CAROL MURPHY

HOW DO CHILDREN COME TO USE A TAUGHT MENTALCALCULATION STRATEGY?

ABSTRACT. This study provides an in-depth analysis of children’s use of a taught mentalcalculation strategy. Three children (aged eight to nine years) who demonstrated contrast-ing spontaneous calculation approaches, were interviewed following the direct instructionof a calculation strategy. Their responses are explored in relation to constructivist and parti-cipation perspectives of learning. In response to a unified view of these theories, hypothesesare generated that ask fundamental questions related to the didactics of mental calculationstrategies.

KEY WORDS: algorithms, arithmetic, calculations, children’s mental strategies, construct-ivist and participation theories, direct instruction, procedural and connected understanding

1. INTRODUCTION

Recent concerns in the development of children’s arithmetic in Englandresulted in the introduction of the National Numeracy Strategy in 1999as a ‘policy lever’ to alter practices in the teaching of mathematics inprimary schools (Earl et al., 2000). The Numeracy Strategy sees mentalcalculation strategies as lying “at the heart of numeracy” (DfEE 1998,p. 51) and the National Numeracy Framework for teaching mathematics(DfEE, 1999a) provides a structured approach to the teaching of mentalcalculation strategies. It promotes the direct teaching of mental calculationstrategies through whole class instruction. This study examines children’suse of a strategy when taught through whole class instruction.

Plunkett’s (1979) account of children’s arithmetic compared the charac-teristics of mental calculation strategies with the characteristics of formalwritten calculation algorithms. He suggested that mental strategies are ‘flex-ible’ as a range can be used to solve calculations. The strategies may alsobe seen as ‘fleeting’ as they are invented ‘on the spot’ by the user for thatcalculation and may not even be remembered for future use. In contrastformal written algorithms have been ‘invented’ by mathematicians in thepast (Ebbutt, 1990). The steps of a written algorithm are unambiguousand can be applied generally to any numerical problem. They constituteprocedures that children can learn how to perform. Plunkett suggested this

Educational Studies in Mathematics 56: 3–18, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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makes their use ‘passive’ and contrasted this to the use of mental calcula-tion strategies that is seen as ‘active’ as they are created by the user to suitthe numbers involved.

Evidence has suggested that children are capable of inventing their ownmental calculation strategies (Plunkett, 1979; Carpenter and Moser, 1984;Steinberg, 1985; Kamii, 1985). Foxman and Beishuizen (1999) have re-ferred to mental calculation strategies as ‘untaught’ and Beishuizen andAnghileri (1998) considered whether informal calculation strategies de-velop ‘spontaneously’. The teaching of mental calculation strategies haspreviously been questioned (Carpenter and Moser, 1984) and the SchoolCurriculum and Assessment Authority discussion paper (SCAA, 1997)considered if “. . .strategies for mental calculation can actively be taughtto pupils, or whether pupils develop them for themselves as a result ofeither maturation or experience” (p. 15).

If children are capable of inventing their own mental strategies needthey be taught or can they be left to spontaneous development? In Englandthere has been a sequence of national reports and initiatives to encouragethe spontaneous development of children’s mental calculation strategies(Cockcroft, 1982; DES, 1989; DES, 1991; DfE, 1995; DfEE, 1999a) butthey have not always been successful (Foxman and Beishuizen, 1999; Earlet al., 2000). It has further been recognised that some children (and adults)do not develop an effective use of mental calculation strategies (Steinberg,1985; Askew et al., 1997). There is evidence to suggest that it is the higherattaining children who employ a range of mental calculation strategies,whereas below average children often rely on inefficient counting proced-ures or taught formal algorithms (Gray, 1997; Steffe, 1983; Askew et al.,1997). Gray (1977) observed that higher attaining children appear able tomake connections with the mathematics they know and use their know-ledge to solve new problems. As such they use deductive mental strategieswhere known calculation facts are used to derive new ones (Askew et al.,1997 and Gray, 1991) and hence “find arithmetic far easier than those whohave to carry out procedures” (Gray, 1977, p. 68).

There is then a possible danger that if deductive mental calculationstrategies are not taught we may deny some children access to them. SCAA(1997) recommended that the development of mental calculation strategies“should not be left to chance” (p. 29) and the National Numeracy Strategy(DfEE, 1999a) stated that some mental strategies may develop intuitivelybut that others “you will teach explicitly” (Section 1 p. 6).

If it is desirable to teach mental calculation strategies, how can they betaught? Plunkett suggested that mental strategies are invented by the userso can direct instruction elicit their use? By categorising and analysing

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 5

deductive mental calculation strategies it is possible to present a processthat can be demonstrated in a mathematics classroom. There have beenseveral attempts to categorise deductive mental strategies for addition andsubtraction (Foxman and Beishuizen, 1999; Thompson, 1999, 2000). Oneof these is the compensation method. Thompson described compensationas a fairly sophisticated but efficient strategy. It involves rounding a num-ber to one that is larger or smaller than is required, usually the next multipleof ten, and then compensating for the rounding. For example: 12 + 9 wouldbecome 12 + 10 – 1. In this way a known fact, such as 12 + 10, is used toderive an unknown fact such as 12 + 9.

The Numeracy Strategy introduces a range of deductive strategies thathave been analysed into possible procedures to support their direct instruc-tion or demonstration via whole class instruction. This study examines howthe demonstration of a mental strategy is interpreted by children. Threechildren who demonstrated contrasting spontaneous approaches to calcu-lations were selected from an opportunity group. The children were theninterviewed following the direct instruction of the compensation strategy.The children’s responses are interpreted according to how they carried outthe taught strategy. By relating the children’s uses of the taught strategyto theoretical viewpoints, critical incidents of learning (Tripp, 1993) arepresented.

2. THE STUDY

The compensation strategy was chosen for this study because it is relativelysophisticated. The user must be able to add a multiple of ten to a multi-digitnumber and realise the compensation needed. Thompson (1999) identi-fied compensation as a strategy not readily invented by children, althoughthere has been some evidence of its idiosyncratic use (Fuson et al., 1999;Foxman and Beishuizen, 1999). The children already had some limited in-struction of the strategy. To find a strategy the children had not encounteredat all during previous instruction was not deemed necessary. The purposeof the teaching session was to highlight or re-introduce the strategy andto provide a common teaching experience which could be referred to indiagnostic interviews, not to introduce a totally new strategy. The com-pensation strategy was also considered appropriate by the class teacherand it is included in the suggested programmes of study for this age rangein the NNS framework.

Three children aged eight to nine years old were selected from an op-portunity group. The group, presented by the class teacher, were con-sidered to be of average ability in mathematics. The study took place in the

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TABLE I

Spontaneous calculation strategies from initial interviews

Caroline Stella Jack

13 + 4 Counting on in onesfrom 13

14 + 4 = 18(because 4 + 4 = 8)18 – 1 = 17

13 + 2 = 1515 + 2 = 17

25 – 3 25 – 2 = 2323 – 1 = 22

5 – 3 = 2Answer 22

Counted on from 3 to25

18 + 7 Counted on in onesfrom 18

7 = 5 + 218 + 2 = 2020 + 5 = 25

18 + 2 = 2020 + 5 = 25

13 – 5 13 – 3 = 1010 – 2 = 8

13 – 3 = 1010 – 2 = 8

Use of counters tosubtract 5 from a setof 13

42 + 10 Counted on in onesfrom 42

40 + 10 = 5050 + 2 = 52

40 + 10 = 5050 + 2 = 52

56 – 20 Counted back in onesfrom 56

50 – 20 = 3030 + 6 = 36

56 – 10 – 10 = 46

26 + 9 Counted on in onesfrom 26

26 + 3 = 2929 + 1 = 30(3 + 1 = 4, so 5 left)30 + 5 = 35

Counting on in onesfrom 26

32 – 9 Counted back in onesfrom 32

32 – 2 = 3030 – 7 = 23

Counted on in onesfrom 9

summer term. The children had received the same mathematics instructionfor at least two terms and the school had been implementing the Na-tional Numeracy Strategy for two years. The selection of the three samplechildren was made through diagnostic interviews to determine contrastingspontaneous use of counting procedures or deductive mental calculationstrategies.

Caroline used counting procedures for six of the eight problems given,including those that involved the addition or subtraction of a multiple often (42 + 10; 56 – 20). Jack used counting procedures for four of the eightproblems but he also demonstrated that he could add or subtract a multipleof ten (42 + 10; 56 – 20). Stella employed a range of deductive mentalstrategies to solve all the problems. She was also able to add or subtract amultiple of ten.

The three children then participated in a group teaching session andwere engaged in a shopping activity where numbers were designed toencourage compensation. For example ‘24p + 19p’ was intended to en-

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 7

courage rounding 19 to 20. The mental strategy was then demonstrated bythe teacher and practised by the children according to the DfEE guidance(1999b) where steps are written as “separate equations underneath the pre-vious one” (p. 40). Hence the recording for the example ‘24 + 19’ wouldbe:

24 + 20 = 4444 – 1 = 43

Post-teaching diagnostic interviews were then carried out a week later withthe three sample children. The children were asked if they remembered thetaught strategy and were given an initial example ‘12 + 19’. They werethen asked to choose their own shopping items and use the compensationstrategy to find the total cost. Other addition and subtraction calculationswere asked according to the child’s responses.

3. POST-TEACHING INTERVIEWS: RESULTS AND ANALYSIS

Stella remembered the taught compensation strategy and explained it withher own example:

Stella: Well if you had 19 you had to change it into a 20 and if it was13, no, like 10 add 19 you would add 20 equals 30 takeaway 1equals 29.

Jack said he remembered the strategy but was not able to explain it in ageneral way. Caroline initially said she could not remember the strategybut then stated:

Caroline: Some were quite easy and some were hard like 52 takeaway 20.They were hard.

The three children were then given numerical problems. Figures 1 to 9show the children’s responses. Where children verbalised the strategiesthese have been included. The children’s responses are interpreted accord-ing to how they carried out the taught strategy. Any modifications to thetaught strategy are analysed.

3.1. Stella

In Figure 1 Stella’s written recording would appear to be an exact copy ofthe taught compensation strategy. Stella’s oral explanation is less clear butshe appears to be adding one to 19 to make 20 and adjusting by subtractingone.

Stella (Figure 2) may have been confused by the intervention of theresearcher but the solution is no longer an imitation of the taught strategy.

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Figure 1. Stella’s response to the calculation 12 + 19.

Figure 2. Stella’s shopping example: 26p + 19p.

Her oral explanation appears nonsensical but one interpretation might bethat she has rounded 26 to 25 and then added 19. She lists a string ofnumbers 25. . .35, 40, 44. This could be:25 + 10 (=35) + 5 (=40) + 4 = 44.She then compensates by adding the one. 44 + 1 = 45.This strategy would seem difficult to categorise. There is a use of roundingand compensating but not as taught. She is not rounding to a multiple often.

In Figure 3 Stella has changed 64 to 65. She then subtracts the 9 bysubtracting 5 and then 4 before she subtracts the 40 but she forgets toadjust at the end. This is not an example of compensation as taught but shehas adjusted a number in order to carry out the calculation. She states that65 is easier than 64 but mathematically this would not seem to make thecalculation easier. If she had subtracted from 64 she could have used the

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 9

Figure 3. Stella’s response to the calculation 64 – 49.

Figure 4. Jack’s response to the calculation 12 + 19.

same complements to 9 but in a different order, that is subtract 4 and then5, rather than 5 and then 4.

3.2. Jack

In Figure 4 Jack has rounded both the numbers in the problem to multiplesof ten, then added the two multiples of ten before adjusting. This wouldstill seem to be a use of the compensation strategy but not an exact imit-ation. In the taught examples only the addend or subtend was rounded toa multiple of ten. Figure 5 shows a similar use of compensation to that inFigure 4. In Figure 6 Jack has continued with the rounding of both numbersto a multiple of ten but he has difficulties in compensating. Jack was thengiven the problem 43 – 9 but he stated he was unable to carry it out.

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Figure 5. Jack’s shopping example: 12p + 38p.

Figure 6. Jack’s response to the calculation 64 – 49.

Figure 7. Caroline’s response to the calculation 12 + 19.

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 11

Figure 8. Caroline’s use of compensation for 12 + 19.

Figure 9. Caroline’s recording of the compensation for 12 + 19.

3.3. Caroline

Caroline (Figure 7) used a standard written algorithm. In an attempt toexamine her use of the taught strategy the researcher talked her through thecompensation strategy by suggesting she rounded the 19 to 20 and then add20 onto 12 (Figure 8). Caroline then attempted to record the strategy (Fig-ure 9) but appeared unable to identify the correct compensation needed.Caroline was not asked to try the compensation strategy further.

Caroline appeared unable to add on a multiple of ten without count-ing on in ones. She had recalled a notion of ‘take back one’ but seemedunaware of which number she should compensate from. She resorted to astandard written algorithm in order to attempt the compensation but madeerrors in her use of the algorithm.

To summarise, all of the three children appeared to show some recall ofthe compensation strategy. Stella was able to provide her own explanation

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indicating rounding to a multiple of ten and then compensating. Jack didnot give an explanation at this stage but his responses to the calculations(Figures 4–6) showed he remembered the principles of rounding and com-pensating. Caroline also gave an explanation that included a multiple often. She also used the term ‘take back one’ in a later response (Figure 8).

Stella used the compensation strategy as taught for the first example(Figure 1). As she carried out the shopping example (Figure 2) she ap-peared to be using other strategies. She commented in one example (Fig-ure 3) that her change of number made the calculation easier but it doesnot appear to be mathematically easier. Jack rounded both numbers inthe examples to multiples of ten (Figures 4–6). This may have made thecalculation easier initially but it then made the compensating adjustmentmore difficult. This was particularly the case with the subtraction problem(Figure 6). Caroline used a standard written algorithm to solve the problem12 + 19 (Figure 7). When prompted to use the compensation strategy sheresorted to a counting procedure to add a multiple of ten (Figure 8). Whenshe attempted to record the compensation strategy she used the standardalgorithm in an attempt to make the adjustments (Figure 9). She did notseem aware of which numbers to adjust to compensate but knew that ‘takeback one’ was involved.

The results and analysis provide a portrait of how the three childrenresponded to the taught strategy and their different interpretations indicatekey points.

• Direct instruction of the strategy as part of a classroom activity provideda method for the children and the researcher to refer to in the inter-view.

• All three children made reference to the use of multiples and com-pensating.

• Stella and Jack appeared able to use the principles of the taught strategy.• The children did not use the strategy consistently as taught and sug-

gest contrasting interpretations.

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 13

4. DISCUSSION

The three children in this study demonstrated contrasting use of the taughtstrategy. Stella may have reverted to her own idiosyncratic use and Jack ap-peared to modify the strategy. Caroline appeared unable to use the strategy.The three children previously demonstrated contrasting spontaneous cal-culation approaches in the pre-teaching interviews. It is possible that theseinitial spontaneous approaches influenced the children’s use of the com-pensation strategy following instruction. It is acknowledged that the studyhas certain limitations. Learning may not be ascribed fully to the teachingsession and it is also acknowledged that the variations in the use of thestrategy could be explained by the children’s inability to remember thestrategy after one week. It is also possible that further shared practicewould develop the children’s use. However the very fact that the chil-dren were not carrying out the strategy exactly as taught is worthy ofexamination. Caroline’s failure to use the strategy might also be rectifiedwith further practice but her misuse at this stage presents another area fordiscussion.

Stella had used a range of idiosyncratic mental calculation strategiesprior to the teaching session and may have reverted to using these. Eventhough she was capable of using the selected knowledge for compensationshe digressed from using the strategy. Jack had shown that he was able toadd or subtract a multiple of ten in the pre-teaching interviews but his useof deductive strategies had been limited. In the post-teaching interviewshe appeared to be using this knowledge to move towards further use ofdeductive mental strategies. Caroline may have remembered some aspectsof rounding and compensating but seemed unable to use the strategy. In thepre-teaching interviews she had used a counting procedure to add a mul-tiple of ten and this may have influenced her inability to use the strategy.From a pragmatic viewpoint if, as a learner, you rely on counting-on inones to add 10 to a number, then rounding 9 to 10 would not make thecalculation easier. It is not possible in a study of this size to make globalgeneralisations but in these instances the three children’s interpretationsof the taught strategy may have been influenced by the experience andknowledge that they brought to the teaching situation.

The direct instruction enabled all three children to participate in a sharedlearning experience. A participation viewpoint of learning would suggestthat learning occurs through participation in a community and that theclassroom can be seen as a community of learners. Lave’s (1988) study ofthe informal arithmetic of adult Americans saw mental strategies as active,flexible responses to problems where the strategies used were related to

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TABLE II

Summary of key distinctions in participation and constructivist viewpoints

Participation viewpoint Constructivist viewpoint

Knowledge Shared activity Personal construction

Learning Becoming a participant Restructuring of schemas

Use of mental Situated in a context Part of an individual’s

calculation schematic development

strategies

Classroom Participating in the learning of Opportunities to construct

practices the classroom as a community personal meaning

Unified view Knowledge influenced and shaped by classroom discourse but built

on the learner’s previous experiences and knowledge.

the situation. The “place of knowledge is within a community of practice”(Lave and Wenger, 1991, p. 100) and situated in a context. The teachingepisode had endeavoured to introduce a contextual situation with the pur-chase of two shopping items and the instruction had used this to examinea deductive calculation strategy that would be appropriate for the numbersinvolved. All three children had made some reference to this learning ex-perience but the participation or situated viewpoint does not provide anexplanation for the individual children’s use following the instruction.

A constructivist perspective, however, would support a more idiosyn-cratic interpretation of the taught strategy as it recognises the developmentof arithmetic from personal knowledge. Several researchers have examinedchildren’s learning of mental calculation strategies from a constructivistperspective (Steffe, 1983; Kamii, 1985; Denvir and Brown, 1986; Fraivilliget al., 1999; Smith, 1999). Steffe (1983) stated the need to take “seriouslythe basic Piagetian principle that children necessarily have to constructtheir own mathematical reality” (p. 110). He supported the notion thatchildren’s mental methods can be seen as schemes where new learningis integrated into an existing schematism or structure. Denvir and Brown(1986) also proposed that individuals “integrate skills which they havealready acquired when these are simultaneously called in to play with theirmental actions” (p. 143).

Several researchers vie for an integration of these two viewpoints. Gooset al (1999) emphasised the “concept of the classroom as a communityof practice” (p. 57) but also acknowledged that mathematics is created“from reflective inner dialogue” (p. 59). They proposed that by elevating

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 15

the “discourse and process of mathematics” students’ concepts are “gen-erated and stabilized through social dialogue” (p. 59). Askew et al (2000)proposed that “participation in some sense precedes acquisition” (p. 64).The classroom can be seen as a community of learners but “knowledge isshaped by the learning opportunities that pupils have experienced” (p. 64).

Sfard (1998) commented that exclusive reliance on one theoretical view-point can lead to “didactic single-mindedness” (p. 11) and proposed that amodel of learning should reflect a ‘dialectic’ between the situation and thelearner but also take into account the learner’s previous experience. In thisstudy the demonstration of a mental calculation could be seen to supportthe sharing of strategies in the classroom practice. Children may be in-troduced to strategies that they have not developed spontaneously but a re-cognition for learners to construct their own understanding of the strategieswould also seem paramount. Children’s use may be influenced and shapedby the classroom discourse but it may also be shaped by their previousexperiences and knowledge. In the use of mental calculation strategiesthis previous knowledge would include the connected use of number facts,number relations and number operations.

Compensation, the example used in this study, more specifically relieson knowledge of the addition or subtraction of a multiple of ten to a multi-digit number. For some reason Caroline had not developed this knowledgeand so was not able to engage in the use of this deductive strategy. Lowerattaining children’s understanding of mathematics may be seen as proced-ural (Gray, 1997). The reliance on procedures such as counting negatesthe need to remember or use numerical facts in a connected way to solvecalculations. It is possible that such “procedural understanding may wellbe a significant cog within the arithmetical, or indeed mathematical entity”and “it may be the reason why some children do not make the links” (Gray,1991, p. 552). Jake, on the other hand, demonstrated some counting pro-cedures but also appeared to have sufficient pre-requisite knowledge. Inthis instance procedural knowledge linked with selected knowledge mayhave lead to the use of a deductive strategy (Gray, 1991). It is possible thatJack was beginning to make further connections.

Gray’s comments suggested that some children may not make the con-nections and links necessary for the use of deductive strategies. The demon-stration and practice of a deductive strategy may not be sufficient to ad-dress this. Participation through instruction in a classroom may allow somechildren to learn “in a connected and coherent fashion” but “. . .in thesame class, others are either failing in their attempts to, or succeedingat, parroting disconnected facts” (Burton, 1999, p. 25). Direct teachingin a classroom situation may introduce deductive mental strategies to chil-

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dren and the use of these methods may support their knowledge of num-ber by gaining “deeper insight into the workings of the number system”(Thompson, 1999, p. 147). However, some children may not have suffi-cient mathematical knowledge to make the connections. As Plunkett (1979)stated mental calculations “require understanding all along [emphasis inoriginal text] . . . they cannot be used to achieve performance in advanceof understanding” (p. 3).

5. CONCLUSION

This small-scale qualitative study has focused on the interpretation of uniquesituations or illustrations of a process. It is recognised that global gen-eralisations are not possible but, through in-depth analysis, insights intothe direct teaching of mental calculation strategies have been explored.It has questioned how children are able to use a taught mental calcula-tion strategy. The unified viewpoint of learning would suggest that mentalstrategies can be introduced to children through whole class instructionbut that their use of the strategies may be reliant on their personal know-ledge. The three children in this study demonstrated a contrasting use ofthe strategy which may have been dependent on the knowledge that theybrought to the teaching session.

This raises the hypothesis that children’s use of a taught mental calcula-tion strategy relies on pre-requisite knowledge that is based on a connectedview of mathematics. One scenario is that the teaching of mental calcu-lation strategies is seen to support children in moving to more flexibledeductive strategies by making links to their existing knowledge. Howeverit is also possible that some children do not make these links and becomemystified by the strategies presented to them. A further scenario is thatsome children already have their own range of deductive strategies whichthey are confident in using.

The National Numeracy Strategy has categorised a range of deductivestrategies that encourage the use of known facts to derive new ones andhas promoted the direct instruction or demonstration of these. However,the strategies, as presented in the National Numeracy Framework (DfEE,1999a), are not explicitly underpinned by the need for children to constructtheir use from their own existing personal knowledge. If a unified view oflearning is followed then this existing personal knowledge would seemparamount in enabling the children to make sense of the mathematics theyparticipate in.

External evaluations of the implementation of the Numeracy Strategy(Earl et al., 2003) have pointed to ‘outstanding’ teachers who are aware

CHILDREN’S USE OF A TAUGHT MENTAL STRATEGY 17

of the understanding that their pupils bring to lessons. “Such teachingis consistent with the implications for teaching of cognitive orientationstoward learning, indicating that children’s learning can be enhanced whenteachers connect learning to what the children already know”. However itis also stated that this is “not the norm” (p. 127).

In the introduction it was recognised that it may be too optimistic toassume that children’s use of mental calculation strategies will necessarilydevelop spontaneously. The question then is how children come to usemental calculation strategies that are taught to them through direct instruc-tion. This study has raised awareness in relation to the personal knowledgebrought to a learning situation and how this may influence the use of ataught mental strategy. Further examination of this personal knowledgein relation to a connected or procedural understanding of mathematics isneeded in order to clarify the didactic approach that will support children’sdevelopment of mental calculation strategies.

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University of Exeter,School of Education and Lifelong Learning,Exeter, EX1 2LUTelephone 01392 264974, Fax: 01392 264792E-mail: C.M.Murphy@ex.ac.uk