The architecture of mental addition and subtraction
Ann M. Heirdsfield and Tom J. Cooper
Centre for Mathematics and Science Education, QUT, Brisbane, Australia
Research has shown that mental computation is a valid computational
method which contributes to mathematical thinking as a whole (e.g.,
Sowder, 1990). It is also a process for which young children have
exhibited a variety of proficient spontaneous strategies contrary to
instruction (Cooper, Heirdsfield, & Irons, 1996a). This paper reports
on a series of three studies on young children's understanding of
mental addition and subtraction and describes the mental architecture
of a proficient mental computer. Analysis of the first study showed
that children's strategy use is idiosyncratic, but influenced by
instructional emphases, experience and presentation forms; particularly
in relation to the strategies underlying pen-and-paper algorithm
procedures. Analysis of the second study identified a relationship
between proficiency in mental computation, number fact knowledge and
computational estimation. Initial analysis of the third study, which
involves detailed construction of mental models, is indicating a more
complex interaction.
The importance of developing number sense as an essential element of
mathematics education has been recognised in the literature (e.g.,
Australian Education Council & Curriculum Corporation, 1991; Reys &
Barger, 1994; Willis, 1992). Trafton (1992) suggested that people who
possess number sense tend to solve computational problems by using
knowledge about numbers, operations and their relationships. Mental
computation may be viewed as a subset of number sense, as people who
are good mental computers use "self developed strategies based on
conceptual knowledge" (Reys, Reys, Nohda, & Emori, 1995, p. 324), and
the ability to compute mentally is an indicator of the possession of
number sense (McIntosh, 1995; Sowder, 1992).
In the past, the focus of primary mathematics computation has been the
traditional pen and paper algorithms. However, more recent research
has suggested that mental computation should play a more prominent role
in number strands of mathematics curricula (e.g., Cobb & Merkel, 1989;
McIntosh, 1996; Reys & Barger, 1991; Sowder, 1990; Willis, 1990).
Reasons for its inclusion are: mental computation enables children to
learn how numbers work, make decisions about procedures, and create
strategies (e.g., Reys, 1985; Sowder, 1990); mental computation
promotes greater understanding of the structure of number and its
properties (Reys, 1984); and it has an utilitarian purpose (Clarke &
Kelly, 1989; Maier, 1977). Further, Kamii, Lewis, and Jones (1991)
recommended that children should be free to formulate their own mental
strategies, as understanding of algorithms is improved if children
construct strategies in line with their own natural ways of thinking.
McIntosh (1996) also agreed that teaching mental strategies the same as
formal pen and paper strategies have been taught in the past is not the
solution to the present lack of attention given to mental computation.
It is within a framework of number sense (flexibility with and
understanding of numbers and operations) that three studies of mental
computation are analysed and reported in this paper. The first study
charted Years 2 to 4 children's accuracy and strategy use for mental
addition and subtraction (Cooper, Heirdsfield, & Irons, 1996a, 1996b;
Heirdsfield & Cooper, 1996). The second study related Year 4
children's mental addition and subtraction proficiency with number fact
knowledge and computational estimation proficiency (Heirdsfield, 1996).
Finally, the third study aimed to generate a description of factors
associated with Year 4 children's proficiency in mental addition and
subtraction (Heirdsfield, in preparation).
Study 1
To explore children's mental addition and subtraction with respect to
accuracy and strategy use, 104 children of varying mathematical
ability (one third each of above average, average, and below average
ability) were selected from 6 schools (representing a variety of social
and cultural backgrounds). All their teachers followed the Queensland
mathematics syllabus (Department of Education, 1991), where the focus
of teaching number and operations is to develop basic number facts and
the traditional pen and paper algorithms.
The children were interviewed three times in year 2, twice in year 3,
and once at the beginning of year 4, using Piaget's revised clinical
interview technique (Ginsburg, Kossan, Schwartz, & Swanson, 1983). The
interview questions consisted of 2 and 3 digit addition and subtraction
word problems, relating to money, and algorithmic exercises, all
presented visually and read to the children. Further, there were three
types of problems (based on Carpenter & Moser, 1984): join addition,
separate subtraction, and missing addend subtraction. The children
were withdrawn individually from the classroom and interviewed in a
separate room, where the interviews lasted no more than 30 minutes and
were videotaped. Paper and pencil and other calculating devices were
not permitted. However, the children were allowed to count on their
fingers.
Detailed results have been reported elsewhere (Cooper, Heirdsfield, &
Irons, 1996a, 1996b; Heirdsfield & Cooper, 1996). In this paper,
general results and trends will be discussed under three areas: general
trends, word problems v. exercises, and additive and subtractive
strategies for subtraction word problems and exercises.
Strategies
To analyse strategy use, a categorisation scheme (Cooper, Heirdsfield,
& Irons, 1996a) based on Beishuizen (1993) was formulated. The
resulting categories of counting, separation, aggregation and wholistic
are described in Table 1.
Table 1
Mental strategies for addition and subtraction
Strategy Example
Counting 28+35: 28, 29, 30, .. (count on by 1)
52-24: 52, 51, 50, .. (count back by 1)
Separation right to left 28+35: 8+5=13, 20+30=50, 13+50=63
52-24: 12-4=8, 40-20=20, 28 (subtractive)
4+8=12, 20+20=40, 8+20=28 (additive)
left to right 28+35: 20+30=50, 8+5=13, 50+13=63
52-24:50-20=30, 2-4=2down, 30-2=28 (subtractive)
20+30=50, 4-2=2, 30-2=28 (additive)
Aggregation right to left 28+35: 28+5=33, 33+30=63
52-24: 52-4=48, 48-20=28 (subtractive)
24+8=32, 32+ 20=52, 28 (additive)
left to right 28+35: 28+30=58, 58+5=63
52-24: 52-20=32, 32-4=28 (subtractive)
24+20=44, 44+8=52, 28 (additive)
Wholistic Compensation 28+35: (28+2)+35=30+35=65, 65-2=63
52-24: 52-(24+6)=52-30=22, 22+6=28 (subtractive)
24+26=50, 50+2=52, 26+2=28 (additive)
levelling 28+35: 30+33=63
52-24: 58-30=28 (subtractive)
22+28=50, 28 (additive)
From the examples in Table 1, it is evident that the wholistic strategy
has less components and steps (less sub-problems) in its solution
procedure than the aggregation strategy, which in turn has less
components and steps than the separation strategy. The separation
strategy requires each number to be separated into place-value
components and which are separately added or subtracted and then
recombined for the solution. As argued in Beishuizen (1993) and
Wolters, Besishuizen, Broers, and Knoppert (1990), less components and
steps means less cognitive load on memory during mental calculation.
This should mean, in line with Sweller and Low (1992), that the
aggregation and wholistic strategies should be more efficient and
accurate particularly for more difficult examples. Similarly, left to
right separation has less sub-problems than right to left separation
and, therefore, should be more efficient than right to left separation
in terms of cognitive load. Thus, the strategies right to left
separation, left to right separation, aggregation and wholistic form a
hierarchy of efficiency.
General trends
Over the 2 years, the percentage of children attempting the questions
increased, as did the accuracy levels. This would be expected due to
maturation. Further, addition was attempted with higher frequency and
accuracy than subtraction. This finding reflects much of the
literature which has reported the difficulties children have with
subtraction (e.g., Fuson, 1984; Thornton, 1990).
Originally, counting was the dominant strategy; although the more
difficult examples were successfully attempted with more advanced
strategies. By year 3, left to right separation had become the
dominant strategy. By the beginning of year 4, right to left
separation was the dominant strategy (particularly for algorithmic
exercises). As reported by Beishuizen (1993). separation strategies
were more popular than aggregation strategies, although aggregation was
more accurate. Overall, a variety of strategies was reported.
In Queensland schools, the traditional pen and paper algorithms for
addition and subtraction are introduced in year 2 and further developed
in year 3. The procedures for these algorithms are symbolic and follow
a set pattern of activity: the numbers are written vertically with
place values aligned, the place values are separated, the ones are
operated on first, and ones and tens are regrouped as required, and
computation proceeds right to left (from the ones to the tens to the
hundreds). The right to left separation strategy may not involve
carrying the ten in addition (e.g., 25+38: 5+8=13, 20+30=50, 50+13=63).
However, the pen and paper procedure for addition and subtraction
algorithms has common aspects with the right to left separation
strategy for mental addition and subtraction (i.e., separate the ones
and tens and work right to left by adding the ones and then adding the
tens) and no other mental computation strategy has these commonalities.
In fact the aggregation and wholistic strategies are in opposition to
the pen and paper algorithm procedure in not separating all place
values. Therefore, it is a reasonable assumption that the teaching of
the pen and paper algorithms, with their attendant right to left
procedure, will have an impact on children's choice of mental
computation strategy. The results from the interviews show that, in
year 3, there is strong increase in popularity for the right to left
separation strategy (as Madell, 1985, also found). Children's
responses to the word problems, and the relationship between these
responses, is a strong indication that there is an instructional
effect, that is, instruction in pen and paper procedures influences
children to choose the right to left separation strategy for mental
addition and subtraction.
Word problems versus algorithmic exercises
Word problems were attempted by more students and with greater accuracy
than algorithmic exercises. This was interesting, as it does not
reflect the sequence for teaching in Queensland schools. Generally,
exercises are presented first; then, if the children are successful at
this stage, they are introduced to word problems. The exception to
this trend was in the last interview where the 3 digit exercises were
attempted with greater accuracy and frequency than the equivalent word
problems. Further, a greater variety of strategies was identified for
word problems than for algorithmic exercises. At first, algorithmic
exercises were attempted with a variety of strategies (though not as
great a variety as for word problems), but this variety diminished in
later interviews.
As argued above, in line with Sweller and Low (1992), the higher
strategies (aggregation and wholistic) should be used more efficiently
and accurately by children, particularly for more difficult examples,
because they are the most cognitively efficient. The children's
responses supported this in the early interviews, but not in the later
interviews where the right to left separation strategy was the most
highly used and most accurate. However, this result can be understood
when familiarity and automaticity with procedures is included in the
cognitive load equation. The instructional focus on the traditional
pen and paper algorithms means that children become sufficiently
familiarised with the right to left separation strategy procedures that
it is cognitively efficient to use them in difficult examples. Hence,
even in cognitive load terms, there appears to be an instructional
effect due to the emphasis on traditional pen and paper algorithmic
procedures in years 2 and 3.
Additive and subtractive strategies
Overall, the children exhibited both subtractive and additive
strategies for separation and missing addend word problems. This was
consistent with the findings of Carpenter and Moser (1984). Further,
both strategies were employed for algorithmic exercises, also reported
by Perry and Stacey (1994) for older students (years 8 to 12). Year 2
children predominantly used a subtractive strategy for separate
problems and an additive strategy for missing addend problems,
reflecting the semantic structure of the problem (similar to findings
of Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993). However, by
year 4, the strategy choice no longer closely reflected the semantic
structure of the problems.
In contrast to findings of Carpenter and Moser (1984), Fuson (1986a,
1986b), Fuson and Willis (1988), and Secada (1982), there was not an
emphasis on additive strategies. Once again, there appears to be an
instructional effect as emphasis on the subtractive traditional pen and
paper subtraction algorithm in Years 2 and 3 means that, by Year 4,
most children had adopted this approach and discontinued additive
procedures for subtraction problems.
Summary and implications
The major finding of study 1 is the instructional effects on mental
computation of the teaching emphasis on traditional algorithms. The
implication is that there should be less emphasis on teaching
traditional pen and paper algorithms, more account of children's
natural preferences and capabilities when designing curricula, and more
emphasis on developing children's spontaneous strategies in problem
solving environment.
However, questions still arise as to why some students are more
accurate and flexible with strategies than others and how their
expertise relates to their knowledge and performance in other
mathematics topics. These questions were the motivation for study 2.
Study 2
During the early stages of study 1, it became apparent that, although a
variety of strategies used by young children had been identified, some
children by year 4 were employing one strategy consistently, while
others employed multiple strategies. Research in the area of mental
computation and number sense has identified computational estimation
and number fact knowledge as contributing or associated factors (e.g.,
Hope & Sherrill, 1987; Resnick & Ford; 1981; Reys, 1984; Reys, Bestgen,
Rybolt, & Wyatt, 1982; Sowder & Wheeler, 1989; Sowder, 1988). Thus,
the focus of study 2 was to compare characteristics of multistrategy
mental computers and unistrategy mental computers in relation to
accuracy in mental computation, proficiency in computational
estimation, and proficiency in number fact knowledge. Strategies for
mental computation, computational estimation, and number facts were
also identified.
The sample of 32 was drawn from the students involved in the
longitudinal study, which was reported as study 1. They were chosen
on the basis of accuracy and employment of a variety of strategies for
mental computation. Each student then participated in two more
interviews: computational estimation and number facts. Both mental
computation and computational estimation tasks were presented in
picture form, accompanied by printed numbers, and the problem
verbalised by the interviewer. The number facts questions consisted of
8 addition and 8 subtraction facts to 20 (presented in written form).
The mental computation, computational estimation and number fact
interviews were analysed with respect to strategy choice and accuracy.
Number fact knowledge was also analysed with respect to speed.
Multistrategy versus unistrategy children
All students who employed multistrategies accurately were proficient at
computational estimation and number facts. These students also
employed wholistic strategies for both mental computation and
computational estimation tasks, where possible. It was argued that the
ability to use such strategies required a good understanding of number.
Thus number understanding was reflected in both mental computation and
computational estimation. It has been argued elsewhere that the
ability to compute mentally and estimate are related skills (Sowder &
Wheeler, 1989). While not all computational tasks could have been
calculated using wholistic strategies, the students who employed
multistrategies accurately also used other efficient mental strategies
(e.g., aggregation).
As well, number facts were recalled quickly and accurately. Where
immediate fact recall was not employed in the number facts test,
advanced derived facts strategies (DFS) were employed, for example, use
doubles, use 10 (c.f., counting). All these students reported using
immediate fact recall to calculate components in the mental computation
tasks. It would appear that accurate and speedy recall of number facts
would aid in mental computation, as more attention can be given to the
overall calculations, rather than partial calculations. Sowder and
Wheeler (1989) and Hope and Sherrill (1987) also posited this.
In contrast, unistrategy students who were as accurate as the accurate
multistrategy users, but employed one strategy consistently throughout
the mental computation (by definition), employed right to left
separation. As argued in study 1, this strategy reflects the
traditional pen and paper algorithm taught in Queensland schools.
These accurate unistrategy users were less proficient at computational
estimation, and, although scored well on the number facts test, did not
use immediate fact recall and derived facts strategies as often; that
is their number facts strategies were not as advanced. It appears that
multistrategy users, although no more accurate than those who were not
flexible, were able to manipulate numbers with more understanding.
Strategy use
Considering that only 32 children were chosen for this study, and only
16 were chosen on the basis of using a variety of strategies, a
substantial diversity of strategies was reported. Further, each
question was answered in a variety of ways. However, the variety of
subtraction strategies exceeded those for addition. It was posited
that the difficulty children in this study had with subtraction (Fuson,
1992) resulted in their generating their own strategies for solving
subtraction, more so than for addition.
With regard to strategy use, separation strategies were used more
frequently than aggregation strategies, although less accurately.
Beishuizen (1993) also reported this. A possible reason for the higher
accuracy was that less load is placed on working memory when
aggregating. Right to left separation was the most popular strategy of
all, in particular with algorithmic exercises. However, it resulted in
the most short term memory errors. Many children reported forgetting
their partial answers, and having to start again when working right to
left.
For subtraction word problems, the semantic structure was not reflected
in the solution strategy; that is, both additive and subtractive
strategies were used for both separate and missing addend subtraction,
although subtractive strategies were more popular.
Summary and implications
Evidence from this study indicated that students had developed mental
strategies (and also estimation strategies) without formal classroom
instruction. Many students resorted to the right to left separation
strategy, without first considering the numbers involved. It was
argued this strategy reflects the school taught pen and paper
algorithm, the teaching of which appears to have resulted in an
overdependence by many students. However, there were some students who
evidently looked at the numbers first, and made a decision regarding
the appropriateness of particular strategies. These students who were
both flexible and accurate were also proficient at both computational
estimation and number fact knowledge, that is, these students exhibited
a propensity for number understanding.
However, questions arose from this study. What allowed these students
to be more inclined to access different strategies? Are other factors
involved, for instance, affective factors? What qualities would
younger children who had not become too dependent on pen and paper
algorithms exhibit? Why are some children better mental computers than
others? As a result of these questions the third study, to be reported
here, was conceived.
Study 3
The aim of study 3 is to explain why some children are better at mental
computation than others. The study is still in progress. The study
was preceded by a pilot to develop instruments.
For the pilot study, sixteen year 3 children from one classroom in an
inner city Brisbane school were interviewed, using mental computation
tasks (similar to those used in study 2), to identify good mental
computers. Because it was of interest to describe characteristics of
not only accurate, but also flexible mental computers, children were
selected on the basis of accuracy and flexibility. This paper will
report on one child, Clare, who was accurate and flexible. Reference
will be made to other students for comparison, particularly two other
children who were also accurate: Emily who was also flexible like
Clare, although there were differences as will be seen in the following
discussion; and, by contrast, Mandy who was not flexible.
To achieve this, some possible aspects were identified from the
literature in order to be able to commence the investigation. These
were: number sense, particularly number facts, computational
estimation, numeration, and properties of number and operation; social
and affective issues including beliefs, values, and social context
(e.g., classroom and home); and cognitive factors such as metacognitive
processes and mental representations.
Connections between mental computation and other aspects
Skilled mental computers use a variety of strategies in different
situations (depending on numbers and context), because they are
disposed to making sense of mathematics (Hope, 1985; Maier, 1977;
Sowder, 1994). Therefore, they must be aware of a variety of
strategies. How do they choose which strategy to use? There is
evidence of awareness of reflection and regulation. Reys, Bestgen,
Rybolt, and Wyatt (1980), Hope (1987), Dowker (1990) reported children
and adults choosing strategies based on their knowledge of number and
operations, and choosing appropriate strategies to deal with the problems.
It is not sufficient to be aware of alternative strategies, but also to
have the confidence to use them. The reasons that some children are
unable to use better strategies than the pen and paper algorithms in
different situations, vary. It may be because of prolonged practice of
these algorithms, and/or being unaware of alternatives. It may also be
because of a lack of confidence in experimentation and lack of belief
in their own ability to choose more appropriate strategies, or lack of
belief in appropriateness of using alternative strategies. Thus, the
study of good mental computers may go beyond cognition and
metacognition, to affects and beliefs (Sowder, 1994).
Connections have also been drawn between mental computation and other
factors, including numeration and place value, number sense,
computational estimation and number fact knowledge. Research has
suggested that mental computation requires an understanding of
numeration (Reys, 1985) and place value (McIntosh, 1996; Sowder, 1992).
McIntosh (1996), Sowder (1992), and Trafton (1992) specifically
mentioned mental computation as an indicator or element of number
sense. It appears that mental computation and computational estimation
may be related (Heirdsfield, 1996; Maier, 1977; Reys, Bestgen, Rybolt,
& Wyatt, 1982; Sowder & Wheeler, 1989). Further, results of research
(Hope & Sherrill, 1987; Sowder & Wheeler, 1989) identified basic fact
knowledge as a related skill to mental computation. Mental computation
has also been linked to number sense (McIntosh, 1996; McIntosh, Reys, &
Reys, 1992; Reys, 1984; Sowder, 1990, 1992). The ability to manipulate
numbers appropriately in different contexts would facilitate flexible
mental computation.
Plunkett (1979) suggested that mental algorithms are often iconic, for
instance, incorporating the use of a number line, or number square.
Reys (1985) stated that mental computation utilises visual thinking
skills, for example, pictorial models. In recent years, some research
has considered young children's mental representations of number
(Thomas & Mulligan, 1995; Thomas, Mulligan, & Goldin, 1996, 1994) and
how the development of children's representations can aid in the
development of number (Bobis, 1993). However, in a study of young
children's representation of the counting sequence 1 to 100 (Thomas,
Mulligan, & Goldin, 1994), it was found that young children do not
naturally view numbers in conventional ways (e.g., number lines, 99
board), but rather, in very idiosyncratic forms; although, in older
children the number line and 99 or 100 chart began to appear (Thomas &
Mulligan, 1995). Further, children with better developed number sense
represented numbers in a dynamic mode; whereas, children with less
developed number sense represented number in a static mode. This
notion of dynamic imagery was also supported by Trafton (1992) when
describing the metaphoric language used by students, for instance,
"chop in half', "knock off', "tack on numbers". Here, students are
assigning meaning to the symbols. It would appear that children's
mental representations of number and operations may be factors in
mental computation.
The interviews
While it is recognised that some of these aspects, described in the
previous section, may be essential components of mental computation,
others may not be as closely linked. With these aspects in mind, an
investigative study of mental computers was initiated.
After the students were selected, they participated in a variety of
indepth clinical interviews. After reviewing the videotaped
interviews, it was often necessary to have the students involved in
further interviews for clarification. Specific items addressing
further mental computation (Table 2), number fact knowledge,
computational estimation, number and numeration, and mental
representations were presented. Other questions relating to self
efficacy, beliefs, and metacognition were included in the interviews.
Mental computation, computational estimation, and number fact responses
were analysed for strategy choice, flexibility, accuracy, understanding
of number and numeration, and metacognition. Number and operations
tasks were analysed for understanding of associativity and inverses,
and relationships (e.g., 69-43=26, (69-44=25). Analysis of students'
responses to numeration tasks were based on Ross's five levels (1986).
Although analysis of individual interviews were undertaken separately,
commonalities across interviews were considered, for instance, whether
understanding of noncanonical partitioning of numbers was used for
mental computation. In order to get a feel for classroom and home
contexts, the children were encouraged to indulge in general
conversation, and the teacher was invited to respond to initial and
general inferences.
Table 2
Number combinations for mental computation word problems
Question type Addition Subtraction
basic fact 6+8 15-8
basic fact & ± 9 9+7 14-9
multiples of 10 64+20 76-20
2 digit w/o regrouping 53+34 58-36
2 digit with regrouping &
including no. fact 46+28 65-28
2 digit, near compatibles 75+28 80-49
2 digit regroup, involving 9 45+19 63-29
bridge 100 76+43 107-15
3 digit, involving 9 246+199 234-99
3 digit, near compatibles 350+52 400-298
Clare's story
Clare was selected for further investigation as she was accurate and
employed a variety of advanced strategies in the selection interview,
e.g., 148+99: 100+99=199, 48-1=47, 247 (wholistic); 52-19: take 2 out
of 9 = 7, 10-7=3, 4-1=30, 37 (this method was also reflected in the
number facts test). She appeared confident in computation, and stated
she liked mathematics, because she finds it easy and is therefore good,
that is, she attributed her success to ability. This type of response
was also elicited by Emily, another student who was both accurate and
flexible ("I like maths, because it's my best subject."). This was in
contrast to Mandy who attributed her success to practice. Mandy was
also accurate in mental computation, but consistently employed a mental
image of the pen and paper algorithm.
When asked how she knew she was correct, Clare replied, "I just think
I'm right. I am usually right." In contrast, Emily and Mandy said
they would check their answers by working through the examples the same
way, and then wait for feedback from the teacher.
Clare stated that she believed she would be able to solve the mental
computation questions, and she could. This was evident when asked at
the beginning of some items, and also in a Student Preference Survey
(SPS) (McIntosh, 1996). Both Emily and Mandy also stated they would be
able to complete the tasks mentally, and could. Results of the SPS
indicated that 5 of the 16 children believed they could complete all
the examples on the survey, mentally. However, 3 or 4 of these
children would not have been able to do so, as evidenced by their
responses in the selection interviews, and when asked to solve some of
the items on the survey. The three children already mentioned
responded with a variety of "yes" and "no" replies to whether they
would calculate mentally or not.
Clare attributed failure to "very foolish mistakes". Further, she
needed to achieve, and only felt confident attempting questions if she
believed she could succeed. After being unsuccessful at calculating
265-99 in the selection interview, she went home and asked her father
how to calculate such examples. She was happy to attempt a similar
question (234-99) in the next interview, because she now knew how to
calculate it. However, she did not know why it worked ("That's what
Dad told me to do."). Her confidence was also exhibited by her stating
that her subtraction method (of levelling) "annoys Miss A...", but she
was determined to continue to use it. However, she did realise that
method was too complex for 3 digit examples. In the follow up mental
computation interviews, when asked to think of another solution method,
she saw no reason to think of a different method, except for the fun of
it (appease the interviewer?). However, once she reasoned that some of
her second methods were better than her first methods, she thought it
was quite a good idea to indulge me. At times, hints had to be given,
e.g., "what is 19 near?". Other times, no hints were given, for
example, after solving 80-49 by 80-40= 40, take another 10, 10-9=1, 31,
Clare then turned 49 into 50 and proceeded 80-50+1. Clare's confidence
in her ability and her reluctance (at first) to try a different method
was reflected across all her classroom work. She had a strong
preference for her own methods, many of which she learnt from her
father (although, not all the time, with understanding). Her later
acceptance of alternative methods and even preference for these came as
a shock to her teacher ("out of character for Clare"). It is suggested
that she had nothing to prove to the interviewer by remaining adamant
about the suitability or otherwise of alternative strategies.
Her ability to manipulate operations in this fashion was not
consistent. In the number and operations interviews, she was not
always sure whether to add or subtract one when taking away one more or
one less (e.g., 73-45=28, 74-46=?). Thus, although her father had
shown her a method based on this principle, there was little
understanding. Emily, (also flexible and accurate) likewise had
problems with this concept. However, she successfully used the idea in
the mental computation interview without prompting. Her success in
both mental computation and number and operations interviews was
inconsistent. On the other hand, Mandy had to be deliberately
encouraged to use strategies other than "calculating operations" (the
term she used for pen and paper strategies). Mandy was successful at
completing such tasks as: 257-100=157, so what does 257-99=? (with a
fair amount of thought), but she stated that she still preferred "using
operations". The three students had no problem with a similar concept
for addition, that is, 234+99=333, because 234+100=334, and take 1, so
333. However, Mandy could not and would not use the concept for the
mental computation tasks. In discussions with Mandy's teacher, it was
revealed that Mandy had high expectations for accuracy and speed when
completing tasks. This could explain her using the same "automatic"
procedure for solutions, and maintaining confidence in this procedure.
For all Clare's confidence, though, when asked to solve subtraction
problems, she replied, "I don't particularly want to. I don't like
doing take away in my head." This was despite the fact that she could.
This attitude towards subtraction was reflected in her response in the
SPS, where she responded positively to calculating mentally for only
the simple subtraction problems. Thus, her preference for written
calculation of subtraction was well founded on her knowledge of her
poorer understanding of the operation. This negative attitude to
subtraction was reflected in the class SPS responses. Four of the 16
students responded with "no" to all subtraction examples, and at least
3 others should have responded likewise, from indications in the mental
computation selection interviews.
Clare admitted that she generally employed the first method "that pops
into my head"; therefore, there were times she chose an arguably less
efficient mental strategy. However, later in the interviews, such
statements as, "why didn't I think of that in the first place?"
indicated she began to consider strategy choice more carefully. Emily
possessed a variety of strategies, but she admitted she also used the
first method she thought of. In contrast to Clare, Emily did not show
evidence of much regulation and monitoring, although she was encouraged
to think of other strategies and decide which strategy she preferred.
Mandy, on the other hand, had employed a mental image of the pen and
paper algorithm in the selection interviews, and stated several times
that she preferred that method and found it easier, as she was "used to
it". Through prompting, Mandy developed a left to right aggregation
strategy, and started to use it later in the interviews, because she
said she wanted to practise the new way which may be easier for mental
calculations. Mandy also was able to use a wholistic strategy for
subtraction with 99, but stated, in all cases, she still preferred the
"old way". In fact, when employing the new strategies, she still
imagined the numbers one under the other, as though setting the
examples out on paper.
Clare's number facts were fast and accurate. In the number facts test,
she used recall (6 out of 16 times), and DFS (build to 10, pattern with
9, through 10 subtraction - like a levelling e.g., 17-9: take 7 out of
9 and out of 17, 10-2=8). The levelling strategy, as already
mentioned, was used same strategy in the mental computation interviews
for subtraction. Emily also used this levelling strategy in the number
facts test and in the mental computation interviews. Both children
stated they had not been taught this strategy, but worked it out for
themselves. This offers support for children who employ derived facts
strategies (DFS) understand relationships between numbers, and are able
to use this understanding of number properties in mental computation.
Emily also employed counting strategies in the number facts test. When
it came to calculating in the mental computation interviews, counting
made it difficult for her, as working memory was taken up with
remembering counts, rather than attending to the calculation as a
whole. One other child who used recall (not always accurately) and
counting appeared to be so disadvantaged by her lack of number facts
strategies, that the interviewer gave her answers to number facts so
that she could complete the mental computation tasks. Mandy also used
counting in both the number facts test and the mental computation
interviews, but did not have the same memory overload problems. Most of
the strategies Clare employed in the number facts test were reflected
in the mental computation interviews. Her agility with number facts
was an advantage in the mental computation interviews, as working
memory was available for efficiently solving more complex problems.
The children's teacher was amazed that the children had formulated such
strategies. She stated that she had used similar strategies when
modelling addition tasks, but did not expect the children to be able to
use them for addition, and in particular, subtraction. It appears that
Clare and Emily had the capacity to build up a rich, interconnected
network of knowledge, and access this knowledge, readily.
Before the indepth mental computation interviews, the children were
presented with the number facts test, in which Clare calculated 15-8 by
levelling (quite a favourite take away method for her). She was the
able to recall this fact for the same question in the mental
computation interview, that is, she had learnt from the experience.
This was not the case with the either Emily or Mandy. They
recalculated the answers to the number facts, although they had already
done so, not 5 minutes before, for instance, use doubles, through 10.
Clare agreed that knowing number facts was important, but didn't know
why, except that her teacher had told her. Emily stated that the
importance of knowing her number facts was to be able to get them
correct in daily tests. Mandy could see a benefit in the future, as
they may be useful in a future profession, for instance, a scientist
would need number facts. She also believed it was necessary to know
them in order to be able to pass pen and paper tests. These responses
surprised the teacher, as she had often used more worthwhile
explanations for the need for immediate fact recall, for instance, ease
of computation.
Computational estimation is poorly treated in the mathematics
curriculum. Clare defined estimation as a "type of guessing", a
definition in common with other children in her class. She stated that
she only estimated when given classroom estimation tasks that were
treated as rounding only. However, Clare did not employ rounding in
the interview. Rather, she used other strategies more appropriate to
the situations, for instance, truncation and wholistic. Because
Clare's mental computation was so good, she attempted to calculate
accurately. This has been reported elsewhere (Heirdsfield, 1996;
LeFevre, Greenham, & Waheed, 1993). It was decided to present Clare
with additional 3 digit estimation questions that were too difficult
for exact calculation. Clare's responses reflected an understanding of
magnitude of number, place value, and the effect of operations. One
example of a successfully completed task was: "Your friend has $152 and
spends $144 on a cassette recorder. You have $156 and spend $142 on
another cassette recorder. Who has more money left?" Response: "I do,
because I started with more and spent less." Emily completed the
computational estimation tasks using similar strategies as Clare. She
also exhibited an understanding of the size of numbers, place value,
and the effect of operations. However, the number combinations did not
have to be altered to prevent her from calculating accurately. In
contrast, Mandy could only relate estimation to measurement, and was
generally unsuccessful at the estimation tasks.
The numeration tasks revealed Clare's understanding of both canonical
and noncanonical representations of number (Ross, 1986). She was
particularly flexible with different representations of such numbers as
560 (5x100 + 6x10 + 0x1; 56x10 + 0x1; 500x1 + 6x10; 55x10 + 10x1; 5x100
+ 3x10 + 30x1) and 209 (2x100 + 0x10 + 9x1; 20x10 + 9x1; 209x1; 19x10 +
19x1). Although MAB (Multibase Arithmetic Blocks) were available,
Clare did not use them. However, there were times the interviewer had
to encourage her to elicit more combinations, although she appeared to
delight in the challenge. In contrast, Mandy was slow at representing
numbers in different ways. She had to be prompted with such questions
as, "What about some ones?", and needed the support of MAB for many
examples. Even with MAB, she did not show a solid understanding of
what she was doing, as she constantly checked and recounted her
manipulations. An alternative explanation could be that her need for
absolute certainty overshadowed her understanding of number. However,
it appears curious that she would have to count and recount tens to
ones, if she truly understood regrouping. Emily also required MAB to
represent alternatives, but she appeared to understand better what she
was doing, as she manipulated the blocks faster and with more
confidence.
Throughout the interviews, Clare, Emily and Mandy were asked whether
they saw anything in their heads while calculating, estimating, and so
on. A very definite "no" was the reply from each child. To
investigate her mental representation of number, they were asked to
close their eyes, think of the numbers between 1 and 100, and then put
on paper what they saw in their heads (Thomas & Mulligan, 1995).
Clare's drawing showed the numbers 1, 2, 44, 99, 100 (possibly from the
rhyme, "1, 2, skip a few, 44, skip some more, 100"). The numbers 1, 2
and 100 were drawn with hands, and 44 and 99 with wings. She revealed
that all double digit numbers would have wings. Clare also revealed
that she didn't normally think of numbers in that way, but wanted to
make them look interesting. Further, the numbers were not doing
anything (not moving), but they were in order. Emily wrote the numbers
1 to 10 on one line, 11 to 20, on the next, 21 to 30 on the next, and
so on, indicating some knowledge of structure of the number sequence.
Further, she indicated that the numbers go across her forehead. She
said she did not use a 99 or 100 board in class. Mandy's drawing
showed the numbers 1, 2, and 3 on a circle. In trying to explain her
drawing, she drew arrows from 1 to 2, 2 to 3, and 3 back to 1. Then
she motioned with her hand that the numbers keep turning as of in a
series of loops. Although she stated the all the numbers are involved,
she only saw the numbers 1, 2, and 3 "going round and round".
During the course of interviews, Clare revealed things about her
thinking, unprompted, for instance, "No, that can't be right", "I'm
lost now", "I'm usually right", "This one's difficult", "This one's
easier", "I like this one, because it has something to do with 99", and
"Seventy-five is easier to use than 76, so I'll use 75". These
statements revealed the existence of metacognitive processes and
beliefs. Clare had access to a variety of strategies, but rarely
consciously chose the most appropriate strategy for the number context.
However, when encouraged to think of other strategies, she made
judgements regarding the suitability of the strategies. Clare was
confident in experimenting with different strategies. She seemed to
disregard what was taught in the classroom, rarely using the taught
algorithm to solve the problems mentally. In fact, Clare revealed that
she often used her levelling strategy for subtraction to solve written
exercises.
Concluding comments
For generations primary mathematics has focused on the teaching of
algorithmic procedures, using one inflexible procedure for each
operation. Changes that have occurred (e.g., the shift from the
'borrow and pay back' to the decomposition subtraction algorithm) have
been in replacing one inflexibility with another. This has interpreted
computation in simplistic terms and assumed children are programmable
computers that can receive and reproduce fixed sequences of procedures.
Left to their own devices, children use a variety of procedures
depending on need, context and number size. Children see computational
situations from a variety of perspectives, for example, some children
see 7-3 as taking 3 from 7 and some as building 3 up to 7. This
complexity is reflected in the real world situations that can be
represented computationally. Hence, as the needs for mathematics turn
from accuracy in computation (now the province of calculators and
computers) to interpreting real-world problem situations, the
inflexible 'do-it-one-way' traditional algorithms become a liability.
Children need the flexibility that comes from constructing their own
procedures for computation that is mental, recorded and estimated. And
as the research above is showing, continuing foci on familiarity with
fixed traditional algorithms is crushing such flexibility
The transition in teaching from inflexible pen and paper algorithms to
self constructed mental procedures is a large step for teachers. In
the first, fixed methods could be applied to all students in a similar
manner. In the second, each student is a special individual case to be
nurtured. Teachers need a repertoire of procedures, teaching
techniques and diagnostic tools.
The three studies have moved from studying the existing situation in
schools to looking at the relationship between mental computation and
other number sense proficiencies. As the complex interaction between
knowledge, affect and proficiency emerges, insight will hopefully also
emerge in how to encourage students to be flexible and creative
interpreters of their world from a computational perspective.
References
Australian Education Council and Curriculum Corporation. (1991). A
national statement on mathematics for Australian schools. Victoria:
Curriculum Corporation.
Beishuizen, M. (1993). Mental strategies and materials or models for
addition and subtraction up to 100 in Dutch second grades. Journal for
Research in Mathematics Education, 24(4), 294-323.
Bobis, J. (1993). Visualisation and the development of mental
computation. In B. Atweh, C. Kanes, M. Carss & G. Booker (Eds.),
Proceeding of the Sixteenth Annual Conference of the Mathematics
Education Research Group of Australasia (pp. 117-122). Brisbane:
MERGA.
Carpenter, T.P., Ansell, E., Franke, M.L., Fennema, E., & Weisbeck, L.
(1993). Models of problem solving: A study of kindergarten children's
problem-solving processes. Journal for Research in Mathematics
Education, 24, 428-441.
Clarke, O., & Kelly, B. (1989). Calculators in the primary school -
Time has come. In B. Doig (Ed.), Everyone counts. Parkville:
Mathematics Association of Victoria.
Cobb, P., & Merkel, G. (1989). Thinking strategies: Teaching
arithmetic through problem solving. In P. Trafton & A. Schulte (Eds.),
New directions for elementary school mathematics. 1989 yearbook.
Reston: National Council of Teachers of Mathematics.
Cooper, T. J., Heirdsfield, A.M., & Irons, C. J (1996a). Children's
mental strategies for addition and subtraction word problems. In J.
Mulligan & M. Mitchelmore (Eds.), Children's number learning. (pp.
147-162). Adelaide: Australian Association of Mathematics Teachers, Inc.
Cooper, T. J., Heirdsfield, A. M., & Irons, C. J. (1996b). Years 2
and 3 children's correct-response mental strategies for addition and
subtraction word problems and algorithmic exercises. In L. Puig & A.
Guiterrez (Eds.), Proceedings of the 20th Conference of the
International Group for the Psychology of Mathematics Education. (vol.
2, pp. 241-248). Valencia: University of Valencia.
Department of Education, Queensland. (1991). Years 1 to 10
mathematics sourcebook. Year 3. Brisbane: Government Printer.
Dowker, A. (1990). The variability of mathematicians' estimation
strategies: Some cognitive implications. Unpublished manuscript.
Fuson, K. (1984). More complexities in subtraction. Journal for
Research in Mathematics Education, 15(3), 214-225.
Fuson, K. (1986a). Teaching children to subtract by counting up.
Journal for Research in Mathematics Education, 17(3), 172-189.
Fuson, K.C. (1986b). Roles of representation and verbalization in the
teaching of multi-digit addition and subtraction. European Journal of
Psychology of Education, 1(2), 35-56.
Fuson, K. (1992). Research on whole number addition and subtraction.
In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and
learning. New York: Macmillan.
Fuson, K.C., & Willis, G.B. (1988). Subtracting by counting up: more
evidence. Journal for Research in Mathematics Education, 19, 402-420.
Ginsburg, H., Kossan, N., Schwartz, R., & Swanson, D. (1983).
Protocol methods in research on mathematical thinking. In H. P.
Ginsburg (Ed.), The development of mathematical thinking. New York:
Academic Press.
Heirdsfield, A. M. (1996). Mental computation, computational
estimation, and number fact knowledge for addition and subtraction in
year 4 children. Unpublished master's thesis, Queensland University of
Technology, Brisbane.
Heirdsfield, A.M., & Cooper, T.J. (1996). The 'ups' and 'downs' of
subtraction: Young children's additive and subtractive mental
strategies for solutions of subtraction word problems and algorithmic
exercises. In P. Clarkson (Ed.), Proceedings of the 19th Annual
Conference of the Mathematics Education Research Group of Australasia.
(pp. 261-268). Melbourne: Deakin Universtiy Press.
Hope, J. A. (1985). Unravelling the mysteries of expert mental
calculation. Educational Studies in Mathematics, 16, 355-374.
Hope, J. A. (1987). A case of a highly skilled mental calculator.
Journal for Research in Mathematics Education, 18(5), 331-342.
Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled
and skilled mental calculators. Journal for Research in Mathematics
Education, 18(2), 98-111.
Kamii, C., Lewis, B., & Jones, S. (1991). Reform in primary
education: A constructivist view. Educational Horizons. 70(1), 19-26.
Maier, E. (1977). Folk math. Instructor, 86(6), 84-89, 92.
McIntosh, A. (1995). Mental computation in Australia, Japan and the
United States. In B. Atweh & S. Flavel (Eds.), Proceedings of the
Eighteenth Annual Conference of the Mathematics Education Research
Group of Australasia. (pp. 416-420). Darwin: MERGA.
McIntosh, A. (1996). Mental computation and number sense of Western
Australian students. In J. Mulligan & M. Mitchelmore (Eds.),
Children's number learning. (pp. 259-276). Adelaide: Australian
Association of Mathematics Teachers, Inc.
McIntosh, A., Reys, B., & Reys, R. (1992). A proposed framework for
examining basic number sense. For the Learning of Mathematics, 12, 2-8.
Perry, A.D., & Stacey, K. (1994). The use of taught and invented
methods of subtraction. Focus on Learning Problems in Mathematics, 16(3), 12-22.
Plunkett, S. (1979). Decomposition and all that rot. Mathematics in
Schools, 8(3), 2-5.
Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics
for instruction. Hillsdale, New Jersey: Laurence Erlbaum Association,
Publishing.
Reys, B. J. (1985). Mental computation. Arithmetic Teacher, 32(6), 43-46.
Reys, B. J., & Barger, R. (1991). Mental computation: Evaluation,
curriculum, and instructional issues from the US perspective,
Computational alternatives: Cross cultural perspectives for the 21st
century. (Unpublished monograph).
Reys, B. J., & Barger, R. H. (1994). Mental computation: Issues from
the United States perspective. In R. E. Reys & N. Nohda (Eds.),
Computational alternatives for the twenty-first century. Reston,
Virginia: The National Council of Teachers of Mathematics.
Reys, R. E. (1984). Mental computation and estimation: past, present
and future. Elementary School Journal, 84(5), 546-557.
Reys, R. E., Bestgen, B. J., Rybolt, J. F., & Wyatt, J. W. (1980).
Identification and characterization of computational estimation
processes used by in-school pupils and out-of-school adults. Final
report, grant no. NIE 79-0088. Columbia, Mo.: University of Missouri
1980. (ERIC Document Reproduction Service no. 197 963).
Reys, R. E., Bestgen, B. J., Rybolt, J. F., & Wyatt, J. W. (1982).
Processes used by good computational estimators. Journal for Research
in Mathematics Education, 13(3), 183-201.
Reys, R. E., Reys, B. J., Nohda, N., & Emori, H. (1995). Mental
computation performance and strategy use of Japanese students in grades
2, 4, 6, and 8. Journal for Research in Mathematics Education, 26(4),
304-326.
Ross, S. H. (1986). The development of children's place value
numeration concepts in grades two through five. Paper presented at the
annual meeting of the American Educational Research Association. San
Francisco, April.
Secada, W.G. (1982, March). The use of counting for subtraction. Paper
presented at the annual meeting of the American Educational Research
Association, New York.
Sowder, J. (1988). Mental computation and number comparisons: Their
roles in the development of number sense and computational estimation.
In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the
middle grades. Hillsdale: NJ: Lawrence Erlbaum Associates.
Sowder, J. (1990). Mental computation and number sense. Arithmetic
Teacher, 37(7), 18-20.
Sowder, J. (1992). Making sense of numbers in school mathematics. In
G. Leinhardt, R. Putman, & R. Hattrup (Eds.), Analysis of arithmetic
for mathematics teaching. Hillsdale, New Jersey: Lawrence Erlbaum
Associates.
Sowder, J. (1994). Cognitive and metacognitive processes in mental
computation and computational estimation. In R. Reys & N. Nohda
(Eds.), Computational alternatives for the twenty-first century.
Reston, Virginia: NCTM.
Sowder, J., & Wheeler, M. (1989). The development of concepts and
strategies used in computational estimation. Journal for Research in
Mathematics Education, 20, 130-46.
Thomas, N., & Mulligan, J. (1995). Dynamic imagery in children's
representations of number. Mathematics Education Research Journal,
7(1), 5-25.
Thomas, N., Mulligan, J., & Goldin, G. A. (1994). Children's
representations of the counting sequence 1 - 100: Study and
theoretical interpretation. In J. P. D. Ponte & J. F. Maors (Eds.),
Eighteenth International Conference for the Psychology of Mathematics
Education, 3 (pp. 1-8). Lisbon, Portugal: Program Committee of the
18th PME Conference, Lisbon, Portugal.
Thomas, N., Mulligan, J., & Goldin, G. A. (1996). Children's
representations of the counting sequence 1 - 100: Cognitive structural
development. In L. Puig & A. Guitierrez (Eds.), Twentieth
International Conference for the Psychology of Mathematics Education, 4
(pp. 307-314). Valencia, Spain: Program Committee of the 20th PME
Conference, Valencia, Spain.
Thornton, C. A. (1990). Solution strategies: Subtraction number
facts. Educational Studies in Mathematics, 21, 241-263.
Trafton, P. (1992). Using number sense to develop mental computation
and computational estimation. In C. Irons (Ed.), Challenging children
to think when they compute. Brisbane: Centre for Mathematics and
Science Education.
Willis, S. (1992). The national statement on mathematics for
Australian schools: A perspective on computation. In C. Irons (Ed.),
Challenging children to think when they compute. (pp. 1-13).
Brisbane: Centre for Mathematics and Science Education, Queensland
University of Technology.
Wolters, G., Beishuizen, M., Broers, G., & Knoppert, W. (1990).
Mental arithmetic: Effects of calculation procedure and problem
difficulty on solution latency. Journal of Experimetnal Child
Psychology, 49, 20-30.