ABSTRACT. The ability to be flexible in mental calculation by using a method that isefficient for calculating the particular problem being faced is an important aim of teachingin this area. Flexibility is commonly seen as arising from a rational choice between mentalcalculation ‘strategies’, based on the characteristics of the problem faced. In this article it isargued that strategy choice is a misleading characterisation of efficient mental calculation,and that teaching mental calculation methods as wholes is not conducive to flexibility. Analternative is proposed in which calculation is thought of as an interaction between noticingand knowledge, and an associated teaching approach to promote flexibility is described.
Two recent changes in the education systems of England and Wales haveraised the profile of mental calculation in the teaching of mathematics inprimary and secondary schools. The first is the inclusion since 1998 ofMental Arithmetic tests in statutory assessments for 11 and 14-year-oldpupils. The second is the introduction of the National Numeracy Strategyto primary schools and secondary schools (DfEE, 1999, 2001), a UK gov-ernment initiative to promote higher standards in mathematics learningthrough an integrated approach to teaching, in which mental calculationhas a strong emphasis. These changes have already had a great impacton what happens in primary schools in England and Wales, with teachersdevoting considerably more time to mental work with number than theyhad before.
The value of having a facility in mental calculation is not doubtedby anyone, but the exact nature of the skills and competencies that aresought is less clear, as is the precise role of teaching in developing them.Thompson (1999a) summarises earlier literature to give four reasons forteaching mental calculation:
1. Most calculations in adult life are done mentally2. Mental work develops insight into the number system (‘number sense’)3. Mental work develops problem-solving skills
4. Mental work promotes success in later written calculations
In the school context these eventual purposes are translated into shortterm curriculum objectives focused on skills, in particular being able tocalculate accurately and reasonably quickly over a range of demands seento be appropriate to the age of the pupil. By the late primary years thisusually includes mental calculations with two and three-digit numbers.Since expectations about skills are readily encapsulated in tests of men-tal calculation (often called ‘mental arithmetic’ in the UK) the purposeof teaching mental calculation can become identified with achieving highlevels of performance on such tests. In countries where tests of mentalarithmetic have become ‘high stakes’ assessments, such as England andWales, teaching which enables children to obtain high marks on these testsand assessments is highly desired.
Yet it is possible to imagine pupils being able to calculate quite quickly,and doing well on tests of facility in mental calculation, without developingthe abilities that would bring enhancements to other number work, im-proved problem solving, better ‘number sense’ and competence as adults.Answers to mental calculation problems can be arrived at in different ways,not all of which are equally suited to the longer-term purposes. Childrencan be correct:
1. By recall of, or ‘just knowing’, a number fact2. By a simple counting procedure, in which the number sequence is
recited (privately) while keeping track of the count3. By making a mental representation of a ‘paper and pencil’ method
(commonly a vertically represented ‘sum’), and working through theprocedure mentally
4. By constructing a sequence of transformations of the number problemto arrive at a solution, for example adding 36 to 28 by first adding 20to 36 (making 56) then thinking of the remaining 8 to be added as twofours, adding the first four to make 60 then adding the remaining 4 toarrive at 64 as the answer.
Any of these could bring success in particular calculations, but approachesof the fourth kind, often referred to as ‘strategies’, are usually considered tobe vital to the broader purposes for mental calculation. Indeed Thompson(2001, p. 75) suggests that the phrase ‘mental calculation’ is used in officialdocumentation in England and Wales to stress the importance of the ‘usingstrategies’ aspect of mental work. SCAA (1997) for example expresses apreference for ‘strategic’ mental methods over counting methods or themental execution of a visualised vertically presented sum.
Thompson (1999b) suggests that mental calculation ‘strategies’ are:
FLEXIBLE MENTAL CALCULATION 31
about the application of known or quickly calculated number facts in combinationwith specific properties of the number system to find the solution of a calculationwhose answer is not known. (p. 2)
A very simple example is of a child ‘working out’ that 5 + 6 is 11 because“I know that 5 + 5 is 10 and 6 is one more than 5”. QCA (1999, p. 14)suggest that strategic methods are those that come about when “numbersare treated in a holistic way, as quantities rather than digits”. The benefitsare that:
• as these methods are more meaningful and conceptually based, theyare more likely to impact positively on other aspects of mathematicaldevelopment; and
• since some methods are more suited to some problems than others,there can be efficiency in calculation.
As an example of the second aspect, the adding of one number to anotherfirst by adding the tens and then the ones separately is more suited to 55+ 24 than to 33 + 29, whereas rounding a number before adding, andthen compensating, is more suited to 33 + 29 than to 55 + 24. A childwho makes ‘good choices’ of this kind is expected to be more effective incalculating. QCA (1999, p. 14) argue for such flexibility to be a key featureof calculating using mental strategies. Anghileri (2001, p. 82) too valuesthe “inclination and ability to match a mental method to the numbers in theproblem”, contrasting this with the use of a mechanistic procedure that canbe used on any numbers but which can be very awkward for some calcula-tions. Thompson (2001, p. 76) reports that “In the Netherlands the idea offlexibility is emphasised in the use of mental strategies” with Beishuizen(1999) citing the raising of pupils’ level of flexibility in mental calculationas a purpose of some recent innovations in Dutch schools.
Flexible mental calculation is thus a highly desired end-point for teach-ing, and as it seems to involve choice between different methods, it is acommon presumption that such flexibility is achieved by teaching pupilsa set of different ‘strategies’, and then teaching them how to choose thebest method for the problem at hand. In QCA (1999), for example, someholistic strategies for addition and subtraction are listed, including com-pensating (example given: 38 + 69 = 38 + 70 – 1), and using near doubles(example given: 38 + 35 is double 35 + 3), and it is suggested that that thestrategies “will need to be taught by teachers” with the intended flexibilitybeing learned through children “discussing which strategies appear moreeffective for particular problems with particular types of number” (p. 19).
With this purpose in mind, Beishuizen (1999, p. 166) argues that it is“important to decide on exactly which methods should be taught and inwhat order”, and Buys (2001, p. 109) describes an approach used in a
32 JOHN THRELFALL
text series in Holland to introduce different types of mental strategies foraddition and subtraction successively through the early years of schooling.
In this article I wish to challenge the presumption that flexibility inmental calculation is achieved by teaching holistic ‘strategies’. I will fo-cusing on issues concerning:
• the inadequacy of descriptions of strategies;• the effects of demonstration on flexibility; and• difficulties with the notion of strategy choice.
I will go on to argue that flexibility in mental calculation is better ap-proached through teaching that focuses on the number knowledge andunderstanding that is drawn on when calculating.
2. THE INADEQUACY OF DESCRIPTIONS OF STRATEGIES
In order for flexibility in mental calculation to arise from the teaching ofholistic strategies, it must be possible to characterise, as general types, areasonably full range of what might be done. A teaching approach thatis intended to foster choice between taught alternatives needs to have acoherent way of thinking about the possible choices, so that they can betaught in an organised and systematic way. In other words, there has to bea classification system that makes sense to the teacher.
There are several attempts in the literature to classify complete solutionsequences in knowledge based approaches to mental arithmetic. Many ofthem focus on the addition and subtraction of two two-digit numbers, per-haps because this level of calculation is a reasonable expectation for almostall pupils, while being unlikely to be done by recall. Thompson (2000, p.24) claims that children are usually expected to “develop general strategieswhich will enable them to add or subtract mentally any pair of two digitnumbers”.
In the following pages, seven contrasting classifications of holistic men-tal calculation strategies for the addition of two two-digit numbers arelisted. Although the authors of many of these did not intend to give guid-ance for teaching, they all attempt to offer a coherent and meaningfulway of classifying the different ways children solve arithmetic problems‘strategically’, and so each represents a possible way to classify strategiesfor teaching purposes.
To test whether any of these different approaches is comprehensiveenough to support direct teaching of strategies towards flexibility in use,each of them will be compared to a set of different methods used by agroup of children to calculate a particular problem. The approaches listed
FLEXIBLE MENTAL CALCULATION 33
arose in a project to explore and promote mental calculation. The studyinvolved children aged from 7 to 11 in two primary schools in the UK.The children were each given a small set of horizontally presented numberproblems and asked to work out each answer mentally and write downhow they had done it. A previous study involving individual interviewswith six children had shown that written responses took the same form asprotocol responses, and so the written format would give a valid indicationof approaches among a larger group. In that study, the children were askedto work out questions presented orally and to verbalise their thinking, withthe responses recorded in note form by the researcher.
A total of 53 children aged 9 and 10 reported on problems involving theaddition of two two-digit numbers. Their replies often took the form “FirstI added 45 and 40, and that makes 85. Then I added 8 to get 93” or similar,although sometimes just symbols were used. The methods are re-presentedhere in a standard format for ease of comparison.
Add 45 and 48
A 40 + 40 5 + 8 80 + 13
B 40 + 40 80 + 8 88 + 5 (perhaps as 88 + 2 then 90 + 3)
C 45 + 40 85 + 8 (perhaps as 85 + 5 then 90 + 3)
D 45 + 8 53 + 40 (or 48 + 5 then 53 + 40)
E 45 + 50 95 – 2 (or 48 + 50 then 98 - 5)
F 45 + 45 90 + 3 (or 48 + 48 then 96 - 3)
G 50 + 50 100 – 7 (perhaps as 100 – 2 then 98 – 5)
H 48 + 2 50 + 43 (or 45 + 5 then 50 + 43)
Although the data drawn on to make this list arose in a particular project,it is important to note that they are not special data, and that diversity ofthis kind is commonplace. Similar data can be collected at any time inany school. Ask a group of children to calculate a problem mentally, andsay how they did it, and it is very likely that different approaches will bemanifested.
In the strategy classifications that follow, each category will be giventhe letter code(s) for the approaches to 45 + 48 above that seem to fit.
1. Thompson (1999) lists what he claims to be the four main strategiesfor two-digit addition:
34 JOHN THRELFALL
The ‘split’ method, of turning each number into tens and ones, then dealing withthe tens and ones separately and re-combining.
A
The ‘jump’ method of turning just one number into tens and ones, and thenadjusting the other number first by the tens amount, and then by the ones,sometimes in two parts, firstly to reach the next ten, and then the rest.
C
The ‘split-jump’ method, combining the ‘split’ and ‘jump’ methods, in whichalthough both numbers are split, and the tens are combined, the ones are dealtwith one at a time, again splitting to bridge the ten if necessary.
B
The ‘over-jump’ or ‘compensation’ method of rounding one number up to thenearest ten, adding it, and then adjusting the result by the amount of the rounding.
E
2. Yackel (2001, pp. 23–4) distinguishes between ‘collections-based’ solu-tions, breaking up both operands into parts, usually tens and units [A andB], and ‘counting-based’ (or ‘sequence-based’) solutions, starting with oneoperand and dealing with the other(s) progressively, piece by piece [C andD].
3. Heuvel-Panhuizen (2001, p. 61) describes three calculation strategies foraddition and subtraction that are taught towards in the Dutch curriculum:
The jumping strategy, which is related to calculating by counting: it implieskeeping the first number as a whole number, e.g. 87 – 39 = . . . 87 – 30 = 57. . . 57 – 7 = 50 . . . 50 – 2 = 48
CD
The strategy of splitting numbers in tens and ones, which is related to calculatingby restructuring: it implies making use of the decimal structure, e.g. 87 – 39 =. . . 80 – 30 = 50 . . . 7 – 7 = 0 . . . 50 – 2 = 48
AB
Flexible counting, which is related to formal calculating: it implies making useof knowledge of number relations and properties of operations, e.g. 87 – 39 = . . .
87 – 40 = 47 . . . 47 + 1 = 48
E
4. Fuson et al. (1997, p. 146) classify methods into four kinds:
methods that begin with one number and move up and down the sequence by tensand ones [C and D]; decompose-tens-and-ones methods in which the tens and theones are added . . . separately from one another [A]; mixed methods in which thetens are added . . . and then a sequence number is made with the original ones anda sequence method is used to add . . . the other ones [B]; and methods in whichboth numbers are changed to make easier numbers [G].
5. Carpenter et al. (1997, p. 4) proffer a three way classification: ‘Sequen-tial’ in which “the sum is kept as a running total” [C and D]; ‘Combiningunits separately’ in which “the tens and ones are [dealt with] separately,
FLEXIBLE MENTAL CALCULATION 35
then combined” [A]; and ‘Compensating’, in which “the numbers are ad-justed to simplify the calculation” [E, F and G].
6. Klein and Beishuizen (1998) list five approaches:
‘1010’ (pronounced ‘ten ten’) in which tens and units are split and dealt withseparately
A
‘N10’ in which the tens of the second number are added to the first number,followed by the ones
C
‘A10’, in which the second number is split to provide a quantity that takes thefirst number to the next ten, and then the remainder of the second number isadded
H
‘N10C’ in which the second number is rounded up to the ten, which is then addedto the first number, with a subsequent adjustment, or compensation
E
‘10s’, in which the two tens are dealt with, then the ones are added sequentially B
7. QCA (1999) include four number ‘partitioning’ strategies in a list tobe taught that includes strategies of a different type, such as counting.
“Using multiples of 10 and 100” by partitioning one or both numbers into tensand ones, e.g. 23 + 45 = 40 + 5 + 20 + 3 or 55 + 37 = 55 + 30 + 7 = 85 + 7
AC
“Bridging through multiples of 10”, by partitioning one of the numbers toprovide the difference between the other one and the higher multiple of ten,e.g. 57 + 14 = 57 + 3 + 11 or 57 + 13 + 1
H
“Compensating”, by rounding one number up to a multiple of 10, adding that tothe other number, then subtracting the amount used to round up, e.g. 38 + 69 =38 + 70 –1
E
“Using near doubles”, e.g. 38 + 35 is double 35 + 3 F
Each of these systems seems in itself reasonable as an approach to clas-sification, yet it is noticeable that none of them is adequate to capture thediversity found in the calculations of a small sample of ordinary primarychildren. It is acknowledged with Beishuizen (2001, p. 122) that there istoo much diversity to count each variant as different, and that categorieshave to be “a little broader to include several methods covering all prob-lems” yet although all the methods are accounted for in the categories ofone author or another, none seems to have room for all of them. It seemsthat general ways of making sense of mental calculation struggle to maponto the variations found in calculating particular problems. This beginsto suggest that more may be involved in flexible mental calculation thanthe choice of an identifiable general strategy. It also means that if it isassumed that a teacher will look for one coherent system as a source for the
36 JOHN THRELFALL
strategies that they teach, they will be teaching a narrower set of strategiesthan the children might use, and as a consequence there seems to be arisk that the calculating approach that would be the most efficient for aparticular problem is not available for use, as it has not been taught.
3. THE EFFECTS OF DEMONSTRATION ON FLEXIBILITY
The potential negative effect on flexibility of teaching mental strategies ofthe kinds described by the authors above is compounded by choices thatteachers make.
This is because, if required to teach such strategies, most teachers chooseapproaches that make ready sense to the children and are easily demon-strated, and as a result, some ‘strategies’ are more likely to be taughtthan others. For example the approaches to addition and subtraction thatinvolve the successive adjustment to the first number can be made senseof by being demonstrated on the number line. The method involving firstpartitioning into tens and ones and dealing with each separately beforere-combining can be made sense of by being demonstrated using base 10materials. If these are preferred by teachers, and other strategies that areless easily demonstrated are not being taught, the children are not beingequipped with the tools to enable flexibility in mental calculation. As Bloteet al. (2000) note, greater familiarity with one method, perhaps from anemphasis on it in teaching, can lead to that approach being the inevitablemethod of choice, or ‘default’ option.
As a kind of confirmation of this argument, Beishuizen and Anghileri(1998, p. 523) report that these two kinds of strategies seem to arise extens-ively among primary aged children in England, the U.S.A. and Holland as aresult of teaching, but to differing extents, as a result of different emphasesin the mathematics programmes of the different countries:
1. The ‘N10’ strategy develops in Dutch Realistic Mathematics Educa-tion contexts, probably through the extensive use of the empty numberline; and
2. The ‘1010’ strategy is widely used in the USA and UK, probably as aresult of an emphasis on the formal place value structure of ‘Hundreds,Tens and Units’.
Thompson (1999) confirms that the method for subtraction of two digitnumbers of turning each number into tens and ones and dealing with eachseparately has been the most common strategy used by children in Englandand Wales – although the National Numeracy Strategy (DfEE, 1999) maychange that, as it emphasises the use of number lines over the use of Base
FLEXIBLE MENTAL CALCULATION 37
10 apparatus. Either way, however, children whose thinking is dominatedby the holistic strategies that they remember best from teaching are notwell placed to develop flexibility in mental calculation.
4. DIFFICULTIES WITH THE NOTION OF STRATEGY CHOICE
Developing flexibility by first teaching children a range of holistic strategiesis also undermined by the issue of what has to happen then – the decisionof which approach to use for a particular calculation. Klein and Beishuizen(1998) invoke a concept of choice:
Because the (number) characteristics vary across problems, students have to ad-just their strategy use according to the features of the problem. For this reason thestudents must be able to employ a range of arithmetic strategies and computationalprocedures among which they can choose flexibly. (p. 449)
Askew (1999) too argues for:
an awareness of different methods of calculation and the ability to choose anappropriate method. (p. 98, italics as in original)
and Thompson (1999) concurs:
Mental calculation places great emphasis on the need to select an appropriatecomputational strategy for the actual numbers in the problem. (p. 147)
If a range of holistic ‘strategies’ are taught and learned, with a view todeveloping flexibility in their deployment, the learner has to be able tomake a good choice when faced with a particular question. However, thenotion of strategic choice is not as straightforward as it may seem. Forexample, a ‘good’ method for the mental subtraction of say 502 – 45 mightbe to round 45 up to the next ten (5), then round up to the next hundred(another 50), and add that to 100 less than the first number (402 + 55 =457).
But when would such a method stop being a good method? How doesone decide from the numbers? Perhaps the larger ‘tens’ number is a cluethat it would not be a good method for 550 – 45, and the proximity of 42 to45 may suggest that there is a better approach to 542 – 45, but what about530 – 45 or 515 – 45?
As another example, 54 – 28 is done by children in a number of ways.Sometimes it involves near doubles – “Half 54 is 27 so if I took 27 offit would be 27 so as it is 28 the answer is 26”, sometimes breaking thesecond number into parts “I took off 20 then 4 then 4” and sometimes byrounding “First I took off 30 and then added the 2 back on again”.
Which of these is the most appropriate? Or will any method do for thisproblem? If there are some problems for which any method is as good as
38 JOHN THRELFALL
any other, what features of the numbers in the problem indicate that this isthe case? QCA’s claim that: “the numbers will provoke the use of differentstrategies” (QCA, 1999, p. 14) seems optimistic.
There is also a problem from the point of view of the alternative strate-gies. Given that, as Beishuizen (2001, p. 122) argues, classifications haveto be broad to include variations, only some of the types identified in theliterature are sufficient to carry the whole solution, and others bring theneed for further ‘decisions’ on solution paths. For example the ‘1010’ and‘N10’ methods for addition and subtraction, once decided, proceed withoutfurther decisions needed, through the several steps involved. On the otherhand, the strategy of changing numbers to make the calculation easier,identified by Fuson et al. (1997) and Carpenter et al. (1997) for example,does not in itself suggest how the numbers should be changed. As thereare several quite different variations in this, including rounding to neartens, to near fives, to near doubles, to near common multiples, and so on,the ‘strategy’ is a generalisation rather than a course of action. As a resultthe strategy of changing numbers to make a problem easier may well beunderstood, but will not be understood in a way that can then be appliedto a particular case. The characteristics of problems that would make thechoice of this strategy a good one cannot be part of the understanding ofthe strategy, as they are so diverse, so how would a decision to ‘choose’ iton the basis of the numbers ever be made?
It seems that there are many circumstances where decisions about ‘goodmethods’ cannot be arrived at in advance of actually moving forward intocalculating the problem. The idea that criteria can be taught to children fordeciding in advance which strategy to use does not seem feasible.
Perhaps these difficulties explain why there are no suggestions in theliterature for direct teaching on how to make good strategy choices inmental calculation. Indirect approaches are always preferred. For example,the teacher chosen by Askew (1999) to illustrate this aspect of teachingsays “I have tried to provide them with a whole range of different ways ofgoing about adding numbers, or taking them away, so that they will be ableto become comfortable with the strategies that they like best.” Personalpreference is not the same as choosing an appropriate method, but the hopemust be that they coincide.
In Holland too, although flexibility is seen as important, there does notseem to be much explicit teaching to guide children in choosing strategies,as it is expected to arise from the ‘mental structures’ developed by theapproach (Blote et al 2000, p. 228) although Blote et al. (2000) do reportthat:
FLEXIBLE MENTAL CALCULATION 39
the students are told to use 1010 exclusively on addition problems, because it canlead to errors on some kinds of subtraction problems. (p. 225)
If the criteria for deciding which method to use are too complex to betaught, the approach of teaching a set of ‘strategies’ as the alternativesfrom which to choose seems risky.
Siegler (1996, p. 14) argues that the idea of children choosing a good‘strategy’ is in any case a misconception. He points out that there have beennumerous constructs trying to account for how children decide what to do,all of which “envision choice as an explicit, mindful, top-down process”but none of which succeed.
As in a number of other decision contexts, it is difficult to determinewhat the criteria for choice of mental calculation method might actuallybe. Even though some problems do seem to suit some ‘strategies’ morethan others, ‘choice’ could not be just about the number characteristics ofthe problem. The knowledge of the individual must also be very important.In written calculations there can be a pause in the execution of a method towork out a number fact as a sub-routine, but mental calculation strategieshave to draw on and adapt what is reasonably well known, and while thereare always ways of figuring out based on what one does know, the calcula-tion sequence cannot become too complex or the execution of it will breakdown. So a child who knows the ‘12 times table’ might calculate 24 × 12by doubling 12 × 12, but another child who does not know the ‘12 timestable’ would be unwise to ‘choose’ that approach. A further dimension tothis factor is the self-knowledge of the child about the knowledge that issecure enough to be used in a particular strategy. The child who does notknow the ‘12 times table’ has to be wise enough not to choose it.
Siegler (1996) says that strategy choice through some kind of consciousmatching of task to alternative approach, mediated by one’s own awarenessof competence, is difficult to imagine:
Do people make explicit judgements about their intellectual capacities, availablestrategies, and task demands every time they face a task they could do in twoor more ways? If not, how do they decide on which tasks to do so? Do theyconsider every strategy that might be applicable, or only a subset of them? Ifonly a subset, how do they decide which ones? How do people determine theircognitive capacity on a novel task, and how do they determine what strategies areavailable to that task? (pp. 14–15)
Siegler (1996) points to deep rooted beliefs about agency that support thesetraditional views of choice, the fact that we commonly use phrases like “Idecided to do X, so I . . .” are seen by Siegler as implying “an inner selfthat reaches a conclusion that then guides further action” and suggests thatthinking of choice as requiring a chooser is misplaced:
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the choices often are at least as much attributable to ‘mindless’ processes that arepart of the basic cognitive architecture as to more ‘mindful’ ones. (p. 15)
As Bisanz and LeFevre (1990, p. 236) put it, “the solution procedures usedin mental arithmetic violate several of the connotations associated with theword ‘strategy’.
5. MENTAL ‘STRATEGIES’ AS TAUGHT PROCEDURES
So although flexibility in mental calculation is a recognised worthwhileaim, the characterisation of it as a ‘mindful’ choice between strategiesseems unwarranted, and the approach to develop it by direct teaching ofholistic strategies seems misguided and unlikely to succeed. What is theproper response to this?
A tempting response could be to question how important flexibilityactually is in mental calculation. After all, as Blote et al (2000, p. 244)point out: “For many problems in [the] maths class, flexibility is not reallyneeded.” Most of the immediate needs of school mathematics can be metby always using the same ‘strategy’ for the same operation, for examplealways adding by breaking the numbers into base 10 elements, and alwayssubtracting by reducing the first number by ‘pieces’ of the second num-ber. Although inferential mental calculation ‘strategies’ are often thoughtof as the antithesis of procedures, this is not necessarily the case. Someof the more easily taught and learned ‘mental methods’ can be used onall problems involving that arithmetic operation – for example the ‘N10’for subtraction and the ‘1010’ for addition. It is feasible to teach them asprocedures.
Indeed it may be argued that if the focus is on performance, one canobtain better short term gains in calculation facility (and therefore suc-cess on high-stakes assessments) by teaching a single mental ‘strategy’ foreach arithmetic operation, to be used with all such problems, whatever thenumbers. This might be further supported by an argument about aptitude– that only the more ‘mathematically minded’ children will be capable oflearning how to make good choices, so flexibility should be abandonedas an objective for the ‘average’ and ‘below average’. In such a context,striving for flexibility may be seen as a mistake.
FLEXIBLE MENTAL CALCULATION 41
6. FLEXIBLE MENTAL CALCULATION
A different response to the problems of teaching strategy choice is toconceptualise flexible mental calculation other than as the application oflearned holistic strategies.
When faced with a fresh problem, the child or adult who follows dif-ferent solution paths depending on the numbers does not do so by think-ing about what the alternatives are and trying to decide which one todo. Rather, he or she thinks about the numbers in the problem, noticingtheir characteristics and what numbers they are close to, and consideringpossibilities for partitioning or rounding them.
For example in a problem involving 63 and 26 some or all of the fol-lowing might be noticed:
63 is. . . 26 is. . .
60 and 3 20 and 6
2 less than 65 1 more than 25
double 30 and 3 4 less than 30
50 and 13 double 13
20 and 20 and 20 and 3 10 and 10 and 6
7 less than 70 just over half of 50
As part of this thinking there are also connections made between whatis being noticed about the numbers separately. If considering 63 and 26together in the context of subtraction, it might be noticed that:
65 is forty more than 25The 3 (of the 63) is nearly the 4 (that the 26 is less than 30)There is a 30 in bothThere is a 20 in bothThere is a 13 in both6 is double 3etc.
The solution path followed by the individual then depends on which ele-ments of what has been noticed chime with their knowledge of feasiblesteps in calculations, and which of the possibilities sits most easily withknowledge they are comfortable with. One child might move forward from65 and 25, adding 3 and subtracting one, where another might proceed bytaking three off 63 and 26, to change the problem to 60 subtract 23, thensubtract the 20, and then count back 3 from 40.
What the individual notices about the numbers in the problem leadson to what is done. If one child notices one or both of the numbers asa composite of different parts, and some of these parts are recognised in
42 JOHN THRELFALL
combination as an item of factual knowledge, he or she may be temptedthat way. If another child notices the proximity of one or both of the num-bers to other numbers, like in rounding, and knows enough about how therounded numbers combine, that kind of approach could take shape. If thechildren have sufficiently well developed knowledge to see the problemthrough, they arrive at a solution. If not, they look again at the numbersand follow a path triggered by what they notice this time.
In this way, flexible mental calculation can be seen as an individualand personal reaction with knowledge, manifested in the subjective senseof what is noticed about the specific problem. As a result of this inter-action between noticing and knowledge each solution ‘method’ is in asense unique to that case, and is invented in the context of the particularcalculation – although clearly influenced by experience. It is not learned asa general approach and then applied to particular cases. The solution pathtaken may be interpreted later as being the result of a decision or choice,and be called a ‘strategy’, but the labels are misleading. The ‘strategy’ (inthe holistic sense of the entire solution path) is not decided, it emerges.That is why the classifications of strategies do not map easily onto thedifferences that there are in practice. They are a post-hoc construct appliedto a different kind of cognitive product.
In this interpretation, the mental ‘strategies’ that can actually occur arenot solution strategies, but analytic strategies. They are ways of think-ing about mental calculations that do not describe the whole sequenceto the solution, but concern just some of the steps, for example ways ofbeginning, ways of thinking about the numbers, and ways of relating thenumbers to other knowledge. Examples of this are: looking for helpfulways of splitting one or both of the numbers (and not always into ‘tens andunits’); looking for doubles or near doubles or other common factors; andlooking for proximities (what ‘easy’ numbers the numbers in the problemare near to). These are strategies because they can be deployed deliberatelyas part of calculating, but they are not alternatives to choose from, as anycan be used on any calculation.
The ‘mental calculation strategies’ listed in the UK government’s Na-tional Numeracy Strategy (DfEE, 1999), include such elements. For addi-tion and subtraction, for example, there is reference to:
• Finding a difference by counting up through the next multiple of 10,100 or 1000
• Counting on or back in repeated steps of 1, 10, 100, 1000
• Partitioning into hundreds, tens and ones
• Identifying near doubles
FLEXIBLE MENTAL CALCULATION 43
• Adding or subtracting the nearest multiple of 10, 100 or 1000 andadjusting
• Using the relationship between addition and subtraction• Using known number facts and place value
For multiplication and division
• Using related facts and doubling or halving• Using factors• Using closely related facts already known• Partitioning and using the distributive law• Using the relationship between multiplication and division• Using known number facts and place value
Some of these approaches can be seen to be complete methods, such asadding or subtracting the nearest multiple of 10, 100 or 1000 and adjust-ing, but others are just elements of calculations, which do not describeall that occurs in calculating but only part of it. For example ‘partitioninginto hundreds tens and ones’ is done, but doing it is not everything thatis needed. Similarly, identifying near doubles may be used in calculating,but it does not characterise how the calculation is completed – the childidentifies a near double, and then has to do something else (and differentthings, depending on the problem). The strategy of changing numbers tomake the calculation easier can also be seen in this way.
As a result, many of the ‘mental calculation strategies’ that the frame-work for the National Numeracy Strategy (DfEE, 1999, 2001) suggests betaught in schools cannot be taught as procedures for mental calculation.Lessons on such ‘methods’ will actually be lessons on ways of analysingand thinking about numbers, and in particular the numbers in the specificproblems that are considered. This will contribute to children becomingflexible in mental calculation even when not intended to. However, it isworth considering how to maximise the contribution of lessons on mentalcalculation to children taking efficient solution paths.
7. TEACHING TOWARDS FLEXIBLE MENTAL CALCULATION
There are many suggestions in the literature about teaching mental calcu-lation, but only a few are focused on flexibility in particular. Thompson(1999) lists four ‘attributes’ that in his view assist in the development offlexible mental calculation:
1. Good knowledge of number facts (appropriate to the age)
44 JOHN THRELFALL
2. Clear understandings of what can be done legitimately with numbers,such as when the order can be changed and when it cannot, how num-bers can be split and dealt with as parts, reverse operations, the beha-viour of zero, and so on
3. Well developed skills (appropriate to the age) such as counting skills,and automated mental calculations at the level below
4. Positive attitudes, such as the confidence to ‘have a go’ and not beingput off by immediate lack of success.
Anghileri (1999, p. 186) also refers to the value of children being able to“see connections that will help them to simplify calculations”, and sug-gests that the role of the teacher is to help pupils to progress by makingmore use of number facts and rules and connections. Additionally Buys(2001) reports that verbalising solution steps is an integral part of a suc-cessful approach used in the Netherlands, and Beishuizen (2001, p. 128)too argues for “verbalising and discussing alternative mental calculations”and “recording the procedural steps of number operation” so that childrenare able to follow the knowledge transformations.
The ‘in the moment’ nature of flexible calculation suggests that theserecommendations for teaching are best pursued within the context of chil-dren calculating and describing to one another how they calculated, withthe teacher pointing out facts and connections, supporting their descrip-tions with explicit language and revealing forms of recording.
Teaching towards flexible mental calculation must include extensivedevelopment of factual knowledge about numbers, so that children willcome to notice a range of different things about the numbers when facedwith a calculation, and are then better placed to develop an ‘easy’ solutionto it. For example, it is helpful to know that the complement of 66 to 100is 34, so that one can add 46 to 66 by taking 34 off the 46, and addingthe remaining 12 to 100; that 100 is five 20s, so multiplying 24 by fiveis just 100 more than four times five; that 67 is not just 60 and 7 but 30and 37, so 67 minus 38 must be one less than 30; that 8 is 2 × 2 × 2 soto multiply by 8 you can just double and double and double; and so on.Some of this knowledge can be developed away from actual calculationdemands: in ‘interactive’ sessions with adults, through work with num-ber patterns; in games; and even in repetitive exercises. However, for thedevelopment of the flexible thinking with numbers necessary for efficientmental calculation it must also be addressed extensively when workingwith actual mental calculations, the intended context of application for theknowledge. It is only in that context that the chains of calculations that arepart of inferential mental calculation can be pointed out and practised.
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Real calculating will provide compelling examples of breaking num-bers into parts, factorising, finding proximities and near doubles, and willalso show what can be done with them, how the parts of numbers can betreated, how the proximities are compensated for, and so on – and thiscan be without any misleading identification of a ‘strategy’ to raise falseexpectations about what might occur in future calculations. So despiterejecting the explicit teaching of whole solution paths as ‘strategies’, itseems important to the development of flexibility to keep the teachingabout mental calculation strongly attached to real attempts to calculateproblems, examining solutions to emphasise the possibilities for numbersthat have been exemplified by what was done.
8. SUMMARY
This article has proposed that if the aim of teaching mental calculation isflexibility, what children and adults do to calculate efficiently should not bedistilled into general descriptions of methods or ‘strategies’ and promotedas holistic approaches to calculation, offered as models to emulate, ortaught as procedures to learn. Rather it suggests that solutions to problemswould be better approached as specific examples of how particular num-bers can be dealt with, how numbers can be taken apart and put together,rounded and adjusted, and so on.
It has offered a model of flexible mental calculation that emphasisespersonal knowledge as a determinant of how a solution path emerges inthe context of particular calculations, by way of what is noticed by theindividual about the numbers in the calculation.
The main focus of the related teaching approach is an analytical engage-ment with children’s ways of dealing with number challenges, graduallyexpanding the range of numbers being dealt with. This approach forms thecore of a teaching series by Frobisher and Threlfall (1998).
The overall perspective is that flexibility cannot be taught as a ‘pro-cess skill’. Flexibility will arise consequentially through the emphasis onconsidering possibilities for numbers rather than by focusing on holistic‘strategies’.
REFERENCES
Anghileri, J.: 1999, ‘Issues in teaching multiplication and division’, in I. Thompson (ed.),Issues in Teaching Numeracy in Primary Schools, Open University Press, Buckingham,pp. 184–194.
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Anghileri, J.: 2001, ‘Intuitive approaches, mental strategies and standard algorithms’, inAnghileri, J. (ed.), Principles and Practices in Arithmetic Teaching, Open UniversityPress, Buckingham, pp. 79–94.
Askew, M.: 1999, ‘It ain’t (just) what you do: effective teachers of numeracy’, in I.Thompson (ed.), Issues in Teaching Numeracy in Primary Schools, Open UniversityPress, Buckingham, pp. 91–102.
Beishuizen, M.: 1999, ‘The empty number line as a new model’, in I. Thompson (ed.),Issues in Teaching Numeracy in Primary Schools, Open University Press, Buckingham,pp. 157–168.
Beishuizen, M.: 2001, ‘Different approaches to mastering mental calculation strategies’,in J. Anghileri (ed.), Principles and Practices in Arithmetic Teaching, Open UniversityPress, Buckingham, pp. 119–130.
Beishuizen, M. and Anghileri, J.: 1998, ‘Which mental strategies in the early number cur-riculum? A comparison of British ideas and Dutch views,’ British Educational ResearchJournal 24(5), 519–538.
Bisanz, J. and LeFevre, J-A.: 1990, ‘Strategic and nonstrategic processing in the devel-opment of mental processing’, in D.F. Bjorklund (ed.), Children’s Strategies, LawrenceErlbaum, Hillsdale, New Jersey, pp. 213–244.
Blote, A.W., Klein, A.S. and Beishuizen, M.: 2000, ‘Mental computation and conceptualunderstanding’, Learning and Instruction 10, 221–247.
Buys, K.: 2001, ‘Progressive mathematization: sketch of a learning strand’, in J. Anghileri(ed.), Principles and Practices in Arithmetic Teaching, Open University Press, Bucking-ham, pp. 107–118.
Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E. and Empson, S.B.: 1997, ‘Alongitudinal study of invention and understanding in children’s mulitdigit addition andsubtraction’, Journal for Research in Mathematics Education 29(1), 3–20.
DfEE: 1999, The National Numeracy Strategy Framework for Teaching Mathematics fromReception to Year 6, DfEE Publications, Sudbury.
DfEE: 2001, Key Stage 3 National Strategy Framework for Teaching Mathematics: Years7, 8 and 9, DfEE Publications, Sudbury.
Frobisher, L. and Threlfall, J.: 1998, Teaching Mental Maths Strategies, Heinemann,Oxford.
Fuson, K.C., Wearne, D., Hiebert, J.C., Murray, H.G., Human, P.G., Olivier, A.I., Car-penter, T.P. and Fennema, E.: 1997, ‘Children’s conceptual structures for multidigitnumbers and methods of multidigit addition and subtraction’, Journal for Research inMathematics Education 28(2), 130–162.
Klein, A.S. and Beishuizen, M.: 1998, ‘The empty number line in Dutch second grades:realistic versus gradual program design’, Journal for Research in Mathematics Educa-tion 29(4), 443–464
QCA: 1999, Teaching Mental Calculation Strategies, QCA Publications, Sudbury.SCAA: 1997, The Teaching and Assessment of Number at Key Stages 1–3. Discussion
paper no. 10, March 1997, School Curriculum and Assessment Authority, London.Siegler, R.S.: 1996, Emerging Minds, Oxford University Press, New York.Thompson, I.: 1999a, ‘Getting your head around mental calculation’, in I. Thompson (ed.),
Issues in Teaching Numeracy in Primary Schools, Open University Press, Buckingham,pp. 145–156.
Thompson, I.: 1999b, ‘Mental calculation strategies for addition and subtraction Part 1’,Mathematics in School November, 2–4.
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Thompson, I.: 2000, ‘Mental calculation strategies for addition and subtraction Part 2’,Mathematics in School January, 24–26.
Thompson, I.: 2001, ‘Issues for classroom practices in England’, in J. Anghileri (ed.),Principles and Practices in Arithmetic Teaching, Open University Press, Buckingham,pp. 68–78.
Van den Heuvel-Panhuizen, M.: 2001, ‘Realistic mathematics education in the Nether-lands’, in J. Anghileri (ed.), Principles and Practices in Arithmetic Teaching, OpenUniversity Press, Buckingham, pp. 49–64.
Yackel, E.: 2001, ‘Perspectives on arithmetic from classroom-based research in theUnited States of America’, in J. Anghileri (ed.), Principles and Practices in ArithmeticTeaching, Open University Press, Buckingham, pp. 15–32.
School of Education,University of Leeds,Woodhouse Lane,Leeds LS29JT,England,E-mail: [email protected]
