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National Curriculum - Selecting and Using Resources for Learning : Key Stage 2 : Mathematics-specific Pedagogy


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Key Stage 2
National Curriculum - Selecting and Using Resources for Learning
Question 1 of 7

1. 1. How confident are you that you are familiar with a range of equipment and practical resources to support mathematics teaching and learning, such as:

a. a. structural apparatus and other models for teaching number?


Example

To support understanding of the number system, it is useful to have a range of different resources available for when you are teaching and also for children to use independently. Structural apparatus is a resource that models some aspect of the structure of numbers.

Different resources embody different levels of abstraction and can help children develop understanding of different aspects of the structure of number. It is important to choose resources that are at an appropriate level of abstraction and which embody the mathematical concept you wise to explore.

Young children use physical objects such as bricks to count with. When they are ready to explore place value they may use structured Base 10 material (Dienes’ apparatus). This draws attention to the way place value works in 10s and 100s. For example, to show 64 we need 6 tens rods and 4 unit cubes:

This is a concrete representation of the number. But if you use arrow cards to explore place value, then you are using a more abstract image of number which allows children to develop a more sophisticated understanding of place value:

A 100 square helps children develop their understanding of place value – rather than the common 1 to 100 square, it can be helpful also to use a 0 to 99 square since each family of tens is then on the same row:

0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99

 

Questions such as:

‘I’m on 54. Where do I get to if I go up a row? Down two rows? Down one row and then one square to the left?’

help children develop their understanding of place value.

Counting sticks and bead strings help children develop an understanding of counting, and lead to an understanding of the use of the empty number line.

A place value chart such as:

10 000 20 000 30 000 40 000 50 000 60 000 70 000 80 000 90 000
1000 2000 3000 4000 5000 6000 7000 8000 9000
100 200 300 400 500 600 700 800 900
10 20 30 40 50 60 70 80 90
1 2 3 4 5 6 7 8 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

 

can be used to build and partition larger numbers and decimals.

Coins (for example just using 1p, 10p and £1 coins) are another resource that can help children develop their conceptual understanding.

What this looks like in the classroom

A teacher used Base 10 material to help children develop their understanding of decimals. Instead of the 100 square being worth 100, she explained to the class that it is now going to be the ‘unit’ i.e. it is worth 1. The tens rod is then one tenth, while the unit cube becomes one hundredth.

In this way the children were able to represent a number such as 1.46 as:

and then were able to use the concrete resource to do addition and subtraction sums with decimals.

A counting stick is a powerful way to help children develop fluency in counting and times tables. A teacher told the children that the left hand end represented 0 and the middle represented 40.

counting stick

From this the children worked out that each section must represent 8 and then she was able to develop the children’s fluency in the 8 times table. She then redefined the left hand end as 2:

counting stick

and so was able to raise the challenge level significantly as children had to think of the 8 times table plus 2.

When working with fractions a fraction wall is a powerful structural resource to help children develop understanding of equivalent fractions:

1
12
12
13
13
13
14
14
14
14
16
16
16
16
16
16
112
112
112
112
112
112
112
112
112
112
112
112

 

Related information and links

Models and Images materials are charts showing a range of different models and images for supporting children’s calculation strategies and understanding of the structure of number.

Using a counting stick has a host of ideas for ways to use a counting stick to develop fluency.

How can practical resources support the development of mathematical understanding? is part of a study module from the Excellence in Mathematics Leadership microsite.

Maths to share - CPD for your school looks at how Dienes’ base 10 material can be used to develop understanding of subtraction using decomposition.

Do we really make the most of our wonderful number system? explores models and images of place value in further depth.

Numicon in the Early Years discusses the use of Numicon to support the early development of number.

Useful Mathemapedia entries:

Related information and resources from other sites

 
 
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