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Discuss BrunerBruner (1966) suggests an important model for depicting levels or modes of representation. One can experience and subsequently think about a particular idea or concept on three different levels: enactive, iconic and symbolic. These ideas have been extended by Lesh (1979) and are discussed in the next section. At the Bruner enactive level, learning involves hands-on or direct experience. The strength of enactive learning is its sense of immediacy. The mode of learning Bruner terms iconic is based on the use of the visual medium: films, pictures, diagrams, and the like. Symbolic learning is that stage in which one uses abstract symbols to represent reality. (Post article)For example, consider the operation "two plus three." From the child's perspective this idea is experienced enactively if the child joins a set of two objects with a set of three objects and determines that there are five objects altogether. This same notion is experienced iconically if the child views a series of pictures. The first might have two objects (birds, children), which are joined with a set of three objects in a second picture. The third picture might show that here are five altogether. Note that at the iconic level the determination of the result, five, is actually made by the developer of the diagram or photo, not by the child. The relationship is symbolically encountered when the child writes 2 + 3 = 5. Bruner contends that all three types of interpretations or modes are important and that there is a common sense order implied by three levels because each requires familiarity with the earlier modes of representation.Implicit in his and later work (Lesh, 1979) is the fact that these modes should be interactive in nature, the child freely moving from one mode to another. For example, given the equation 2 + 3 = 5, the child could be asked to draw a picture of this situation. This would in effect be a translation from the symbolic (2 + 3 = 5) to the iconic mode (pictures).Your representation of 5. Was it enactive, iconic or symbolic – or a combination> How do the resources fit in to this? Which use symbolic only? Which use enactive? A combination of more than one mode? How could you represent in that way. Think about this when you look at the next problems.Bruner and Skemp and two names that we will refer to in most sessions and are key to our beliefs around primary mathematics teaching.
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 bulletThis ensures number understanding is secure – quantity value and relative size/position emphasised in mental calculation. These are important precursors for understanding written methodsMental calc strategies do not spontaneously arise in all childrenNew calc methods shoudl be shown alongside a previously well understood method to support transition from one to anotherJust because we know how to add in columns, doesnt mean it is the only one we use. See new NC, emphasises mental calc throughout KS2Problem 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 multiplication18 x 23What about decimals?
DivisionModel this as an approach to differentiationWhen 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 estimationSee separate menu on Word doc
Ei505 maths 1
EI505 Computing and Contemporary Developments
NC 2014 Update
Primary Mathematics 1
11th Feb 2014
• Revisiting EP104 and reflections on teaching
maths on 1st placement
• Key features of primary mathematics in the
new 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 will come into force from
(Current NC has been disapplied for these
• The new national curriculum for Year 6 will
come into force from September 2015.
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 there in the introduction.
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
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?
Sort the cards, putting the images in order of
• Is any stage missing?
• How each method relates to the National
Curriculum 2014 calculation strand
• Where are the tricky transition points? Why?
• Which key resources support understanding by
clearly modelling the method?
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
New ‘Mathematics Education’ Area of
(within ‘My School’ area)
• A range of resources to support teaching and
• E.g. Screencasts
• NS ‘Strand’ documents
Problem solving with the grid method
Which two numbers have been
multiplied together in each grid.
How do you know?
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
245 642 563 126
246 487 623 399 280
450 266 511 188
• Look at the numbers in the yellow cloud and the
numbers in the blue cloud.
• Choose a number from each cloud and create a
• Solve the calculation by chunking on a number line
or using written chunking
• See next slide for options
Division practice options
• 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
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
Follow up from this session:
• Revisit and update your primary maths tracker –
especially the action plan (and upload this to your eportfolio, 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.