Knowing your place
ny primary mathematics curriculum for the 21st century must develop fluency in mental calculation based on understanding of the number system. This is needed for sensible use of calculators and to develop more advanced mathematics. Mental methods vary, but consider how many people will find the answer to 23 + 45 in this way: "Twenty-three and forty-five . . . twenty and forty makes sixty . . . three and five makes eight . . . sixty and eight makes sixty-eight. " Important features of this procedure are: o Understanding of language is used to decompose, combine and recompose the numbers o Knowledge of number bonds is used, without resort to counting o The order for calculating corresponds to that for reading, namely from left to right.
As we shall see, the prevailing model for teaching number is weak in all three aspects. There is also a problem at the very beginning of schooling, in misguided "pre-number" activities.
Counting: a natural beginning Sorting and matching activities have disguised a more natural beginning to number, namely counting. Sorting is a process of handling data, which should link with purposeful activities such as classifying materials.
Devised matching activities are of dubious value. Their use derives partly from interpretations of Piagetian experiments that questioned children's ability to "conserve" number.
We have come to realise that children agreeing that they see "more" when a set of objects is rearranged may reflect a stage of language development, rather than some misconception (a colleague pointed out that to small children "more" often means "less", as in "please may I have some more dinner?").
Number should begin with counting, initially to learn number names in sequence, later counting sets of objects by reciting the sequence in co-ordination with successive "pointing" at objects.
Note that learning number names first enables undivided attention to be given to co-ordinating the counting. Time is saved by counting obiects informallv, delaying insistence on correctness until later. Synchronising these two sequential actions, pointing to each object once and once only, is more complex than simple one-to-one matching. It is best learned when children are motivated to count things, with an associated physical action, such as going down stairs, replacing objects in a container.
Notation: linking the oral and written language of numbers There are two major difficulties with teaching place value. First, although oral counting will naturally extend the "tune" of the number sequence beyond l0, there are problems when written numbers are also taught sequentially: the sounds of English words for numbers between l0 and 20 are irregular, disguising, at a vital stage, the regularity of written numbers.
Second, attempts to explain place value by grouping and exchanging objects in tens do not "make an abstract idea concrete". Place value is not a property of any physical material. It's simply a convention of language and notation, and has to make sense in itself, not through external reference. Manipulation of base l0 blocks and abaci is not essentially connected with number notation. Translating from one context to the other is not obvious - beads look different from digits and have to be counted, ten-rods are different from unit cubes and have no positional significance - whereas "place value" refers to the fact that the same mark (digit) is read differently according to its position in a number.
An alternative approach treats place value as a part of language learning, by working directly and systematically on the notation. This can be done by building up a "tens table" in stages, each practised to fluency before moving on, so that it eventually looks like Figure 1 (top).
Using a wall chart and pointer, first teach the class to say the numbers from 1 to 9. Unconventionally, then add the third row, with the additional sound "hundred" enabling a lot more numbers to be spoken, for example by successively pointing "five hundred and seven". Add the sound "ty" with some slight alterations (such as "twenty" rather than "two-ty") then complete the middle row, except for "10". Build up to examples like "eight hundred and sixty-seven". Finally, add "10", perhaps temporarily adopting the sound "one-ty".
Beginning with the component sounds of numbers overcomes the objections raised by exposing the structure hidden in our compact notation (for example 537 = 500+30+7). It sets the irregularities of spoken English in the context of large numbers, where regularity predominates. Also, it is purely a language exercise, with no attempt to abstract place value as a property of materials.
Exercises for reading and writing numbers might use a set of 27 "arrow" cards, corresponding to the numbers in the tens table. The following cards, placed on top of each other (with arrow heads together) will show 537: The cards make visual what lies hidden in the written form. Pupils need to internalise this notation, reading the "3" as "thirty", not "3 in the tens column".
Number bonds: moving away from counting Computation requires not only command of the language for naming numbers, but also quick recall or, making efficient use of memory, rapid reconstruction of number bonds. Again, there are two major weaknesses: o In oral or class work, there may be a lack of rich activities to develop fluency through exploration of relationships.
o In individual work, use of concrete materials which permit inefficient counting methods.
Systematic and rigorous oral work is needed - not just chanting tables or random testing. A structured approach helps to "fix" certain facts such as 5+5=l0 (two hands) and establish others by quick mental transformations (for example 9 plus 8 10 plus 7 17).
With hands spread out and four fingers folded down, the facts 4+6=10, 10-4=6 and 10-6=4 are simultaneously available. There is no counting involved, rather the imprinting of an image which holds the fact.
Multiplication facts need to be tackled by emphasising the underlying structure. For example, after exercises on doubling and halving, this table could be used: Pointing systematically to sequences of blank cells triggers responses with the appropriate product, while developing awareness and use of relationships in the table.
When exploring number relationships in individual work, much time can be wasted with inappropriate materials and tasks. Pupils who find 5+8 by setting out blocks of 5 cubes and 8 cubes, combining and counting the cubes, are simply learning bad habits. "Counting on" using a number line is also inefficient.
Pupils progress by developing more advanced methods - having begun with counting, it becomes necessary to discourage it! They need to be able to treat numbers as undivided entities, which is why digital representations are useful.
Contrary to popular belief, calculators can greatly enhance learning - posing problems with larger numbers, where estimation, trial and improvement are needed, promotes insightful exploration.
And Cuisenaire rods can improve dramatically on counters and linking cubes because they offer an unsegmented image for numbers.
Computation: connecting with mental processes Understanding of notation and knowledge of number relationships, together enable computation with larger numbers. With calculators available, it is no longer necessary to perform complex calculations compactly on paper. But it is useful to make jottings of intermediate results in mental work, and to understand how to break down more complex calculations. Pencil-and-paper methods are worthwhile only as an extension of mental methods.
The major block to progress is the persistence of traditional written methods of working from right to left, with "borrowing" and "carrying" of detached digits. These do not relate to the rich variety of mental methods which involve breaking numbers down into component parts. They are poorly understood, extraneous explanations based on manipulating materials - often more of a hindrance than a help.
A better approach involves pupils in discussing their own methods. Often they will work as they read, starting on the left with the most significant digit. In practical contexts where no operation is specified, they will use ad hoc methods. For example, when asked how many 57-seater buses would be needed to carry 200 children and 15 adults on a trip, pupils find ways without using division. Research on arithmetic in the work-place confirms the value of, and preference for, personal "back-of-an-envelope" methods.
Pupils will eventually be ready to discuss some more organised formats, based on the principle of decomposing numbers as they are read. They should adapt their preferred methods, rather than using a single imposed method. The first example below reads like this ,"300 and 200 is 500, 60 and 70 is 130, 5 and 8 is 13, 500 and 100 is 600, 30 and l0 is 40, and 3 gives 643". This mental "patter" is an important link to understanding what is written. Subtraction can be done in many ways - pupils who use calculators from an early age often invent a method which employs negative numbers: When traditional formal methods are abandoned, grouping and exchanging using concrete materials becomes a redundant activity, except possibly for division. "Borrowing" and "carrying", mysterious processes for many pupils, then have only historical interest.
Towards a new model for teaching number Effective teaching of number needs structured exploratory work. Traditional teaching required repetition of teacher-given processes. Supposedly more insightful methods have involved "explaining" processes using concrete materials, often avoiding large numbers in the early years. Both approaches have been found wanting.
An alternative model would induct pupils into the language of number and reinstate the social practice of chanting but in a richer, more meaningful way. Exploring relationships and drawing on understanding derived from using the language, pupils would then develop a flexible range of mental methods which they explain to each other and refine together.
After 20 years of calculators in schools, old theories and practices have an insecure yet tenacious hold on classrooms. Changing our approach could raise standards and save precious time.
Alan Wigley is mathematics adviser in Wakefield. This is a condensed version of an article which appeared in Mathematics Teaching, the journal of the Association of Teachers of Mathematics.