A student's understanding of numeracy concepts generally forms in stages, with each subsequent and more sophisticated understanding building on the last. New concepts should be introduced to students using this process. Students should be taught:
- how to act on quantities in particular ways
- to describe these actions using language and imagery (this is very important for students with maths learning difficulties and dyscalculia)
- how to use algorithms and mathematical symbolism.
Throughout this process students are also taught how and when to use each procedure. At first, they will require scaffolding but with sufficient practise they will eventually be able to use the procedures independently and automatically.
Teach students to act on quantities
When demonstrating a new concept or skill, concrete materials or manipulatives (for example, MAB blocks) are a useful tool to show students how to act on quantities. These tools help them to think about the numeracy actions they need to apply to written problems.
For students with learning difficulties, it may be necessary to model the physical actions first and explain what they represent explicitly. Students can then copy these actions.
Concrete materials include:
Sets of items
A collection of individual items such as blocks or counters can be used to form sets, count the number of items in a set, match quantities, group items according to colour or size and use these groups to form patterns, add or takeaway items from a set and count how many are left.
Number ladders and paths
This allows students to act on the numbers to learn counting strategies. You can use vertical number paths and ladders to teach counting up and counting down, counting in twos or threes and so on. Put a marker, such as a clothes peg, on a number and ask the student to count up or count down from that number. You can show the order of the decades (10, 20, 30) and how to cross each decade when counting.
Students can be taught place value concepts by arranging icy pole sticks into bundles of 10 and then 100, and by talking about the number of bundles and the number of loose sticks in a quantity. They can act on a quantity of bundles and loose sticks and rearrange the tens into ones. You can also use the bundles to illustrate two-digit addition, subtraction and division.
Students can use bead chains to learn how to count in different increments, multiplication tables, division, and factors.
This introduces the concept of equality. The student can also investigate subtraction and multiplication by using beads or blocks to act on the scale. For students with learning disabilities, such as dyscalculia, this can also be used to consolidate their understanding of concepts such as less than, more than and equal to. For example, a student can learn that 3 + 2 blocks on one side is equal to 4 + 1 blocks on the other side.
Multi-base arithmetic (MAB) blocks
Use this to teach place value concepts and computations. A MAB block is made up of longs and cubes. A cube is one unit. Each ‘long’ comprises 10 units. The square or ‘flat’ is 100 and the block is 1,000. These materials can be used to represent quantities and to teach decimal concepts and procedures. For decimals and fractions, the block can represent one whole: the flat is one-tenth, the long is one-hundredth and each cube is one-thousandth
These kits come in all shapes and sizes and are for demonstrating equivalent fractions. For example, showing that four eighths and two quarters are both equal to one half. They can also be used to demonstrate how to add, subtract, multiply and divide fractions in a visual way. This is a useful learning aid for all students.
These are rectangular boards with nails or pins arranged in formation. Students use rubber bands to create and investigate the properties of 2D shapes. They can also investigate the effect of rotations and translations in two dimensions.
As students begin to consistently demonstrate proficiency when acting on quantities, you can begin to ask them to imagine the outcome without using the concrete materials. This is when students will begin to visualise the numeracy idea.