Download The focus is not on rote procedures, rote memorisation or tedious

Problem solving is at the heart of mathematics.
Here at St. Andrew’s CE (C) Primary and Mary Howard Primary School we have adopted the philosophy of a Singapore Maths system that uses a CPA approach to learning (Concrete, Pictorial,
Abstract).
When solving problems, pupils are encouraged to follow ‘Solve It’ steps to success (G. Polya – 1957):
1.
Understand The Problem

What information do you have?

What is known / unknown?

What is the whole and what are the parts?

What would the answer sentence look like?

Write the answer sentence.
2. Devise a Plan

What similar problems have you solved?

What strategies do you know?

Is it a part-whole, comparison or before/after problem?

Is there a diagram you can draw?
3. Carry Out the Plan

Model the problem

Write a mathematical sentence

Put the answer in the answer sentence
4. Look Back

Reread the question

Did you answer the question?

Is the answer reasonable?
Problem solving is at the heart of mathematics
The focus is not on rote procedures, rote memorisation or tedious calculation but on relational understanding.
Pupils are encouraged to solve problems working with their core competencies, in particular:
1. Visualisation
2. Generalisation
3. Decision Making
How lessons are taught
Lessons typically are broken into three parts and can last one or more days.
Pupils master topics before moving on.
The three parts to a lesson are:
1. Anchor task —the entire class spends a long time on one question guided by the teacher
2. Guided practice —practice new ideas in groups guided by the teacher
3. Independent practice —practice on your own
Use of the ten frames
The 10 frame appears early in year 1.
Initially it is used as pictorial model to represent numbers up to 10
It reappears later when we get to counting to 20.
It is used to help pupils recognise that 10 ones can be renamed to 1 ten,
introducing the concept of place value.
It is used for adding numbers up to 20
Alongside the number-bond diagram it helps build a mental model to develop a strong number sense which is later used as a platform for making decisions.
Number bonds
A lot of emphasis is put on number bonds and in particular a recognition that numbers can be split up and put together in different ways.
This formulates a basis for number sense which is applied as a platform for decision making.
The number bond diagram is consistently used as a visual way of showing how numbers can be split into their component parts, often as a means of describing what children are thinking.
Simple Adding
Adding with Renaming
Add 278 and 349
Add the ones.
8 ones + 9 ones = 17 ones
Regroup the ones.
17 ones = 1 ten+ 7 ones
H
T
O
2 17 8
3
4 0
_________
7
Foundations for the Bar Model
1. Concrete —modelling with real objects
Based on Jerome Bruner’s theory of representations, students first start by acting out problems with the real objects. When the problem is counting apples, the children should be counting real
apples.
Foundations for the Bar Model
Moving away from always dealing with the real objects we can now start to model situations with other objects, for example counters, or with pictorial representations.
2. Representation —modelling with different objects
Bar Model
Transition to the bar model
Students are introduced to the bar model method in year 2, first as a means to
solve word problems in length and mass.
Although students will only see the bar model in year 2 for the first time, the foundations
have been laid in previous years with the ground work done on part-part-whole relationships and comparisons using concrete objects and more ‘concrete’ pictorial
representations.
The bar model is used to solve problems using addition, subtraction, multiplication and division to begin with.
CPA approach (Concrete, Pictorial, Abstract approach) -Jerome Bruner
Bruner’s theory: Knowledge representation develops in 3 stages
Enactive- based on hands-on sensory experiences of physical objects together with the consequences that go after.
Iconic-Knowledge can now be represented using models and pictures. Learners know how to make mental images of their world.
Symbolic-Learners can think in abstract. So abstract terms and symbol systems can be used to represent knowledge like numbers, mathematical symbols, letters and language.
Progression in Bar Modelling
Reception
Modelling with real things
Modelling with plastic copies of real things
Modelling with pictures
Linking problems with stories
Year 1
Modelling with counters/ multilink
Drawing using squares with 1 square representing 1 object (discreet counting)
Making up own stories to represent problems
(Up to this point, teaching and learning will be developing the numerical aspect of the objects)
Year 2
Single step problems
Part-Whole – recognising parts of the whole problem
Comparison (addition and subtraction: ‘more than’/’less than’ problems)
Multistep problems
Year 3 and Year 4
All operations in all formats
Before / After problems
Model unknowns
Year 5
Fractions/ Percentages and Ratios
Year 6
All complex problems