
another 10 on each table which adds up to 180, 6 were left so we put 1 on each table.”
Brandon, Brittany + Chris H
Drew out 6 tables and put tens then units on each table, counted them, which was 31
each.
[This group were prompted by the assistant helping Chris H]
3/4. Help Mrs Ingham with tables and coffee
Ashley and Jade
Got 81 units, drew tables and put 6 at each till there were none left, got 14 tables.
Thought they needed 13 pots, no logic for this and then started to work out 7x13 but
couldn’t.
Shannon C + Sam
Drew 2 tables with 6 at each and just kept going till none left. Said they needed 13
and a half tables, I asked,
Can you have half a table?”
“Oh, right.” replied Shannon. “Then we’ll need 14.”
With the coffee pots they started with 10x7=70, then kept going till they got to 84,
got 12 pots.
Ewan + Shane
Added 6+6 till they got to 48, then 48+48=96 [they didn’t realise this added up to 96
till I pointed it out then they took away 16 then added 1, then took tables away.
They didn’t get on to the coffee pots.
Cameron + Edith
Drew 81 spots then circled them into groups of 6, got 14.
Then they just used the units already drawn and circled them in groups of 7.
Brian and Zoe
Started from 9x9=81, then 81÷6=16
They did not want to draw everything out as it would take time but came to a
standstill and didn’t get a solution.
Sophie + Jack
6x4=24, drew 4 tables, then another 4=48 and then doubled it =96.
They didn’t realise this was 96 till I pointed it out so they took 15 away. [2 tables]
Coffee pots 81÷7=14
Ewan Mc M+ Sean
6x12=72 Then 6x13, see what that made then 6x14=84
81÷7=14 but couldn’t tell me how they got this.
Rory and Amy
Started off with 8 tens and 1 unit practical materials but couldn’t split them up so
abandoned that idea.
Drew tables with 6 on each but counted up in 3’s because this was easier.
Then drew 7 cups on each table till they got to 13 tables, they miscalculated 13x7
which was 91.
Brandon and Shona
Drew 3 tables with 6 at each= 3x6=18 then carried on till 81 was used up but didn’t
keep drawing tables stopped after 8 then drew a table with number 6 in the middle.
For coffee pots, 81 circled in groups of 7= 11r3
Christopher and Grant
Took 81 units, separated into groups of 6, when we’ve figured the answer we’ll draw
the tables with [number] 6 on them, it’s 13 tables with 6 and 1 with 3 = 14 tables.
Its 7 cups and you’ll need 12 pots.
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Brittany and Daniel P
Drew 5 tables to start with, put 6 at each 5x6=30.
5 tables come to 30, another 5 tables =60 and 21 left which is 4 more tables.
They drew lots of coffee pots with 7 on each but hadn’t yet worked out how many
they would need.
This week was a breakthrough week because most of the pairs were looking for
shortcuts to avoid drawing out all the groups. They are still starting with what they
know 5x6=30, counting in threes etc
The next lesson will interesting, I want to see if they get to the point of seeing the 6
or 7 times tables as the starting point.
Appendix 3
Our
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learning challenges:
Be a team player
Share ideas and strategies
Find smart ways to multiply big numbers
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Video Transcript
Steven Lumb, Bank Manager HBOS
and S2 Dalbeattie High
Steven introduces himself and says that he has been asked to speak to the pupils
about what they have been doing recently and what he has been doing in his job in the
Bank of Scotland, Dalbeattie.
The girls tell Steven about the games and booklets they have been making and
demonstrate how the ‘Pairs’ game works.
The boys tell Steven about another game of ‘Matching Clocks’ they made and Steven
gives them a chocolate egg as a prize as it’s Easter!
Steven asks them what else they have been doing. One of the girls tells him about
one of the worksheets she made where you have to change a number into a month.
She says, ‘Number one would be January.’ Steven asks what ten would be and another
girl wins an egg for answering October!
Steven asks them what skills they have learned during their project and one of the
girls tells him that they needed to co-operate with each other. Another girl tells him
that they had to work as a team. He asks them how they found it working as a team,
and a boy tells him it was fun.
Steven asks them what they needed to do when making the worksheets and one of
the girls tells him that they needed to solve problems.
Steven pulls this together by emphasising the three key skills that the pupils had
mentioned:
 Communication skills or talking to each other.
 Team working skills or working together as a team.
 Problem solving.
He tells them that the above are the three most important skills that he looks for
when he’s trying to employ someone to work at the branch in Dalbeattie. He says that
he is the manager there and when they (S2) finish school and ‘fancy a wee job up
there’, that the three attributes that he’d be looking for are: people who can talk well
to each other, people who can work well as a team and people who can solve problems
within that.
Steven asks them to give him an example of when they have walked into a bank and
communication would be important. He tells them that the person you talk to in the
bank has to be really clear.
He asks them why they think that problem solving would be important when working in
a bank. One of the boys replies by saying that, ‘When someone robs a bank you have
to be able to see how much is pinched!’
Steven asks them if anyone knows why banks exist. One boy answers, ‘To save up
cash.’ Steven asks them what they might save cash for and a boy mentions a car. He
then asks him what kind of car he would like if he could have any car in the world and
the boy answers,’ A Nissan Skyline.’
Steven goes on to say that banks can be used for saving for lots and lots of things,
e.g. a car, house ,holiday. He asks who is going on holiday this year and the group tell
him where they hope to go.
Steven asks them why they think teamwork would be important when working in a
bank. He gives them the scenario of coming into school in the morning when they are
feeling really tired and asks them how they would motivate each other. He asks them
how they would feel the morning after a football match if Celtic lost. One boy says he
would feel ‘gutted’ and ‘not chuffed’ and Steven asks him what would make him feel
better. The boy replies by saying that he would play on his Play Station.
Steven goes on to say that being able to motivate each other and talk as a team to
help each other, are fantastic skills that he needs to use every day in the bank. He
has to motivate his staff and colleagues and they need to motivate each other. He
explains that passing exams are not the ‘be all and end all’ as the basic skills of
communication (talking to each other), team work (how to motivate), and problem
solving are really important skills that you will need to have.
Steven concludes the session by answering questions and asking the pupils what they
would like to do when they leave school.