Advice for Maths teachers
Maths is frequently a tricky subject after ABI. It is probably the most conceptual subject tackled in schools which makes it extremely difficult for pupils with an ABI, who find abstract ideas difficult. Maths is also a subject where sequencing is important. The steps in algorithms are very precise, yet these pupils have trouble thinking sequentially.
In many cases Maths is taught as a procedure, with pupils having to remember sequences of instructions to say solve an equation, rather than understanding what it is about. Some techniques, such as long division, have to be taught this way, and are really only understood if you pursue studies to a higher level. In addition some ideas have no real explanation - it is just the way it is understood.
Put with this the way language is used in Maths and you have a really difficult subject to understand. Many of the common words in maths have more common uses in everyday life. The idea of difference is one which confuses not only pupils with ABI, particularly in the lower years. But with ABI pupils the confusion will persist well into KS 3 and 4. However careful teaching can overcome some of these problems.
Language
The explanation of the meaning of symbols must go further than just a list on the classroom wall, although this is very helpful
Telling the history of the symbols can help to remember their meanings eg =
Making symbols into a picture can also help some pupils, eg = becomes the base and beam of a balance scales.
Be aware of simplistic meanings provided in younger years which have stuck, eg = means the answer!!!
Explain that mathematical equations can be read right to left, as well as left to right!
It will probably need to be pointed out that it is not always possible to equate a word in a problem to an algorithm.
Ask the pupil to imagine the situation being described by the problem to find out what to do
Sometimes words are omitted from questions and need to be inferred which is very tricky after ABI
If they cannot think what to do, ask them to imagine a calculator and what they would do if they had one
Ask the pupil to rephrase what they have to do, so that they have to dialogue with themselves
Ask the pupil to chunk up the problem into units of meaning and tackle each bit separately. Ask the pupil to explain all the mathematical language or give an example
Ask the pupil to identify words which are causing confusion, look for clues in the question or example given
Keep a glossary of terms in the back of their exercise book - a mathematical dictionary is usually too much. Some phrases denote a particular method, eg ‘trial and improvement’, examples of these need to be kept in the glossary
Memory
Mnemonics can help to memorise orders, eg run before you jump, along the corridor and up the stairs for grid references
Encourage use of fingers to hold onto numbers being manipulated
The calculator chosen should retain the input while showing the result, eg Casio fx-85WA and similar models
Ask the pupil to explain to you what they have been taught. If it is then just memorised explore why you do the steps in that order or what the problem is about.
Many of the formal Victorian algorithms are very heavy on the memory, and earlier algorithms may be easier to manage, eg gelosia, but make sure the result is read left to right.
Dealing with the abstract
As much as possible, apply it to a concrete situation eg 6x=12, becomes 6 T-shirts cost £12
Stay with iconic representation of ideas as long as possible, before introducing the symbols. Build in as much practical work as possible, try to get the pupil to record the idea symbolically. Explain that words are symbols too and that the word multiply labels a process in the same way that x does.
Explain that many of the symbols and ideas used actually label short cuts, eg multiplication is a short cut for repeated addition, index notation is a shortcut for writing out the full form, eg a x a x a x a = a3
Explain uses for the concepts, try to give different examples so that the pupil does not latch onto just one use
After an ABI pupils want to take everything literally, including mathematical diagrams, point out that they are not to scale. The pupil should habitually look for the ‘not to scale’ indication in questions and touch it to emphasise it.
The introduction of index notation is often problematic; the pupil has probably accepted the natural number system and will want to follow the same rules, rather than see it as a different system.
Direction & Orientation
This can cause difficulties after and ABI and maths is a good place to teach it! However you need to make sure the pupil understands clockwise and anticlockwise - by reference to an analogue clock or watch - points of the compass, right angles (square corners/90°)
Tracing paper is useful to show how an angle is a turn, or use an angle measurer where one circle is turned above another.
Tracing paper can also be used to look at symmetry
Placing a protractor is often problematic and will need lots of practice
Establishing techniques
takes longer than for other pupils. You will probably need to repeat them daily for at least a week, maybe more
the order of the stages for a longer technique may need a mnemonic, eg MESSS for simultaneous equations (make the same, eliminate, solve, substitute, solve)
putting stages on post it notes and asking the pupil to sequence them correctly may be helpful
once techniques become routine you won’t change them!
Planning investigations
making a plan is quite difficult after ABI and will need a lot of support, possibly using post it notes to contain ideas or as a concept map for ideas they may explore.
frameworks are best provided as a Word document, with the headings already in place, together with a written list of instructions expressed in the imperative, eg ‘write down what you plan to do’. The pupil then has to fill in their plan, method, investigation and evaluation in the report document. A header could already be activated with an instruction for the pupil to fill in their name and other details, and a footer in place with the page numbers. In this way some of the headaches of organisation have already been dealt with and this acts as a model for future tasks. A sample IT and support advice document is available from SHIPS.
© SHIPS Project August 2009
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