EI505 Computing and Contemporary Developments
Primary Mathematics 1
19th Feb 2015
• Revisiting EM402 and reflections on teaching
maths on 1st placement
• Key features of primary mathematics in the
2014 National Curriculum
• Progression in arithmetic (NC 2014)
• Implications for EI505 Assignment
Looking back at Year 1…………
• Key learning from EM402
• Key learning from practical experience of
teaching maths on placement 1
Relational and Instrumental Understanding
• Relational understanding is….”what I have always meant by
understanding: knowing both what to do and why.”
• “Instrumental understanding I would have until recently not
have regarded as understanding at all. It is what I have in the
past described as ‘rules without reasons’ without realising
that for many pupils and their teachers the possession of such
a rule, and the ability to use it, was what they meant by
Skemp, R. (1976) “Relational understanding and Instrumental understanding.”
Mathematics Teaching, 77, pp 20-26.
Bruner’s modes of representational thought
The child needs
experience at all
only the two
Delaney, K, (2001), ‘Teaching Mathematics Resourcefully’, in Gates, P (Ed), Issues
in Mathematics Teaching, London: Routledge Falmer (available as an e-book)
Where are we now?
• The new national curriculum for Year 3, Year 4
and Year 5 came into force from September
(Year 6: KS2 SATS are unchanged for 2014/15)
• The new national curriculum for Year 6 will
come into force from September 2015.
(New KS2 SATS for 2015/16 are currently in
N.B. Assessment and tracking pupil progress is a major challenge for schools
The ‘old’ National
• Ma1: Using and
• Ma 2: Number
• Ma 3: Shape, space
• Ma 4: Handling data
The ‘new’ National
– and place value
– addition and subtraction
– multiplication and division
– fractions, decimals (Y4+) and
– Ratio and proportion (Y6+)
- properties of shape
- position and direction
Some key difference between mathematics in the old and the
• More detailed – and now set out in Year groups.
• A ‘mastery’ curriculum. NC ‘levels’ have gone!
• More ambitious expectations, especially for number.
• Greater emphasis on arithmetic – especially formal
• Almost no mention of problem solving, reasoning or
communicating in the Programmes of Study – although
these elements are explicit in the introductory aims.
The new National Curriculum: Aims
The national curriculum for mathematics aims to ensure that all pupils:
• become fluent in the fundamentals of mathematics, including through
varied and frequent practice with increasingly complex problems over
time, so that pupils develop conceptual understanding and the ability to
recall and apply knowledge rapidly and accurately
• reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and developing an
argument, justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of
routine and non-routine problems with increasing sophistication, including
breaking down problems into a series of simpler steps and persevering in
Catering for the needs of all pupils….
The expectation is that the majority of pupils will move
through the programmes of study at broadly the
same pace. However, decisions about when to progress
should always be based on the security of pupils
understanding and their readiness to progress to the
next stage. Pupils who grasp concepts rapidly should
be challenged through being offered rich and
sophisticated problems before any acceleration
through new content. Those who are not sufficiently
fluent with earlier material should consolidate their
understanding, including through additional practice,
before moving on.
Excerpt from NC 2014 mathematics programme
of study (p100)
Information and communication
Calculators should not be used as a
substitute for good written and mental
arithmetic. They should therefore only
be introduced near the end of key stage
2 to support pupils’ conceptual
understanding and exploration of more
complex number problems, if written
and mental arithmetic are secure. In
both primary and secondary schools,
teachers should use their judgement
about when ICT tools should be used.
What does this
Are we allowed
and if so when
and what for?
What is the progression through the four
operations, from mental strategies to
compact written methods?
Key principles in supporting the development of
written calculation methods
• Mental calculation confidence should be established before
written methods are introduced
• Mental calculation strategies need to be specifically taught
• We need to carefully structure progression into written
methods to ensure each new method builds on
• Children need to be encouraged to make decisions about
which method to use and when
• Opportunities to apply and problem solve with calculation
skills and strategies should run alongside practice of them
• We need to ensure that we use resources to support
understanding of how methods represent number
‘Mathematics Education’ Area of
(within ‘My School’ area)
• A range of resources to support teaching and
• E.g. Screencasts
• NS ‘Strand’ documents
Which two numbers have been
multiplied together in each grid.
How do you know?
Problem solving with the grid method
Multiplication grid ITP
Shuffle some digit cards and make a stack. Turn over one card at
a time and decide together where to put it. Will your product be
more than 300? Five points if it is. How do you know? When do
you know? Use grid method to calculate the answer.
Play against or with a friend
• With a partner, choose one number from the yellow cloud
and one from either the green cloud or the orange cloud to
create a division calculation
• Estimate the answer
• Solve the calculation using an efficient method
• Could you have used a more efficient method?
• Do you always get a remainder?
245 642 563 126
246 487 623 399 280
450 266 511 188
Division practice options
(to practice and refine ‘chunking’ method – using just the green cloud)
• What remainder do you get when
you have divided your numbers?
• Put a cross or counter on the grid
to match this remainder. Can you
get three crosses in a row?
Work with a partner to solve some
division calculations using the cloud
What do you notice?
How can you make your chunks efficient?
(use the smallest number of chunks)
What makes the answer smaller or
Can you predict if you will have a
remainder and how much this remainder
1 4 3 2
3 1 5 6
2 3 2 1
You have won a prize in a competition – a free
meal at your favourite pizza restaurant! You
want to gain the most possible from your £20
prize but cannot spend more than this amount.
Which choices would you make if you choose
one each from the following:
• Main course
Considerations for your assignment?
Which areas of the NC are suitable for this task?
How might you include mathematical reasoning
and problem solving as well as fluency?
Follow up from this session:
• Revisit and update your primary maths tracker –
especially the action plan (and upload this to your e-
portfolio, tagging it to TS3)
• Familiarise yourself with the resources in the
Mathematics Education area of studentcentral
• Familiarise yourself with mathematics programme of
study for KS2 in the 2014 National Curriculum and
English specialists – consider possible content choices
for your assignment.
NC documents – and ‘strands’ for multiplication and division
Check studentcentral links
In groups – 5 mins to discuss and then feedback and discuss
For me, this is a key theory
Evidence of this on 1st placement?
Bruner and Skemp and two key names and are key to our beliefs around primary mathematics teaching.
Or where will you be when you next teach KS2 maths? Either next year and/or in your induction year – and you may not know yet!
Where are we now re NC for KS3? NC 2014 is now statutory for KS3 – but even next year, Yr 7 pupils will have been taught and assessed against the old NC - so will need stop back-filling.
How do they compare in terms of progression? Expectations
Give example of cross curricular topic work where big numbers involved and we are interested in the thinking not the calculating
First bullet This ensures number understanding is secure – quantity value and relative size/position emphasised in mental calculation. These are important precursors for understanding written methods
Mental calc strategies do not spontaneously arise in all children
New calc methods shoudl be shown alongside a previously well understood method to support transition from one to another
Just because we know how to add in columns, doesnt mean it is the only one we use. See new NC, emphasises mental calc throughout KS2
Problem solving gives reason for doing this work, and need to find ways that skills practice involves thinking as well as doing to avoid tedium (see next slide and then next group activity)
Place value a key issue in compact methods and will always need to consider place value when errors arise.
Reveal on grid multiplication
18 x 23
What about decimals?
Model this as an approach to differentiation
When they have engaged with the task – where is the differentiation?
Where is the U&A: Making reasoned choices about method, numbers, initial chunk to be subtracted
Addition / subtraction with decimals – includes estimation See separate menu on Word doc