Structured learning all down the line
The key models advocated by the National Numeracy Strategy for the teaching of number in primary schools are the number line and the related 100-square, with base-ten materials relegated to a more lowly position. The Framework for Teaching Mathematics and the Qualifications and Curriculum Authority's booklet of ideas for teaching mental calculation also recommend the use of an "empty number line". This is a very powerful and sophisticated model for stimulating and supporting the mental addition and subtraction of one and two-digit numbers, and it is introduced in the Framework in Year 3 for this purpose. In a section concerned with informal pencil and paper methods, or jottings, the empty number line is recommended for helping children develop the addition strategy of counting on in multiples of 100, 10 or 1 and the subtraction strategies of compensation and complementary addition.
As the Dutch invented, researched and developed the empty number line it is inevitable that they should take this as their main calculation model and spend a substantial amount of time presenting it to children and developing it slowly in a carefully structured manner. In Year 2 (the first year of formal school), teachers work on extending children's facility with the counting sequence, helping them recognise the regularities and patterns in the number word sequence. They then develop their facility with operations on numbers to 20 using an arithmetic bead frame - not unlike a baby's counting frame - which is structured in fives with adjacent sets of five white and black beads in two parallel rows.
They also introduce a numbered number line from 0 to 20 in order to familiarise children with written numbers presented in an ordered sequence. Numbers to 100 are later introduced via a bead string the width of the blackboard which contains 50 black and 50 white beads in adjacent tens. Children count in tens from different starting positions, and learn to place a peg after, say, 56 beads. They learn to add on or remove beads and then carry out simple calculations on the bead string.
In Year 3 the empty number line is introduced, initially to 20 with guiding marks at 5, 10 and 15, and then later to 100 with marks denoting the decades. The important relationship is made between the bead string, with its emphasis on quantity (the cardinal aspect of number), and the number line, with its emphasis on sequence (the ordinal aspect of number). This gives meaning to the relative positioning of numbers on the line, and helps children to appreciate, for example, that the decade numbers are evenlyspaced, that 28 is nearer to 30 than to 20, and that, just as 5 comes immediately after 4, then 95 comes directly after 94.
A video containing edited 20-minute sequences of two different Dutch teachers teaching the same lesson to their Year 3 children has recently been made available (with subtitles!). These lessons involve an oralmental introduction; an interactive whole-class discussion dealing with different strategies for subtraction; and finally an over-the-shoulder, end-of-lesson section showing children working from their books. In the first lesson we see children coming to the board to solve a two-digit subtraction in different ways; naming their strategies; explaining their methods; and choosing the one they consider to be the best. These strategies include sequencing (jumping in tens and ones) and compensation (subtracting more and then adding back). We also see an example of a child who has "progressed" from the empty number line to using arrow notation to support and explain his strategy.
In the second sequence we see "love hearts" (where the two sides sum to 10) used to revise complements in 10 and subtractions from 10, but the main part of the lesson focuses on the introduction of "difference problems" and the development of the complementary addition strategy, which involves calculating a difference by adding up from the smaller to the larger number. Children illustrate their different methods at the blackboard, and there is an interesting sequence illustrating one aspect of the Dutch teacher's questioning technique. One particular child, obviously very confident and proficient, is challenged to justify her strategy at every stage of the explanation with the teacher playing devil's advocate, reaching incorrect solutions which anticipate possible errors and misconceptions of the other children in the class. This part of the lesson almost becomes a peer-tutoring situation guided by the teacher.
The video succeeds in giving the watcher a feel for the approach to teaching mental calculation in the Netherlands, and illustrates the power of the empty number line to evolve into an internalised mental representation which can be used when children have matured sufficiently to be in a position to dispense with the actual line. These different stages are clearly illustrated in Figures 1 and 2.
Ian Thompson is senior lecturer, Newcastle University.For details of how to obtain a copy of the video 'Lessons on the Empty Number Line' send a stamped self-addressed envelope to Barbara Thompson, Department of Education, University of Newcastle upon Tyne, St Thomas Street, Newcastle upon Tyne NE1 7RU