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From Informal Strategies to Structured Procedures: mind the gap!

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This paper explores written calculation methods for division used by pupils in England (n = 276) and the Netherlands (n = 259) at two points in the same school year. Informal strategies are analysed and progression identified towards more structured procedures that result from different teaching approaches. Comparison of the methods used by year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach, which is based on careful progression from informal strategies to more structured and efficient procedures. This success is particularly not able for the girls in the sample. For the English pupils, whose written solutions largely involved the traditional algorithm, the discontinuity between the formal computation procedure and informal solution strategies presents difficulties.
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JULIA ANGHILERI1, MEINDERT BEISHUIZEN2and KEES VAN PUTTEN3
FROM INFORMAL STRATEGIES TO STRUCTURED
PROCEDURES: MIND THE GAP!
ABSTRACT. This paper explores written calculation methods for division used by pupils
in England (n = 276) and the Netherlands (n = 259) at two points in the same school year.
Informal strategies are analysed and progression identified towards more structured pro-
cedures that result from different teaching approaches. Comparison of the methods used by
year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach,
which is based on careful progression from informal strategies to more structured and
efficient procedures. This success is particularly notable for the girls in the sample. For
the English pupils, whose written solutions largely involved the traditional algorithm, the
discontinuity between the formal computation procedure and informal solution strategies
presents difficulties.
KEY WORDS: algorithms, division, arithmetic, strategies, written computation
BACKGROUND
In recent years there has been widespread publicity for results of inter-
national testing of arithmetic in schools with countries like England per-
forming less well than other countries in Europe and some Pacific Rim
countries. Contributing to these variations in performance will be a di-
versity of factors including different attitudes towards education, different
social pressures and different teaching approaches, as well as the con-
tent, timing and emphasis given to arithmetic teaching in the school cur-
riculum (Macnab, 2000). Although comparisons are complex, children’s
written solutions for selected problems can shed light on some reasons for
differences in attainment and this study identifies critical differences in
calculating approaches in England and the Netherlands.
DIFFERENT TEACHING APPROACHES
As close neighbours in Europe, England and the Netherlands share many
cultural characteristics but approaches to mathematics teaching have been
subject to different pressures over the last two decades (Brown, 2001: van
Educational Studies in Mathematics 49: 149–170, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
150 JULIA ANGHILERI ET AL.
den Heuvel-Panhuizen, 2001) and this has resulted in contrasting teach-
ing approaches to written calculations (Beishuizen and Anghileri, 1998).
Different national requirements for the school curriculum put pressure on
teachers to introduce specific written methods. In England, “understanding
of place value is central to pupils’ learning of number.... Progression in
understanding about place value is required as a sound basis for efficient
and correct mental and written calculation” (SCAA, 1997). In Dutch RME
approaches, on the other hand, calculating “is not based on the teaching
of the place value concept in the first place but develops more gradually
through the extension of counting strategies” (Beishuizen and Anghileri,
1998; Thompson, 1997).
In England, the National Numeracy Strategy places emphasis on men-
tal calculations in the early years and proposes that working with larger
numbers will necessitate the introduction of “informal pencil and paper
jottings” that become “part of a mental strategy” (DfEE, 1998, p. 51). By
the age of 11 years, children are, however, required to know a standard
written method for each operation as “standard written methods offer reli-
able and efficient procedures” (DfEE, 1998). The standard written methods
illustrated in the Framework for Teaching Mathematics (DfEE, 1999a)
are little different from the traditional algorithms that have been taught
to successive generations. It has been acknowledged that “many children
do not reach the stage of recording calculations the traditional way (by
the age of 11 years)” (Cockcroft, 1982, p. 77) and calls have been made
for pupils to be encouraged to develop alternative methods (Thompson,
1997; Anghileri, 2000). However, the only documentation to omit explicit
reference to the traditional algorithms is the new National Curriculum for
England, which refers to children having “efficient written methods” for
calculating (DfEE, 1999b).
In the Netherlands the Realistic Mathematics Education (RME) move-
ment (Treffers and Beishuizen, 1999; van den Heuvel, 2001) has intro-
duced some radical changes in the teaching of calculating methods with
early focus on mental methods and later a development of different levels
in written calculating. Research has led to the proposal of ‘trajectories’
whereby learning evolves as a process of gradual changes as “students
pass various levels of understanding: from the ability to invent informal
context-related solutions to the creation of various levels of short cuts and
schematisation” (van den Heuvel, 2001). A fundamental aspect of learning
written calculations is “guided development from informal to higher-level
formal strategies” which involves “reflection on strategy choice” in whole
class discussion (Beishuizen and Anghileri, 1998).
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 151
The RME approach asks children “to solve many real-world problems
guided by interactive teaching instead of direct instruction in standard al-
gorithms” (Beishuizen, 2001, p. 119). Central are contextual problems that
“allow for a wide variety of solution procedures, preferably those which
considered together already indicate a possible learning route through a
process of progressive mathematization” (Gravemeijer, 2001). The Dutch
approach places emphasis on the development from naïve skills such as
counting and doubling, and involves holistic approaches to numbers within
a calculation in contrast with the place value approach developed in the
English curriculum (Beishuizen and Anghileri, 1998; van Putten, Snijders
and Beishuizen, in preparation).
THE ARITHMETIC OPERATION OF DIVISION
By focussing on pupils’ strategies for division in late primary school (10
year olds), it is possible to highlight progression from mental methods
and informal strategies to the more structured approaches that are adopted
when written calculating procedures are introduced.
Two models for division, normally referred to as quotitive division (how
many sevens in 28?) and partitive division (28 shared between 7) have
formed the basis for analysing the division operation for whole numbers
(Greer, 1992). Related to these models are two distinct procedures for writ-
ten calculations: repeated subtraction of the divisor (becoming more effi-
cient by judicious choice of ‘chunks’ that are multiples of the divisor) and
sharing based on a place value partitioning of the number to be divided
(used efficiently in the traditional algorithm). There are many informal
strategies that will be built upon and Neuman (1999) includes counting,
repeated addition, chunks (performed in different ways), reversed multi-
plication, dealing, estimate-adjust, repeated halving, repeated estimation,
many of which will be incorporated into structured procedures for division
calculations.
Whether the approach is informal or reflects a taught procedure, struc-
turing the recording becomes beneficial as more complex problems are in-
troduced. In considering pupils’ use of written recording, Ruthven (1998)
identifies two distinct purposes: “to augment working memory by record-
ing key items of information” and “to cue sequences of actions through
schematising such information within a standard spatial configuration”.
The former may be identified with informal solution strategies that are
often idiosyncratic and give little consideration to efficiency or ease of
communicating to others. The latter suggests a taught procedure that will
152 JULIA ANGHILERI ET AL.
“direct and organise” (Anghileri, 1998) children’s approaches and has as
priorities efficiency and clarity of communication.
Formal written procedures for calculating can be difficult to recon-
cile with intuitive understanding (Fischbein et al., 1985; Anghileri and
Beishuizen, 1998) and can lead to mechanical approaches, which are prone
to errors (Brown and VanLehn, 1980). Ruthven and Chaplin (1998) refer
to “the improvisation of malgorithms” to describe pupils’ inappropriate
adaptations of procedures for the algorithm.
There is also evidence of “conflict” between computation procedures
and context structure (Anghileri, 2001a) and it is suggested that there is one
primitive model for division in children’s thinking, the partitive, and that
the quotitive model is acquired with instruction (Neuman, 1999). Where
problems are set in a context this may influence the solution strategy but re-
search suggests that the quotitive model appears to influence more strongly
written approaches with calculations such as 42 ÷6 interpreted as “How
many sixes in 42?” (Anghileri, 1995; Neuman, 1999).
COMPARING SOLUTION METHODS
This study considers pupils’ written methods for solving ten division prob-
lems, using five word problems that vary in their semantic structure to-
gether with five ‘bare’ problems expressed only in symbols. Comparisons
are made between the strategies used by English and Dutch pupils and their
success for different problem types. By identifying the pupils’ solution
strategies at two points in the school year (January and June) changes in
approach are identified and related to instructional approaches in the two
countries. Dutch pupils are introduced to written methods for division of
large numbers in the second term of Group 6 (Year 5: 10 – 11 year olds) and
this would be common in all schools where mixed ability classes are taught
mathematics by the class teacher. There was not such consistent practice in
the English schools where, although many pupils work on division prob-
lems in the same year group, at the time of this study there was no common
curriculum and experiences varied with teachers and textbooks and across
different groups, which were often streamed according to age and ability.
SAMPLE
Nine and ten year old pupils (n = 553) in twenty different schools were
involved in the study. Ten English and ten Dutch schools with average
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 153
class sizes were selected in and around small university cities in England
and in the Netherlands.
Although comparison is complex, the nature of the populations in the
two localities appeared to share many common characteristics such as sta-
bility of population and general nature of employment in the area. Further
criteria for selection of schools were high scores on standard national as-
sessments (in the case of English schools) or use of specific textbooks (in
the Dutch schools) related to their involvement in implementing a Real-
istic Mathematics Education (RME) curriculum. The English schools all
had their most recently published Standard Assessment Test (SAT) scores
in mathematics (average 72.5% at level 4 or above) well above the local
(LEA) average of 54.3% and the national average of 53.2%. In the Dutch
schools, teachers were using approaches to mathematics teaching centred
on the use of RME textbooks. All schools were selected so that the pupils
were likely to have confidence to tackle novel problems and the ability to
show some working to reveal their strategies.
Average ages of the children (n = 553) in January were similar for the
two cohorts (English: mean = 9.79, s.d. = 0.28; Dutch: mean = 9.90, s.d. =
0.44). The distribution was, however, different due to national policies of
the two countries. In England pupils’ ages determine the class/grade they
will join and it is rare to find any variation (Bierhoff, 1996; Prais, 1997).
In the Netherlands the age range in most classes will be wider, reflecting
a national policy for accelerating able pupils and holding back, for one or
sometimes two years, those who do not reach the required standard.
Another difference that is not evident from the statistical data is the
policy for pupils with Special Educational Needs. In England there is a
policy to integrate such pupils into mainstream classes whenever possible
while in the Netherlands many such pupils will attend special schools. In
the results of this study such influences need to be taken into account when
interpreting the differences in performance of the two cohorts.
All pupils completed a test of mental arithmetic but a reduced cohort
(n = 534) completed division tests in January and also in June. Only pupils
who were present for both division tests were included in this analysis
[English (n = 275) and Dutch (n = 259)]. This reduced cohort showed no
significant difference from the larger sample in age distribution and was
evenly balanced for gender with almost exactly 50% girls/boys in each
population.
154 JULIA ANGHILERI ET AL.
METHODOLOGY
Pupils were tested twice, in January and in June of the same school year, so
that changes would be evident in the calculating methods used. In the first
round of testing each of the twenty classes completed a short, timed ‘speed
test’ of mental calculations in addition, subtraction and multiplication,
based on Dutch national tests. This was followed by a written test with no
time limit so that all the pupils could complete it. The tests were designed
collaboratively by the English and Dutch researchers and administered by
the researchers. Pilot tests were administered in schools, which were not
involved in the final testing, and modifications were made where necessary.
Problems were presented in workbooks and pupils were invited to com-
plete the problems in any order and try another problem if they were stuck.
The teacher and the researcher assisted with reading the problems where
necessary but gave no further guidance. When the pupils were tested for
the second time, in June of the same year (5 months later), only the written
test was used.
THE SPEED TESTS
Pupils’ division strategies may be related to their ability in mental arith-
metic. In order to assess the pupils’ performance they were given a short
speed test, which involved 5 columns each with 40 mental calculations
of progressive difficulty. Column 1 involved addition from 1+1 to 54+27,
Columns 2, 3, 4 and 5 involved subtraction, multiplication, harder mul-
tiplication and harder subtraction, respectively, each involving a progres-
sion from easy to more difficult questions. After attempting some practice
questions, pupils were timed for one minute each for columns 1–3 and 2
minutes each for columns 4 and 5. The number of questions completed and
the number of errors were scored. The overall numbers of correct responses
were used to select and compare the written tests of better and weaker
pupils, which are reported elsewhere (Anghileri, 2001b).
THE DIVISION TESTS
Two practice items were presented one at a time to the class and, after a
minute of thinking, solution strategies were invited from the pupils. The
researcher wrote pupils’ suggestions clearly on the board so that at least
three different strategies, including informal/intuitive approaches, were il-
lustrated and these illustrations were left for the duration of the test. Pupils
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 155
TABLE I
Ten problems used in the first test
number type bare problem context problem (context)
problem type
2-digit divided
by 1-digit
no remainder
6. 96÷6 1. 98 flowers are bundled
in bunches of 7. How
many bunches can be
made?
grouping
2-digit divided
by 2-digit
no remainder
7. 84÷14 2. 64 pencils have to be
packed in boxes of 16.
How many boxes will be
needed?
grouping
3-digit divided
by 2-digit
remainder
8. 538÷15 3. 432 children have to be
transported by 15 seater
buses. How many buses
will be needed?
grouping
3-digit divided
by 10
remainder
9. 802÷10 4. 604 blocks are laid
down in rows of 10. How
many rows will there be?
grouping by
10
4-digit divided
by 1-digit
remainder
10. 1542÷5 5. 1256 apples are di-
vided among 6 shopkeep-
ers. How many apples will
each shopkeeper get? How
many apples will be left?
sharing
then worked individually on the problems, each with space to show their
working and an answer, and were encouraged to record ‘the way they think
about the problems’.
Between the tests in January and June, all pupils will have had further
experiences in arithmetic learning including some work on multiplication
and division.
THE DIVISION TEST ITEMS
The written test consisted of ten division problems, five illustrated word
(context) problems followed by five symbolic (bare) problem with similar
numbers. Problem types included ‘sharing’ and ‘grouping’ models, and
involved single-digit and two-digit divisors, with and without remainders
(Table I). There were more grouping (quotition) problems as the pupils’
156 JULIA ANGHILERI ET AL.
Figure 1a. Scores for the ten questions in test 1
Figure 1b. Scores for the ten questions in test 2
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 157
TABLE II
Results for the speed tests
Speed test Column 1 Column 2 Column 3 Column 4 Column 5
addition subtraction multiplication multiplication subtraction
mean s.d. mean s.d. mean s.d. mean s.d. mean s.d.
Dutch 23.3 4.7 20.8 4.5 19.9 4.7 12.8 6.6 11.0 5.1
(n = 262)
English 18.7 5.4 13.6 5.5 13.9 5.7 7.5 6.0 4.2 4.2
(n = 293)
strategies were sought and sharing (partition) is already known to be a
well established intuitive strategy for division (Fischbein et al., 1985). The
numbers were selected to encourage mental strategies and to invite the use
of known number facts so that it would be possible to approach all the
problems using intuitive methods. Some numbers were selected to include
the potential for the common error of missing a zero in the solution.
In June, the problems involving 96÷6, 84÷14, 538÷15, 802÷10 and
1542÷5 which were ‘bare’ in test 1 were given the contexts used in the
first test. Again, the context problems were the first 5. The problems 98÷7,
64÷16, 432÷15, 604÷10 and 1256÷6 were now presented in ‘bare’ format
as problems 6–10.
RESULTS
Performance in the mental arithmetic speed tests
Prerequisite knowledge for learning division in school includes mental
computation in addition, subtraction and multiplication. The scores were
higher in every type of problem for the Dutch pupils who not only com-
pleted more questions but made fewer errors in their attempts. The mean
numbers of questions completed correctly for the different calculations in
the timed test are recorded in Table II.
The highest scores were similar for both cohorts but the standard devi-
ation shows greater variation among the English pupils. The better success
of the Dutch pupils is not surprising as the emphasis given to mental arith-
metic has for some time been greater in the Netherlands and speed tests of
this type are familiar in schools. More recently there has been a growing
emphasis on mental arithmetic in England but the pupils in this year 5
cohort will have experienced more focus on written calculation.
158 JULIA ANGHILERI ET AL.
Figure 2. A sharing strategy for division.
Performance in the written tests
In the first test in January the number of items with correct solutions was
similar for the Dutch (mean 4.7: s.d. 2.9) and the English (mean 3.8: s.d.
2.7) but average Dutch scores were higher on all but one of the ten items
(Figure 1a).
In the second test differences between the performances of the Dutch
and English pupils were more marked with the English (mean 4.4: s.d. 2.6)
and the Dutch (mean 6.8: s.d. 2.6). On all ten items the average score for
the Dutch pupils was greater than that of the English pupils (Figure 1b).
The biggest difference appears for items 3 (538 children have to be
transported by 15 seater buses. How many buses will be needed?) and
particularly for item 8 (432÷15). Both items involve a two-digit divisor,
which is not normally encountered in Year 5 in English schools where
the emphasis on formal procedures means that the long division algorithm
would need to be introduced. Dutch pupils in Group 6 are taught a written
method that is equally appropriate for single-digit and multi-digit divisors.
These differences will be discussed further in a later section.
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 159
Strategies for solving division problems
The pupils’ written methods ranged from inefficient strategies such as
tallying or repeated addition to use of a standardised written procedure.
There were marked differences in the ranges of strategies in the different
countries and in the ways the pupils organised their calculations on paper
and this led to complex initial classifications in order to represent important
variations. Most of the strategies identified by Neuman (1999) were evid-
ent to some extent but the larger numbers involved in the study meant that
such naïve strategies were sometimes adapted to include efficiency gains.
Neuman’s category of ‘dealing’ one at a time, for example, was similar to
dealing/sharing using multiples of the divisor (Figure 2).
In the English sample there were examples of complex procedures where
the recording was difficult to follow while many Dutch pupils showed
clearer organisation in their recording methods that could be associated
with a taught procedure based on repeated subtraction. Progression was
evident in the Dutch strategies from inefficient strategies, through struc-
tured recording, to more formalised and efficient procedures and the Dutch
approaches illustrated how similar procedures were used at different levels
of efficiency by individual pupils (Figure 3).
In the English methods, some informal strategies showed sound ap-
proaches but were disorganised in their recording. There was no clear
progression from this idiosyncratic structuring to the standardised proced-
ure.
RESULTS
Features associated with different strategies
Naive strategies such as tallying, repeatedly adding or subtracting the
divisor, and sharing generally showed pupils had understanding of the
nature of division as these approaches were often correct and could lead
to a solution. Such strategies were sometimes successful for the smaller
numbers (98÷7 and 96÷6) but where larger numbers were involved (e.g.
432÷15) few pupils worked through to an answer.
By low level chunking pupils showed some attempt to gain efficiency
with repeated addition particularly in the problems involving division by
15 where subtotals of 30 or 60 were used. High level chunking showed
good understanding of the relationships between numbers. Chunking 96
into 60 and 36, for example, related to division by 6 while chunking 98 as
70 and 28 related to division by 7.
160 JULIA ANGHILERI ET AL.
Figure 3. Progression in Dutch solution strategies.
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 161
Figure 4. Errors with the algorithm.
Some strategies showed a place value approach based strictly on (thou-
sands, hundreds) tens and units. This sometimes led to complex calcula-
tions, for example where pupils attempted to solve 1000÷6, 200÷6, 50÷6
and 6÷6 adding the results.
Amental strategy, where an answer was given with no working, was
most widely used when dividing by 10 where errors mainly involved an
incorrect number of zeros or a wrong remainder, for example, 802÷10 = 8
rem 2 or 802÷10 = 82.
The traditional algorithm (for short division) was widely used by the
English pupils and provided a structure to the written recording but led to
a variety of difficulties. Errors were evident where pupils missed a zero
in the solution or worked with separate digits (Figure 4). In tackling the
problem 64÷16, the standard algorithm was used by some pupils to divide
64 first by 10 and then by 6, adding the answers. Sometimes both numbers
were separated, for example to give 6÷1and4÷6oreven6÷1and6÷4
with division reversed for the units (leading to the answer 61r2). [It appears
in this case that commutativity was not understood and 6÷4 was selected
as an easier option than 4÷6].
Many errors with the algorithm involved wrong procedures for using
remainders, for example, in 1256:6 a ‘remainder’ of 6 was carried for-
ward. Such procedural errors were evident in the second test where formal
procedures were used more widely, and perhaps more mechanically. Some
answers were quite bizarre, for example, 1256:6 led to an answer 0101011
(Figure 4) by a pupil who had correctly solved the problem in the first
test. This example shows the consistent use of a wrong procedure with no
162 JULIA ANGHILERI ET AL.
account given to ‘number sense’, which would have indicated that this was
an inappropriate solution.
A notable characteristic of much of the work of the Dutch children was
the way solutions were formally structured whether they involved small
chunks and long calculations, or introduced efficiency gains by using larger
chunks (see Figure 3). In many cases Dutch pupils started by listing mul-
tiples (2×,4×,8×,10×) they could use in the calculation. Where written
recording was poorly structured, particularly characteristic of some of the
English working, pupils lost track and this appeared to lead to confusion.
Some working showed correct calculations which pupils were unable to
use appropriately to find an answer.
Classification of strategies
Classification of the different strategies was somewhat different for the
two cohorts of pupils as progression within Dutch strategies meant that
methods that were essentially the same involved different levels of effi-
ciency. Initially fourteen different categories were identified which were
then grouped into 8 types:
1) – Using tally marks or some symbol for each unit;
– Repeated addition of the divisor;
– Repeated subtraction of the divisor from the dividend;
– Sharing with images of a distribution;
These four strategies involved long calculations with no evident attempt
to gain efficiency despite the large numbers involved and were grouped
together as 1(S).
2) – Operating with the digits independently (e.g. 84÷14 using 8÷1and
4÷4);
– Partitioning the dividend into (thousands) hundreds, tens and units (e.g.
1256÷6 calculated as 1000÷6, 200÷6, 50÷6and6÷6);
Both strategies involved ways to ‘break down’ the numbers using ideas of
place value and were classed together as 2(P).
3) – Low level ‘chunking’ e.g. adding small subtotals (30 instead of 15)
within long procedures sometimes using doubling, or repeated doubling of
the divisor;
Working with small multiples of the divisor gained some efficiency but
generally led to long calculations. Some such chunking involved doubling
and halving within the calculation. Such strategies were classed together
as 3(L).
4) High level ‘chunking’ strategies using efficient subtotals (for example,
150 for division by 15) and shortened procedures were classed as 4(H).
For some calculations where halving and doubling were very efficient (for
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 163
TABLE III
Percentage of pupils using each strategy and success rates (in brackets)
English Dutch
test 1 test 2 test 1 test 2
attempt correct attempt correct attempt correct attempt correct
1(S) 17% 7% 11% 6% 10% 4% 1% 1%
2(P) 5% 0% 3% 0% 7% 1% 6% 2%
3(L) 6% 2% 8% 2% 16% 7% 6% 5%
4(H) 8% 5% 7% 5% 41% 28% 69% 51%
5(AL) 38% 18% 49% 25% 4% 1% 3% 1%
6(ME) 9% 5% 11% 6% 9% 6% 11% 7%
7(WR) 3%0% 2%0% 5%0% 1%0%
8(UN) 4% 1% 3% 0% 2% 0% 1% 0%
o 9%0% 8%0% 8%0% 2%0%
total 100% 38% 100% 44% 100% 47% 100% 68%
example, 64÷16) the strategy was classes as high level chunking.
5) The traditional algorithm involving formal layout was classed as 5(AL).
Although the traditional algorithm sometimes involved informal jottings to
support the calculations it was classed as a separate category because the
solutions were structured by this approach. The algorithm was taken to be
a strategy identifying a procedure that involved other strategies.
6) Mental calculation showing an answer but no working was classed as
6(ME).
7) A wrong operation (for example, 98 – 7 = 91) was classed as 7(WR).
8) An unclear strategy was classed as 8(UN).
No attempt (missing) was classed as o.
Comparison of success rates associated with different strategies
The relative success rates for each of the strategies were compared
(Table III).
There was improvement in facility for both cohorts and a general trend
towards use of more efficient strategies although these are not altogether
more effective. The most popular strategy for the Dutch pupils involved
identification and use of large chunks, 4(H), usually in a structured proced-
ure of repeated subtraction which was used for 69% of items in test 2 with
51% successful. This contrasts with the traditional algorithm 5(AL) used
in 49% of the items in test 2 by English pupils with success in only 25%
164 JULIA ANGHILERI ET AL.
TABLE IV
Success rates for the problems involving division by a single digit
96÷6 1256÷698÷7 1542÷5Average
English test 1 69 22 60 31 45.5
Dutch test 1 73 27 62 27 47.25
English test 2 74 (+5) 24 (+2) 81 (+21) 41 (+10) 55 (+9.5)
Dutch test 2 81 (+8) 56 (+29) 84 (+22) 63 (+36) 71 (+23.5)
The figure in brackets shows the % gains from test 1 to test 2.
of all attempts. In the written recordings of the Dutch pupils, progression
was evident with reduction in the use of low level strategies 1(S) from 10%
in test 1 to only 1% in test 2. A similar change is evident for the English
pupils but 22% persist with the low level strategies 1(S), 2(P) and 3(L) in
the second test with low (8%) success rate. Working mentally 6(ME) was
generally associated with problems involving division by ten and the table
shows similar frequency of use with better success rates among the Dutch.
Repeated subtraction may be viewed as an intuitive approach to division
but it was evident only in the Dutch children’s methods suggesting that it
is learned rather than used spontaneously as a strategy. In the second test
repeated subtraction did not persist as an informal strategy but appeared
as a structured procedure with the introduction of ‘chunks’ to improve
efficiency. The accessibility of this procedure as a direct progression from
more naïve methods could account for no Dutch pupils using tallying, shar-
ing or repeated subtraction in the second test. English pupils, in contrast,
used repeated addition in both the first and second tests and many (3%)
of their attempts were impossible to decipher, 8(UN). General confidence
appears to be better in the Dutch cohort in test 2 as only 2% of items were
not attempted compared with 8% of items for the English cohort.
Comparing the English and Dutch facilities for division by a single digit
Better results for the Dutch pupils may be explained by the fact that they
meet division by a 2-digit divisor in group 6 (Year 5) while most English
pupils will meet only 1-digit divisors. There were, however, differences in
those items involving only a single digit divisor. Improvements are similar
for the items, 96÷6 and 98÷7, but for the 4-digit numbers, 1256÷6and
1542÷5, the Dutch improvements were higher (Table IV).
Scores in the first test (January) were close for the English and Dutch
sample with averages of 45.5 and 47.25 correct solutions over the four
problems. Both cohorts of pupils were more successful in dividing a two-
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 165
TABLE V
Percentage use of most popular strategies for test 2
Strategy 96÷6 1256÷698÷7 1542÷5
English traditional algorithm 66 (51) 67 (21) 66 (52) 70 (34)
test 2
Dutch repeated subtraction 78 (69) 72 (50) 76 (69) 71 (52)
test 2 of large chunks
The figure in brackets is the percentage of correct attempts.
digit number than in dividing a four-digit number. In three of the four items
the score was higher for the Dutch children while the English children were
more successful with the problem 1542÷5. This could be due to English
pupils greater familiarity with 5 as a divisor because of its relevance in
place value teaching but the change in test 2, where the Dutch pupils did
better, shows any advantage does not appear to persist.
In the second test (June) improvements are similar for problems in-
volving the division of a two digit number, 96÷6 and 98÷7, with Dutch/
English improvements +8/+5 and +22/+21 respectively for the two ques-
tions. For the problems involving division of four-digit numbers, how-
ever, the Dutch improvements are much higher than those of the English
children with increases +29/+2 and +36/+10 respectively.
Looking at the most popular strategies used for these problems, English
pupils used the algorithm with low success rate for the 4-digit numbers.
The Dutch pupils used repeated subtraction with large chunks and although
the success rate is not as high for 4-digit numbers, differences are less
marked.
Errors by the English pupils included missing digits in the answer, but
also many confused attempts often leading to impossible (and sometimes
bizarre) answers.
IMPROVEMENTS OF BOYS AND GIRLS
When considering improvements from test 1 to test 2 there is a significant
difference in the performance of Dutch boys and girls but great similarity
between English boys and girls. The Dutch girls made bigger gains (mean
= 2.6) than the Dutch boys (mean = 1.5). An unpaired t-test for the Dutch
cohort shows this is significant with t = –3.14 and p = 0.0018. For the
English cohort there is some difference with mean gains of 0.64 (girls)
and 0.50 (boys) but this difference is not significant. When comparing the
166 JULIA ANGHILERI ET AL.
TABLE VI
Strategies used by boys and girls in the first and second tests
Strategy Dutch girls Dutch boys
test 1 test 2 test 1 test 2
attempt correct attempt correct attempt correct attempt correct
1(S) 11% 5% 2% 1% 8% 4% 1% 1%
2(P) 8% 1% 7% 2% 6% 1% 4% 2%
3(L) 16% 7% 6% 5% 16% 7% 6% 5%
5(H) 37% 24% 70% 53% 45% 31% 68% 50%
7(AL) 5%2% 3%1% 2%1% 2%1%
8(ME) 6% 3% 8% 5% 12% 8% 14% 9%
9(WR) 4%0% 1%0% 5%0% 2%0%
10(UN) 1% 0% 1% 0% 2% 0% 1% 0%
missing 12% 0% 2% 0% 5% 0% 2% 0%
total 42% 68% 52% 68%
English girls English boys
1(S) 17% 7% 12% 7% 15% 6% 9% 5%
2(P) 5% 0% 4% 0% 3% 0% 2% 0%
3(L) 6% 2% 9% 2% 6% 2% 7% 2%
5(H) 8% 5% 6% 5% 8% 6% 7% 4%
7(AL) 38% 18% 48% 25% 39% 16% 49% 24%
8(ME) 9% 5% 8% 5% 12% 6% 14% 6%
9(WR) 3%0% 2%0% 3%0% 2%0%
10(UN) 4% 1% 3% 0% 4% 1% 2% 0%
missing 9% 0% 7% 0% 11% 0% 9% 0%
total 38% 45% 37% 42%
Dutch pupils’ strategies and facilities, in test 1, the Dutch boys not only
used high level chunking 4(H) more often but had more success with all
the strategies they used and were successful in 52% of the items compared
with the girls success in 42% of the items. In the second test the girls were
still using more lower level strategies overall but showed greater use (70%
of all items) of high level chunking 4(H) and greater success with this
strategy (53% correct). The girls have ‘pulled up’ to the success level of
the boys with both successful in 68% of the items.
The Dutch boys showed no working in 14% of items in test 2 compared
with Dutch girls (8%). The English cohort show very similar results with
14% of items attempted by English boys showing no working compared
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 167
with English girls attempts (8%). About two thirds of all attempts were
correct except for the English boys who were correct in less than half of
these items.
Overall improvements for English and Dutch cohorts
In test 1 Dutch pupils solved 47% of the items compared with 38% solved
by the English pupils. In test 2 the results were 68% and 44% respectively.
When individuals’ scores were compared for test 1 and test 2, Dutch pu-
pils showed better improvements with 69% improving their score while
almost half of the English pupils (49%) showed no improvement or a
deterioration.
Despite the fact that the English pupils had more scope for improve-
ment than the Dutch, when actual improvements (number of items correct
in test 2 which were not correct in test 1) were compared with possible
improvements (total number of items which were not correct in test 1) the
Dutch were more than twice as successful.
CONCLUSIONS
Learning is most effective where written methods build upon pupils’ intu-
itive understanding in a progressive way. Informal solution methods may
be inefficient, but support in structuring such approaches in a written re-
cord appears to lead to better efficiency gains than replacing them with
a standard procedure. Application of taught methods can become mech-
anistic and unthinking where pupils are unclear about the links between a
taught procedure and the meanings they can identify. Application of taught
methods becomes the first imperative and appears to inhibit more thought-
ful approaches that take account of problem structure and the numbers
involved.
The Dutch approach to written division calculations, involving repeated
subtraction using increasingly large chunks, builds progressively on an in-
tuitive strategy and retains whole numbers at all stages. The success of the
Dutch pupils reflects their mastery of an increasingly efficient approach
that has the flexibility for individuals to use the knowledge of multiplica-
tion facts that they have. On the other hand, the traditional written format
extensively used by the English children, introduces a schematic approach
that focuses on separate digits with their true value implicit, rather than
explicit.
Many English pupils, at the end of year 5, continued to use low level
strategies that are inefficient and prone to errors but these informal strategies
168 JULIA ANGHILERI ET AL.
show a holistic approach to the numbers and an understanding of appro-
priate working. ‘Messy’ recording often involved a good strategy with
a written record that became too complex. Application of the structured
standard procedure, however, appeared to exclude return to the more intu-
itive approaches. A problem such as 64÷16 caused difficulty to the English
pupils because it does not respond readily to the traditional algorithm that
was used in preference to informal approaches. Instead of recognising
the number relationships involved, pupils used a procedure cued by the
operation. The results for the English pupils show discontinuity between
their informal strategies and the traditional algorithm, which was widely
used, but often in a procedural and unthinking way. It was evident that the
algorithm replaced more intuitive strategies rather than enhancing them.
It is clear from this study that the Dutch approach, which develops
and standardises the informal strategy of repeated subtraction, leads to a
procedure that pupils are confident to use and that they use effectively.
Because this procedure can be used at different levels of efficiency an ele-
ment of choice is retained so the pupils continue to have some ownership
of the thinking within the structured approach. This appears to achieve a
smooth transition from an intuitive strategy to a more formalised procedure
avoiding the mechanical application of taught rules.
It is possible that the Dutch procedure will not reach the concise ef-
ficiency of the traditional algorithm but the benefits include the fact that
whole numbers are used throughout and that the same procedure will work
with single-digit and multi-digit divisors.
The Dutch progression appears to suit girls particularly well and this
could indicate that they benefit from the structuring of a written record that
supports a developing procedure. The Dutch girls appear to be less able to
sustain low level strategies to a successful conclusion than the Dutch boys,
which would perhaps suggest that their tenacity in problem solving or their
confidence is less well developed. At this stage such suggestions are only
speculative and further research will be necessary to identify the reasons
for greater improvements among the Dutch girls.
At a time when mental calculation and pupils’ own informal strategies
are being encouraged there is a need to consider how efficient calculating
skills are to be achieved for larger numbers. When a standard procedure for
calculating is taught in school it appears to take precedence over informal
methods and implementing the procedure can be at the expense of making
sense of a calculation. If children are to retain confidence in the strategies
they understand, and see mathematical problem solving as a progression
towards procedures that are efficient, it is necessary that structured written
recording is introduced to complement and guide their informal working.
FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES 169
Where intuitive approaches are built upon to gain efficiency, “powerful
and correct theorems in action, and clear signs of metacognition” emerge
(Murray, Olivier, and Human, 1991). It is this metacognition that is key
to the development of mathematical thinking and to the development of
number sense.
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1Author for correspondence:
Faculty of Education,
Homerton Site,
University of Cambridge,
Hills Road,
Cambridge, CB2 2PH
Telephone (44) 1223 507280, Fax (44) 1223 507137
E-mail: jea28@cam.ac.uk
2University of Leiden,
3University of Leiden
... Most research on this topic has been undertaken with primary school students [3,5,6,[8][9][10][11][12][13] and conducted by Hickendorff and Fagginger-Auer and their colleagues in the Netherlands [3,5,6,[8][9][10]. In that country, the Realistic Mathematics Education (RME) reform has heavily influenced school teaching by promoting the learning of flexible strategies and introducing the traditional division algorithm in the last year of primary school [6]. ...
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... Χζηφζν, είλαη ν δπζθνιφηεξνο θαη ν πην πεξίπινθνο απφ φζνπο παξνπζηάζηεθαλ θαη απνηειεί κηα δηαδηθαζία πνπ δπζθνιεχεη αθφκα θαη ηνπο καζεηέο ηππηθήο αλάπηπμεο (Λεκνλίδεο & Παπιίδεο, 2003;Musser et al., 2010). Απηφ πξνθχπηεη απφ ην γεγνλφο φηη απαηηείηαη πνιχ θαιή γλψζε ησλ πξάμεσλ ηνπ πνιιαπιαζηαζκνχ θαη ηεο αθαίξεζεο, νη νπνίεο πνιιέο θνξέο πξέπεη λα εθηειεζηνχλ λνεξά θαη ελαιιαζζφκελα (Anghileri & Beishuizen, 1998;Anghileri, Beishuizen & Van Putten, 2002;Αγαιηψηεο, 2011α). Δηδηθά γηα ηελ αθαίξεζε, είλαη δχζθνιν λα γίλεη θαηαλνεηφο ν ξφινο ηεο ζηε δηαίξεζε, ελψ αθφκε, ην γεγνλφο φηη νξηζκέλα ςεθία αγλννχληαη θαη δελ ιακβάλνληαη ππφςε θαηά ηελ εθηέιεζε ηεο πξάμεο, πξνθαιεί ζχγρπζε ζηνπο καζεηέο ζρεηηθά κε ηε ζεζηαθή αμία (Leung, Wong & Pang, 2006). ...
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... Division in general is very demanding for students, even in higher grades (Fagginger Auer, Hickendorff, van Putten, Béguin, & Heiser, 2016;Hickendorff, Heiser, van Putten, & Verhelst, 2009;Neuman, 1999;Schulz, 2015). Often, traditional, digit-based algorithmsonce they have been introduced in the classroomreplace a more flexible use of (informal) whole-number-based solution strategies (Anghileri, Beishuizen, & van Putten, 2002;Clark & Kamii, 1996;Heirdsfield, Cooper, Mulligan, & Irons, 1999;Hickendorff, 2013;Kamii & Dominick, 1997;Selter, 2001). But digit-based algorithms do not prepare children for the later learning of fractions and algebra, which can be better achieved by whole-number-based solution strategies (Clarke, 2005;Confrey, Maloney, & Corley, 2014;Schifter, 1997;Vermeule, Olivier, & Human, 1996). ...
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... Strategy use and learning are influenced by the characteristics of the learning environment, especially by the kind of instruction implemented by the teacher (Ellis, 1997;Torbeyns et al., 2009;Verschaffel et al., 2009) and by the textbooks and tasks used in the classroom (Anghileri et al., 2002;Fagginger Auer et al., 2016;van Putten et al., 2005). ...
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The present study aimed to examine the role of pictures in solving verbal div ision problems and to investigate second grade students' translation ability and its relation to problem solving ability. For the first aim of the study, two tests were administered-Test A consisted of 4 verbal division problems (2 partitive and 2 quotitive) and Test B consisted of the same problems accompanied by informational pictures. At a second phase, Test C was administered, which consisted of 6 translation tasks between different representations-verbal, iconic, symbolic. The results showed that informational pictures had no significant effect on students' performance. It was also found that there is no semantic congruence between the different representations. The translation tasks from the verbal to the symbolic mode were the easiest for the students, while translation tasks from the iconic to the symbolic representation were the most difficult. The results also showed that translation ability is related to problem solving success.
... A number of influential reports published just after the turn of the century highlighted concerns in the teaching and learning of mathematics in the United Kingdom. In particular, the report Making Mathematics Count (Smith, 2004) recommended the increased use of applications of mathematics, and a number of research papers (Anghileri, Beishuizen, & Van Putten, 2002;Brown, Askew, Millett, & Rhodes, 2003;Hodgen, Küchemann, & Brown, 2009) reported that although there had been improvements in students' end of school assessments, longer term conceptual understanding and the ability to apply mathematics remained an issue. It was against this background that our interest in RME evolved. ...
... Por otro lado, hay estudios enfocados en analizar aspectos específicos del sentido numérico; por ejemplo, aquellos en los que se identifica la habilidad o flexibilidad para realizar cálculos mentales (Reys et al., 1995;Blöte, Klein y Beishuizen, 2000;Swan y Sparrow, 2001;Berch, 2005), diferencias entre métodos utilizados por estudiantes de distintos países al hacer cálculos mentales (Beshuizeny Anghileri, 1998), generación de estrategias para sumar y restar números enteros de varios dígitos , relación entre el entendimiento de las cantidades discretas y continuas (Leivobich et al., 2017), entendimiento de los números racionales y el razonamiento proporcional (Lamon, 2007), o análisis de métodos escritos para efectuar divisiones (Anghileri, Beishuizen y Van Putten, 2002). Otros trabajos han intentado caracterizar niveles de entendimiento respecto de componentes particulares del sentido numérico, tales como el valor posicional (Jones et al., 1996), la flexibilidad al realizar cálculos mentales (Threlfall, 2002), procedimientos para realizar sumas y restas de números de un dígito (Fuson, 1992), habilidad para estimar el resultado de sumas (Case y Sowder, 1990), el uso de representaciones equivalentes de los números (Pitta-Pantazi, Christou y Zachariades, 2007), o cálculo de múltiplos y potencias de 10 (Anghileri, 2006), entre otros. ...
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