Primary Mathematics SPRING 2019 Volume 23 Number 1 - Maths Week London
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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
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
2 Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
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
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.
4 Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk
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
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
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
Primary Mathematics – Spring 2019 • The MA website www.m-a.org.uk 7
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.ukFigure 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 9an 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.ukTeaching 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 11this 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 13Figure 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.ukFigure 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 15to 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.
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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.ukProfessional 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 17used 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.ukYou can also read
