Interactive Mental MathematicsPeter Patilla
This article is based upon the address given at the SMC conference held at the University of Stirling on 25 April 1998
Introduction
For a number of years I have been working with teachers both in the UK and in European International Schools on ways of improving mental mathematics. Making whole class mental mathematics lessons interactive, participative and enjoyable has been an important part of this INSET work. I accept that it is not normal to see the words 'enjoyable' linked to 'mental mathematics' but shall attempt to show during this article how it is possible to raise pupil self confidence and willingness to participate in what, historically, has been an ordeal for many. How we question pupils, the way in which we allow thinking time for some questions and how we build up pupil self confidence and number knowledge clearly influence their competence and comfort in mental mathematics.
Show me
One strategy in particular helps overcome many of the difficulties we experience in whole class mental maths lessons. This strategy is called 'Show me' where pupils hold up a simple piece of apparatus in response to teacher questioning. What this technique does is ensure that everyone in the class takes part, they are all holding up their response. It stops answers being called out by individual pupils. It allows thinking time, especially if pupils are told to hold their response close to their chests until told to 'show me'. It encourages the more insecure - they can quickly look around and see others with the same answer. It allows for teacher assessment speedily and efficiently - a quick glance round will show who is having difficulties.
Band Aid Strips
One example of 'Show me' which has enormous potential across a wide age and ability range is the use Band Aid Strips. These are simply strips of scrap card onto which is threaded an elastic band, taut enough so as not to drop off. The length of the card strips is fairly arbitrary.
band aid - strip of card with an elastic band
I developed this piece of apparatus and activities to improve sophisticated counting skills across a wide age and ability range of pupils. It is excellent for work on position, estimation and approximation of numbers. Their use is simple. Each end is given a number and pupils slide the band along to indicate the positions of nominated numbers. For example, name the ends 0 and 10 then ask:
Show me the position of: 5, 9, 3, ¼
Change the number range to, say, 0 and 100.
Show me the position of: 40, 75, 88, ¼
What if the number ranges were: 0 to 20? 0 to 1000? 0 to 50? -5 to +5?
Band Aid Strips can be used for decimals or fractions using the range 0 to 1
Show me the position of: 0.8; 0.54; 
, 
, ¼
I also use strips which have 10 divisions marked on them.
These divided Band Aid Strips are used in exactly the same way as the unmarked strips. The divisions bring into play a different set of counting and estimation skills for pupils to apply. Both types should be mixed and matched during 'Show me' activities.
Fan Numbers and Number Generators
Other pieces of apparatus I have developed as part of 'Show me' include Fan Numbers and Number Generators. These are used when pupils need to hold up numbers to show the result of some mental calculation or problem. Fan numbers are digits fastened at the bottom with a fastener. These are used to make a set of numbers which have a specific total or product, such as:
Show me a set of numbers which has the product 160.
They are also used to show 2-digit and 3-digit numbers.
Show me the difference between 94 and 59.
Adding a decimal point to the set creates a Decimal Fan.
Show me: 
; 
; 
.
Show me 10 multiplied by: 0.5; 1.02; 0.03.
Fan Numbers can be used for pupils to show the answers
to a range of mental maths questions with whole and decimal numbers.
Pupils can work in co-operating groups, each group
Show me a set of numbers which has the product 160.
They are also used to show 2-digit and 3-digit numbers.
Show me the difference between 94 and 59.
Adding a decimal point to the set creates a Decimal Fan.
Show me: ; ; .
Show me 10 multiplied by: 0.5; 1.02; 0.03.
Fan Numbers can be used for pupils to show the answers
to a range of mental maths questions with whole and decimal numbers.
Pupils can work in co-operating groups, each group
member making a different number on their fan. The group decides who hold up fan numbers to answer the question posed. This co-operative group work strategy encourages all pupils of whatever ability to take part.
Number generators are bands of digits threaded through a slotted strip to display different numbers. They can be used as part of 'Show me' activities where large numbers are required such as multiplying by 100 and 1000 and rounding to the nearest 10, 100 or 1000.
Show me: 40 ´ 700; 36000 ¸ 40; ¼
Round off to the nearest 100: 24556; 561988; ¼
A decimal point can be placed on the strip to create a
decimal generator. As with Fan Numbers individual responses and
co-operative group responses are expected.
Number generators are bands of digits threaded through a slotted strip to display different numbers. They can be used as part of 'Show me' activities where large numbers are required such as multiplying by 100 and 1000 and rounding to the nearest 10, 100 or 1000.
Show me: 40 ´ 700; 36000 ¸ 40; ¼
Round off to the nearest 100: 24556; 561988; ¼
A decimal point can be placed on the strip to create a
decimal generator. As with Fan Numbers individual responses and
co-operative group responses are expected.
This illustrates a few of the varied activities which encourage all pupils to participate in mental activities through 'Show me' activities. Obviously other simple materials such as number cards showing the digits and fractions are just as useful. The teacher needs to try and create 'Show me' activities for as many situations as possible. They take very little time and effort to create and are hugely beneficial.
Unison responses
In addition to 'Show me' activities, participation in mental maths lessons can occur through what I call 'unison responses'. Unlike chanting, 'unison responses' have a visual point of reference or are linked to rhythm and physical actions. This difference is significant. These examples have helped pupils develop sophisticated counting skills such as counting on, counting back, counting in multiples, estimation, approximation. They include work with integers, decimals and fractions.
Counting Stick
A
counting stick is simply a length of wood marked off into 10 divisions
- a metre stick marked into decimetres is ideal. The stick is used as
an unnumbered number line in different ways.
Which number goes here?
The teacher names each end of the stick, initially 0 and 10, then points to each division in turn asking 'Which number goes here?' The class responds in unison. The finger then starts at 10 and pupils count back to 0. Eventually point to random positions on the stick and ask 'Which number goes here?' For each of the following number ranges the technique is just the same, count forwards, count backwards then name random positions. The ends of the stick can be given numbers to change the number range being operated upon. For example: 0 to 100; 0 to 1000; 0 to 1; 56 to 86; -5 to +5; ¼
Another example of unison responses is called 'thigh clap snap snap'. On the thigh action pupils gently slap the top of their thighs with both hands followed by a clap of both hands. On the snap, snap action they snap first with one hand then with the other. Once a rhythm is steady pupils count on from different starting numbers. For example counting in sixes from 45
thigh clap snap snap thigh clap
45 51 57 63 69 ¼
If this proves too challenging count on the 'snap snap' or just the 'thigh' actions to allow more thinking time.
thigh clap snap snap thigh clap snap snap thigh
7 14 21 28
Pupils can count back from different numbers in different sized steps. They can count in eighths, remembering to cancel as they go. Counting in squares, primes and Fibonacci Sequence are all possibilities. The point of the activity is that everyone takes part, thinking time can be speeded up or slowed down and it can be used to develop a whole range of counting skills.
Conclusion
I believe we lose pupils to a love and enthusiasm of numbers at one of three aspects in their mathematics education. These three aspects form the basis of an ability to juggle numbers in the head, to mentally compute and to feel really comfortable with calculations. They are:
developing sophisticated counting skills,
developing immediate recall of number bonds in addition, subtraction, multiplication and division,
developing a thorough understanding of place value.
Each of these three aspects support each other and interrelate. For example using counting skills to count forwards and back in multiples of 6 and 7 which links to learning the multiplication and division bonds for 6 and 7. Eventually using knowledge of place value allows pupils to mentally compute sums such as 60 ´ 70 and 600 ´ 700. These aspects form the basis of mental mathematics and lead into efficient mental strategies in later work. The more interactive and participative the lesson the more likely it is that pupils will assimilate what is being taught. 'Show me' activities and 'unison responses' are as appropriate in the secondary maths lessons as they are in the primary school classrooms.
References
Interactive Mental Maths, Peter Patilla, Heinemann (pub 1999)
Longman Primary Maths, Peter Patilla, Paul Broadbent & Anne Montague Smith, Longman (1997)
Mental Maths, Day by Day, Peter Patilla & Paul Broadbent, Longman (1997)
Maths Challenge, Peter Patilla, BBC publications -book and tapes (1998)