Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London

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Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Primary
Mathematics
SPRING 2019 • Volume 23 • Number 1

in this issue   l Teaching multiplication tables l Explicitly connecting mathematical ideas l Teaching
                with Challenging Tasks l Professional learning through research l Primary Mathematics
                Challenge at Balgowan Primary school l Tackling Tables revisited l Book Review
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Editorial
                   Cherri Moseley
                                                               Primary
                                                               Mathematics
                                                           SPRING 2019  •  Volume 23  •  Number 1
Welcome to the Spring 2019 edition of Primary
Mathematics. This year sees the introduction of the

                                                                            Contents
national voluntary pilot of the online multiplication
table check (MTC) to year 4 pupils in June 2019. The
MTC becomes statutory in June 2020. With this in mind,
you will not be surprised that most of our articles have
a focus on multiplication and multiplicative thinking.         Teaching multiplication tables                            3
   With many teachers expressing some concern
                                                               Gemma Parker
about the check, the joint MA/ATM Primary Group
spent some time discussing various ways of teaching
multiplication tables well. Gemma Parker, Vice Chair           Explicitly connecting mathematical
of the group, distilled the group’s thinking into              ideas: How well is it done?                               7
an article for this issue. The group concluded that            Ray Huntley and Chris Hurst
making connections and spotting patterns are key to
conceptual understanding and shared several ideas on
                                                               Teaching with Challenging Tasks:
how to represent those relationships and patterns to
                                                               Experiments with counting patterns 11
develop understanding. Ray Huntley and Chris Hurst
make us think a little more deeply about multiplicative        James Russo
thinking while James Russo looks at using Challenging
Tasks to develop pattern recognition. The tasks use skip       Professional learning through
counting to explore prime and composite numbers,               research: Planning for success and
at the same time consolidating the patterns of the             identifying barriers                                    17
multiplication tables. Dennis Brown updates us on the
                                                               Ruth Trundley
impact of Tackling Tables in one school, while a parent
reviews using this approach at home. This is just one of
many approaches to learning tables. We would love to           Primary Mathematics Challenge at
hear about what works for you in your school.                  Balgowan Primary school                                 22
   Ruth Trundley looks at using action research to             Joyce Lydford
develop professional learning. This approach could be
used by colleagues in your school to explore planning          Tackling Tables revisited                               24
for success and identifying barriers to developing
                                                               Dennis Brown
multiplicative reasoning. And finally, Joyce Lydford
gives us a quick look at how she uses the Primary
Mathematics Challenge (PMC) to develop children’s              Book Review                            16, 26 and 27
toolkit of strategies for problem-solving, relevant
across the mathematics curriculum and beyond. PMC is           Cover picture: Teaching with Challenging
aimed at children in Year 5 and 6 but the Mathematical         Tasks. A student works through the third
Association is currently developing a Junior Mathematics       Challenging Task, trying to determine which
Challenge for Year 3 and 4 children. We will bring you         numbers will survive.
more news on this as it becomes available.
   Come and join us at the joint MA/ATM conference
at Chesford Grange, Warwick, 15–18 April 2019
www.m-a.org.uk/conference-2019. Many of the
joint primary group will be there. Alternatively, email      PLEASE GET WRITING!
Gemma Parker and come and join us at one of our
                                                             Share your expertise, experiences, reports, reviews, hints, tips,

                                                                                                                ✍
termly meetings.                                             tales and howlers with others in your profession.

 Cherri Moseley, Senior Editor, Primary Mathematics          Please email to the editorial team at:
                      primarymaths@m-a.org.uk                primarymaths@m-a.org.uk

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                                                1
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
2   Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Teaching                                                              Gemma Parker
                                                                        summarizes the Joint
  multiplication                                                        MA/ATM Primary
                                                                        Group discussion on
  tables                                                                this topic

In May 2018, the joint primary group of the                  The group identified that making connections
Association of Teachers of Mathematics and The             and spotting patterns are key to conceptual
Mathematical Association discussed the agenda              understanding and the following sections illustrate
item of how to teach times tables well. This was           how these factors might translate to primary
considered a high priority due to the introduction         classrooms and whole school policies. Examples are
of statutory testing for Year 4 children from 2020         given of how varied representations (including visual
(DfE, 2018), which has propelled fluent recall             images), contextual situations and manipulatives can
of multiplication facts to the top of the agenda.          be embedded in teaching and learning sequences.
Whilst the group outlined their objections to the          Suggestions are also made as to how to encourage
proposed test in the assessment consultation, it           children’s reasoning around multiplication facts, as
hopes that this article will provide a useful resource     this helps to develop confident, competent children
for teachers as they prepare their children. This is       who are building a firm foundation of known facts.
considered particularly important by the group as
they recognise the importance of children knowing
their multiplication facts yet are wary of the way in
                                                           Connections
which the proposed test emphasises rote learning           An ‘important characteristic of understanding is
and rapid recall over understanding of mathematical        that it involves connections between different
structures. This article has been written to support       ideas or concepts’ (Barmby, Harries and Higgins,
teachers to continue to teach multiplication facts         2010:46). Through emphasising connections within
in a sustainable way which focuses on children’s           mathematics, pupils develop deep learning that
understanding. The group’s discussion is shared            can be sustained (NCETM, 2016). This is because it
here, and it is hoped it will be a useful talking point    relieves the pressure on memory which, as we can all
and call to action for all those involved in teaching      attest, can be fallible! As Holt (1982:10) suggests,
times tables.                                              if children simply learn multiplication facts parrot-
                                                           fashion, devoid of understanding or connection,
                                                           when memory fails, a child ‘is perfectly capable
Overarching principles
                                                           of saying that 7 × 8 = 23…or that…even when he
The highest priority identified by the group was           knows 7 × 8, he may not know 8 × 7, he may say
the development of conceptual understanding                it is something quite different.’ For confidence-
alongside fluent recall of multiplication facts. The       building, accessible learning which is long-term and
group were mindful of the risk rote learning poses         deep rooted, a focus on connections is key!
to this, so advocated a thoughtful balance between            For multiplication facts, this means making
conceptual understanding and recall-focused                explicit the link with other operations. Can children
activities. Members were supportive of the National        explain multiplication as repeated addition? Are they
Curriculum aims of problem solving, reasoning and          confident with deriving division facts from known
fluency regarding multiplication facts and believe         times tables? Can they solve
that embedding them is beneficial to children’s            questions such as [ ] × 6 = 42?
conceptual understanding in mathematics. Indeed,           Employing representations
the NCETM (2016) propose that ‘procedural fluency          such as this visual image of
and conceptual understanding are developed in              the inverse relationship of
tandem because each supports the development of            factors and multiples via
the other.’                                                triads can be very powerful:

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                             3
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Similarly, showing jumps along a number line                    How would you use 3 × 7 to work out 6 × 7?
highlights the link between repeated addition and                 How would you find a number which is both a
multiplication:                                                    factor of 64 and 40?
                                                                Confidence with links between multiples breeds
                                                                fluency as it unlocks the potential for known facts to
                                                                reveal so much more and as Cotton (2013) states,
                                                                one of the most useful ways of using known facts
                                                                is to derive new facts. Frequently using sentence
  Calculators ‘provide rapid and accurate feedback              starters such as ‘If I know ___, then I know___’
about the number system’ (Hopkins et al., 1999:34)              encourages children to be creative and seek out
and can reinforce this link. Cumulatively adding 3              their own connections, which is a great example of
to zero will produce on the screen a sequence of                learning at greater depth.
multiples of 3. The ease with which such a pattern                 Flexible, autonomous use of number facts can
can be generated makes it accessible to those                   develop confident learners who are resilient and
children at the early stages of learning multiplication         able to think creatively. For example, if children are
facts. Challenging them with questions such as ‘how             unable to recall 7 × 9, they could recall 7 × 10 and
many times do you need to add 3 to generate 15 on               subtract 7. Employing Cuisenaire rods (or sticks
the screen?’ before encouraging them to check, is a             of multilink) to represent this provides a physical
worthwhile activity. If they can convincingly explain           experience which can become internalised. Creating
why, deeper learning is occurring.                              arrays from squared paper can work similarly. By
  Making links extends to making connections                    creating an array of 5 × 8 and folding it in half to
between multiples. At the simplest level, under-                show 5 × 4, children are stimulated to conclude that
standing of the commutative law can help those                  5 × 8 is double 5 × 4. Using squared paper arrays
children who struggle with 7 × 3 because they don’t             can exemplify the distributive law of multiplication
know their ‘sevens’, to easily find the answer when             too. After creating a squared paper array for 6 × 9,
they flip it to 3 × 7. Arrays are fantastic representations     children can fold to see that this array comprises of
that illustrate this:                                           three lots of 6 × 3 – another derived shortcut.
                                                                   Equipping children with investigative skills and
                                                                tools can imbue them with confidence. Arrays which
                                                                illustrate square numbers demonstrate the aptness
                                                                of their name and manipulating them can reveal an
                                                                elegant pattern which opens a world of possibilities!
  In general, thoughtfully planned learning sequences
which encourage children to explore and exploit links
will be far more impactful than repeatedly testing
of multiplication facts. Essentially, a focus on the
process, instead of the answer, can be extremely
                                                                  Ask children to spot the pattern in this table and
valuable, and a great stimulus for discussion.
                                                                see how reorganising an array illustrating the square
                                                                number in the first column can reveal the answer to
                                                                the calculation in the second column.

                                                                 2×2=4                      1×3=3

                                                                 3×3=9                      2×4=8
                                                                 4 × 4 = 16                 3 × 5 = 15
                                                                 5 × 5 = 25                 4 × 6 = 24

                                                                  And challenge them to now answer 39 × 41.
  Challenging children to explain to an alien who               Whilst this is clearly not part of the 12 × 12 known
doesn’t know her tables how to work out 6 × 3 is                facts range, piquing curiosities and exploring elegant
an excellent check of understanding. Other useful               solutions can engage and enthuse even the most
challenges might include:                                       reluctant of children.

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Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Connections with real life can be very powerful
and there are plenty of examples which marry
everyday objects and times tables. Pairs of socks,
5p coins, puppy footprints and octopus tentacles
provide accessible representations of multiples and
children’s own interests should be a driving force.
Singing has long been a favourite for teaching tables
and familiar songs which count in multiples can be
enjoyable and memorable.
  Focussing on connections removes the risk that
children will forget isolated facts. It supports the
notion that everyone can do maths because children
are being equipped with the skills and mindset to
work out what they do not yet know. The foundation         the tens too, which can help highlight other patterns,
this builds for fluency and confidence is fundamental      for example within the nine times table where the
for success, and an indisputable pillar of teaching        tens digit increases as the ones digit decreases. For
times tables.                                              a greater depth challenge, can children convincingly
                                                           explain (perhaps using manipulatives/drawings) why
                                                           this is the case?
Pattern                                                    Using structured sentences can help ‘children to
Pattern is an integral element in primary                  communicate their ideas with mathematical precision
mathematics. It is critical for fluent recall of           and clarity’ (NCETM, 2015) and ‘sentence structures
multiplication facts as it can lighten the memory          often express key conceptual ideas or generalities
load. For example, if children know that multiples         and provide a framework to embed conceptual
of 2 have a repeating pattern of 0, 2, 4, 6 and 8 in       knowledge and build understanding’ (NCETM,
the ones digits, then it is easy to work out that the      2015). For example, ‘3 multiplied by 6 is 18. 3
next multiple of 2 from 14 is 16. Indeed, ‘asking          multiplied by 12 is double 18’ is a sentence structure
children to explore the patterns in numbers in the         which children could use to highlight relationships
times tables is a good way of encouraging them to          between multiplies and derive unknown facts. Here,
get a feel for the properties of numbers’ (Cotton,         connections are key and posing the open question,
2013:92). However, it is important to be wary of           “What’s the same, what’s different between the
over-generalisation as children may invent plausible,      three times table and the six times table?” really
but incorrect, new rules such as every odd number          requires children to explore and make connections
is in the x3 table (Cotton, 2013). Challenging them        for themselves. Using manipulatives collaboratively
to disprove their conjectures with counter-examples        to do so could be a fantastic learning experience.
can be a powerful learning experience.                       The humble number line is one of the most
   Visual representations of patterns can help secure      important resources to support children in noticing
them in children’s minds, for example, highlighting        pattern (Cotton, 2013) and skip counting (Askew,
multiples on a 100 square is an explorative opportunity    2009) along with a counting stick can be a great
to help children begin                                     group activity. Whilst repeatedly saying multiples
to generalise. Using                                       in sequence using rhythm helps embed them in
dials to join up the                                       memory, moving along a counting stick illustrates
ones digits of a times                                     repeated addition thus reinforcing the structures of
table is another great                                     multiplication. Developing this strategy provides a
way      to     illustrate                                 tool for children when faced with a question about a
pattern. Ask children                                      fact that they are unable to immediately recall.
to work out which                                            Finally, it is recognised that practice is a vital
table this dial shows.                                     part of learning, and intelligent practice that both
   Challenge children to identify which different times    reinforces pupils’ procedural fluency and develops
table could be overlaid in another colour to miss every    their conceptual understanding is the most valuable
other digit in this pattern. This simple representation    (NCETM, 2016). Practise which elicits and highlights
focuses on the ones digit of multiples but using base      pattern in multiplication facts is intelligent and
10 equipment to represent sequences incorporates           worthwhile, and integral to the Shanghai mastery

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                              5
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
1×1 =1

 1×2=2     2×2=4

 1×3=3     2×3=6     3×3=9

 1×4=4     2×4=8     3×4=12    4×4=16

 1×5=5     2×5=10    3×5=15    4×5=20    5×5=25

 1×6=6     2×6=12    3×6=18    4×6=25    5×6=30    6×6=36

 1×7=7     2×7=14    3×7=21    4×7=28    5×7=35    6×7=42    7×7=49

 1×8=8     2×8=16    3×8=24    4×8=32    5×8=40    6×8=48    7×8=56    8×8=64

 1×9=9     2×9=18    3×9=27    4×9=36    5×9=45    6×9=54    7×9=63    8×9=72    9×9=81

 1×10=10   2×10=20   3×10=30   4×10=40   5×10=50   6×10=60   7×10=70   8×10=80   9×10=90    10×10=100

 1×11=11   2×11=22   3×11=33   4×11=44   5×11=55   6×11=66   7×11=77   8×11=88   9×11=99    10×11=110   11×11=121

 1×12=12   2×12=24   3×12=36   4×12=48   5×12=60   6×12=72   7×12=84   8×12=96   9×12=108   10×12=120   11×12=132   12×12=144

approach. The above table highlights pattern as                   Barmby, P. and Harries, A.V. and Higgins, S.E. (2010)
well as the importance of the commutative law for                     ‘Teaching for understanding/understanding for
decreasing the number of known facts to be learned.                   teaching’, in Thompson, I (Ed.) Issues in teaching
  To summarise, pattern spotting can help nurture                     numeracy in primary schools, Berkshire, Open
enjoyment and curiosity in primary mathematics                        University Press, pp. 45-57.
(Gifford and Thouless, 2016), and it can support                  Cotton, T. (2013) Understanding and Teaching
children’s developing fluency with multiplication                     Primary Mathematics, Harlow, Pearson
facts. Through a focus on pattern spotting, supported             DfE (2018) Multiplication tables check trials to begin
by manipulatives and drawings, children develop an                    in schools, Available at: https://www.gov.uk/
invaluable sense of number and their innate feeling                   government/news/multiplication-tables-check-
of whether an answer is right or wrong matures. If                    trials-to-begin-in-schools [accessed 12th June
they can explain why 17 cannot be a multiple of 3,                    2018]
they are heading in the right direction!                          Gifford, S. and Thouless, H. (2016) Using pattern
                                                                      to inspire rich mathematical discourse in mixed
Conclusion                                                            attainment groups, Available at: https://www.
                                                                      atm.org.uk/write/MediaUploads/Resources/
This article aims to provide support for teachers                     MT254_Using_Patterns.pdf [accessed 12th
building a long-term, whole school approach to                        June 2018]
teaching multiplication facts. By prioritising children’s         Holt, J. (1982) How Children Fail, Pitman Publishing
conceptual understanding through a clear focus on                     Company, USA
drawing connections and pattern spotting, teachers                Hopkins, C., Gifford, S. and Pepperell, S. (1999)
can develop an approach which supports children                       Mathematics in the Primary Schools, A Sense
to be competent in recalling multiplication facts.                    of Progression, London, David Fulton Publishers
Through using resources, discussing, exploring and                NCETM (2015) Calculation Guidance for Primary
understanding, the sought-after dyad of conceptual                    Schools, Available at: https://www.ncetm.org.
understanding and procedural fluency is within reach                  uk/public/files/24756940/NCETM+Calculati
of all, and that is something to be celebrated.                       on+Guidance+Oct+2015.pdf [accessed 12th
                                                                      June 2018]
References                                                        NCETM (2016) The Essence of Maths Teaching for
                                                                      Mastery, Available at: https://www.ncetm.org.
Askew, M. (2009) On The Double, Available at:
                                                                      uk/files/37086535/The+Essence+of+Math
    http://mikeaskew.net/page3/page2/files/
                                                                      s+Teaching+for+Mastery+june+2016.pdf
    LearningMultplicationFacts.pdf [accessed 12th
                                                                      [accessed 12th June 2018]
    June 2018]

 Dr Gemma Parker is a Vice Chair of the MA/ATM Primary Committee.
 She works in schools across London to help improve primary mathematics. Please email her at
 gemmaparker@reflectivemaths.co.uk if you are interested in joining the primary group –
 they are always welcoming to new members!

6                                                     Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
Explicitly connecting                                                Ray Huntley and Chris
                                                                       Hurst explore place
  mathematical ideas:                                                  value and the distributive
                                                                       property within
  How well is it done?                                                 multiplicative thinking

Introduction                                               from knowing about the distributive property is
                                                           that multiplication strategies based on partitioning
Multiplicative thinking is one of the ‘big ideas’
                                                           and the distributive property are more advanced
of mathematics and underpins many important
                                                           than those based on other ideas such as repeated
mathematical concepts required beyond primary
                                                           addition.
school years. Multiplicative thinking could be
described as a complex set of interrelated concepts.
The development of multiplicative thinking
                                                           Study
depends largely on knowing about the links and             This report is from an on-going study into
relationships between ideas in order to understand         multiplicative thinking of children from 9 to 11
why procedures work as they do. Siemon et al.              years of age. The original study has been conducted
(2006) defined multiplicative thinking as a capacity       for over three years in Western Australian primary
to work flexibly and efficiently with an extended          schools and has gathered data from over 1000
range of numbers (larger whole numbers, decimals,          children in eight schools. A Multiplicative Thinking
common fractions, ratio and percent), an ability to        Quiz, and a semi-structured interview have been
recognise and solve a range of problems involving          developed and refined and are used in this study
multiplication or division including direct and            involving two primary school classes at a school in
indirect proportion, and the means to communicate          Plymouth in the south-west of the United Kingdom.
this effectively in a variety of ways (materials,          The quiz was administered to both classes on the
words, diagrams, symbolic expressions and written          same day under identical conditions. The framework
algorithms). If students are to work ‘flexibly’ with       for analysis of data is based on connections between
a range of numbers, we believe that there must be          place value partitioning, the distributive property of
explicit teaching of the many connections within           multiplication and the standard written algorithm for
the broad idea of multiplicative thinking. Here we         multiplication, to determine if students understand
explore the link between partitioning based on place       and articulate those connections. The framework is
value and the distributive property.                       in Figure 1.
  The distributive property of multiplication                 In the Multiplicative Thinking Quiz (MTQ), students
could be considered as the basis of the vertical           were asked a total of 18 questions, 5 of which are
multiplication algorithm that is taught in a range of      based on aspects of the framework (see Table 1).
ways by teachers. The importance of this property          We wanted to find out the extent to which students
cannot be under-estimated and the importance               demonstrated understanding of partitioning, were
of partitioning, which is the first step in moving         able to identify when the distributive property
beyond repeated addition and using the distributive        was correctly applied and whether they were able
property to make sense of multiplication. The              to explain why the property worked in terms of
array is crucial in developing an understanding of         partitioning. In short, we wanted to see the extent to
the distributive property which helps students             which they connected the ideas and then how they
understand what multiplication means, how to               used the written algorithm during the interview.
break down complicated problems into simpler
ones and how to relate multiplication to area by           Results
using array models. Part-part whole reasoning with
groups also enables children to use the distributive       Table 1 shows the responses to the relevant
property of multiplication over addition. The quality      questions from the MTQ. Class A was a Year 5 class
of understanding about multiplication that results         (n=29), Class B, a Year 6 class (n=27). The table

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Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
were able to use a written algorithm to solve 9 × 15,
                                                                 based on the standard place value partition (Question
                                                                 3). In the analysis of the quiz responses for Question
                                                                 3, students needed to indicate that they had ‘carried
                                                                 a 4’ to qualify as a correct response. Second, a much
                                                                 smaller proportion of students were able to identify
                                                                 both correct responses to the question about the
                                                                 distributive property. The interesting aspect of this
                                                                 observation is that the mathematical understanding
                                                                 that underpins Questions 2 and 3 is the same as
                                                                 for Questions 4 – partitioning based on place value.
                                                                 Third, a comparatively small proportion of students
                                                                 could explain their choices of answers (Question
                                                                 5) in terms of what they had already seemed to
                                                                 understand from their responses to Questions 1,
                                                                 2, and 3. In other words, the majority of students
                                                                 were able to use place value partitioning either
                                                                 mentally or in a two by one digit algorithm, but
                                                                 many of them were unable to connect the same
                                                                 idea of partitioning to identify when the distributive
                                                                 property was correctly applied, and even fewer
                                                                 could explain that in terms of partitioning. All of the
                                                                 seven students who explained the fifth question in
         Figure 1 Framework for analysing data from              terms of partitioning used partitioning to explain
                 Multiplicative Thinking Quiz                    their answers to Questions 2 and 3.
                                                                    The following samples from Student Wesley are
shows the percentage of each class that responded                indicative of responses for the MTQ questions.
correctly for each question. Several observations                   Wesley appears to have an understanding of place
can be instantly made.                                           value partitioning and has given sound examples of
   First, while approximately two thirds of the total            it for the first two questions. However, when the
sample were able to mentally calculate the answer to             question is presented in a different context, he
6 × 17 (Question 1), a smaller percentage were able              seems quite confused and has mistakenly identified
to explain their calculation in terms of place value             all options as being correct. Wesley has also confused
partitioning (Question 2), which is the basis of the             the idea of ‘inverse operations’ a term that he would
written algorithm. However, a similar proportion of              have heard at some stage but not fully understood.
students who performed a correct mental calculation              As well, Wesley did not seem to trust the idea of

                  Table 1 Summary of responses to selected questions from the Multiplicative Thinking Quiz

    Question from Multiplicative Thinking Quiz                                                   Class A     Class B

    1.    Used mental computation to obtain correct answer for 6 × 17                            62%          67%

    2.   Explanation of mental computation for 6 × 17 is based on place value partition           45%          59%

    3.    Use of standard algorithm is correct and shows place value partitioning                66%          70%
         (i.e., the ‘carried 4’) to solve 9 × 15

    4.    
         Identifies both (80 × 3 + 9 × 3) and (90 × 3) – (1 × 3) as the only correct              34%          26%
         options giving the same answer as 89 × 3 (Distributive Property)

    5.    
         Explanation  of above question (about Distributive Property) is based on place           10%          15%
         value partitioning

8                                                     Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
Figure 2 Samples from Student Wesley

partitioning as he has used an algorithm to work           There seem to be a couple of possible explanations
out the answer to 89 × 3 when there was really no          for this, as exemplified by the sample from Student
need to do so, if he understood how the property           Izzy (Figure 4). First, it could be that students who
works. During the interview, Wesley used a four line       did that did so as a matter of course or habit, in
algorithm to solve 29 × 37. This seems to indicate         that they accept that they need to use an algorithm
that he understands how to apply the distributive          for such calculations irrespective of whether they
property as he has identified that there are four          actually need to do so or not. Second, it may be
elements to the multiplication.                            that their understanding is not sufficiently robust –
   In contrast to the explanations of students who         perhaps they need to calculate with an algorithm
were unable to explain the fifth question in terms         to prove to themselves that the partition actually
of partitioning, the following sample from Student         works.
Callum is presented as an example of a satisfactory          It is worth considering the work of a student who,
explanation. Student Callum also displayed some            in general, did not respond well to the five MTQ
flexibility in his thinking by solving the first example   questions, as shown in Table 1. Student Francis made
with non-standard partitioning as shown in the
second part of the sample.
   Another point of interest is how some students
who used place value partitioning for both the
questions about 6 × 17 and 9 × 15, and who also
identified the correct choices for the question about
the distributive property, still found it necessary to
calculate the answer for (80 × 3) + (9 × 3), despite
saying that it would give the same answer as 89 × 3.

        Figure 3 Sample from Student Callum                         Figure 4 Samples from Student Izzy

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                             9
an incorrect calculation for the question about 6 ×         2. Students who understand place value partitioning,
17, did use an algorithm to correctly work out the             use it when calculating answers to multiplication
answer for 9 × 15, but was unable to identify the              examples (either mentally or written), correctly
correct choices for the question about the distributive        identify examples of the distributive property,
property. During the interview, the following exchange         but do not trust the partitioning and need to
occurred [with notes by the interviewer]:                      calculate a product as proof.
                                                            3. Students who understand place value partitioning,
I: [Francis said that (80 × 3) + (9 × 3) would give            use it when calculating answers to multiplication
   the same answer as 89 × 3 but when explaining               examples (either mentally or written), but do not
   how it worked, he had to actually work out the              apply it to explain how and why the distributive
   two parts and took prompting to arrive at the               property works.
   correct answers for each part. He wrote it as a          4. Students who demonstrate a partial understanding
   vertical addition]. “Do you need to work it out to          of aspects of the above three characteristics
   prove it?”                                                  but whose understanding is incomplete and not
F: “Yes”.                                                      consistently applied.
I: [He was shown the example (50 × 6) + (3 × 6)]
   “What would it be the same as?”                          Hence, we believe that there are some clear
F: “Fifty-three times … twelve … no … times six”.           implications for teaching. First, teaching should focus
I: [Francis was shown (70 × 4) + (6 × 4)] “Do you           on establishing the link between standard place value
   need to work them out or are you happy that              partitioning and the distributive property and this
   they will give the same answer as 76 × 4”?               could be successfully developed through the use of
F: “Yes”.                                                   the multiplicative array. Second, the written algorithm
                                                            for multiplication needs to be developed from the
There is a considerable degree of uncertainty about         grid method, which is based on standard place
the answers offered by Francis. While he made a             value partitioning and the array. Third, the specific
computational error in the 6 × 17 question, he did          mathematical language related to ‘partitioning’
use place value partitioning for that question and          should be incorporated when developing students’
also the question about 9 × 15. However, he was             understanding of the distributive property. As well,
unable to apply that knowledge to the questions             we think it is important for teachers to encourage
about the distributive property, both in the MTQ and        students to trust the fact that ideas like the distributive
the interview. This suggests that he has developed          property will work when applied correctly. Helping
partial understanding of the mathematics involved           students to make such connections should situate
but has certainly not been able to connect the idea         them better when learning how the distributive
of place value partitioning to the explanation of how       property informs aspects of algebraic reasoning.
and why the distributive property is applied.

                                                            Reference
Conclusion                                                  Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J.
On the basis of the analysis of data from the MTQ               (2006). Scaffolding Numeracy in the Middle Years
and the interview, it would seem that there are                 – Project Findings, Materials, and Resources,
several levels of understanding shown by students               Final Report submitted to Victorian Department
in the sample. These could broadly be described as              of Education and Training and the Tasmanian
follows:                                                        Department of Education, Retrieved from http://
                                                                www.eduweb.vic.gov.au/edulibrary/public/
1. Students who understand place value partitioning,            teachlearn/student/snmy.ppt
   use it when calculating answers to multiplication
   examples (either mentally or written), understand
   the distributive property and explain the latter in
   terms of partitioning.

 Ray Huntley is a mathematics education                     Chris Hurst works at Curtin University, Perth,
 consultant in the UK rayhuntley61@gmail.com                Australia.

10                                               Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
Teaching with Challenging                                                             James Russo
                                                                                        explores
  Tasks: Experiments with                                                               teaching with
  counting patterns                                                                     Challenging
                                                                                        Tasks
In this article, I briefly overview how to teach with        Generally, the idea is for the teacher to strive to
Challenging Tasks. I then demonstrate how three            organise the whole-class discussion in a meaningful
Challenging Tasks exploring counting sequences can         manner, determined by both their in-lesson
be used to expose young students to prime and              observations of students working on the task, as well
composite numbers.                                         as their prior knowledge of the key mathematical
                                                           concepts embedded in the task. For example, the
                                                           teacher may structure the discussion by tacitly
What are Challenging Tasks?                                getting students to share their solutions in order
Challenging Tasks are complex and absorbing                from least to most mathematically sophisticated,
mathematical problems with multiple solution               whilst endeavouring to make connections between
pathways, where the whole class works on the               different student solutions (Stein et al., 2008).
same problem (Sullivan & Mornane, 2013). The task
is differentiated through the use of enabling and
                                                           Differentiating the task through
extending prompts. Teaching with Challenging Tasks
                                                           enabling and extending prompts
enables all students to work on a similar core task,
and therefore encourages them to engage with, and          Enabling prompts are an integral aspect of
contribute to, the subsequent discussion around the        Challenging Tasks. They are designed to reduce the
relevant mathematics. Consequently, Challenging            level of challenge through: simplifying the problem,
Tasks provide an appropriate means of inclusively          changing how the problem is represented, helping
differentiating mathematical instruction (Sullivan et      the student connect the problem to prior learning
al., 2014).                                                and/ or removing a step in the problem (Sullivan,
                                                           Mousley, & Zevenbergen, 2006). Students should be
                                                           encouraged to access enabling prompts at their own
How to teach with Challenging                              initiative. Enabling prompts should be a student’s
Tasks                                                      first point of call if they feel they need some
Generally teaching with Challenging Tasks involves         assistance to make progress with the problem (i.e.
a three-stage process: launch, explore, discuss (and       rather than asking for support from the teacher). As
summarise) (Stein, Engle, Smith, & Hughes, 2008;           part of this process, the teacher should ensure that
Sullivan et al., 2014).                                    all students know where the enabling prompts are
  The teacher begins by launching the challenge,           in the room, and that there is no stigma associated
which involves presenting the problem, engaging            with accessing an enabling prompt (e.g. an overly
students in the relevant mathematical mindset              competitive classroom climate, where it is implicitly
and highlighting resources students have at their          or explicitly assumed that ‘good mathematicians
disposal (e.g. enabling prompts, concrete materials        don’t need help’) (Russo, 2016).
such as bead-frames and hundred charts). After                By contrast, extending prompts are designed
the challenge is launched, students explore the            for students who finish the main challenge, and
task, either individually or collaboratively, and the      expose students to an additional task that is more
teacher encourages students to develop at least one        challenging, however requires them to use similar
potentially appropriate solution. The next stage of        mathematical reasoning, conceptualisations and
the lesson involves the teacher facilitating a whole-      representations as the main task (Sullivan, Mousley
group discussion, which provides students with an          & Zevenbergen, 2006).
opportunity to present their particular approach to           In my classroom, I call the enabling prompts the
solving the task.                                          ‘hint sheet’ and print one prompt on each side of

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                            11
this sheet. During each Challenging Task, I include       Context for the three Challenging
a pile of hint sheets up the front of the classroom       Tasks
on a chair, so students know exactly where they are.
By contrast, I call the extending prompt the ‘super       These three interrelated Challenging Tasks encourage
challenge’ and generally place the extending prompt       students to make connections between different
on the flip-side of the Challenging Task.                 counting sequences and all involve the use of a
   Challenging Tasks are often developed with             hundreds charts as the primary representation.
multiple learning objectives in mind, however, in           The tasks are appropriate for students in Grades
most instances, there is a primary learning objective     1 and 2 to extend student understanding of skip
at the heart of the task. Consequently, when              counting patterns. The tasks could also be used
developing enabling prompts for Challenging Tasks,        with Grade 3 or Grade 4 students to launch a unit
it is critical that they do not undermine the primary     of work on counting patterns and to begin exploring
learning objective of the lesson by ‘giving too much      the notion of common multiples, factors and prime
away’. By contrast, enabling prompts may modify,          numbers. All three tasks have the same primary
and even remove, secondary learning objectives, in        learning objective, that is, for students to recognise
order to allow students who find the initial task too     patterns in number sequences. More specifically for
complex to focus on the primary learning objective        students to appreciate that:
(Russo, 2015).

         Figure 1 Enabling prompt A: A hint to students about the relevant skip-counting patterns.

12                                             Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
●● Overlaying multiple skip counting sequences             ●● Next, I skip counted by 5’s to 20, again placing
   will result in some numbers being covered more             a counter on all the numbers I landed on.
   than once (i.e. numbers with many factors) and          ●● Finally, I skip counted by 10’s to 20, again
   some numbers not being covered at all (i.e.                placing a counter on all the numbers I landed
   potential prime numbers).                                  on.
                                                           ●● What are the numbers with three counters on
Each of the tasks also contains additional secondary          them – the numbers I landed on three times?
learning objectives relating to the exploration of
specific skip-counting sequences. For example, the         Extending prompt
first Challenging Task requires students to count by       Tackle the task again, but instead of skip counting
2’s, 5’s and 10’s from zero. Creating these skip-          from 0, start skip counting from 6. How does
counting sequences without teacher assistance or           starting from 6 change the counting patterns? What
prompting effectively becomes a secondary learning         numbers do you land on three times?
objective for this task. However, these secondary             Without actually doing the skip counting, can you
learning objectives are effectively removed if             predict what numbers I would land on three times
students access enabling prompt A, which provides          if I tackled the task again but started skip counting
a hint about the counting patterns potentially             from 9?
relevant to the Challenging Task. Note that this
same enabling prompt can be used in relation to all
                                                           Challenging Task 2: Fourth time
three tasks.
                                                           luckier
   Each task also contains a second, individualised
enabling prompt (enabling prompt B), which                 The secondary learning objective for this task is:
provides students with a simpler task that has
the same primary learning objective as the main            ●● For students to be able to skip count by 2’s, 3’s,
challenge. Undertaking this simpler task is of value          5’s and 10’s beginning at zero without teacher
in and of itself, whilst also providing struggling            assistance or prompting.
students with an ‘in’ so that they can better navigate
the core Challenging Task.                                 Challenging Task
                                                           ●● Starting at 0, I skip counted by 2’s to 50, placing
                                                              a counter on all the numbers I landed on.
Challenging Task 1: Third time                             ●● Next, I skip counted by 3’s to 50, again placing
lucky                                                         a counter on all the numbers I landed on.
The secondary learning objective for this task is:         ●● After that, I did the same thing counting by 5’s,
                                                              and then 10’s.
●● For students to be able to skip count by 2’s,           ●● There is only one number with four counters on
   5’s and 10’s to 100 beginning at zero without              it. What is that number?
   teacher assistance or prompting.
                                                           Enabling prompt B: Easier problem
Challenging Task                                           ●● Starting at 0, I skip counted by 3’s to 20, placing
●● Starting at 0, I skip counted by 2’s to 100,               a counter on all the numbers I landed on.
   placing a counter on all the numbers I landed           ●● Next, I skip counted by 5’s to 20, again placing
   on.                                                        a counter on all the numbers I landed on.
●● Next, I skip counted by 5’s to 100, again placing       ●● What is the only number with two counters on
   a counter on all the numbers I landed on.                  it?
●● Finally, I skip counted by 10’s to 100, again
   placing a counter on all the numbers I landed           Extending prompt
   on.                                                     ●● What if I continued skip counting to 100 instead
●● What are the numbers with three counters on                of 50? How many numbers would I have landed
   them – the numbers I landed on three times?                on four times? What are these numbers?
                                                           ●● List all the numbers I would land on four times
Enabling prompt B: Easier problem                             if I continued counting to 1000. Do you notice
●● Starting at 0, I skip counted by 2’s to 20, placing        any interesting patterns with these numbers?
   a counter on all the numbers I landed on.

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                               13
Figure 2 A student solution to the task Third time lucky.

Challenging Task 3: Twos, threes,                          ●● Next, starting at 0, I skip counted by 3’s to 20,
fours and fives; which number                                 crossing off the numbers as I went.
will survive?                                              ●● Some numbers were crossed off more than
                                                              once, but some numbers survived – they weren’t
The secondary learning objective for this task is:            crossed off at all. Can you guess which 7 numbers
                                                              survived? Now check if you are right.
●● For students to be able to skip count by 2’s, 3’s,
   4’s and 5’s beginning at zero without teacher           Extending prompt
   assistance or prompting.                                  What if I also skip counted by 6’s, 7’s, 8’s, 9’s and
                                                           10’s? Would all 10 numbers still survive? How many
Challenging Task                                           more numbers would get crossed off?
●● Starting at 0, I skipped counted by 2’s to 40,            If we kept our skip counting patterns going (2’s,
   crossing off the numbers as I went.                     3’s, 4’s, 5’s, 6’s, 7’s, 8’s, 9’s and 10’s) all the way
●● Then I did the same thing, but instead skip             to 100, how many numbers do you think would
   counted by 3’s.                                         survive? Can you list these numbers?
●● Next, I did it by 4’s.
●● Finally, I skip counted again, but counted by 5’s.
●● Some numbers were crossed off more than                 Relevant questions for post-task
   once, but some numbers survived – they weren’t          discussions
   crossed off at all. Can you guess which 10              During the post-class discussions, students should
   numbers survived? Now check if you are right.           be encouraged to describe their various approaches
                                                           to the task(s) and the conclusions they reached.
Enabling prompt B: Easier problem                          Part of the discussion should be focussed on
●● Starting at 0, I skip counted by 2’s to 20,             getting students to recognise that there is overlap
   crossing off the numbers as I went.                     between different skip-counting sequences and

14                                              Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
Figure 3 A student works through the third Challenging Task, trying to determine
                                       which numbers will survive.

that this overlap occurs in a regular way. This can           covered more than once and other numbers are
be explored with students using an interactive 100’s          not covered at all?
chart      http://www.primarygames.co.uk/pg2/              ●● For example, why is it that unless we skip-count
splat/splatsq100.html During these discussions,               by 13’s, we never land on 13? What about 12?
and depending on the age group of the students,               We often seem to land on 12 when we are skip
the teacher may also consider introducing key                 counting. Why is 12 so easy to land on and 13
mathematical terminology such as multiples,                   impossible? (Note: If students cannot move past
factors, composite numbers and prime numbers.                 the concept that 12 is even and 13 is odd, and
  Examples of questions to stimulate discussion               that this is solely responsible for the difference,
when using the interactive 100’s chart include:               encourage them to next compare 22 and 24;
                                                              why is it that 22 is so much harder to land on
●● If you skip-count by any even number (e.g. 4               than 24 when skip-counting?).
   or 10), how often do you think you will land on
   a number that is part of the 2’s skip-counting          Beyond exploring the counting patterns themselves,
   pattern (i.e. a multiple of 2)? Why do you think        these discussions should also reinforce the idea
   this is the case?                                       that there is benefit in approaching such tasks
●● If you skip count by any number ending in zero          systematically (particularly if the task is being used
   (e.g. 100, 270), how often do you think you will        with Grade 3 or Grade 4 students). Prompting
   land on a number that is part of the 5’s skip-          questions may include:
   counting pattern (i.e. a multiple of 5)? Why do
   you think this is the case?                             ●● What do you think was the most efficient way
●● Why do you think it is that when we skip-                  of approaching the challenge?
   count by different amounts, some numbers are            ●● Having listened to other student’s approaches

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                             15
to the challenge, how would you tackle this task                    Friendly Giant. Australian Primary Mathematics
     differently next time?                                              Classroom.
                                                                    Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K.
                                                                         (2008). Orchestrating productive mathematical
Concluding thoughts                                                      discussions: Five practices for helping teachers
In my experience, students thoroughly enjoy                              move beyond show and tell. Mathematical
exploring the interrelationships between different                       Thinking and Learning, 10(4), 313–340.
counting sequences and the hands-on nature of                       Sullivan, P., Askew, M., Cheeseman, J., Clarke, D.,
these Challenging Tasks, where students can play                         Mornane, A., Roche, A., & Walker, N. (2014).
the role of the scientist, making predictions and                        Supporting teachers in structuring mathematics
running ‘experiments’ with number patterns.                              lessons involving Challenging Tasks. Journal of
Moreover, unpacking these tasks in the subsequent                        Mathematics Teacher Education, 18(2), 1–18.
discussion provides opportunities for students to                   Sullivan, P. & Mornane, A. (2013). “Exploring
develop insights into the properties of prime and                        teachers’ use of, and students’ reactions to,
composite numbers. I hope readers find these three                       challenging mathematics tasks.” Mathematics
Challenging Tasks to be of use in their classrooms.                      Education Research Journal, 25(1), 1–21.
                                                                    Sullivan, P., Mousley, J., & Zevenbergen, R. (2006).
                                                                         Teacher actions to maximize mathematics
References                                                               learning opportunities in heterogeneous
Russo, J. (2015). How Challenging Tasks optimise                         classrooms. International Journal of Science
    cognitive load. In K. Beswick, Muir, T., & Wells,                    and Mathematics Education, 4(1), 117–143.
    J. (Ed.), Proceedings of 39th Psychology of
    Mathematics Education conference (Vol. 4, pp.
    105–112). Hobart, Australia: PME.
Russo, J. (2016). Teaching mathematics in primary                     James Russo is a lecturer in the Faculty of
    schools with Challenging Tasks: The Big (not so)                  Education, Monash University, Australia.

                                                   BOOK REVIEWS

Best of the Best – Feedback                  Since the                                   and reward, and peer feedback. Each
Authors:   Isabella Wallace and            seminal work                                  chapter covers around 7–9 pages and
                                           on feedback                                   includes a theoretical piece with further
           Leah Kirkman
                                           by Paul Black                                 reading followed by some practical
Publisher: Crown House Publishing,         and Dylan                                     strategies about that aspect.
           Carmarthen                      Wiliam in the                                   I particularly enjoyed reading Paul Dix’s
           www.crownhouse.co.uk            1990s – ‘Inside                               chapter on ‘wristband peer feedback’,
ISBN-10: 1785831879                        the Black Box’,                               and I’ve been a fan of Shirley Clarke’s
ISBN-13: 978-1785831874                    feedback                                      work for many years, and her chapter on
                                           practice has been continually developing      ‘getting underneath the understanding
Price:     £9.99
                                           in schools and is now regarded as             and acting on it’ is highly recommended.
This pocket-sized (12.5cm × 15cm)          one of the most significant aspects of        The book ends with a really good
volume is a collection of short chapters   teaching and learning. In this ‘Best of       chapter by the Teacher Development
about different aspects of feedback.       the Best’ volume, it is only right that the   Trust on how to move forward and plan
It is part of a growing, wider series on   opening chapter is by Dylan Wiliam, in        your next steps using this little treasure
teaching topics such as engagement,        which he sets out the case for formative      trove of material on feedback.
progress and differentiation. The          assessment as a basis for the rest of the       Best of the Best is a really simple yet
authors/editors have gathered a series     book.                                         effective idea for a series of books, I
of pieces from the very best names           In subsequent chapters in the 150           found this one to be really readable and
in educational theory and practice to      pages that follow, various aspects of         helpful to further inspire and inform
produce a collection of inspiring and      feedback are explored. For example,           teachers.
practical chapters to help teachers with   giving and receiving feedback, the notion
this very critical element of being a      of feedback against ‘feedforward’,
teacher in a 21st century classroom.       time for feedback, targets, praise                                        Ray Huntley

16                                                    Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
Professional learning                                                             Ruth Trundley
                                                                                    explores using
  through research:                                                                 action research
                                                                                    as CPD
  planning for success and identifying barriers

Working in a team of maths advisers, I spend a lot         Professional development should
of time thinking about and planning professional
development opportunities which are intended to
                                                           have a focus on improving and
have an impact on learners (teachers and pupils).          evaluating pupil outcomes
As a team, one of our preferred ways to do this
                                                           The project was set up with a clear focus on
is to engage teachers in action research designed
                                                           vulnerable children and improving their mathematics.
to reflect the five elements of the standard for
                                                           Teachers were asked to identify three focus children
teachers’ professional development:
                                                           for the project; closing the gap was one of the key
                                                           drivers, and teachers were encouraged to identify
1. Professional development should have a focus on
                                                           disadvantaged and vulnerable pupils as their focus
   improving and evaluating pupil outcomes.
                                                           children.
2. Professional development should be underpinned
   by robust evidence and expertise.
                                                               s teachers our job is really to disrupt the
                                                              A
3. Professional development should include collab-
                                                              trajectories of students who haven’t had
   oration and expert challenge.
                                                              challenging experiences and to provide all
4. Professional development programmes should be
                                                              students with the richest and most challenging
   sustained over time.
                                                              environment possible.                  Boaler (2014)
And all this is underpinned by, and requires that:
                                                           The intention of the project was to disrupt the
5. Professional development must be prioritised by
                                                           trajectories of the focus children using the tools of
   school leadership. 			             (DfE 2016)
                                                           pre-teaching and assigning competence. These
                                                           children were the focus throughout the year in all
For this article I am focussing on one such project
                                                           elements of the project. The adviser supported
which involved thirty-nine teachers in pairs or trios
                                                           data collection with these children at the start of
from seventeen schools across Devon, supported by
                                                           the year: their needs were discussed at meetings,
five maths advisers from Babcock Education (LDP).
                                                           the collaborative lesson research cycles focused on
The project ran from September 2016 to July 2017
                                                           planning for these children, teachers kept reflective
and included: introductory webinar, project launch,
                                                           journals that included observations of the children
staff meeting, data collection, journals, learning
                                                           throughout the year and wrote case studies
partners, cluster meetings, research readings and
                                                           detailing the impact on the children at the end of
cycles of collaborative lesson research (Takahashi
                                                           the year. All teachers identified positive impact on
and McDougal 2016).
                                                           the focus children as a result of using pre-teaching
   The participating teachers were informed that
                                                           and assigning competence.
they would also be the subject of research as part of
the project and that data collected on them would
include: reflections on shared activities, audio and       Professional development
video recordings of discussions, questionnaires (pre-      should be underpinned by robust
and post-project) and observations at meetings. At
the end of the project this data was used to identify
                                                           evidence and expertise
themes and evidence of professional learning and it        Evidence-based research informed the setting
is an analysis of this, including identifying barriers,    up of the project. The research question ‘How
that I will explore here.                                  can pre-teaching and assigning competence be

Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk                                              17
used to effectively support children to access
age-appropriate mathematics and be active and
influential participants in maths lessons?’ was a
response to findings from a 2015/6 research project
run by Babcock Education (LDP). The focus on pre-
teaching was informed by wider readings (including
Minkel 2015) whilst the research of Cohen and Lotan
(1997) around status interventions in classrooms
and the work of Cohen et al (1998) on assigning
competence as an important tool for addressing
status, further informed the project.
   The project was led by five maths advisers
with experience in supporting action research,
collaborative lesson research and mathematics. The
role of the adviser included introducing readings
relevant to issues discussed by the teachers at the
cluster meetings, ensuring that existing research
was used to inform throughout the project. Teachers
found the use of research beneficial in a number of          Discussion between learning partners and adviser
ways:
                                                             Whilst the focus for the project and the research
  I have found it increasingly powerful to access         question were established by the maths advisers,
   research readings alongside my planning. This is        the teachers involved made most of the decisions
   something that I have not been doing as much as         throughout the project, including those related to
   I should and I feel it has had a real impact on my      the selection of focus children, the structures for
   teaching and professional development.                  pre-teaching sessions, the content of pre-teaching
   					 (Y3 teacher)                                      sessions and how to assign competence in lessons.
                                                           These were the aspects being researched, and they
  I found the research that we read on the first day
                                                           were the focus of the discussions that took place
   re: the model of pre-teaching and then teaching,
                                                           in schools, in local cluster meetings and within the
   as opposed to the ‘teaching, failure, intervention’
                                                           whole group meetings. Teachers all valued the role
   idea probably the most powerful idea and I
                                                           of their learning partner and the maths adviser in
   have often reflected on it throughout the year.
                                                           furthering their own thinking and understanding:
                                         (Y4 teacher)

  I have also found research reading a great asset           he most useful aspect of the project, I feel, has
                                                             T
   to my professional development this year. I was           been the dialogue and support that I have shared
   aware in class that there were children who were          with my colleagues who are also involved in the
   perceived as brighter or more popular by other            project.                              (Y4 teacher)
   children but have never assigned a name such
                                                              ime to work alongside a colleague was the most
                                                             T
   as perceived high and low status children to this
                                                             valuable – it was great to have time to plan/talk/
   before. Reading research done on this has made
                                                             share experiences and reflect on the project.
   me consider my own practice and ways of tackling
                                                                                                (Y3/4 teacher)
   perceived status in my own class. It has been
   invaluable reading the research.      (Y6 teacher)         eetings with the adviser and getting her
                                                             M
                                                             support and feedback have been the most useful
Professional development should                              things.                              (Y4 teacher)

include collaboration and expert
                                                           Most teachers also found the cluster meetings
challenge                                                  useful as they allowed a wider sharing of experience
The teachers worked together in pairs/trios as             and an opportunity to learn from others:
learning partners in their schools and were also
grouped to form five clusters. Each cluster was               cluster meetings have helped my practice
                                                             …
supported by a maths adviser.                                as it is invaluable to talk to other practitioners

18                                              Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
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