Tandi Clausen-May shows how multiplication facts can be made more memorable for pupils who need visual cues.
The National Numeracy Strategy: Key objectives, Year 5 ... Know by heart all multiplication facts up to 10 x 10" - The National Numeracy Strategy: Framework for teaching mathematics from Reception to Year 6.
So children must learn the multiplication tables "by heart". How are we to teach them? There are a number of different approaches - but no one method will work equally well for all children. Teachers need a range of activities, reflecting different thinking styles, and the different types of memory their pupils use most effectively.
Reciting magic spells This approach relies heavily on aural and symbolic memory. It is very effective for many - perhaps for most - children. Learning the spells involves lots of practice and repetition. First the children learn each spell forwards:
"One gru is gru Two grues are flum" ... and so on.
Then the spells are mixed up:
"Seven grues are...
Three grues are..."
... and so forth.
It is the way most of us were taught, and it worked for us (probably, more or less). And it will work (more or less, probably) for many of the children we teach. But it is less likely to be effective for children who have specific learning difficulties, or who have problems with short or long-term aural memory.
Using computers A lot of programs just replicate the "magic spells" approach. They may be dressed up with fancy graphics, and they can be tailored to the needs of individual children - Terry can practise the three times table while Chris is learning the seven times. But essentially, this approach depends on learning the multiplication facts individually, "by heart". The inter-relationships between those facts are not brought out.
A new piece of software from the Association of Teachers of Mathematics, Developing Number, goes further. The program "Tables" is structured around the completion of a conventional multiplication square, but children are encouraged to build on facts they already know in order to work out new facts. The four times table, for example, is double the two times, and the eights are double the fours. The 6s can be worked out by adding the 5s to the 1s - so six 6s, for instance, are five 6s, plus another 6.
However, this approach is still depends heavily on numbers. The eight times table is double the four times table because the number 8 is double the number 4. Recognising and understanding such inter-relationships between multiplication facts is very valuable. But what about the children for whom "four 6s are 24" does not mean very much? "Eight 6s are double 24" will not mean much either. What can we do for children who, as Einstein did, think more readily in pictures than in words and numbers?
The Slavonic abacus This is for the spatial thinkers in the class. These are the children who have found all the nets of a cube while you are still explaining to the others just what a net is. The Slavonic abacus can be especially helpful with the calculation of products over five times five.
A Slavonic abacus is an ordinary counting frame, with the usual 10 rows of 10 beads. What makes it special is the colouring of the beads. Half of each row of 10 beads is in one colour, and half in another. Furthermore, the top five rows are coloured one way, and the bottom five another. You can find a detailed discussion of the Slavonic abacus in Eva Grauberg's Elementary Mathematics and Language Difficulties: A book for teachers, therapists and parents. (1988, London: Whurr Publishers.) With practice, the eye can take in up to four beads at a glance, without counting them. The way the beads are arranged by colour on a Slavonic abacus makes it easy to distinguish five beads. This makes it possible to see a row, or a column, of up to 10 beads on the Slavonic abacus: you can see up to five beads in one colour, and another one, two, three or four beads in the other. You can recognise llllllll for example, as 8, without having to count the individual beads.
The Slavonic abacus is not easy to find in the UK. But for the purpose of multiplication, a grid representing the beads, and an L-shaped shield, may be used instead of a real abacus. You can photocopy the grid on this page and use the scale diagram to make the shields. You may want the children to colour in the beads before they start using their grids.
The rectangle whose edge lengths are the numbers to be multiplied is isolated with the shield. For example, to multiply 5 by 8, a 5 by 8 rectangle is isolated: The 5s can be grouped into pairs to make 10s, and the total found: To multiply a pair of single digit numbers greater than 5, the rectangle whose edge lengths are the numbers to be multiplied is again isolated. For example, to multiply 8 by 7, an 8 by 7 rectangle is isolated: Once the shield is in place, the product can be read off the grid: Using the grid and the shield takes practice. With time, however, pupils can build up a mental image of the arrangements of the beads. When they are able to visualise the grid, to see it in the mind's eye, they can use it to multiply any pair of single digit numbers mentally.
The "Gypsy" method This last method is not new, but it deserves to be more widely known. With its appealing name it is again a kind of spell. It is a way of finding the harder multiplication facts (the six, seven, eight and nine times tables) by carrying out a simple routine. But the routine is something to do, not something to say: it involves kinaesthetic memory - the awareness of movement and body position - so for children who can remember movement more easily than words it may be just what is needed.
Each hand must be labelled with the numbers 6 to 10.
The tips of the two fingers whose numbers are to be multiplied are brought together, so they are just touching. For example, to multiply 8 by 7, the tip of the middle finger (labelled 8) of one hand is put with the tip of the fore finger (labelled 7) of the other.
Next, the two touching fingers, and all the fingers (and the thumbs) above them, are counted, giving 3 on one hand, and 2 on the other - a total of 5. This is the number of 10s in the answer.
But there are some fingers left over, below the touching pair - two on one hand, and three on the other. These two numbers are now multiplied together, and the product, 6, is added to the 10s that have already been calculated. So 8 multiplied by 7 is five 10s, plus 6.
It takes a little practice but, with time, actually labelling the fingers becomes unnecessary: pupils just know that the thumb is "6", the forefinger is "7", and so on. The method works less well for the six times table - six times six, for example, gives only two "10s" (the two thumbs), plus four times four for the number of 'units'. But 20 plus 16 is 36, so the result is correct.
If you are concerned about the theory which underlies the method, then the formula you want is: It is unlikely that many pupils will be able to follow the algebra - but if you have a child who simply cannot remember "seven 8s", try a little Gypsy magic.
Tandi Clausen-May works at the National Foundation for Educational Research, Slough and is a member of the General Council of the Association of Teachers of Mathematics. She is co-editor of 'Spatial Ability: a Handbook for Teachers', which suggests ways in which pupils with high spatial ability may be supported in a number of different curriculum areas
References Clausen-May T, Smith P . Spatial Ability: A Handbook for Teachers. 1998. Slough: National Foundation for Educational Research.
Department for Education and Employment. The National Numeracy Strategy: A Frame work for Teaching Mathematics from Reception to Year 6. 1999. London: The Stationery Office.
Grauberg E. Elementary Mathematics and Language Difficulties. 1998. London: Whurr .
Association of Teachers of Mathematics. Developing Number (software). 1999. Derby: ATM.