Paper presented at the Annual Conference of the British Educational Research Association,
University of Exeter, England, 12-14 September 2002
Children’s experiences of mathematics
PETER KELLY, University of Exeter
ABSTRACT There is universal acceptance that mathematics learnt in school should be
useful to the learner, both in other school situations and away from the educational context.
However, despite much advice about how to achieve this, little success has been shown to
result from schooling. This article reports the findings of a study of primary school children,
which describes the phenomenon of knowledge utilisation as the children experience it. The
study adopts a low intrusion stance in the collection of data, attempting to capture children’s
experiences as faithfully as possible. Findings, in part, parallel those of other studies,
providing an indication of their generality: in almost all cases primary children do not see
mathematics as being a significant feature of the world beyond school. Further findings
suggest that if we are to improve children’s mathematical knowledge utilisation, we must
promote their movement towards more productive beliefs about mathematics in relation to
utilisation. A categorisation of metaphoric descriptions of such beliefs is provided, ranging
from those that are unhelpful in promoting mathematical knowledge utilisation to those in
which this is central and explicit.
Using mathematical knowledge
The transferability of mathematical learning to new and unfamiliar contexts has long been
recognised as a major goal for primary educators (for example, in the Australian context:
Ainley, 1997; in the UK: DFE, 1995; DFEE, 1999a; and in the USA: NCTM, 1989; 1999).
Thus the latest version of the UK national curriculum describes mathematics as a “uniquely
powerful set of tools to understand and change the world” (DFEE, 1999b, p60), and in a
review of mathematics teaching contexts across the countries participating in the TIMMS
project, Robitaille (1997) suggests:
“Amongst the goals most frequently mentioned [in their curricula] were …
increasing the relevance of mathematics through a focus on real-world applications,
and teaching the children the skills and processes needed for successful problem
solving [in novel contexts].” (Robitaille, 1997, p32)
That such a goal has been difficult to achieve has also been recognised, both in the UK
(Askew and Wiliam, 1995; DFEE, 1997; NCEE, 1983; OFSTED, 1991; 1992; 1993; 1999;
SCAA 1994; 1995) and internationally (Ellerton and Clements, 1989; Halpern, 1992;
NCEST, 1992).
Much well documented research has supported the conclusion that the teaching and learning
of useable mathematical knowledge in school is not effective. Research on everyday
mathematical problem solving reveals that people rarely use school methods to solve
problems encountered out of school (Carraher et al., 1985; Lave, 1988; Nunes et al., 1993;
Saxe, 1988). Similarly Saljo and Wyndham (1990) found that children had considerable
difficulty in deciding which aspects of their school knowledge they could apply to ‘real life’
problems.
1
Verschaffel and his colleagues (1999) have suggested three areas which many upper primary
children do not master in order to approach mathematical application problems in an efficient
and successful way:
 They lack of domain specific knowledge and skills (DeCorte et al., 1988; 1996;
Greer, 1992);
 They have shortcomings in the heuristic, metacognitive and affective aspects of
mathematical competence (DeCorte et al., 1988; 1996; Greer, 1992; Lester et al.,
1988; Schoenfield; 1985; 1992);
 They hold inadequate domain-related beliefs about and attitudes towards
mathematics learning, teaching and problem solving (DeCorte et al., 1988; 1996;
Lester et al., 1988; McLeod, 1992; Schoenfield; 1985; 1992).
The study described here focuses on the third of these areas, exploring the range of
mathematical beliefs which upper primary children hold, from those that are unhelpful in
promoting mathematical knowledge utilisation to those in which this is central and explicit.
Previous studies of children’s mathematical beliefs
Research providing insight into children’s mathematical beliefs falls into three areas:
 Studies from the perspective of particular learning theories;
 General descriptive studies of learners’ experiences and understandings of learning;
 Specific studies of learners’ experiences and understandings of mathematics.
Research from the perspective of particular learning theories
Research within the information processing tradition has concentrated on identifying
differences between the ways in which experts and novices view problems within their field.
Experts’ representations of phenomena are accurate and inclusive. They recognise how each
of the relevant dimensions interacts within problems, together with their typical patterns of
interaction. Experts are therefore better able to plan ahead (Chase and Simon, 1973; DeGroot,
1965). They classify problems according to their underlying principles, and those experts with
the soundest grasp of such underlying principles are more able to solve problems which are
superficially different but structurally similar (Chi, Feltovich and Glaser, 1981; Larkin,
McDermott, Simon and Simon, 1980). Novices, however, adopt representations of
phenomena which are shallow, relying heavily on superficial features such as physical
appearance without consideration of underlying principles (Chi, Feltovich and Glaser, 1981;
Larkin, McDermott, Simon and Simon, 1980). They recognise only the individual elements of
phenomena and see these elements as operating unidimensionally, without interaction
(Ferretti et al., 1985; Siegler, 1976).
Social constructivist research has identified that context is an issue in how learners experience
and understand phenomena. Context influences the use of cognitive skills (for example, see
Cole et al., 1971), and individuals’ responses to similar tasks in different contexts depend on
context (for example, see Lave, 1979). Further, children’s experiences of particular
phenomena within the same context vary widely (Abreu, 1995).
These two research traditions are theory led: the theories and assumptions of researchers
about the process of utilisation guide the collection of data, and this data is subsequently
analysed within the same theoretical framework. Because of the hypothetico-deductive nature
of this research, theory rather than data leads the way: a theory is proposed; the theory is
tested; the theory is modified in the light of test findings; a new theory is proposed. The
present study seeks new insights, by building an understanding of learners’ attempts at
knowledge utilisation from data whilst attempting to adopt a neutral position regarding
theories of knowledge utilisation.
2
General descriptive studies
Phenomenography is a research approach which focuses on describing peoples’ experiences
and understandings of particular phenomena (Marton, 1981). Phenomenographic studies
indicate a limited number of distinct ways in which people conceptualise the phenomena they
experience (Dahlgren, 1979; Prosser and Miller, 1989; Ramsden et al., 1993; Saljo, 1979),
and these conceptualisations are context dependent (Gibbs et al., 1982). Little research within
this paradigm has focused on primary age children. Of the few studies, one involving 8 year
old children ascertained that they see learning in three ways: as becoming able to do, to know
and to understand (Pramling, 1983). Further they believe that they come to be able to do
something as a result of personal experience, to know something as a result of being told and
to understand as a result of both personal experience and reflection (Pramling, 1986).
Research has identified six conceptions which adult learners have of learning (Marton et al.,
1993): increasing one’s knowledge; memorizing and reproducing; applying some knowledge
or procedure; understanding which involves coming to see something in a particular way;
changing understanding; and finally changing as a person. Consideration has also been given
to higher education students’ approaches to learning across a wide variety of contexts,
including reading, attending lectures and writing essays. Approaches were classified as deep
where the learners intention is understanding, surface where the learners intention is
completion of task and strategic when the student made a choice between being the first two
(Marton et al., 1997).
Some studies have briefly explored the range of experiences learners, both young and old,
have of the purpose of school learning. Some report that is simply for subsequent school use,
whilst others indicate wider usage (Dahlgren and Olsson, 1985; Doverborg, 1987; Francis,
1982). However, these studies have provided little exemplification of these views.
Unlike research described in the previous section, the phenomenographic research described
here attempts to adopt a neutral position regarding theoretical frameworks. However, none
has addressed the issue of mathematical knowledge utilisation either directly or indirectly.
The present study seeks to do this.
Specific studies of mathematical beliefs
There are few studies of primary children’s conceptions of mathematics generally, rather than
their understandings and conceptions of particular aspects of mathematics. Of those few,
several studies have indicated that children’s success in mathematics is affected by their
beliefs about mathematics (Cobb, Yackel and Wood, 1992; Dweck, 1986; Kloosterman, 1988;
1996; Nickerson, 1992). These beliefs are relatively stable over time (Kloosterman, Raymond
and Emenaker, 1996) Further, children will not be highly motivated at school unless they
believe what they are learning will be of value to them and that effort will help them learn
(Eccles et al., 1993; Schunk, 1991; Stipek, 1993).
Carl Bereiter’s research (Bereiter, 1990) suggests that children relate to school work in two
ways. The first marginalizes learning, making it no more than coincidental to school work,
with children focusing on work, activities, completion and production rate. However, in the
second learning is central and understanding is pursued. These beliefs have been confirmed in
numerous studies (Bristow and Desforges, 1995; Cotton, 1993; Desforges and Cockburn,
1987; Frank, 1988).
Several studies have found that children view mathematics almost exclusively as calculation
and number, and the usefulness of mathematics in terms of shopping (Cotton, 1993; Frank,
1988; Kouba and McDonald, 1991).
3
The studies described in this section use two approaches in the collection of data: written
questionnaires and dialogues gathered through discussions, interviews and explanations. It is
possible that the data provided by a questionnaire is restricted and shaped by the preconceived
notions of mathematics implicit within the statements of the questionnaire. Indeed, the ways
in which dialogues are managed by researchers similarly limits and shapes the range and
depth of the data collected for analysis. Thus the present study attempts to use a largely
neutral approach in gathering data about children’s experiences and understandings. It is to
this method that I will now turn.
Exploring children’s experiences of mathematics
Method
This study uses a discursive phenomenographic method. Phenomenography is a research
approach which attempts to uncover people’s ‘taken for granted’ understandings of particular
phenomena, describing without attempting to explain such understandings, and mapping the
range of these, both within each individual and across groups of individuals. Descriptions of
particular phenomena are developed from an empirical base (rather than a conceptual area of
inquiry) using what Marton (1981) describes as ‘a rigorous method’ which aims to enhance
replicability.
In the discursive method (Marton, 1981), a semi-structured interview is used to explore
participants’ conceptions of a phenomenon by eliciting descriptions of their experiences and
understandings relating to the phenomenon in various contexts. Categories of description,
representing these conceptions, are then constructed from this data through a process of
analysis.
In the present study, a discursive phenomenographic research method is used to collect and
describe children’s qualitative experiences and understandings of mathematics in a variety of
mathematical and non-mathematical school contexts. Marton (1981) asserts that it is possible
to construct a range of conceptions in relation to a particular phenomenon within a relatively
small number of individuals, and similar studies have based their analyses on 20 hours of
interview data (eg Dahlgren, 1987; Larsson, 1983; Stalker, 1993). Thus, ten high attaining
and articulate 11 year old children, balanced in terms of gender, in their final year of primary
school were selected, and each participated in two hour long interviews: the first relating to
their experiences in mathematical contexts and the second to their experiences in nonmathematical contexts. Children in their final year of primary school were chosen as these are
an under-researched group in relation to mathematical knowledge utilisation (Desforges and
Bristow, 1992), and primary education is currently the focus of a major mathematics teaching
initiative, the National Numeracy Strategy (DFEE, 1999b) in the UK. Descriptions of
experiences were subsequently returned to the children in a process of participant validation
(Kelly, 2002). Generalisations were then developed between individuals and across
mathematical and non-mathematical contexts, by developing descriptive categories of
common experiences and understandings. A judge confirmed the reliability of these
categories by using them to re-categorise all data (Kelly, 2002).
In response to criticisms of the discursive phenomenographic method (eg Hasselgren and
Beach, 1998; Saljo, 1996), a relativist approach to validity was adopted:
“Researchers must accept that the findings of discursive phenomenographic studies
describe what interviewees say about their conceptions of particular phenomena in
particular contexts, but not necessarily what they think about these. Thus our
findings describe the limited number of qualitatively different ways in which
people talk about the phenomena they have experienced.” (Kelly, 2002, p13)
4
Findings: children’s conceptions of mathematics
This paper provides an overview of the derived category system for children’s conceptions of
mathematics. A description of each derived conception is provided, identifying ‘what’
constitutes mathematical experiences, ‘how’ mathematics is acquired, and its ‘use’. The
category system is summarized in Table 1.
Table 1. The final category system for children’s derived conceptions of mathematics.
Category
Orientation
‘What’
Focus
1. Labourer
Work
Doing without
understanding
Doing,
knowing,
beginning to
understand
2. Mechanic
Task
3. Performer
Task
Expert
performance
4. Craftsperson
Thinking
5. Academic
Analysis
Expert
performance,
understanding,
solving
problems
Understanding,
describing,
investigating,
and solving
problems
Follows
instructions
Putting
procedures into
practice, little
flexibility and
easily set off
course
Tricks, puzzles,
using secret
knowledge
Creative action,
skilful tool use,
care and pride
Generalising,
search for
patterns,
describing
situations
mathematically
‘How’
‘Use’
Rote
Exact repetition
Externally
taught, showed,
told, with some
internal reaction
Similar
contexts,
although very
‘schooling’
Initiated by
expert, practice
leading to
understanding
and expertise
Shared
responsibility,
social learning
and personal
reflection
Much personal
reflection
Exact repetition
to entertain
Similar contexts
although more
‘real world’
Separate
strategies and
skills useful in
other contexts
Conception 1. Labourer
In this approach the children orientate themselves towards the work aspects of the activity
rather than the task itself, or their thinking about the task, and the children’s emphasis is on
doing mathematical work. They do not believe that they have to think about their work
I wasn’t thinking anything, I was just doing it. (Anna)
Sometimes the children work without personal purpose, and they focus on following
instructions. The children can suggest that they have to do the work because they have been
told to.
Then we had to do the five times table down there, I don’t know why, it’s just
maths – it was just some maths we had to do. (Anna)
At other times the focus for the children is on getting all the right answers, and the focus on
following instructions is less prominent. When this is the focus, there can be a desire to finish
quickly with the children in competition with each other. Thus work can be individual and
secretive.
I did the numbers in my head because I didn’t want anyone else to know the
answers. I like it because I am always the first and I get them all right (Adam)
Throughout there is little understanding of mathematics, and little or no recognition of the
mathematical knowledge required, apart from simple number names, counting, addition,
subtraction as ‘take away’, multiplication as ‘times’ and division as ‘sharing’. Work includes
doing rote algorithms.
5
When we first learned to do these I was in Class 4. I just remembered what to do
each time and then I’d be able to do all of the sums without thinking about them.
(Ben)
There is a belief that to make work harder you use bigger numbers and to make it easier you
use smaller numbers. The children express an enjoyment for doing easy work and talking, but
tend not to enjoy doing hard work by themselves.
It was quite enjoyable because we were allowed to talk and it was quite easy,
because if it is really hard you wouldn’t get through it as quickly. (Barbara)
Learning is predominantly by telling, practice, rote and drill.
You just have to go over it so many times that it sticks. When you haven’t done
them for a while you need to be told what to do again, but you remember quickly
and it’s easy to do them again, and the more you do the quicker you get. (Carol)
Having rote learned algorithmic procedures, doing mathematics involves the repetition of
these until fluency.
The children have a very restricted view of use: the described mathematics is used directly in
other mathematics which is exactly the same, or in tests where the tested mathematics is
exactly the same. It is not seen as being useful anywhere else.
I’ve used these ideas in my maths book, there are lots of questions like these. I
haven’t used it anywhere else, but it is useful to know so you get them all right.
(Colin)
This view of use relates closely to the nature of ‘what’ is learned and ‘how’ it is learned in the
conception. In working without understanding, children are easily led to make links based on
superficially similar but actually unrelated features. For example, sometimes the children
latch on to words to associate use with other contexts. On these occasions the only link is the
actual word: its meaning and use in the two contexts are otherwise unrelated.
I used the brackets in English. When we are writing we use brackets for something
and if we didn’t have enough room in the story you’d put brackets. (Denise)
At times the metaphor for mathematics is mental exercise: doing mathematics is compared to
mental work. To make mathematics harder you use bigger numbers. After a rest (when you
play and have fun) you feel refreshed, but need to get used to doing mental work again.
Sometimes after a long rest you need to warm your brain up, doing easier work first. Your
performance on work often depends on how you feel.
We did it to get your numbers working because you might have lost your numbers
in your head over the holiday because you need to refresh your memory. I had to
refresh my memory because over the holiday I wasn’t doing much number work, I
was just doing like fun things because it was quite sunny. It got harder as you got
further on, the bigger the number the harder it gets. You make it harder by putting
higher numbers. On the division it was harder because there was more to do.
(Adam)
Similarly, in relation to learning, the harder your brain works, the more good it will do you.
This is because you will become more able to do work easily and thus become quicker at
working. This mirrors directly physical exercise.
You could have made it a lot harder and then that would have been more mindbending and it would make your brain work a lot harder. If you always do easy
work then it isn’t really going to get you anywhere, and if you make it harder it
helps you learn things more quickly. (Daniel)
6
Conception 2. Mechanic
In this approach the children orientate themselves towards the task. The emphasis is on doing
mathematics tasks, but the children also recognise that they know some mathematics and are
beginning to understand it. Thus it is distinct from the Labourer conception because the focus
is on doing mathematics and not on doing work.
At times the focus for the children in this conception is somewhat mechanically putting a
procedure or algorithm into practice. This differs from the Labourer conception because here
the children choose when to use an algorithm or procedure, and their choice relates to
completing the specific mathematical task.
I think of dividing as sharing it out, and sometimes I don’t write it and sometimes I
write it down. It depends if you have a lot of time or not because you can spend
ages drawing 63 spots and then dividing them by 7, and if I really was stuck on it
then I’d do it that way. (Carol)
However, in making such a choice (as in typical school mathematics word problems
involving addition, subtraction, multiplication or division) they often use clues (which may or
may not be appropriate) to identify the correct mathematical procedure.
A number story is like it says somebody pays an amount for something to eat or a
particular toy and then asks how much would they pay for a particular number.
They’re easy because I always do timeses. (Adam)
At other times, and depending on the nature and presentation of the task, the focus in this
conception can be putting procedures or algorithms thoughtfully into practice, with some
understanding.
With brackets you need to know that for (27+13) x 2 you just don’t go and do 13
times 2 first, because … that the bit in the brackets you do that first … and so what
you’re left with is 40, 2 times 4 is 8 so that’s 80. You need to know that you need
to look at the sum properly before you write it down so you don’t get it wrong.
(Denise)
This differs from a later conception, the Craftsperson, because, although both approaches
focus on completing the mathematical task, in the mechanic conception the children are still
following taught procedures and rules inflexibly, whereas in the Craftsperson conception the
approach is flexible, and that flexibility is based on a greater understanding.
The children do their work to find out the answers. They want to get their work all right, but
at a personal level they want to go on to do harder sums and get better at them. There is a
slight element of performance in both of these purposes.
Yesterday we did some work on perimeter and area to learn what the area of some
things are, to learn how much area it takes up. If we can do these then we can go on
and do some harder ones, if we show Mrs Jones we are getting better at them.
(Daniel)
However, this differs from the Performer conception because with the Mechanic the teacher
recognises and praises success, moving the child on accordingly, whereas with the Performer
the audience is simply entertained, there being no consequent next step for the child to follow
on and learn.
In terms of what is learned, the mathematical knowledge recognised includes simple number
(nominal, cardinal and ordinal), addition, subtraction, multiplication, division, symbols,
measures and money. Throughout there is some understanding of mathematics. However, to
begin with this is often dependent on features of the presentation of the task (such as the
7
layout or wording) which are not helpful in novel contexts, and so the children can become
easily confused and set off course.
We’ve used some textbooks, you had to copy the timetable into your book and …
answer questions like what’s the difference between 8am and 3pm, and you just
added from the first to the second. We answered the questions and Mr King said
are you sure that’s the right way of going about it or is there an easier way. (Ellen)
The children’s mathematics is seen as being externally taught, received knowledge. This is
distinct from the Labourer conception because there is recognition of some internal reaction
to this external knowledge, although learning is seen simply as getting more knowledge.
When we first got this I thought I don’t know how to do this so I went and asked
Mrs Lee and then I think I struggled a bit on it and I went home one night and said
“Dad, I’m really stuck on these” and he really helped me. He said, “Well you’ve
got to get better at your tables”, and I started to understand it a bit, and then I don’t
really know what he did. (Carol)
Sometimes the children focus on and identify links between ideas in a hierarchy of
knowledge. Here they see learning as getting more knowledge in the hierarchy.
You needed to know what came first, like what number they were in, like were
they tenths, fifths, really going into fractions now. If you don’t know hardly
anything about fractions then you won’t be able to find out hardly anything about
decimals because fractions are really decimals and things, because if you add
fractions it’s like adding decimals. (Barbara)
Mathematics is seen as being used in exactly the same way in different contexts, be they
mathematical or otherwise. Thus it is used in exams or tests where there may be a slight
change in presentation, wording or context. Similarly the children identify the unproblematic
and direct application of basic number, money and measures to situations very similar to
those they have encountered in school mathematics. Occasionally somewhat unrealistic
school mathematics problems are identified as being possible real-life situations.
When we went down to the shop the other day Paul was beginning to buy three
Refreshers and I said hang on a minute Paul because he was only allowed to spend
30p and they were 12p each and I added them all up and I said that’s 36p and he
said mum won’t mind if I spend 6p extra and I did 12 times 3 is 36. (Denise)
In the same terms the children identify basic number, money and measures as being
potentially useful in future life.
You may have to measure things up when you go to college, and the more you
learn at school the better job you’ll get. (Daniel)
This is distinct from the Craftsperson because the context of application is less flexible, being
like school problems rather than real everyday problems. The latter are more evident in the
Craftsperson conceptualisation.
Conception 3. Performer
In this approach children orientate themselves towards the task, and their emphasis is on
expert performance. The purpose of the children’s work is to move towards expert
performance and then to entertain with this performance.
My dad showed me how to do algebra and then I practiced a lot at home and then
when I got good I showed my friends at school and Mr Moore. They thought it was
good and Ben wanted me to tell him how to do it. (Eric)
The emphasis on task is distinctive to that of the Mechanic conception because, with the
Mechanic the task is completed to show success in learning mathematics, whereas with the
Performer the task is completed because it is entertaining and will become a trick in a
performance.
8
In terms of what is learned, the children give a sense that mathematics is about slightly secret
knowledge. Examples given by the children focus, on the one hand on short cuts, and on the
other on particular meanings and ways of working. The former includes people who can work
out numerical problems quickly.
I saw this man on the TV, Record Breakers, and he could add up really big
numbers really quickly in his head. He had a way of doing it. I thought, I wish I
could do that. (Ben)
The latter focuses on algebra, where the secret knowledge is that the letters represent
numbers. However these letters tend to represent single numerical values rather than a
variable range of values.
My dad taught me how you could make z if it was z+3=y, if z=3, then y=6. You
would put the 3 inside and the next question you’d get the answer. You’ve got to
work out the values of each of the letters. I’ve got to find a way of making what y
means. (Eric)
Children identify someone having taught or showed them how to do this mathematics, like a
trick to perform, and they are still a little unsure of it, needing constant prompting. There is a
need to remember what to do next throughout the whole process, and practice makes perfect.
As such, doing often precedes understanding, but this comes with expertise. The ‘trick’ is
tried out frequently and refined.
Well my dad showed me this sum and he told me how to do it and he told me to
work out the value of x you need to see the difference between two numbers, and if
it was y+6=7, y must be 1. (Eric)
There is also, at times, a view that some people can have a natural ability to, for example,
work out numerical problems quickly.
He could always do things like that. I expect he started at school. He could
probably just do it. (Ben)
Each of these views, that coaching and practice produces expert performance on the one hand
and that there are some natural performers on the other, exist side by side.
Finally, the children see mathematics as being used to put on a show, to perform and to
entertain. In these situations it is always a direct repetition of the practiced ‘act’. This is
distinct from the Labourer conception because the purpose of the direct repetition is to
perform and entertain rather than complete mathematical work or tests.
Conception 4. Craftsperson
In this approach the children orientate themselves towards the thinking aspects of the activity,
with the emphasis being on expert performance involving working out and understanding.
This is different from the Performer because the expert performance is used, not to entertain,
but to solve mathematical problems.
The children have considered opinions, although they use procedures they do not
mechanically carry them out, and show understanding rather than rote learning. They use a
thoughtful and flexible approach of their own aimed at successfully completing the task or
solving a problem.
If you don’t estimate numbers it might get really hard when you get to do high
sums like you have to estimate it to the nearest 5 or 10 then you would work it out
in your head. It makes it easier to do when you do the 5, 24-10 I did 25-10 so I
reasonably put it up to the nearest 5 so I put the estimate 15 and it was 14 and if
you don’t know estimates you might get very confused because estimating numbers
gets very confusing if you’re older because you might have 121 so you’d
reasonably put that to 120 and 131 so you’d put that to 130 so it would be 250 as a
reasonable estimate then you add on 2 and it makes 152. (Adam)
9
Sometimes this involves trying out and drafting.
When I have a complicated problem to solve I try out lots of ideas on scrap paper
and see which way fits the best. (Carol)
The children show a pride in their work. They also recognise the need to be careful and
patient, and are able to reflect on and appraise their work. Here the understanding aspect of
mathematics is emphasised.
I didn’t get through it as quickly as some people. I just took my time to make sure
that I got it all right, and I felt good when I had finished it. (Barbara)
You had to be patient, so that you could work it out. It took me a long time and
there were loads of problems and I coloured it in first but what would have been
better would be, and what I’d do now would be to write the first letter of each
colour down and then colour it in at the end. (Carol)
At times there is a slight sense of performance, with the children showing what they can do.
All you really think about is getting it right first time and this is really showing Mr
Norman what you can do, showing him what you know about it. And if you don’t
know very much about it then he will try to help you. (Barbara)
No one else could do it and then I saw that if you said that it was half a square then
you could work out the area of the square and half it. Then you’d have the area of
the triangle. So I told Mrs Owen. (Adam)
However this differs from the Performer because the emphasis is on the elegance and ease of
a competent performance rather than the performance of a trick. Thus it is like watching a
craftsperson at work rather than a magician.
The children have a good understanding of number, addition, subtraction, multiplication,
division, measures and money. They have a more sophisticated view of how work might be
made harder other than simply by making the numbers bigger.
[To make finding area harder] you can use different shapes that we haven’t used
yet, irregular shapes. You could use a triangle and that would make it harder
because it’s got half centimetre squares. (Daniel)
Generally they find the mechanical aspects of tasks easy but skilful tool use (such as
measuring temperature accurately harder. They like the challenge of harder work and aspire
towards being able to do it.
It was really good when I did this work because I didn’t think I would be able to do
it when I started but I wanted to do it because it looked interesting. (Carol)
With regard to learning, the children try to see their work in the wider context.
We hardly did any work about perimeter and area last term and this was like a
starter because we had to measure it first and then write it down, and work out the
perimeter and area each time. This got us ready for the harder work later. (Barbara)
Thus they can provide an overview of the whole rather than just focusing on one task, and
have an understanding and vision of the wider issues. They also show some account of
previous learning.
The children recognise that the teacher will help, not tell you what to do. They see the onus
for doing and learning as being very much on the learner.
All you really think about is getting it right first time and this is really showing Mr
Norman what you can do, showing him what you know about it. And if you don’t
know very much about it then he will try to help you. (Barbara)
Further, there is a sense of shared responsibility for learning, and children often adopt a social
and cooperative approach with peers, and learning often takes place through working
together.
10
They were put on the board and I thought this looks quite difficult and I did the
first one and I thought no it isn’t this is easy and then I took off and I did them all
and then I helped some of my friends. (Carol)
Thus the children see learning as involving what you bring with you, what you are taught,
what you read in books and personal reflection.
At times the children think there is also a general mathematical ability, and that this may be
inherited.
I don’t know why I could do it easier than my friends. My dad is really good at
maths as well. (Carol)
However, unlike the performer where there is, for some, a natural way of doing tasks, with
the Craftsperson this natural ability links to the learning of skills and involves having a
disposition to do and understand particular things.
Adam is really good at maths and he just picks things up really quickly and he only
has to do things once and then he can do them. (Barbara)
The children identify simple application in measures and money in a variety of everyday
contexts. The focus is on expert tool use.
Dad has put in a new carpet in Helen’s room and I helped dad measure it all up and
we had to measure really carefully so that there would be no gaps in the carpet. If
you want to find the perimeter of a room you can do one side and then another side
and work out the rest. That’s what Mr Norman told us. Dad told the people at the
carpet shop so they could get the right measurement for Helen’s room. (Barbara)
The children also show evidence of a wider view of number, for example, they know when to
use multiplication.
It’s like when you want to make things bigger, say five times bigger, so if it’s three
then five times bigger is 15, and if it’s four centimetres then five times bigger is 20
centimetres, so to make a picture five times bigger you’d times all the
measurements by five. (Ben)
This differs from Mechanic because with the Craftsperson it is understanding which leads to
use in everyday problems rather than clues in the layout or wording of school mathematics
problems.
Conception 5. Academic
In this approach the children orientate themselves towards analysing the activity, and their
emphasis is on understanding, describing, investigating and solving problems. This differs
from the Craftsperson as the children adopt a systematic approach, look for and identify
patterns, and recognise each of these as being useful in mathematical tasks. Even in
mechanical tasks the search for pattern is implicit in the child’s approach. Generalisations are
used to conclude the task, and these are sometimes tested.
There were 24 carriages, and if you double them there would be 48 different ways
and with 12 there would be 24 different ways, and next we worked it out for 6
carriages and there were 12 ways. Each time I counted them and it was double the
amount of carriages. (Eric)
The formula was the number of carriages times 2 is the number of patterns you can
make because if there’s six carriages then there’s six ways and if you do the
opposite that’s 12 ways, and so on. I tested the formula and it worked each time.
(Ben)
11
The purpose of mathematical tasks is to find mathematical descriptions of situations.
Well, when you know the formula then it doesn’t matter how many carriages there
are, you know how many ways the trains can be arranged and you don’t have to
actually do it. (Ben)
In terms of their learning, the children have a good understanding of the tasks and concepts
involved. Sometimes they use this understanding to develop short cuts (by searching for
meaningful patterns) which are then used. However the children are also able to identify
possible errors in their use of these short cuts, and do not use them mechanically. An example
of this was provided by Ben who saw that to calculate a percentage of a number you could
multiply it by the percentage represented as a decimal. He was then able to use this in a
variety of contexts, and realised his error when finding 9% and mistakenly multiplying by 0.9.
The children identify searching for mathematical description and solution of the problem as
being both challenging and rewarding, whereas they identify mechanical work as being
boring.
The first bit, when we had to find out how many patterns there were I found that
quite easy and it was fun to do. I liked trying to work out the formula. The other
work was boring because you just had to count how many squares were around it
and write it on the page. (Ben)
The children see the source of their knowledge as being from others, through personal
reflection (that is by thinking or working it out) and from some innate ability. They see these
as interrelating with each other. However, although they often recognise that they worked out
the pattern in a problem solution, they suggest that they didn’t learn to recognise patterns,
rather they developed this ability.
I worked some of it, like the short cuts, out myself, and some Mrs E told me, and
my dad did as well. I learned the patterns stuff myself, I thought about it. As I grew
up I got better at seeing where the patterns were. I don’t know how you do it, you
just look for things that repeat themselves or so that you know what is going to
happen next. (Ben)
They suggest that generalising is a skill which they have been taught. In relation to learning,
this conception differs from the Craftsperson because of the strong role of personal reflection
in the process of learning.
In terms of use, identifying patterns and generalising are seen as separate strategies and skills
that can be applied in other tasks and contexts.
I have used these ideas when we have done another sheet, and we’ve done pattern
as well, and I’ve drawn tables. When playing football games I need to draw out a
table. I’ve used other ideas in a piece of work we did a few days ago when we had
to make a formula. (Ben)
The value of describing situations mathematically is also recognised.
When you look at something then, if you look for patterns and then find a formula,
you can say what will happen if it’s a bit different, like you’ve got more trains or
cars or something, so it will help you see what will happen. (Ben)
Occurrence of conceptions of mathematics within this study
In 88% of instances in mathematics contexts, high attainers in mathematics in their final year
of primary school indicate that their experiences of mathematics are those described in the
‘Labourer’, ‘Mechanic’ and ‘Craftsperson’ conceptions. In these conceptions, use centres on
counting, simple number operations, money and measures. Further, in 72% of instances, these
children indicate that their experiences of mathematics are those described in the ‘Labourer’,
‘Mechanic’ and ‘Performer’ conceptions. In these:
 use is inflexible, being seen as being a direct repetition of that which has been taught;
12


use can occur either in further mathematical exercises, in performing or in very
similar out of school contexts such as adding money;
when the context is different, clues are often used to determine the approach chosen,
rather than understanding. These clues might be inappropriate and mislead.
In only 19% of instances did children indicate that their experiences of mathematics are those
described in the ‘Craftsperson’ conception. In this conception understanding determines the
mathematics used, and mathematics is used flexibly in appropriate contexts. Further, there is a
very low occurrence of instances of children indicating that they have experienced
mathematics as suggested by the ‘Academic’ conception: as a useful way of describing the
world.
Conclusion
The findings presented above provide a somewhat depressing picture: in almost all cases,
high attainers in mathematics in their final year of primary school do not see mathematics as
being a significant feature of the world beyond school. Instead, they have a superficially
descriptive view of mathematics, rather than seeing mathematics as a “uniquely powerful set
of tools to understand and change the world” (DFEE, 1999b p60).
Returning to the suggestions made by Verschaffel and his colleagues (1999) and cited at the
start of this paper, curriculum reform in the UK has to a large extent focused on improving
students’ domain specific knowledge and skills. Regardless of how successful educational
reforms have been in achieving this, the findings of the present study suggest that there will
have been little impact on students’ success in approaching mathematical application
problems because students still hold inadequate domain-related beliefs about and attitudes
towards mathematics learning, teaching and problem solving. In order to improve application
we must promote more productive beliefs in terms of mathematical knowledge utilisation.
What might we do? If we accept that such beliefs result, in part, from different pedagogical
practices (Winograd, 1991; Carey and Franke, 1993), we might adopt approaches which
emphasise:
 links between understanding mathematics, recognising when it might be useful and
using it;
 a view of mathematics as a useful way of describing the world in terms of
relationships and not simply superficial features;
 the ability to see and search for mathematics in the world around.
So, for example, for the UK’s National Numeracy Strategy (DFEE, 1999a) to result in
utilisable mathematical knowledge:
 children should have considerable experience of the practical use of all aspects of the
mathematics curriculum, not just numerical operations, money and measures;
 children should be taught to use strategies and numerical concepts and operations
with understanding as well as with pace;
 children’s reasoning for their approaches to word problems should be explored with
them;
 children should have considerable experience in searching for mathematics and
relationships in the world around them;
 children should be helped to see themselves as mathematicians and mathematics as a
useful way of describing the world.
Indeed, it may be that derived categories describing children’s experiences of mathematics in
the present study are also useful in helping learners engage with the metacognitive and
affective aspects of mathematical competence.
13
Without such action it is likely that improvements will remain superficial: any improvement
in children’s attainment as measured by their performance in tests and examinations will not
be matched by an improvement in their achievement of mathematical knowledge utilisation
beyond the mathematics classroom.
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Dr Peter Kelly is an Honorary Fellow of Exeter University. He would welcome comments
about this article, and can be contacted via email: P.A.Kelly@exeter.ac.uk .
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