Scenario: Chloe teaching a mental arithmetic strategy

Country: UK

Grade (student age): Year 1/2 (age 5-7)

Contributed by: Tim Rowland, University of Cambridge, UK

Context – national, curricular, professional, other

At the time when this lesson was recorded, mathematics in primary schools was prescribed by a National Numeracy Strategy and a related curriculum framework. In the early years, in keeping with Dutch RME principles, arithmetic emphasises children’s mental methods, and introduces a number of related strategies. The focus for this lesson is subtracting near-multiples of 10 (specifically, 9, 11, 19, 21) by subtracting 10 (or 20) and then adjusting by 1.

Chloë was a graduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her one-year teacher preparation.


The objectives of the lesson were taken directly from the English National Numeracy Strategy (NNS) Framework (DfEE, 1998) teaching programme for Year 2:

Add/subtract 9 or 11: add/subtract 10 and adjust by 1. Begin to add/subtract 19 and 21: add/subtract 20 and adjust by 1. (p. 3/10)

These objectives are clarified by examples later in the Framework; such as

58+21=79 because it is the same as 58+20+1; 70-11=59 because it is the same as 70-10-1

24-9=15 because it is the same as 24-10+1; 35+19=54 because it is the same as 35+20-1 (p. 4/35)

Chloë’s planning, following the procedure and some of the examples in this official NNS guidance for teachers, involved symbolic exposition and recording. The lesson plan included, for example, “24-9=15 because it is the same as 24-10+1”. Chloe’s lesson plan indicated that she had intended to teach the lesson by giving the pupils ‘rules’, expressed symbolically, for the subtraction of 9, 11, 19 and 21, based on examples such as these.  However, just as the lesson was about to begin, Chloë noticed a large 1-100 square lying on a table in the classroom. She mounted it on a vertical board at the front of the class, and proceeded to demonstrate the compensation strategies spatially, by reference to horizontal and vertical ‘moves’ on the grid. However, she was not happy when the pupils described the sub-routines involving subtraction of 10 or 20 in terms of moves ‘up’ and ‘down’. For example, at one point in her exposition she placed a counter at 70 on the 1-100 square, and asked the class what they should do to subtract 19. One pupil responded “We go up one[1]”. Chloë rebuked the child with “Don’t tell me what we’d go up, tell me what we’d take away”. She used the same corrective formulation “Don’t tell me what we go up/down” at other times in the lesson.

Chloë’s efforts to teach these particular mental strategies – especially the ‘compensation’ strategies with 9 and 19 – in a meaningful way, with the 100 square, meet with very limited success. In the end she falls back on:

Chloë:             Let’s write these rules on the white board so we remember. Then we’ll have something to follow when we’re doing our sheets.

– and Chloë lists a menu of ‘what to do’ rules on a whiteboard at the front of the class.

The children are forbidden to use 100 squares when they do the worksheet exercises. Chloë refuses a request from one child for a “number square”, saying, “I want you to work them out all by yourselves”. In fact, there is nothing in Chloë’s lesson plan to indicate that she had intended to use the 100 square in her demonstration.

Knowledge Quartet Coding Commentary

Contributed by: Tim Rowland, University of Cambridge, UK

Knowledge Quartet Dimension: Contingency

Knowledge Quartet Code: Responding to (un)availability of tools

Scenario: Chloe teaching a mental arithmetic strategy

This scenario exemplifies a striking clash between two representations – spatial and symbolic – and the associated strategies and pedagogies, including choice of examples. This clash was directly attributable to the late, spontaneous adoption of a pedagogical tool – the 100 square – which had not featured in Chloë’s thinking at the planning stage. Although the 1-100 square invited directional language (up, down, left, right), Chloë’s attention was on the symbolic version in the National Numeracy Strategy (NNS) documentation, and in her lesson plan, which  involved adding and subtracting by adjusting digits rather than “going” up, down, and so on. Moreover, she aimed for the children’s written records of their calculations to conform to the NNS templates. Chloë’s ‘what to do’ rules consisted of an example of each of the four problem-types (subtracting 9, 11, 19, 21) on the worksheets that she had prepared for their group work. She wrote four ‘model’ solutions for reference on the board, e.g.

70 - 9 = ?

70 - 10 + 1 = 61

From this point, her earlier demonstrations on the 100 square became irrelevant, especially since the children were forbidden to use 100 squares when they did the worksheet exercises.

The ‘clash’ has consequences for her Chloë’s choice of examples: in fact, to demonstrate subtraction, Chloë chose to subtract 19 from 70, in accordance with ther planning. Now 70 is on the extreme right boundary of the 1 to 100 square, so after moving up two squares to 50, there was no ‘right one’ square, so she had to move down and to the extreme left of the next row, so that the neat ‘knights move’ was obscured, and the procedure unnecessarily complicated. We note that one of the NNS Framework examples was 70 - 11, and that all four of Chloë’s whiteboard template examples were of the form 70 - n.

There was nothing in Chloë’s lesson plan to indicate that she had intended to use the 1-100 square in her demonstration. What we see, in this episode, is her response to the unanticipated availability of this resource, the unresolved tension between the language and modes of thought supported by two contrasting representations, and her apparent lack of awareness of the problematic nature of this tension. Even at the end of the lesson, many children remained uncertain about how to use and apply the strategies. Chloë adopted the 1-100 square voluntarily as a resource, but without recognising how it would perturb her planning.

A more extended account and analysis of this lesson is given in Rowland and Turner (2007), downloadable from


DfEE (1999) The National Numeracy Strategy: framework for teaching mathematics from Reception to Year 6. Sudbury: DfEE Publications.

Rowland, T. and Turner, F. (2007) Developing and using the Knowledge Quartet: a framework for the observation of mathematics teaching. The Mathematics Educator 10(1), pp. 107-124



[1] This is corrected later to effect subtracting 20 rather than 10, but the children’s grasp of the intended procedure, and its variations, is tenuous here and throughout the lesson.

RAT: Scenario 3
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